Calculus For Professionals [I] 9788195396207, 8195396208

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CALCULUS FOR PROFESSIONALS

List of other books Other Books on Calculus Calculus for professionals (Volume II) Calculus for XI-XII (Volume III)

A to Z Math Series Books KG-VIII Reading Book KG-VIII Explained Examples (2 Volumes) and Exercise Book (2 Volumes) IX-X Reading Book (3 Volumes) IX-X Explained Examples (3 Volumes), Exercise Book (3 Volumes) XI-XII Reading Book (2 Volumes) XI-XII Explained Examples (2 Volumes), Exercise Book (2 Volumes)

New-genre Math Books for KG I-II Volume I – Pre-Quantification Language Theme I Diversity, Theme II Opposite, Theme III Similarity and Difference Volume II – Pre-Number Quantification Approximate Number System, Conceptual Subitising, Perceptual Subitising Volume III – Numbers Counting Part I, Part II Measurement Part I, Part II

A to Z Science Series Books Biology (2 Volumes) Chemistry (3 Volumes) Physics (4 Volumes)

A to Z Social Science Series Books Science of History (3 Volumes) Physical Geography (3 Volumes) Human Geography (3 Volumes) Story of Nations (Civics)

Foundational Books on Critical Thinking Foundations of Addition Foundations of Subtraction Foundations of Multiplication Foundations of Division

Book on How to Educate Children You the Unsung Hero

CALCULUS

FOR PROFESSIONALS Volume I

CALCULUS FOR PROFESSIONALS Volume I Copyright© Sandeep Srivastava, Dr. Garima V Arora, 2022 The moral right of the author has been asserted. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form, or by any means, without the prior permission in writing of the publisher, or be otherwise circulated in any form of binding or cover other than that in which it is published and without a similar condition, including this condition being imposed on the subsequent purchaser. ISBN: 978-81-953962-0-7 Managing editor: Saloni Srivastava Domain editor: Nikita Editors: Nikita Todarwal, Nidhi Gandhi, Gaurav Gupta Cover design: Saloni Srivastava Publishing team: Dipti Chauhan (Lead), Manish Kumar, Tanu Gaur, Karan Anand Published in India in 2022 by Nextgen Books Private Limited, the publishing partner of IYCWorld Softinfrastructure Private Limited. In partnership with

www.sandeepsrivastava.online A-43, II Floor, Zamrudpur, G.K 1 New Delhi 110048 India

Dedicated to My daughter, Shreya Sahai, the most modest mathematician, Garima, and the 21 year old editor Nikita. Sandeep

Dedicated to A joyous association with mathematics – the idea, and the language, and its grammar! Garima

Content

.Preface ................................................................................................01 Rendezvous with Calculus ................................................................08 Welcome to zero, the hero Math to the rescue of science Welcome to the reality of change Changing positions – The visible motion Rate – A relationship between two quantities Mathematics of speed – The gross or average speed Instantaneous values of (changing) quantities Functions – The gateway to instantaneous world Derivative of Function Anti-derivative Mathematical expressions using Derivatives (Differential equations) A new narrative for Limit and Continuity Origin of Calculus Algebra versus Calculus The nature of Math Function .............................................................................................88 The need of instantaneous values Function – Capturing realities in mathematical expressions Graphing Functions for fuller understanding The Parent Functions Derivative and Anti-derivative of f(x) = x Derivative and Anti-derivative of f(x) = x2 Derivative and Anti-derivative of f(x) = x3

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Derivative and Anti-derivative of f(x) = |x| Derivative and Anti-derivative of f(x) = 1/x Derivative and Anti-derivative of f(x) = x Derivative and Anti-derivative of f(x) = ex Derivative and Anti-derivative of f(x) = log x Derivative and Anti-derivative of sin x Graphical representation of Combined Functions Differentiation .................................................................................211 Beyond change – The rate of change The concept of Differentiation Derivatives in real life How are derivatives represented? Ways to find the Derivative of Functions Implicit Functions Non-differentiable Functions Applications of Derivatives Differentials Maxima, Minima and Point of Inflexion Extreme value theorem Mean value theorem Rolle’s theorem High Order Derivatives Partial Differentiation Appendix Limit and Continuity........................................................................315 The role of Limit and Continuity One-sided Limits Limits at Infinity Understanding Continuity Continuous Functions Discontinuous Functions Acknowledgment .............................................................................393

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In the hope that all the readers shall read this first sentence of the book, it is emphasised that the preface, the first chapter, and the second chapter must be serially read, paragraph after paragraph. The book offers a comprehensively new narrative on calculus and the idea of mathematics in general; the first two chapters are the necessary foundation for mastering calculus. The (teaching and) learning of calculus has been waiting for real-world, learner-centered, and inventive conceptual narrative; calculus is intimidating even for the overwhelming majority of those who formally study calculus at the undergraduate level, including mathematics majors. We sincerely believe that this book presents a peerless story of the fundamental nature of calculus. Language is the key human capability Mathematics is a language just like all other natural languages (mother tongues), implying that it is to be used in everyday life, in the real-world of thinking, communication, and actions. However, the nature of natural languages is such that it allows significant latitude to speakers/writers, as well as the listeners/ readers, in interpreting sentences and words. This room for personalisation of languages makes understanding easier to learn and use, and allow interesting usages of words and sentences for more affect, and effect; making mother tongues a source of such immense inclusion (and thus, division, too), joy, creativity, personal bonding, and deliberate play. 1

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By the same token, there is room for a language that allows us to be precise, without deliberate play, to communicate exactly one meaning/scenario/situation/condition; a language that cannot be misconceived (‘seeing something, describing something else’), miscommunicated (no ‘transmission losses’, for example), or misunderstood (misreading the message). Mathematics is that language. There is no two-way about any mathematical expression. We need such a language for expressing scientific phenomena, facts, laws, situations, and conditions; physics, for example, is all mathematics. It cannot be visualised and articulated without mathematising observed realities. Mathematics is the language of the ‘real world’ (and the universe) There is another very interesting and important need for precision, certainty, and uniqueness in communication – qualifying everyday situations, in quantitative terms. For example, the paintable surface (area) of a wooden plank, the quantity of rice in a heap of rice, or the instantaneous pressure at a point on the surface of an inflated balloon. While the first situation is a rather familiar one, and algebra/ geometry is what we harness for computing area of definite geometric shape (such as the rectangular wooden plank), the other two situations defy the applicability of algebra. Indeed, the algebra and geometry we study in schools are not meant to be applied in very many situations. The most natural and man-made shapes can not be strictly categorised as square, rectangle, circle, sphere, etc., and most relationships among properties of things are not strictly linear, at least over a broader range of consideration. Calculus – The mathematics of the real world We need a special kind of algebra – ‘algebra of non-linear (nonstandard) shapes and relationships’; for instance, the area of the skin over our bodies is typically not amenable to be computed using algebraic expressions and manipulations. Calculus is

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that domain of math that deals with the ‘real world’, the world as is, the world of natural shapes and relationships, and one that changes over time; we use calculus to find area of the skin covering our bodies. To that extent, mathematics in general, and calculus in particular, is the (only) language to precisely quantify real-world and scientific situations and events. In fact, without the knowledge of calculus, the knowledge of most of our world would be incomplete, incomprehensible, and even undefinable Mathematically. No wonder calculus education is not only part of every K-12 curriculum, but it is also the crown of the math curriculum – taught in XI-XII years. Calculus for Professionals Calculus is like looking through a lens which affords us a very different view of almost everything. For example, the quantitative understanding of the second law motion, Force applied on a body = Mass of the body × Acceleration of the body,

is introduced in the middle school curricula as F=m×a implying that acceleration is kind of a constant quantity over a period. The reality cannot be farther from the truth. Holding acceleration to be constant over a period is a very substantive assumption, almost impossible to be true in any motion (except free fall). However, introducing the law as F = m × dv/dt rightly conveys the highly probable reality of changing (instantaneous) velocity and acceleration and also a sense of instantaneous values of quantities. Calculus, as a language, best captures the reality of change being the only constant in our lives and the universe. It must be introduced as a pre-secondary school knowledge, experience, and observation. On the contrary, nothing is right about calculus education in schools and higher educational institutions. This plays havoc with due appreciation of mathematics, as well as science, and kills the urge, passion, and capability to study

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the patterns and logical structures of many situations we are routinely part of. The unfortunate reality of Calculus (education) While fractions, algebra, mensuration, etc., are not easy, trigonometry and calculus take the phobic hysteria around math to a crescendo. The introduction to calculus on the back of the concepts of limit and continuity, and resting that on the reference to the method of exhaustion, actually takes us back to the times and thoughts of the founders of calculus over 300 years in the past. We can do better than that, also because calculus is now the language of all of the real world, way beyond the language of physics developed by Newton and Leibnitz. Expectedly, limit and continuity have been taken to the appendix to offer a uniquely alternate conceptual and physical understanding of the beauty of calculus, and yet retain the academic rigour of limit and continuity. This is the first step in democratising, mass-scaling success in appreciating applications of calculus (and in a confident understanding of the world around us). As of date, success in learning math at schools and universities is unfairly, unnaturally, and unethically elitist, geekish, and sexist. This must reverse now. The undue, hurtful eliteness of mathematical modelling Research data presented by Qi Dan, from the book ‘Trends in Teaching and Learning of Mathematical Modelling’ show a strong positive correlation between mathematical modelling and creative thinking skills. On the other hand, happily, there is an insignificant correlation between students’ mathematical modelling skills and the scores they achieved in basic mathematical courses. Although some patterns of relationships do exist. No ‘extra intelligence’ is required for being able to model physical realities in mathematical terms. Mathematical modelling must be an easily accessible skill for all – not only engineers, scientists, and educators (and

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students), but also creative professionals (no one should be left behind in mathematics). Data modelling as a substrate for inorganic intelligence While we are still nowhere close to deciphering how our brain works, how exactly do we learn, we are sure-footedly getting closer to human-like artificial thinking and action. At the core of the difference between the knowledge of organic and inorganic thinking and intelligence is a very interesting reality – the ‘reverse engineering’ of intelligence. The only, and as yet the primary way of learning about anything is what we call research, it is to discover the knowledge by studying the targeted ‘behaviour’ in the present and continue the same in the future. The digital capturing of data points has ushered us into a completely novel capability where we can learn about anything by looking at its relevant past ‘behaviour’ data, map the data onto an explicitly expressible ‘model’, and fine-tune the model to fit with the observed behaviour in the present (and future). The ‘model’ can be physical or mathematical; obviously, a mathematical model is better for fine-tuning and mastering the relevant knowledge. Data and mathematical modelling are at the heart of the fourth industrial revolution, offering unprecedented innovations in all spheres of life, limited only by our ability to capture and mathematise the data. Calculus is a key component of mathematical models because calculus is how we define and quantify change – the rate of change and the quantum of change (over a specified period). The dynamics of change are the reason for capturing data, and learning about newer things/ events. There is no reason to learn about things that would not change. Differential equations are humanity’s best interpreter, for example, a simple differential equation around the NavierStokes equations (mentioned just for the record) can be used to model the normal blood (a non-Newtonian fluid) flow within a reasonable range of certainty.

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Preface

Welcome to a new reconceptualisation of mathematical modelling Together with linear algebra (vector and its manipulations) and probability (with statistical tools), calculus is the common denominator for mathematical modelling. The book attempts an audacious vision – offering the entire spectrum of mathematical logic for modelling in its most conceptual form. Over two volumes, the book contains the following chapters – Chapter 1 is ‘Rendezvous with Calculus’, it explores the theme Z of the A-Z series, Z representing zero. It kicks off the foundation of the calculus was laid. The concepts of calculus have also been explained in an intuitive way, which makes it easy for the readers to visualise and work the otherwise hard concepts. Chapter 2 is ‘Functions’, which is a unique feature of this book. It talks about some of the common parent functions and how applying the concepts of calculus to these functions makes it easy to understand the infinite functions that we have. Chapter 3 is ‘Differentiation’. The formal definitions and properties of differentiation has been explained in this chapter. Limit and Continuity have been moved to the Appendix. Volume II of the book is dedicated to the other concepts of calculus – integration and differential equations. The other domains of math (apart from calculus) – vector calculus and probability which are applied in Artificial Intelligence have also been introduced and explained. Chapter 4 is thus ‘Integration’, which focuses on the applications and properties of integration on various geometrical figures. Chapter 5 is ‘Differential Equations’ and models some real-life situations of changing quantities. It introduces and exemplifies the significant applications and solutions of differential equations. Chapter 6, ‘Vector Algebra and Calculus’, which contains the basics of calculus applied to vectors, once the grounding in the idea of vectors and using them for ‘n × n’ system of linear

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equations is established. It also contains portions of linear algebra, which forms the basics of AI. Chapter 7 is ‘Probability’, consisting of probability axioms, distributions and the famous Bayes rule, which lays the foundation to deep learning algorithms. The DNA of the book The book is what it is due to a peerless collaboration of a math educator – Sandeep, and a mathematician – Garima. The educator kept the spotlight on the learner/reader, raking up all possible ‘doubts’, and leaving no stone unturned to chip off (all the) weakest links in our story of calculus – the much-needed new narrative. The mathematician kept creating ‘doubt-free’ explanations and the academic integrity of the story. As a team, we had set our sights on offering conceptual breakthroughs (restatements) in appreciating and working with calculus. We have worked hard, and dedicatedly to ensure that the graphs, equations, and symbols are correct and according to the standard mathematical syntax. We have also lived the best publishing practice of multiple, and specific rounds of editing. Errors, if any, would be entirely inadvertent, minor, and not affect the meaning of the essential communication in the context. We must also acknowledge that in our zest to create a narrative benchmarked to be relatable even by teens (who formally know the concepts of velocity and acceleration, for instance) we might have over generalised, or over sliced the mathematical rigour and exactness. Dr. Garima V Arora Sandeep Srivastava

Rendezvous with Calculus

Welcome to zero, the hero Zero, above all, is a powerful and unique concept and idea. It is the simplest yet foundational and the most important quantity (and quantification) in math! Unlike all other digits, it represents no quantity. For example, we use zero to quantify the apples (for that matter, any other fruit or something else) in a basket of pears – zero apples. Zero is also a placeholder. For example, in 302, 0 is placed at the Tens place to imply ‘No Tens’, and ‘0’ holds the place for the ‘Tens’ packets that are not there in the numeral 302 – three hundreds and two ones.

Visualising numeral 302 with block diagram

The parity of zero is even. Zero is considered an even number. Zero is followed and preceded by an odd number. It is neither positive nor negative. Recall that negative and positive numbers are defined with respect to zero itself. It is foundational because compared to the other 9 digits (1–9), it is most instrumental in making possible the positional number system we use. Zero is a part of all the number systems, such as – binary (0, 1 are the digits used in binary numbers), octal (0, 1, 2, 3, 4, 5, 6, 7), decimal (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) 8

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and hexadecimal (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F) – positional systems. In a decimal number, the quantity represented by 1 unit at a position is ten times more than the position on the right. For example, 743 in the decimal number system can be expanded as (7 × 102) + (4 × 101) + (3 × 100). In the binary system, we use two symbols 0 and 1. All the numerals can be constructed using these two symbols. A list of numerals in the binary system can be represented as 0, 1, 10, 11, 100, 101, 110, 111, 1000, etc. Since, in a binary system, each position on the left is twice the previous position, the number 10110 in a binary system can be expanded as (1 × 24) + (0 × 23) + (1 × 22) + (1 × 2 1) + (0 × 20) which is equal to (16 + 0 + 4 + 2 + 0) = 22 in the decimal system. The computers majorly use the binary system. For example, a byte is 8 bits which is 23 MB, a 1 GB RAM is 1024 MB which is 210 MB, a 16 GB RAM is also 24 × 2 10 MB. This pattern applies to both octal and hexadecimal number systems as well. 8 digits (0–7) are used to represent numerals in the octal system. The numeral next to 7 is 10. Whereas, in the hexadecimal system, the numeral next to ‘F’ is 10. No wonder the A–Z series of math books end with a chapter founded on zero. Zero – The most important part of any number system Zero is a source of expressing a unique reality. Take a box with nothing in it. This is an empty box. It physically represents zero. Now take another empty box, and place it in the first one. How many objects are there in the first box now? One. We can (touch, point to, and) count the one box inside the first box. This represents a unique reality – zero is not countable. It is not used for counting; we cannot count anything that is quantified as zero. All other digits and numerals are ‘counting numbers’ (ordinals) as well as ‘quantity numbers’ (cardinals); zero is only a quantity number.

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Zero showcases of the real power of abstraction in math. For instance, the ability of zero to quantify in the most ‘mathematical way’, i.e., (very precise manner) when used in a specific context; 31 becomes ten times bigger as 310 (when zero is appended at the newly created last place – the unit place). No other digit show any such definite property, for example, 31 becomes 311 when 1 is appended at the end of the digit, but the two bear no neat and certain relationship, just as the arithmetical relationship between 102 and 1022 is not the same as the arithmetical relationship between 111 and 1112. Zero ‘combines’ with any other numeral to always form numerals upto ten times larger than the given numeral. Zero represents ‘nothing’ We have already referred to how it is impossible to specifically show zero of anything, such as zero person, zero tree; except that we can show a person and then take him off sight and explain the absence as being zero person. More interestingly, to the question ‘how many deer may be found in a school classroom?’ Ideally speaking, we may not respond using zero for quantification; deer are not expected to be found in the school classrooms. Zero is not, or should not be used to numericalise in situations that are not possible. Zero is best used for situations where it represents ‘nothing of something that is possible’. Zero is not used when ‘no thing like that is possible’. The arithmetical wonder that is zero To the point, zero is at the heart of the magical simplification of mathematical operations, such as division and multiplication. Recall, ‘trailing zeros’ are used in these operations. While the idea of trailing zeros is useful in all number systems, we use the familiar decimal number system to recall the simplification. For example, the product of 7 and 40000 is simply calculated by, finding the product of 7 and 4, which is 28, and then placing as many zeros as in the numerals. Thus,

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7 × 40000 = 280000. In a similar way, 400 ÷ 2 = 200. This is obtained by dividing 4 by 2 (which is equal to 2) followed by two 0’s from 400. Now, consider 4 ÷ 200. In this case, 4 ÷ 2 gives the quotient 2. 4 2 = . There are two zeros in 200 100 the denominator, we simply shift the decimal to two places to the left, making 4 ÷ 200 as 0.02.

On simplification, we have,

Technology and zero Zero is of fundamental importance to technology and science. It has made vital contributions in every sphere of life. For example, zero and one made the base of Morse code communication, which is a very accessible and easy communication code; still used for emergency communication to send specific messages, such as by using flash lights to signal. Morse code usually assigns (a sequence of ) dots and dashes to each of the English alphabets. For example, the letter ‘E’ is given the symbol ‘–’ whereas the letter ‘B’ is assigned ‘–...’ designed in such a way that the most frequently occurring letter in the English language text is assigned the shortest symbol. Due to such nature of assigning the shortest symbol, it had revolutionised long-distance communication in the form of telegraph, before telephone communication. Why does zero make a good base for communication? It is easy to signal – it is no signal for the agreed duration, e.g., no sound, no light for a second, or the blank spaces in a message. In practice, it would mean ‘no action’, whatever the action is for other numerals. Due to the simplicity of the formation of Morse code, it was used in aviation and maritime shipping and is still used for amateur radio communication. Since Morse code uses only two symbols, these can also correspond to 0 (for dot) and 1 (for dash), thus giving us the Morse code in binary form.

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Similar to Morse Code, binary numbers (base 2), consisting of ‘0’ and ‘1’, are at the foundation of the digital revolution – since computers use binary numbers to store data [since 0 and 1 connect Boolean algebra (the foundation of binary circuits) and binary arithmetic in a simple fashion]. Zero is central to science, too, with specific reference to physics, the idea of zero helped us understand the details of motion in ways that were not possible otherwise. Zero is considered the lowest possible temperature (called the absolute temperature), and we talk about zero-point energy, the lowest possible energy that any physical system may possess. Zero was invented Zero was not a discovery. It is not that we kind of ‘found zero’; it was invented to best explain our world, everyday observations, and living. Zero was created to make quantitative communication easier and simplify the language of quantity. We lived in a world without zero until the beginning of the last millennium (i.e., around 1000 CE). Till this time is that no quantification or computation was done using zero. The Greeks and the Egyptians, in ancient times, used various symbols for numerals such as 8, 80, 800, and so on without having any specific symbol for zero. This made computations tough. For example, to multiply 2 with 80 and to multiply 20 with 80, an altogether different set of symbols and processes were deployed. Whereas, in the current perspective, simple arithmetic just multiplies 2 with 8 and places the trail of zeroes, as occurring in the two numerals at the end of the product. However, the Greeks eventually adopted the symbol for zero as a placeholder from the Babylonians for their work in astronomy; at the level of expressing quantities, it is not anywhere close to how we now use zero as a placeholder. Roman numerals (these are the counting numbers and start from I, first) were also a popular means of quantification, and even now, schools wrongly use roman numerals for operations.

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For example, the statement II + IV = VI is not correct. Mostly, the Roman numeral system was used for the purpose of trading, where they used ‘nulla’ to describe zero. It had no place or recognition of zero until a few centuries ago. Fascinatingly, the idea of ‘nothing’ was felt early in our evolutionary journey, but ‘nothing’ as any quantity/count was thought to be illogical and even irreligious. Unlike the Greeks and the Babylonians, Indians considered zero more than just a placeholder. They, too, represented numbers using symbols but soon realised that zero represented the absence of any quantity. This thought was big enough for mathematicians who began to use zero for calculations. Zero or ‘shunya’ then spread to the Middle East, where it was called ‘sifr’ (meaning empty) and was adopted along with the other Hindu numerals. These numerals are the ones that are used now and are called the ‘Indo-Arabic numerals’. Upon its discovery, zero gave people the arithmetic power to do calculations without needing an abacus. Math became more accessible, versatile, and commonplace in its own right. The pre-zero world – Abacus, the saviour No discussion on zero would be complete without the mention of the abacus. It must be emphasised beforehand that the abacus partly mitigated the challenges presented by the lack of zero; it worked without using zero, its design facilitated simple arithmetic operations. It worked without a symbol/bead for zero. However, the abacus did not resolve the need for zero. The details of how it worked without zero are not relevant for the discussion in this book; after all, the book is about zero and the human revolution that followed its creation. The detail is also being avoided in the interest of all of us not exposed to an abacus. The (European) renaissance (also called the scientific revolution) and enlightenment followed the acceptance of zero in the European scientific and trading communities (and steadily, the larger population). In the early thirteenth century,

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Fibonacci brought zero to Europe, and within the next two centuries, the renaissance was catalysed (in the fifteenth and sixteenth centuries). Fibonacci was drawn to North Africa (a part of the Arab, Islamic world that enjoyed a scientific edge for a few centuries starting the eighth) to quench his zest for learning, where he was introduced to zero. Zero helped him understand mathematics; surprisingly, advanced calculations could be done without using an abacus. These facts about zero were seen as valuable by people in Europe. The era of abacus and roman numerals started to fade just as zero started to shine. Thus, by the fourteenth century, the ‘Indo-Arabic numerals’ which included zero, became the most important way to express any quantity and mathematical ideas. Interestingly, school education still tacitly approves the use of abacus (the best of the schools have no opinion about the right place for an abacus in current times and future). Briefly, the need for zero was felt in many civilisations. For example, the Babylonian system (where symbols were used to describe ‘nothing’) was efficient while working with smaller quantities. However, the lack of zero persisted while dealing with very large quantities. All the non-zero, symbol-based systems of different times became obsolete when numbers had more use than just simply counting the number of bread loaves sold in a day. Post-zero world – Accelerated change We paid the price for the late arrival of zero in our manmade world because the scientific revolution and renaissance literally waited for the discovery of zero. The scientific world had stagnated without the ability to duly quantify observations and manipulate observed values for obtaining more and deeper significance. In fact, the language in which we explore and expound physics is mathematics; Einstein, perhaps the greatest

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physicist of all times, was a theoretical physicist who did ‘mathematical exploration’ of physical phenomena for his path-breaking theories. More specifically, in 1202 AD, Fibonacci started spreading the concept of zero across Europe. He influenced merchants and bankers that any number can be written using the nine Indian numerals 1, 2, 3, 4, 5, 6, 7, 8, and 9, along with 0 for their accounts. The next European to promote the use of zero in the late 1600s was Frenchman Rene Descartes. He used (0, 0) as origin of graph coordinates for X and Y axes in the middle of the 1600s. Then British mathematician Isaac Newton, and German mathematician Gottfried Leibnitz, made further advances. They used zero in a kind of mathematics called calculus, which operated on near zero quantities (such as the speed at an instant). Without calculus, we would not have physics, engineering, and many more domains of knowledge about how our world works. Not surprisingly, the name calculus is derived from Latin root word which means ‘small pebbles’, small quantities. The aforementioned ‘stagnation of science’ must be qualified – the use of metals, their extraction, alchemy to make mixes such as bronze and brass, etc., has a recorded history of nearly 5000 years. The notion that the earth revolving around the sun was proposed as early as the third century BCE (and it could be traced even earlier), but the first mathematical modelling of the same was offered in the sixteenth century by the renaissance scientist Nicolaus Copernicus; his work changed the way physics and astronomy would be used to study the universe. Science was not really hypothesised and then ‘academically’ established/proven in the pre-zero era. Hit and trial and the experience were the basis of science (not bad, but highly local in context, for example, the iron age was not universal, nor even around similar times across the world.)

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Also, the first ‘scientific revolution’ may actually be in the Islamic golden age, beginning in the eighth century, almost entirely coinciding with the birth of the Indo-Arabic positional decimal number system (founded on the key digit zero). It marked a period of rapid growth in scientific understanding. Math to the rescue of science – The source of acceleration Let us further explore the relationship between math and science, for a lot of math evolved due to the need to express and extrapolate relationships observed in nature. This chapter also develops on the back of understanding the science of motion. Science is the study of nature – physical (everything that happens due to the action or absence of forces), chemical (everything that is reflected as a change in the nature of matter), and biological (everything that is about living things) – a domain of knowledge that involves a change in the state of things. Further, nature, and thus, science is very ‘predictable’ – strictly follows a chain of causes and effects. In other words, nature is much patterned (precise, repetitive, similarly organised). Math proved to be just the language that science needed to observe and explain nature. Math is the language to capture and communicate patterns (precisely express real-world, scientific relationships), it is the only language to uniquely express conditions and real-world situations (no two such situations are mathematically written the same way), as well as simplify our universe through codifying a set of sub-patterns that ‘add up’ as needed to match all kinds of real-world situations and conditions (a set of parent functions can take care of most complex real-world situations). The following picture shows how math uniquely captures and expresses physical realities.

Calculus For Professionals

Mathematically, this is 2 × 3 crayons

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Mathematically, this is 3 × 2 crayons

No less importantly, math can also express changes very efficiently math is the only way we can see changing outcomes for changing (input) variables of situations. The mathematical concept called functions mathematise changing state of things and become the medium of a world of mathematical manipulations to visualise, simulate infinite scenarios of interest. Physics particularly benefitted from math because it is the kind of knowledge that organised our understanding of the world powered by force (and motion). Physics is the only domain of knowledge that is universally valid, and that is why many see physics, and math, in particular, as the language of the Gods (if the universe is the work of the Gods). Physics is wholly expressed in mathematical terms. But there is another important relationship between physics and math – a lot of math advanced and developed due to the work of the physicists; Newton’s invention of calculus is just one example. To solve the mysteries of the world, physicists developed new insights and created new mathematical concepts and operations. Other physicists who developed the field of mathematics are William Playfair, a Scottish engineer who founded graphical statistics. Leonhard Euler, Swiss mathematician, and physicist, founded pure mathematics. Their contribution has made significant development in the field of mathematics. One of the most important ways math helped physicists, and they reciprocated, is the challenge of instantaneous measure of (fast and slow changing) physical conditions. For example, physics could not talk of instantaneous velocity, or acceleration,

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i.e., when time and distance measurements are nearly zero; it is assumed here that not much distance can be traversed in a very small amount of time (though this discussion equally holds even if this assumption does not hold to be true). The challenge is the physical measurement and quantification of such a short time and distance. However, math does not have the limitation of ‘smallest quantification/number possible’ because we can always find a new real number between any two real numbers; mathematically expressing scientific realities allow much powerful analysis and computations. Thus, in this context, math gifted science with a completely new possibility; division by infinitesimal or ‘nearly zero’ numbers and being able to address problems that involve ‘instantaneous’ change. Of course, there are almost infinite maths applications in physics. Welcome to the reality of change ‘Change’ is the law of nature. Not many things are static in nature. Our body keeps renewing itself by the second. Living beings are constant works of change – all life processes work 24 × 7, creating a different state of the body every other second (or less). The skin cells undergo regular birth and death cycles, and so are the cells in all other organs; the position of planets and stars continuously shift; the state of health/illness of a patient varies in response to medicines; weather conditions hardly remain the ‘same’ even for a few minutes, and nor is the condition of air quality. And all these constant changes are natural. And same is true for the man-made world – most things change their position, shape, pressure, strength/stress, etc. For example, the pressure of water in a pipeline (on the walls of the pipes), the temperature of the engine or the tyres of a moving car, and the stress in the cable of a moving lift are just a few physical situations with changing conditions.

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The precise way to explain most situations around us is by defining how they are changing, i.e., by knowing their quantitative state as they change at different instants of time. Thus, the need for knowing instantaneous conditions of things is important as change is the commonest reality of life and the universe. Wool pulled over our eyes Does this introduction of math as the language for understanding changing quantities somewhat conflict with the algebra, geometry, polynomials, and mensuration studied till Grade X? Not really. The scholastic math till Grade X, and the math we use in our daily lives, is more about discrete quantities. The latter is about quantities that change in expected/known amount/magnitude, at expected/known times, for example, the average marks obtained by a student in an examination, the number of students present in a class on a particular day, or the number of cars owned by a family. Mathematically, discrete quantities are easily expressible using integers or fractions; discrete quantities can be counted. The graphs of discrete quantities would be a collection of discrete points, i.e., specific values at certain points. The changes in discrete quantities are easy to calculate. Why? For example, in the graph below, the difference between marks obtained in Maths and Hindi can be easily calculated as 20 (= 100 – 80).

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On the other hand, there are quantities that change so that the difference between their magnitudes before and after the change could be any amount, these quantities are called continuous quantities. As these quantities vary, we find a way of knowing their specific values as they change. Often, time is the variable with respect to which we calculate the specific values of the changing quantities. For example, the amount of medicine in our blood at different points of time after taking medicine will vary indeterminably (it depends on many factors). The concentration of pollutants in the air in a location would also vary, and the changing height of a child up to the age of 5 years also varies in a way that is unique to every child. Mathematically, such quantities are expressed as real numbers that can have any specific numerical value. The graphs of such quantities show the variable (shown on the x-axis) and the specific value that quantity assumes at different points of the variable (shown on the y-axis). The following graph is an example of a continuous quantity – the average monthly temperature of a place. The temperature values vary in a day, over days in a month hence they are averaged. The graph is also connected, uninterrupted, because it shows continuity in temperature. The temperature at any point in a day could have any specific numerical value (within a range).

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However, it is difficult to calculate changes in these graphs compared to the discrete graphs, as in discrete data, every value is distinct and can be changed or manipulated. However, when values are given for different ranges in continuous data, any deviation in the values cannot be easily ascertained as values are for a certain range. We adequately know the math that deals with discrete quantities. We need a ‘new math’ for working with continuous quantities. There is a need for a language to understand the natural order of things that are continuous and can change at any point in time and in any magnitude. The real-world It is important enough to re-emphasise that situations around us are always in a state of flux (one which is continuously changing). For example, the traffic that flows on the road displays continuous variation in the density of vehicles at different points on the road and at different times. Most simply, all things in motion are in a continuous state of change – slow, fast, or periodic, but changing nonetheless. Speed, direction, or both, may change for things in motion. Even in circular motion with ‘same speed’ there is a continuous

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change in direction. In general, things in motion routinely experience varying acceleration and velocity. Constant speed or direction (speed in free fall) is rare or misperception. It is easy to visualise that things in a state of continuous change pose a significant challenge to accurately quantifying the state of the changing values. For instance, for a car in motion, the most apparent changing feature of the car is its position at different instants of time. As a clock ticks ahead, a moving car will be at different position as it moves. The change in position could be fast (changing every second, for instance) or slow (changing every month). Changing positions – The visible motion Changing positions is the appropriate basis to uniquely define a motion. All things in motion can only be at one position/place at any given instant of time; reporting a (change in) motion in terms of the trail of changing positions is the way to identify and describe every motion. However, is changing position at given instants an easy way of describing motion? Let us explore if the calculation of position is practically easy. Positions, or ‘points’ on a plane, are specific physical quantities, but we need two points to register a motion, the starting and finishing point of a motion. Here lies the first challenge – between two points, there are infinite pathways and unique motions, as shown in the picture given below. Evidently, two positions cannot define the ‘chosen path’ of a motion, the exact distance, and the directions of the motion.

Possible path of motion between two points (infinite paths)

Possible path of motion between few points (infinite paths)

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Can a few more than two positions define a complete path of a motion? Apparently, not. We would need infinite points of position between the start and finish point of a motion to uniquely, characteristically define a motion, a totally unwieldy situation. A

B A B Very many points of position show a unique path of motion

Positions – Not a practical anchor for motion Positions are not a practical way of defining the change in motion. But whatever the practical way, it has to be based on a change of position. The latter is the correct and obvious way. Distance is the physical quantity expressed simply as one number, and it is the name given to a set of infinite positions – the actual path of motion – that lies between the start and finish position/point.

Distance – The practical choice for quantifying motion In general, the change in positions over time is recognised through the physical dimension called distance, the length of the ‘chosen path’ between the positions. The (total) distance

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travelled by a body is the most visible physical quantity continuously changing when a body is in motion. Thus distance become the only feasible, practical primary descriptor of motion. The quantity of distance travelled may be immeasurably small, or several light years, in just a few seconds. Investigating distance as a measure of change in motion Accepting that distance (travelled) is the best way of measuring the change in motion, we must ask ourselves if the description of motion in terms of distance does best capture the dynamics of motion – the flux, the ‘intensity of motion’, the ‘heat of motion/change’. For instance, what inferences may be made out, for certain, regarding the motion of two cars if one of them travelled a distance of 300 km in a day and the other a distance of 600 km on the same day (or another): • The first car travelled for less time to reach 300 km. • The second car travelled for more time to reach 600 km. • The first car must be an older car, or the driver was older. • The second car must be a new car, or the driver was younger. • The first car is closer to home (or the starting point). (Hint: think of displacement)

• The second car is farther from home (or the starting point). (Hint: think of displacement)

We can think of more descriptors of the two motions. However, nothing can be inferred with certainty on the aforementioned six conditions. The only fact we are sure of is that second car travelled a longer (double) distance compared to the first car. Surprise as it may, the distance by itself hardly describes the quality of motion, the exact nature of the change taking place. We need more than the distance travelled to know enough about motion.

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Distance and time – A better story? Time is the other feature of motion – a motion happens over a period of time; the change of positions in motion happens within a non-zero time lapse. Motion can not be accurately defined without mentioning time. Let us explore if the inclusion of time, over and above the distance travelled, gives us any better information about the quality of a motion. The situations and conditions presented earlier has additional information. One of the cars travelled a distance of 300 km in 2 hours, and the other covered a distance of 600 km in 6 hours. Which of the inferences may be more assertively responded to: • The first car was faster to the destination. • The second car was faster to the destination. • The first car must have travelled for 150 km in 1 hour. • The second car must have travelled for 100 km in the first hour. • The first car must have travelled 150 km in the next 1 hour too. • The second car must have travelled 100 km in each of the next 5 hours. • The first car was always faster than the second (at all moments during the 300 km journey). • The second car was always slower than the first (at all moments during the 600 km journey). More descriptors could be explored, but these eight descriptions duly support our current scope (a text of math rather than physics). The outcomes of the analysis of these conditions are: the first two descriptors are correct and come straight out of the fact that time of travel is known; the third, fourth, fifth, and sixth descriptors are possible but highly unlikely; the seventh and the eighth descriptors are not necessarily true at all, but quite likely. What does this analysis say about time and distance as the known quantities of a motion? Clearly, the knowledge of the duration of motion, besides the knowledge of the distance

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travelled, does not throw any extra light on the nature of motion, except for what the time and distance say about a motion independently. Recall time and distance are the only two unique, directly measurable physical quantities about a motion (on a plane surface). To be right, if we want to know more about a motion, we have to go beyond the discrete time and distance quantities of that motion. For instance, analysing the third to eighth descriptors requires additional knowledge beyond time and distance. Yet, let us also know that we do not have another physical quantity that comes out as a measured outcome of a motion. We must be able to use time and distance to create additional descriptors of motion. How time and distance can be deployed to create any additional descriptor(s) of motion? Can the two be linked together to study their joint effect in describing a motion (that is the only new possibility to use the two)? Can we find a relationship between the time and distance of a motion? Relationship of time and distance (of motion) Mathematics can come to our rescue here! Recall that math is the language we use to quantitatively define precise relationships, especially those that show any specific pattern. Relevantly, in all instances of motion, the time of being in motion and the distance travelled in that period show one basic relationship – they both increase together (more time in motion means more distance covered even for the slowest possible motion). Prima facie, time and distance are closely related, change in time changes the distance travelled by the moving body. How may math describe such a relationship? Rate is that specific mathematical relationship that quantify change. Rate – A relationship between two quantities Mathematically, rate is a special kind of ratio but a very commonly used ratio. It helps us to compute the change in one

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quantity due to the change in another quantity, expressed as a ratio of the first and the second quantity. Some of the simplest examples of rates are – price per dozen (any change in the ‘number of dozens’ will change the price), minimum wage (per hour, day, week, or month), heart rate, two apples per 100 (half of two apples – one apple – for half of 100). Rate quantifies how much something changes with respect to any other quantity; time is just one quantity to measure the change against. The rate can be expressed for any quantity, for example, price per seven dozen, the minimum wage for three days, heart rate per hour, and 5 apples per ₹250; after all, it is a relationship. It makes more practical sense to express it in terms of (one) unit quantity of one of the two quantities, for example, price per dozen rather than price per seven dozen. In the case of motion, ‘rate of motion’ can help us know the distance travelled in a given period of time. Further, it helps that the independent quantity is expressed in one unit, it makes the computation of the dependent quantity easier; in the rate ‘price per dozen’, ‘dozen’ is the independent quantity, and price the dependent one, and in the rate ‘distance per hour’, hour (time) is the independent unit, and distance the dependent unit. Rate of change between two changing quantities Rate can also be seen as a measure of change, a mathematical expression that captures how something changes. It is used to quantify change. However, it quantifies the change in relative terms. It defines a change in one quantity with respect to a change in another quantity. Specifically, this relative change is what we call ‘rate of change’, and to be precise, the ‘rate of change’ quantifies how much one quantity changes due to one unit of change in the other. For instance, the rate of change of the area of a rectangular field due to every unit of change in the length of the field or the rate of change of the (simple) interest amount due to every unit of change in time period. The rate

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of change is expressed as a quantity (of something) per unit of some other quantity. It may be clarified again that the rate of change of any quantity may not always be with respect to change in one unit of time; it could be with respect to any kind of changing quantity, as exemplified above (though time is a common quantity used in defining rate of change). Speed – Mathematical relationship between distance and time As the motion continues, distance and time change. We can quantify the rate of change of distance with respect to change in period of time in motion. Speed is the name given to the ‘rate of change of distance, over time’! Quantitatively expressed as speed, this relationship of distance with respect to time tells us more about the nature of the motion of which we know the time and distance traversed. For example, if we define the speed of the motion of the two cars, the following descriptors could be affirmatively analysed and responded to: • The first car must have travelled for 150 km in 1 hour. • The second car must have travelled for 100 km in the first hour. • The first car must have travelled 150 km in the next 1 hour too. • The second car must have travelled 100 km in each of the next 5 hours. • The first car was always faster than the second. • The second car was always slower than the first. Knowing the speed of the two cars will help us respond confidently. For instance, if the speed of the two cars comes out to be uniform at 150 km/hr and 100 km/hr in the first hour, respectively, then the conditions are true; the third and fourth conditions would be true if the calculated speed is indeed 150 km/hr, or 100 km/hr, as the case may be (though the first car will have travelled at 150 km in the second hour). Interestingly, however, the fifth and the sixth conditions can also be analysed by computing speed, but we will soon discover a challenge in doing so.

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Mathematics of speed as rate – Gross or average speed Speed is a rate of change of distance over time, and arithmetically, Distance Speed = Time In this calculation, distance is the total distance travelled in the given time. For example, the calculation of the speed of a car over 30 minutes if the distance travelled is 60 km is 2 km 2 km 60 km = 2 km / min = = = 120 km / hour Speed = 30 min 1min (1 / 60 ) hr What is the significance of speed in terms of 2 km per minute? It is not that the speed is constant at 2 km/min for each of the 30 minutes, or speed is such that 120 km will be uniformly travelled for 1 hour, and the travel will be abandoned halfway through (i.e., at the end of half-hour), at 60 km. 2 km/min is the average speed, a gross measure of speed. In fact, the speed of 2 km/min may never be an actual speed achieved by the car. It is a kind of ‘unreal speed’, a speed that may not be true at any instant of the motion of the car. Pertinently, this unrealness of the average speed could become a source of a mishap. For example, a vehicle skidding away at sharp turns on the road, a vehicle brakes suddenly, and is hit by another vehicle behind it, or even being challaned for slow driving on a fast-moving lane. What explains these situations linked to the speed of a vehicle for which the average speed is known and does not warrant such miscalculations? The idea and computation of average speed is not good for microdetails of the motion; it is not anyway similar to the numerous instantaneous speed. Indeed, all the common arithmetical computations of the rate of change of distance, over time, is gross – average. It is not the real speed, and it may not be the actual speed at any time during the entire period of a motion.

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Gross to particular We need a new way to measure speed to know the actual speed of motion at any instant or set of instants. For this we will study the nature of change in motion and find a way of computing the speed at any instant. The instantaneous values of quantities that change is important to study, record the exact nature of change, and extrapolate the values to predict the change in times ahead. Hopefully, it is very evident that across the dimensions of everyday living and nature, average as a means of quantification of quantities that are changing is highly deficient, it misses the changing nature of things, and by implication, misses the dynamics and reality of things. We know the rate is the only mathematical implication of (quantifying) ‘rate of change of distance, over time’, or speed when talking of changing motion, but the rate offers the ‘average’ of the distance traversed over a period of time. The idea of ‘average’ is embedded in the idea of rate – it ‘evens out’ the distance traversed over a period of time. Average assumes the ‘evened out’ rate of change – the one value of the rate of change to be the rate of change during the entire period. But the reality of changing motion is that the average is not the same as the actual values at different instants. Instantaneous values of (changing) quantities Revisiting the earlier situations in motion, the following cannot be explained except by knowing the instantaneous speed at all the times in the motion: • The first car was always faster than the second. • The second car was always slower than the first. The rate could well be used for computing any instantaneous value by reducing the unit of time to be near zero (instantaneous), i.e., by taking the duration of a time period to be near zero and measuring the distance traversed in such a duration.

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The challenges in computing instantaneous speed using the idea of rate are obvious, as under: • Physically, and in most everyday situations, it is impossible to accurately record time which is less than a few seconds and to correctly record the distance traversed in very small durations (by accurately pinpointing the start and end points of motion in that duration); any error in computation will get greatly pronounced because the measured quantities can only be small in magnitude for such small measurement windows. • Mathematically, rate involves a division operation by almost zero (at an instant) when computing instantaneous values. The solution to this division by almost zero evaded centuries till a new mathematical foundation for infinitesimal quantities was invented. The advantage of math is that it knows no limit on the magnitude/quantum and these infinitesimal quantities, however small, can be quantified/derived. Real numbers are just for that; math can indefinitely create smaller and smaller intervals to give us the most accurate measure of the instantaneous value of changing quantities. • Conceptually, what is ‘near zero’, what number may be nearly zero, is it 1 second, 0.1 second, 0.01 second, or even less, when measuring the time period of changing motion; how small is really as small as possible? There is also the issue of the ‘real’ quantitative difference between any infinitesimal and zero. For example, how does one really visualise the difference and distinction between 0.0001 and zero? But let us not forget that smaller the infinitesimal, the more accurate the computed instantaneous value. • Importantly, the idea of infinitesimal also implies infinite instantaneous values within the smallest time intervals (or any other changing quantity). This understanding will be required to appreciate certain transformations truly. And then infintesimal would also vary with situations with the kind of changes being studied.

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These challenges were specifically addressed. For example, the physical challenge (to find the distance travelled in small time periods, given an object in motion) was addressed using the idea of derivative that avoided such difficult measurements the idea of limit addressed the mathematical challenge (the mathematical formulation of finding the distance traversed in a small time interval) and the conceptual challenges (that such small quantities do exist) were addressed using the idea of continuity. Avoiding measurement Given the challenges of accurate physical measurements of infinitesimal time and distance traversed in that time in motion, we need to invent a non-physical way/method of finding instantaneous values. Howsoever long the distance may be, accurately figuring out the start and end points of motion during the given infinitesimal time is next to impossible. Measurement almost always comes with some or the other kinds of error. It implies that we must develop a new concept, an indirect way of computing instantaneous values. Indeed, this is exactly what we do – we have defined a new meaning that can be derived from something else and not computed out of any instantaneous measurements. To be exact, we ‘convert’ a motion, or any other given changing quantity, into what we call functions and then use the function to get the instantaneous values of the state of motion. Functions are mathematical expressions that represent the relationship of quantities; in the case of the everyday motion of things, the function of such a motion will express the relationship between time, speed, and distance. Functions – The gateway to instantaneous world We ‘functionalise’ situations everyday to mathematically capture the changing nature of the relationships embedded in

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such situations. The above examples show real-world situations following expressed as functions. Force = Mass × Acceleration Distance Speed = Time Velocity Acceleration = Time Weight = Mass × Acceleration of gravity y = ax + b where (x, y) is the coordinate of a line. Most importantly, and expectedly, using functions to find the instantaneous values of the variables is among the greatest mathematical inventions. It is at the heart of what calculus is all about. We use functions of the real situations to derive instantaneous values at all instants of the changing quantity (without any measurement of any kind), by the process called derivation, and the instantaneous value is called the derivative. By a mathematical process called derivation, we use the function of motion to derive speed of the motion at a point in motion (it is the derivative of the function of motion at that point in motion). The importance of functions could be understood in terms of converting harsh physical changes, i.e., real changes, to completely virtual changes, for example, testing the integrity of a valve to the increasing pressure of the fluids to flow through it. One of the most amazing realisations of a better understanding of the real world using the language of math is that the infinitely colourful, diverse, and finely differentiated things and their interactions can be accurately represented by (mathematical) functions. The behaviour of well-conceived and expressed functions closely resembles the realities. Importantly, and obvious as it may be, the use of functions in place of actual situations involving physical actions and objects not only saves costs and

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make the study of changes simple, and it also makes the study of many situations feasible; the impact of a car crash on the head and neck of human passengers cannot be tested in reality, and there is more; using functions also increases the accurate prediction of the changes. Above all, functions and the mathematical operations that calculus works on have expanded our ability to understand our world beyond what we can learn out of purely personal experiences, way beyond the physical touch and feel of the ways of the world. We have abstracted the real world into functions and operations on them. To be true, derivation, limit, and continuity form the foundation of calculus – the domain of math focused on understanding and predicting the behaviour of all real-world situations. To reiterate, it redefines these situations in very small steps/parts/moments and then studies them with very high accuracy. In fact, the idea of calculus is a milestone in critical and creative thinking abilities. It is something all children need exposure to and training to hone. Yes, the ‘mechanics’ of calculus is tedious, but we hope to simplify that, too, in a very inventively presented functions chapter. Derivative of function Derivative of a function is the mathematical operation that works on a function to ‘derive’ indirect meaning(s) out of the function. We know the direct meaning of functions is in the primary relationships they hold; for instance, interest amount is the direct meaning out of the function that relates interest earned on a principal amount deposited in a bank for a time. It is the quantification of some new aspect of a changing quantity, for example, knowing speed and acceleration of a motion whose time and distance traversed is known. To be precise, derivative gives a very specific new knowledge about something in change - the instantaneous value of ‘something’ about the change in ‘something else’. For example, in the case of motion, the rate of change of distance (the ‘something else’) at

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any instant in motion is the value of the speed (the ‘something’) at that instant. The conceptualisation of derivative of a function can be visualised as detailed hereunder: • Something is continuously changing (slow, fast, regular, irregular, …); for example, a car in motion is continuously changing its position. • The change in position is measured by a (physically measurable) quantity (distance). • The quantity that reflects the change is motion (distance) changes. • The rate of change of that quantity could also be changing (the change in distance, with respect to time, could be variable). • The instantaneous value of the rate of change represents something else (speed). • The value of ‘something else’ is the value derived out of another quantity (speed is a derived quantity out of distance). • The derived quantity is, thus, called a derivative (speed is a derivative). • The derived quantity is not a primary measurable observation (speed is not a directly measurable quantity). • The derived quantity also changes in tandem with the primary quantity (speed varies as the distance varies). • The nature of the derived quantity is not given. It takes an intensive understanding of the thing that is changing to describe and define the derived quantity (to appreciate the nature of speed, we must command an understanding of motion, the real-world motion). In general, derived quantity is a new, special function derived from another function. The derived quantity is the value of the function (that is changing) at an instant. Theoretically, it must be possible that derived quantities can be used to derive another quantity (acceleration is derived out

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of velocity – speed and direction), and we can also ‘un-derive/ anti-derive’ the primary quantity (distance from speed). And it is possible. To sum, derivative measures how fast or slow something changes, and that rate of change is the value of something else (different from the thing that is changing). Finding derivatives of changing quantities is about measuring the nature of change of the rate of change of all kinds. And derivatives are about a specific value (or point) lying in the domain of a function. Primary Derived Quantity Quantity Energy/Work Power

Explanation

The rate of change of energy or work per unit time is the power of the body at an instant. Momentum Force The rate of change of momentum of an accelerating body is the force on the body at an instant. Electric Electric current The rate of change of electric charge charge is electric current at an instant. Flux Electromagnetic The rate of change of magnetic force flux density is the value of induced electromagnetic force at an instant. Change of Quantity The rate of change of reactant’s of product reactant’s quantity at any quantity formed point in the reaction gives the quantity of the product formed at that instant. Consumption Marginal utility The change in total utility rate per unit change in the goods consumed gives the marginal utility.

Calculus For Professionals

Amount of heat

Heat flow rate

Cost function Marginal cost

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The amount of heat transferred per unit time is the heat flow rate. The rate of change of cost function at any production level gives marginal cost at that production level.

Time is not the only dimension The rate of change may not be with respect to time alone. There may be other dimensions as well. For instance, we can consider a change in the volume of a spherical balloon with respect to its radius or its surface area. Here, the rate represents surface area or radius. Generally speaking, the change can be with respect to any quantity. Anti-derivative As the name suggests, it is mathematically the opposite of the idea and the operation of derivative. Let us construct the understanding of anti-derivative one dimension at a time, out of the definition of derivative. We know that a derivative is a function that gets created from another function. Thus, the anti-derivative must also be a function created by another function; (both, ‘input and output’ of derivative is a function, so the reverse view of derivative will also be ‘input and output’ as functions) for example, the function of speed gives the function of motion itself, the function of distance (motion is about a change of positions, distance) to be precise. Function of Distance

Derivative Anti-derivative

Function of Speed

Next, we know that a derivative is a rate. The implication of being a rate is that it is a slice of the action, a ‘part of a whole.’

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For example, the speed at an instant in a motion. What may be the anti (or opposite) of a ‘part of a whole’? ‘The whole’ itself. For instance, what may be the anti-derivative of the (derivate) speed? Let us take a second to scrutinise speed; it is the distance travelled in a unit time (whatever be it), it is a part of the (total) distance travelled. Indeed, a speed of 54 m/s implies that each slice of 1 second of motion is a distance of 54 m. The anti-derivative of speed is, in fact, distance, ‘the whole’. Of course, in the story of anti-derivative, ‘the whole’ need to be identified because ‘a whole’ could also mean the universe. ‘A whole’ as anti-derivative needs to be specifically limited, and that is what is next explored. We also know that derivative is the value of something at an instant. What may be the opposite of an instant? A period of time, an interval of time. Indeed, anti-derivative qualifies (i.e., further explains) the quantity discussed previously (total distance travelled). Indeed the opposite of an instant is a period of time. And to the extent that anti-derivative operates over a specified range of the values of the changing quantity, the specification of that range is an input in finding anti-derivatives. Thus, the anti-derivative quantifies how much something has changed over a time period. Anti-derivative is kind of a difference between two quantities – one at the start of the period, or a state of things, and the other at the end of the period, or state of things. Another way to see it is the summation of the changes over a period or the range of change of another thing. Common examples of the application of anti-derivatives, to find out the amount of change in something (that is changing) are – the growth of a tumour inside the body of animals which can be derived from the rate at which the tumour grows in the body over a period, the speed slowdown of a self-driven vehicle while plying on the road to stop hitting the vehicle in front (in the face of usual traffic ahead, for instance), the impact of a head-on collision of two cars in the sense that the impact

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is a series of (very fast happening but) small changes after the moment of collision (and that cumulative change is the anti-derivative amount of the function that represents the crashing parts of the car). Here are a few more examples of the deployment of the idea and operation that is anti-derivative: Average/ Function Function

Area

Density function

Magnetic flux

Anti-derivative

Explanation

Average value of Average value of a function over a function a range is the anti-derivative of the function. Volume Volume is the anti-derivative of the area; over a dimension; similarly, area is the antiderivative of one dimension. Mass Mass of an object is the antiderivative of a density function (which is mass per unit volume) over a given volume. Magnetic force

Acceleration Average velocity

Magnetic force is the antiderivative of magnetic flux over an area Average velocity of a moving object over a period is the antiderivative of the acceleration in that period.

But how do we get the distance traversed over time, given the speed of a car in motion? Very interestingly, it is not very intuitive and the best way to appreciate how we get the distance from speed and time of motion is to visually explore the same using a graph.

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Graphically exploring anti-derivative of velocity.

The above graph represents a car in motion having a linear speed, or we can say that the velocity increases linearly with respect to time, that is at t = 1 min, it has a speed of 1 km/min; at t = 2 min, it has a speed of 2 km/min. Since displacement = velocity × time, it can be best interpreted by the shaded region. The anti-derivative of velocity is the area of the shaded region, which is the displacement of the car in motion. Infact, the anti-derivative of a function is a quantity that is the area under the curve of function. Mathematical expressions using derivatives (Differential equations) Recall that algebraic expressions are combinations of constants and variables which are put together using mathematical symbols, and algebraic equations are expressions that are set equal to zero. It is interesting to think that equations can also have expressions that incorporate changing conditions quantified through the rate of change. Such expressions are common, we mathematically express them every day and when used under scientific conditions, for now, they are called derivative equations.

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Do not be startled if we say that the idea and application of ‘differential equations’ is a primitive biological, animal instinct. We are pretty adept at using them. For starters, maybe winning a race is such an adrenaline producer because it uses the brain’s raw/god-gifted logical abilities of calculus – gauging the increasing or decreasing distance with the runners up in the heat of the sun, as things change dynamically. The conscious and engaging assessment and extra push mid-air when jumping across a ditch ensures that the jump is successful; it is truly a mathematical decision based on the derivative (rate of jump) and anti-derivative (the extent of jump). It is also a common animal instinct and ability. Wherever there are changing quantities in the ‘equation’ of thing, the situation is mathematically expressed as differential equations. These equations can be used to configure everyday life to rocket science. The laws of nature, and dynamism in science and maths can easily be explained through differential equations. Think of the way we go across a busy road in many of the cities across the world – constantly juggling with the estimated speed (rate of approach), the closing distance and the fast-approaching vehicles, the distance left to be crossed, the personal speed to crossing over, as well as the obstructions on the way (the other people crossing over from the opposite direction, for instance); it is a fairly complex situation of changing dimensions, but most of us have gotten it right every time. Some examples of differential equations in real life: 1. Any change in human body temperature is a response to changing conditions outside and within the body, such as ambient temperature, the type of food eaten, the type of clothes worn, the type of activity performed during a particular time, and more. For example, if a person is doing an exercise, this would increase the heart pumping and blood circulation rate, thereby increasing the temperature. So, ‘in the equation’, the temperature of a human body responds to various changing conditions underlying it.

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2. Calculating the time required to drain a tank full of fluid is where the differential equation comes into play. Draining time depends on various factors like the volume of fluid, the air pressure, the height of the tank, the density of the fluid, the rate of flow, the shape of a draining hole, and more. Any change in the above factor may impact the draining time of the tank. For example, if water and petroleum are put in the same type of tanks, their drainage time would be different due to the difference in density. Similarly, if the shape of the draining hole is small, it would impact the draining time. ‘In equation terms’, factors like the shape of the hole and height of a tank are constant terms for a specific tank for all fluids, while density, the volume of fluid, and air pressure are differentiated with time to estimate the draining time. 3. The value of the National income of a country is dependent on various factors like general price level, aggregate demand, aggregate supply, compensation to employees, saving rate, government policy, and more. These factors are dependent on the inflation rate, total production of goods and services, wage rate, marginal propensity to consume, etc. All the factors are interlinked and are dynamic in nature. For example, the general price level of a country increases due to an increase in the inflation rate, which may change due to government policy or a change in supply. ‘In equation term’, compensation of employees and saving rate are constant terms for a period while general price level, aggregate demand, and supply are differentiated with time to know the value of national income. 4. The growth of bacteria is exponentially fast. To calculate its growth, we need to know the amount of bacteria present at that particular point. If the bacteria present is higher, this will lead to multiple fission. But the growth rate would depend on factors like oxygen level, temperature, light requirement, and pH level. As these factors are dynamic, any favorable change would enhance the growth of bacteria. For example,

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certain bacteria grow well in cooler temperatures and die at higher temperatures. So, ‘in equation terms’, the amount of bacteria present is considered constant at a certain point in time while other factors are dynamic. 5. Hot coffee cools and comes down to room temperature. Differential equation explains it. Hot coffee cooling depends on the temperature difference between that of coffee and its surroundings. If the temperature of the surrounding is cool, then the rate of cooling will be faster. This, in turn, is affected by the quantity of the coffee. If it is a large quantity, then there will be a decline in the rate of cooling. So, ‘in equation terms’, the quantity of coffee is kept constant while the temperature is a variable factor. 6. The differential equation is used in a video game to determine rate of motion of an object. For example, consider the static force diagram for a ball rolling down a ramp. Knowing the time duration in which it will roll down depends on various factors like gravity vector, mass of the ball, and the angle of the ramp (its normal vector).These factors are further dependent upon the net force applied on the ball and the acceleration of the ball in that frame. So, ‘in equation terms’, the mass of the ball and gravity vector is considered constant, while other factors are variable and differentiated with respect to time. 7. Radioactivity is the spontaneous emission of particles or electromagnetic radiation from the unstable nuclei of an atom. Various particles (protons, neutrons) are emitted from nuclei to become stable nuclei. To calculate the rate of emission of particles, a differential equation comes into play. The emission of particles is further dependent upon environmental factors such as pressure, temperature, chemical form, magnetic field, etc., which are changing. For example, when unstable nuclei emit particles to become stable, the rate of emission depends upon temperature. The more the temperature, the rate of radioactivity would be

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higher. So, ‘in equation terms’, the emission rate, magnetic field, and pressure at a particular point are constant terms, while temperature and number of particles in nuclei are variable terms. 8. The regular to and fro movement of the pendulum in a clock is an interesting sight to look at. Wonder how this movement takes place? As the bob of the pendulum is attached to a light inextensible string and suspended from a fixed point, its movement depends on gravity, the string’s length, the bob’s mass, and the pendulum’s position in relation to the earth. The to and fro motion can also be examined in a child’s swing. The more the length of the swing, the more the oscillatory movement of the swing from its mean position. The more the child’s weight, the lesser the swing’s oscillatory movement. So, any change in variable factor affects the to and fro movement of the pendulum. ‘In equation terms’, the gravity of earth and position of the pendulum in relation to earth are constant terms, while the length of the string and mass of the bob are variable terms. 9. To invest money in equity (stocks), one must know the financial market conditions. Various tools are made to predict the fluctuation of stock prices. This fluctuation is dependent upon factors like inflation rate, technological advancement, level of trust in the legal system, corporate performance data, government policies, and more which are further dependent upon aggregate demand and supply, monetary policy, natural disasters, etc. For example, if it is predicted that the price of the stock of company X will increase, this prediction would be based on the company’s performance data which in turn would be dependent upon the inflation rate, the company’s policy, the profitability of the company, etc. So, ‘in equation terms’, legal system policies, level of trust in the financial market, confidence index, and fiscal and monetary policy are constant terms while a company’s performance, inflation rate, supply and

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demand, news on war, conflicts, etc. are variable factors. 10. Earthquake-resistant buildings designed by civil engineers are modelled using systems of differential equations. During an earthquake, vibrations of the ground cause a large horizontal force to be applied to structures leading to the displacement of structures. The magnitude of this displacement depends on the earthquake’s frequency and building characteristics, which are further dependent on the amplitude of the earthquake, depth of the earthquake, underlying soil, and their interaction. So, ‘in equation terms’, underlying soil and building characteristics are constant terms, while proximity to the fault, earthquake depth, magnitude of displacement, and frequency of the earthquake are variable terms. It is invigorating and confidence-boosting to figure out the extensive use of differential equations around us. Here are ten such domains of science and technology where differential equations are the only way to conceive and create meaning and applications. An interesting aspect of differential equations is that, unlike algebraic equations and the math we know, the ‘solution’ of differential equations is not a quantity (or a set of quantities) but another function. Such solution might be expected because when we deal with derivatives, we essentially deal with functions. It means that differential equations give us a ‘modelled’ behaviour of things, not a particular instance of behaviour. And there is often a set of solutions for a given differential equation. A few famous equations in physics which depict the rate of change are: Force = Mass × Rate of change of velocity Power = Voltage × Rate of change of charge Momentum = Mass × Rate of change of distance The conceptual exploration of derivative (and the related

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idea of ‘anti-derivative and derivative equation’) concludes here. We will now study the idea of limit and continuity. We also need a degree of a revisit of certain background concepts before limits and continuity could be easily visualised. A new narrative for limit and continuity We must acknowledge that the conversations on limits and continuity are mispositioned in the education of calculus. It is much like the trigonometry nightmare in secondary curricula, but all due to poor introduction to triangles in middle school. Limit is indeed at the foundation of derivatives and anti-derivatives, but, as we have already witnessed in the introduction of derivatives and anti-derivatives, the latter two may be appreciated in a way that is conceptually independent of limits and continuity. The way of conventional teaching and textbooks makes limits and continuity a gateway to learning derivative and anti-derivative, which significantly contributes to calculus phobia. In this book we avoid that route and we have placed limit and continuity in the appendices, out of the ‘main string of calculus’. Even otherwise, limits and continuity are not directly used in the computational processes of derivatives and anti-derivatives. We use limit and continuity as the universal basis for finding derivatives (and by inversion, anti-derivatives); for example, we can find the derivative of sin x as being cos x using the concept of limits. Pertinently, we use cos x as the derivative of sin x in computations and do not dip into limits for computing the same whenever we have to find the derivative of sin x. Prepping to appreciate limit and continuity To the extent that limit and continuity are conceptual foundations for finding derived quantities using a rate of change of quantities, the slope of the functions becomes an important element in appreciating limit and continuity. Consequently, tangents to curves become relevant to understanding limit and

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continuity, as the slope of the curves varies along the curve, and the slope of the curve at a point is best approximated as the slope of the straight line at the point(the tangent). It will help if we recap slope, tangents for curves, and how the slope of the tangent best approximates the slope of the curve at a point. Once these foundational elements are refreshed, we will explore limit (the science of infinitesimal quantities that ensure the slope of the tangent is indeed the slope of the curve at the point of tan- gency) and how the conception of continuity of a function at a point is computed using the idea of limit. Recalling slope We know that it is easy to find the rate of change of linear motion. For example, where the rate of change of the distance travelled is a constant, as shown in the following graph.

Let us calculate the slope (steepness) of a straight line and check whether the slope varies with the change in points. To calculate the slope of any function, we need to choose a point of interest. Let (x1, y1) be the point where slope of the line is to be determined.

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Let us take any random point, say (x2, y2) on the same line. Then, the slope of the straight line is given by, slope, m =

change in y ( ∆y ) change in x ( ∆x )

=

y 2 − y1 x 2 − x1

Graphical representation of calculation of slope

Calculating slope Consider a line of y = 2x. The line y = 2x would look like the following, as shown in the graph. Let us take any random point Q (1, 2) and a specifically chosen point P (2, 4) on the line y = 2x to find the slope.

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Graphical representation y = 2x

The slope of QP is given by, slope =

y 2 − y1 x 2 − x1

(Formula to calculate the slope of a function)

4−2 =2 2 −1 Similarly, take another point, R (–1, –2), on the given line and find the slope of RP. The slope of RP is given by, y − y 1 4 − (−2) 6 slope = 2 = = =2 x 2 − x1 2 − (−1) 3 =

From the above calculations, it can be interpreted that both points have the same slope. Thus, a straight line has the same slope irrespective of the chosen point. There is more to slope. Let us consider the example of a line y = 5x to visually experience the effect of a bigger slope. The line y = 5x would look like the following, as shown in the graph.

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Let us take any random point P(0.5, 2.5) and a specifically chosen point Q (1, 5) on the line y = 5x. Graphical representation of y = 5x

Now, the slope of PQ is given by, y − y 1 5 − 2.5 slope = 2 = =5 x 2 − x1 1 − 0.5 Hence, the slope of the line y = 5x is bigger than the slope of the line y = 2x. Visually, this is reflected in the graph as the bigger the magnitude of the slope, the closer the line towards the y-axis, and the lesser the magnitude of the slope, the closer the line towards the x-axis. Recalling curves and tangents The ‘curvilinear functions’ pose a challenge when it comes to find the slope of the curve as the slope is not constant and keeps changing along the points on the curve. In other words, the rate of change (slope) of the curved function keeps on varying; the following graph illustrates the changing rate of function

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as brought out by the different slopes of the three (straight line) tangents on the curve at three different points – P, Q, and R. Each tangent is a straight line and has a unique slope (as illustrated below). Graphical representation of tangent at three points on the curve

A tangent at a point on a curve is a straight line that best approximates the slope of the curve near that point. Tangent best approximates the slope of curve Let us get a real-life feel of the continuous change of the slope by exploring the surface linearity of curved objects such as a table tennis ball and a soccer ball. A flatter surface will be associated with the soccer ball. In general, the larger the curved surface, the more the sense of flatness of the surface. It can be visualised that parts of the surface of 3-dimensional things may look rather flat for a very large curved surface.

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For 2-dimensional graphs, we can visualise a similar reality – how enlarging a curve flattens its curvature around a point, and this flatness keeps increasing with every enlargement of the curved surface. On a page, we cannot enlarge anything beyond the size of the page. Thus, to show the enlargement, we will enlarge a smaller and smaller curved portion. We will start with a full curve with a tangent at a point where we may want to find the slope of the point on the curve. Slope of a tangent and slope of a curve Let us see for ourselves how the slope of a curve at a point is best approximated by the tangent at that point (there can be only one tangent at a point on a curve). As drawn below, we start with an ellipse with the point Q and the tangent PR. From this picture, it is not evident that the slope of PR is the same as the slope of the ellipse at Q. P

Q

R

We minimise the size of the tangent PR, as drawn below, and we still cannot see the relationship of the slopes of PR and the ellipse at Q. P

Q

R

We further reduce the size of the tangent and we enlarge the diagram to view the tangent and curve. P

Q

R

Interestingly, as we keep zooming in or reducing the size of the tangent to know the preciseness of the slope of the curve at

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point Q, it will be noticed that a very small area of the curve (or a point) coincides with the tangent line. P

Q

R

Importantly, looked the other way round; each enlargement of the curved surface meant having a closer look at the point of touch between the tangent line PR and the curve. P

Q

R

As we have already discussed, it is easy to calculate the slope of a straight line. This ease of using the slope of a straight line as the best approximate value of the slope of a curve is how we find the slope of a curve at a point on them. In the diagram, the slope of the tangent PR is the slope of the elliptical curve at Q, as the tangent PR coincides with the curve at Q. Thus, the slope of a tangent PR gives the best approximate value of the slope of the curve at Q. Limit Let us get back to the challenge of finding the instantaneous value of a changing quantity and speed as an example of the same. The discussion eventually boiled down to overcoming the limitation of the mathematical expression of average, which is the best conception possible for finding the variation of something over something (or some time). We know how we overcame this challenge by developing the idea of a derived quantity, the derivative – the instantaneous value of speed at a point was seen as the rate of change of distance travelled in an instant (technically, within zero duration of time). Conceptually this was a breakthrough, but computationally, finding the rate of change at an instant remained the challenge (strictly, a ‘zero’ duration of the observation, as well as the distance in that ‘zero duration’, in the case of computing speed). The next best thing is to make the denominator so small that we say it is non-zero but tends to be as close to zero as possible. We need a non-zero denominator for the computation

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of instant change to be possible, but to reflect instantaneous values, the non-zero denominator must be the smallest possible; this property of the varying quantity being non-zero, yet approaching zero is called as ‘the limit of the quantity is zero’. This non-zero, but closest possible to zero, approach calls the importance of the limit of a function f(x) where the rate of change in the value of x is non-zero but near to zero. Now let us take an example of a curve, parabola y = x2, to see how we actually get to infinitesimal quantities around a point on a curve to find its slope at that point on the curve. We will try to find the slope by continuously reaching closest to the point of interest Q (2, 4), on the curve and thus, getting the best approximate value of the slope of a curve by drawing an infinitesimal tangent line around Q.

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Take any random point on the curve, say, P (0.5, 0.25) and R (4, 16), and draw a line to know its slope. The slope of PR is given by, y − y 1 16 − 0.25 slope = 2 = = 4.5 x 2 − x1 4 − 0.5 Now we move P and R further close [i.e., P (1.98, 3.9204) and R (2.02, 4.0804)] to know the better approximate value of the slope of the line PR.

The slope of PR is given by, y − y 1 4.0804 − 3.9204 0.16 = = 4 slope = 2 = 0.04 x 2 − x1 2.02 − 1.98

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Similarly, take another set of points, say, P (1.9, 3.61) and R (2.1, 4.41), which are further close to knowing the slope of a line.

The slope of PR is given by, y − y 1 4.41 − 3.61 0.8 slope = 2 = = =4 x 2 − x1 2.1 − 1.9 0.2 As the points P and R get closer and closer to the chosen point, Q, the line becomes steeper. The steepness of the line at the chosen point (where the line becomes the tangent) gives the best approximate value of the slope of the curve at that point. Thus, taking small infinitesimal values near the point of interest on the curve give the best approximate

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value of the slope of the curve at the chosen point. Another way to look at it is as follows.

y 2 − y1 x 2 − x1 For a curve, x1 is taken very close to x2 and y1 is taken very

As seen, the slope of a line =

close to y2. Thus, ∆x = x 2 − x1 and ∆y = y 2 − y 1 are very small. Also, tan θ =

∆y ∆x

Thus, a slope can be characterised using tan θ , which is the angle made by the tangent to the curve at the given point and the horizontal axis. How do we find the slope/instantaneous rate of change of the curve at different points? Obviously, we cannot keep computing the slope/rate of change at each point, we need a simple way, or ‘formula,’ for the same, wherein we may place the value of the x-coordinate to get the value of the slope at the desired point. Before we aim to find a formula for the slope of curves, let us explore the big deal about the slope of curves. Why do we care so much about the slope of the functions that curves represent?

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The slope gives the following information about a function: • It tells about how steep a function is. • It tells the direction of change of the line. • For a function y = f(x), the slope describes the degree of sensitivity of the dependent variable (y) on the independent variable (x), i.e., the quantum of change in the former due to a small change in the latter. For example, a slope of 4 at a point means the y-axis will grow 4 times the (small) change in the x-axis. • Compare any set of functions to know if they are parallel, perpendicular, or converging and the rate of convergence. • The maximum and minimum value of a function – local (within a limited range of the variables) or global (over the entire range of values). • Using the slope, we can find whether the function increases or decreases after the location of the point of maxima or the minima. Indeed, the most important characteristics of nonlinear functions are their slope. Now we know that slopes are important. Why is the limit so called? The word ‘limit’ is the best descriptor of what the value of the function is when there is the smallest change in the value of the variable, say x. Formally, ‘limit of the function f as x goes to c is t’. It can also be rephrased as ‘As x approaches c, the value of the function f gets arbitrarily close to t’. In real life, when a chemical reaction takes place consisting of two chemicals, a new compound is formed as time passes. So, here, the new compound is the limit of a function as the time approaches infinity. Similarly, tossing a coin gives a head or a tail. To know the probability of the outcome, we may flip a coin many times, making repeated trials. Here, as the time approaches a larger period, the number of heads becomes equal to the number of tails in general. So, the limit of tossing a coin is

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the probability of getting an equal number of heads or tails as time approaches infinity. Continuity – Making sure the infinitesimal is valid The entire concept of limit hinges on how effective is the chosen infinitesimal value in detecting the rate of change or slope of a function at the chosen instant. For example, while finding the best approximate value of the slope of y = x 2 at the point Q (2, 4), the points P and R are moved as close as possible to Q. The reliability of the computed rate of change, at an instant,

is measured in terms of how consistent is the value of the rate, that is, how close are the slopes of the tangent P and R. One of the more obvious ways and means of seeking consistency is to look for the values of the rate at instants very close to the chosen instant. It is easily appreciable that the consistency of the rate of change of a function would be considered higher if at the two instants around the chosen instant – before and after – the computed values of the rate (before and after) are the same as the rate value at the chosen instant. Thus, the infinitesimal value should be such that it can detect sharp variations in the values of the rate, closest to the point of interest (for finding an instantaneous value). The chances of capturing any sharp variation increase as the infinitesimal becomes smaller (and comes closest to the value of the instant). Any detected sharp variation declares a lack of continuity at that point. Continuity is an important consideration for finding derivatives, it helps to know if a function may not have a derivative at an instant (non-continuous functions do not have derivatives, without having to attempt a computation of the derivative), but it is not a necessary condition for computing the non-derivative value of a function over a range. A digital recording of a song is an example of a continuous function. The digital recorder records little bits of sounds several times a second which may contain enough information for a computer to reproduce more or less what you sounded like the whole time you were singing.

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The growth of nails in human hands and feet is an example of continuous function. The nails grow at an average rate of  3.47 millimetres (mm) per month, or about a tenth of a millimetre per day. It grows and slides along the nail bed (the flat surface under the nails), giving strength to the nail. This process continues until the death of a human being. However, some factors that affect this continuous growth of function are age, location, season, hormones, health, etc. Ascertaining continuity at a point Let us graphically see how important is the choice of infinitesimal in appreciating the concept of continuity of a function (as evident in its curve) at a point. For a function to be continuous at a point it is obvious that its slope before and after the point must be near same. This is ensured by checking the slope of the tangent just before and after the point. The two slopes must be nearly the same.

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Take any random point on the curve y = x 2, say, P (0.5, 0.25) and R (3.5, 12.25), and draw a line such that both the points P and R are joined on a curve. The slope of PQ is given by, y 2 − y 1 4 − 0.25 3.75 = = = 2.5 x 2 − x1 2 − 0.5 1.5 The slope of QR is given by, y − y 1 12.25 − 4 8.25 slope = 2 = = = 5.5 x 2 − x1 3.5 − 2 1.5 slope =

Now we move P and R further closer to Q to know the better approximate value of the slope of the curve at Q. Let us choose P (1, 1) and R (3, 9), which are closer to Q (2, 4):

The slope of QP is given by, y − y 1 1 − 4 −3 slope = 2 = = =3 x 2 − x1 1 − 2 −1 The slope of QR is given by,

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slope =

y 2 − y1 9 − 4 = =5 x 2 − x1 3 − 2

Similarly, take another set of points, say, P (1.8, 3.24) and R (2.2, 4.84), which are further closer to point Q (2, 4).

The slope of PQ is given by, slope =

y 2 − y 1 4 − 3.24 0.76 = = = 3.8 x 2 − x1 2 − 1.8 0.2

The slope of QR is given by, y − y 1 4.84 − 4 0.84 slope = 2 = = = 4.2 x 2 − x1 2.2 − 2 0.2 Now we move very close to Q. Take points P (1.95, 3.80) and R (2.05, 4.2025).

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The slope of PQ is given by, y 2 − y 1 4 − 3.80 0.20 = = =4 x 2 − x1 2 − 1.95 0.05 The slope of QR is given by, slope =

y 2 − y 1 4.4025 − 4 0.4025 = = 4.05 slope = = 0.05 x 2 − x1 2.05 − 2 As the points P and R get closer and closer to the chosen point, Q, the lines PQ and QR will coincide, eventually forming a tangent at Q. The closer the points near the chosen point, the closer are the values of the slopes at these points, suggesting a continuous function. Other examples of continuous change beyond motion Average or indicative rate of reaction: One important characteristic of chemical reactions is their rate of reaction. It is

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an essential parameter in the case of large-scale manufacturing of chemicals, drugs, and household chemicals such as cleaning products. For instance, knowing how products are being produced and the bottlenecks in the production (which may mostly be due to the lower than anticipated speed of reactions) to fine-tune the production process. The change in concentration of any reactants or any product per unit time (such as second, minute, or hour) over a given period of time is called the average rate of reaction. And the rate of change of concentration of any of the reactants or products at a particular instant of time is the instantaneous rate of reaction at that specific instant of time. An interesting feature of the rate of reaction is that it continuously changes during every reaction – it depends upon the residual concentration of the reactants (which decreases with each passing instant of the reaction). A reaction never proceeds at the average rate of reaction. To really understand a chemical reaction, we need to go beyond the average rate of reaction. Stock market: In Intraday–trading, the trader buys and sells financial instruments based on fluctuating prices on the same day. The trader makes a profit or loss based on the instantaneous stock price. This signifies a continuous process. Whereas, in long-term investment, we look at the average of the stock prices and then invest in those companies’ stock, which gives a good average return on the stock prices.

Average share price =

Total cost of the shares purchased Number of shares

Simple and compound interest (amounts): Interest rates are a matter of common knowledge (if not understanding) and experience that could be easily extended to broaden the appreciation of the difference between the basic idea/concept of average and instantaneous values of quantities that frequently or continuously change in time or space (i.e., change with change in position). It may also be added that when we talk

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about the ‘real world change,’ we also imply that the change cannot be completely predicted. Simplistically, simple interest rates are kind of ‘average’ of interest rates. The same flat interest rate will be applied for computing the interest amount on a principal over a period of time. The interest amount is assumed to be the same amount every day in that period. On the other hand, the compound interest rate resembles the idea of instantaneous interest amount – which varies by the day, week, month, or whatever period of compounding – over the deposit period. This is to bring out that the instantaneous value of the interest amounts would behave differently under simple and compound interest situations. The surprise may be not: The non-living nature is no exception to change – they undergo autonomous, ongoing, and ‘scientifically determined’ change, ignoring the human-induced changes. One key difference between the living and man-made world is that the the living world period of change is far longer than we are used to. And one key similarity – the average expected change over a period, and the actual change at any point of time are different; for example, the average erosion of a rock surface due to any of the natural agents – air, water, ice, or chemicals – may be 0.1 mm per year (the mentioned amount is just for illustration, that it is not possible to generalise rock erosion to any such specific amount), but the actual in a particular year may be very different, for example, 0.01 mm, or 0.005 mm. There are four primary means of measuring change due to erosion: weight, surface elevation, cross-section perpendicular to the erosion agent’s action, and sediments collected from around the eroded object. Newton and Leibnitz Mention must be made of these two founders of calculus. Also, the fight for the credit of ‘inventing the science of infinitesimal’ was among the ugliest in the history of math (and science). To

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be true, evidence point to the fact that Newton left no stone unturned to discredit Leibnitz (and the latter also campaigned hard against Newton). As per records, Newton had been using the idea of calculus privately about a decade earlier than the younger Leibnitz. He was not the first to go public with the concept. Newton was indeed very secretive about ‘his calculus’ and used coded writing to communicate on it (and that is why much of the symbols used in calculus are attributed to Leibnitz, though Newton is given credit for being the first to invent it). More interestingly, Newton and Leibnitz approached the fundamental entities of calculus in very different ways; the use of ‘infinitesimal’ (infinitely small but not zero) and graphs was the common link between the two, and they both did not think in terms of functions. It took nearly two centuries for calculus to develop to the level of formalness we see in its application in our times. Both used infinitesimal quantities to seek instantaneousness in their computations, but it later evolved into the idea of a limit that was about some value being very close to another value. The need for infinitely small quantities was removed and replaced by ‘limit.’ What is also interesting is that the formal education system is still locked in the historical mode of introducing and teaching calculus, for example, introducing the examples of Newton, Leibnitz, or both, the limits (and continuity), and then differentiation, and more. And we know the outcome of such lesson planning – calculus is the nemesis for almost all of us (irrespective of the marks/grades in school exams). We hope this book will comprehensively demolished that lacunae and learning and applying calculus are truly democratised. Origin of Calculus The seminal contribution of Newton and Leibnitz must not convey a message that calculus was completely a brainchild of the two. The need for a body of knowledge like calculus was apparent to the ancient Greeks; Newton and Leibnitz only

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responded to long pending challenges in scientifically and mathematically understanding our world. Others who enriched Calculus (and how) The mention of Newton befits this section, too, for Newton is regarded more as a physicist, yet his contribution to mathematics is no less revolutionary. To be true, he developed calculus to achieve his research ends in physics. In general, mathematics has grown with the significant contributions of non-practicing mathematicians. To cite another example of a non-mathematician’s contribution, Pierre de Fermat may be the best name; math was a hobby for him. Beyond the daily grind as a lawyer, he found joy in exploring unresolved mathematical quests. He was responsible for using the idea of limit, wherein two quantities approach equality but never realise it, and it morphed into the law of limits. His work on finding the maximum and minimum values of quantities, such as surface area, and volume, using varying dimensions of the quantities, places him as a key inventor of the world of differential equations (the equations he worked on have changing quantities as integral to the equations (such as the changing surface area). In the early nineteenth century, Augustine-Louie Cauchy was credited with being the first to lay a sound base for limits and calculus. Cauchy’s definition of an infinitesimal is interesting – when the successive absolute values of a variable decrease indefinitely to become less than any given quantity, that variable is called an infinitesimal. It has zero for its limit. He founded the math domain called Analysis, which is the theoretical side of limits and calculus. It was Karl Weierstrass who, in the nineteenth century, gave the first rigorous definition of continuity of a function f(x) at a point ‘a’. He stated, ‘a’ function f(x) is continuous at a point ‘a’ if x-values that are close to ‘a’ get mapped by the function to y-values that are close to f(a)’. He penned a function that was

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continuous all along, but at no point, the derivatives existed. The graph of the function was drawable without lifting the pen, but a tangent could not be drawn on it at any point on it! Graph of continuous function with no derivative.

Rene Descartes must also be known in discussions on calculus, for he lived in the first half of the seventeenth century; Newton and Leibnitz lived in the second half of the same and were quite influenced and helped by Rene’s work, especially the analytic geometry (i.e., use of Cartesian graphs). For example, the area under a curve of a function f(x) (drawn as a curve) needs the Cartesian plane. In physics, domains like mechanics, electricity, and thermodynamics are also best based on Cartesian coordinates. More directly relevant to calculus is the ‘lost calculus’, or the ‘algebraic calculus’, primarily developed and used by Rene in the early seventeenth century, and it avoided concepts like infinitesimals. Broadly, it could do all of what calculus could do today but operated only on algebraic functions and approximations. Algebra versus Calculus One of the foundational principles of learning is that new knowledge is built upon the foundations of familiar knowledge.

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We all broadly appreciate a domain of math called algebra. Thus, a comparison of calculus with algebra simplifies the internalisation of the former. Algebra is a branch of mathematics that deals with solving algebraic equations (linear, quadratic, or cubic) for unknown variables. For example, solving x + 9 = 11, x2 = 4 and 2x3 = 16. Calculus is the study of the ‘rate of change/slope’, that is, how the change in one variable (the independent variable x) would lead to a change in the function (or the other dependent variable y, which is dependent on the variable x). However, to know the concepts in calculus, we need to have prior knowledge of algebra, which in turn uses arithmetic results. For instance, to sketch the graph of y = x2, we need to first find the values which ‘y’ would take as the values to ‘x’ are given, which is all about knowing the algebra right, that is, when x = 1, then y = 1, when x = 2, then y = 4, and every value between these two values which is how the graph is sketched. Thus, algebra is the prerequisite for calculus. Graphical representation of equation y = x2

Following are the various dimensions on which the two branches (algebra and calculus) are similar/dissimilar. • Though both the branches help to find slopes of various geometrical figures, that for straight lines falls in the domain of algebra, whereas calculus deals with finding slopes for curved figures.

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• Algebra helps to find the area and volume of definite geometrical figures like polygons (n-sided figure), for which we simply divide the polygon into small triangles and then add them to get the total area. However, calculus deals with finding area and volume for curved shapes and objects like a circle or an ellipse. We will read about the area of curved shapes later. • Algebra covers up the rectilinear motion and talks about average speed/acceleration. We talk of average speed if something travels at a fixed and steady pace, such as in the case of the distance-time graph being a straight line.

Whereas calculus is all about curvilinear motion and instantaneous speed/acceleration a more realistic model, as we cannot be moving at a constant speed throughout any travel; the speed keeps changing. Thus, knowing the speed and acceleration at a particular instant is important.

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• Algebra primarily talks about the value of a function at a point. For example, the value of y = 1 + x (which is a linear equation) when x = 2 is 3, or when x = 0 is 1.

Likewise, we can calculate other discrete values of y as the values of x are substituted. Whereas the need to study calculus is to know about the rate of change of a function at x2 a point, that is, how the function y = f(x) = changes at 2 1 x= . 2

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y = f(x) =

x2 2

(0.5, 0.125)

Thus, discrete functions are studied in algebra and continuous functions are studied in calculus. The above discussion can be presented in a tabular form as under. Dimensions Measures slope of geometric figures Measures length, area, and volume Measures direction, speed, distance, acceleration Measures work done by a force, a mass of a body Deals with functions

Algebra and Calculus Similarity

Differences

Both do

Algebra is for straight lines and calculus for curves

Both do

Algebra is for straight lines, polygons, and some definite geometries, and calculus is for curved shapes and objects

Both do

Algebra is about rectilinear motion, average speed and calculus are about curvilinear motion and instantaneous speed, etc.

Both do

Algebra is about constant density and force situations, and calculus is about varying density and force situations

Both do

Algebra is about finding the value of the function at a point, and calculus is its slope at any point

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The complementary relationship between the two is quite large. No wonder algebra is the only true prerequisite for calculus. For example, to appreciate the idea and measurement of area, volume, acceleration, etc. of any kind of shape, it would be either through algebra or calculus. Functions make the soft dictate the hard Simply put, software translates a rigidly logical understanding of a domain of activities or knowledge into instructions for the hardware to process certain input data or process flow inputs (such as the readiness of the network printer) and offers steps of actions or inputs for further processing. The secular trend is increasing softwarisation of computing systems. For example, specialised hardware like routers (a critical hardware device that enables computer networks) are now a piece of (complex) software resting on any node in the network of hardware like computer. The software dramatically multiplies the powers and potential of hardware. In this context, the software is impossible without functions; functions ‘logicalise’, encoder real world or scientific situations for universal application/understanding (mathematics is the only universal language for us and the only language for machines, including digital machines). Whatever can be faithfully expressed as functions can also be written as software. The nature of math No discussion on calculus must miss a brief encounter with math itself. Calculus is now quite at the heart of the power of math, as also the core of the emerging inorganic intelligence (linear algebra and probability are the other two critical domains needed for it). Despite being a domain of knowledge that has continuously engaged humans for over twenty-five hundred years, arguably starting with the Greek Thales of Miletus (624–548 BCE), mathematics is still undefinable in a sentence. It may be the

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language of the universe, the Gods; the motion of the planets in the slope system, and the solar system itself in the milky way is precisely defined using mathematical concepts. Everything that is patterned, definite and predictable is best expressed mathematically. And everything is patterned by its very nature (else, there will be vicious chaos and descent to nothingness). Mathematics is too vast, too foundational, just like the natural languages. The math of Thales of Miletus, and later Pythagoras and others, was deductive in nature. Math was built on general principles that were supposed to be true, and those believed truths were used to create newer conclusions and logical inferences. Of course, deductive reasoning was vetted against the observations and inductive findings (the general principles formed out of some relevant observations, the opposite process of deductive reasoning). For example, Pythagoras theorems and Euclid’s theorems were proved deductively. A bit of serious thought is enough to extrapolate the essence of deductive logic – that (some) foundational mathematical knowledge was a prerequisite for making sense of the observations, experiences, and ways of the world. And that math was a superior knowledge; for Pythagoras, ‘everything was number’ and for Plato (fourth century BCE) ‘God was a Geometer’. Aristotle wrote about the idea of infinity: “It is always possible to think of a large number of things, for the number of times a magnitude can be bisected is infinite. Hence the infinite is potential, never real. This understanding of infinity had to be top down, deductive, offered as a broad reality.” Method of Exhaustion Similarly, the idea of infinitesimal was also unavoidable in some form or the other. The Greeks started the method of exhaustion (as named during the Renaissance) that used infinitesimal

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lines to find the area of curved shapes, such as circles. It was so named to reflect the nature of the mathematical process – that involves successive approach to find the area of figures such as polygon. It did have the slopes of calculus, finely invented over 2000 years later. We will now explore how the area of a circle is found with the best possible accuracy by approximating it to the area of a polygon that is enclosed in the circle. It is essentially a series of approximations in which a polygon gets closer and closer to the circle. The starting polygon is a square.

The area of the nth inscribed polygon, An, converges to the area, A, of the bounding circle of radius r as n increases without bound. First approximation: A square is inscribed in the circle. The first approximation of the area of the circle will be the area of the square. To find the area of the square, we divide the square into four equal triangles.

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Area of △AOB =

1 1 1 × base × height = × r × r = r 2 2 2 2

Thus, the area of the inscribed square, 1 A1 = 4 × area of △AOB = 4 × r 2 = 2r 2 2 Thus, the first approximation of A1 = 2r2 A 1 It can also be seen as, 1 = r 2 ⇒ A1 = 2r 2 4 2 Second approximation: Let us consider an octagon as the next inscribed polygon. Here, we divide the octagon into 8 equal triangles.

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The inscribed octagon of area, A 2: A2 1 2 = r sin(45°) ⇒ A 2 = 2 2r 2 8 2 Third approximation: Let us consider an hexadecagon (16-gon) as the next inscribed polygon. Here, we divide the hexadecagon into 16 equal triangles.

The inscribed hexadecagon of area, A3: A3 1 2 = r sin(22.5°) ⇒ A 3 = 4 2 − 2r 2 16 2 Fourth approximation: Similarly, we will find the area of a triacontadigon (32-gon).

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The inscribed 32-gon of area, A4 : A4 1 2 = r sin(11.25o ) ⇒ A 4 = 8 2 − 2 + 2 r 2 32 2 Continuing this way, we find the area of a polygon with 64 sides (A 5)

A5 1 2 = r sin(5.625o ) ⇒ A 5 = 16 2 − 2 + 2 + 2 r 2 64 2 Thus, from above, we can say that A n 1 2  360o  = r sin  n+1  2n+1 2 2 

 180°  2 n 2 ⇒ A n = 2 .sin  n  .r → πr as n → + ∞ 2   Where,  180o 2n .sin  n  2

 (n − 1) . 2 − 2 + 2 + 2 + .......... → π =2 

To sum, using, by the area of the best-approximating polygon, we literally exhausted the area of the circle. In other words, (almost) nothing is left unaccounted for in the area of the circle the area inside the polygon. In general this method of exhaustion was used to prove the ‘formula’ regarding the areas and volumes of geometric figures. Interestingly, by the seventeenth century, the idea of infinitesimal started to get to the centre stage in solving some of the mathematical challenges. For instance, the regular geometric shapes could be seen as an infinite collection of infinitesimal lines. For example, a line could be seen as made of infinite very small lines (nearly the same as points), and a solid 3-D shape could be seen as a stack of infinitely thin planes of the same shape, all placed infinitesimally close together, one upon the other. And it was on this foundation that calculus was founded where the length, area, and volume of any kind of shape could be calculated, which was impossible using the

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‘Euclidean geometry’. And then, as discussed earlier in reference to Cauchy and Weierstrass, the domain of ‘Analysis’ developed, and the notions of limit and continuity were placed at the base of the calculus edifice, and it was rationalised. Of course, as we all know, approaching derivatives and anti-derivatives through limit and continuity has been a blunder in teaching calculus. Newton and Leibnitz were aware of the (‘theoretical’) weaknesses in their infinitesimal-based approach, but it worked and got them amazing results. Let us reiterate that this book takes a critically well-woven and uniquely detailed yet intuitive approach to derivatives and anti-derivatives, as well as limits and continuity. Calculus was never easier to learn and teach. A backgrounder Slope of a line or a curve, is its most important characteristics. It tells us all about the steepness and direction of lines. Characteristics of a slope: Let us recall slopes. The slope of a function gives the rate of change of the function and is best evaluated at a given point. Consider the graph of three straight lines as follows:

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The following observations on the slope of the three lines can be made from the graph: • On moving from left to right, line B is steeper than line A. • Line C as steep as line B, but in a different direction. Positive slope: Consider a line joining the points (3, 4) and (–3, –2).

The slope of the line (m) is given by slope m =

change in y 4 − (−2) 6 = = 1 = change in x 3 − (−3) 6

Here ‘m’ is positive. This means that as the x-coordinate increases by a unit, the y-coordinate also ‘increases by 1 unit’. Mountains are an example of landforms with positive slopes.

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Negative slope: Now, consider a line joining (0, 4) and (2, 1).

1 − 4 −3 = , and m is negative. This means that 2−0 2 with a unit increase in x-coordinate, the y-coordinate ‘decreases Here, m =

3 units’. 2 A valley has a negative slope because its y-coordinate decreases as x-coordinate increases (considering the origin at the base of the top of the valley). Zero slope: Consider the edges of ruler which are parallel to each other. Now, consider a line joining (0, 3) and (2, 3). by

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In this case, m =

3−3 = 0. 2−0

This shows that for a unit increase along the x-coordinate, there is ‘no change’ in y-coordinates. Infinite slope: Consider a vertical line x = 2.

The point (2, 1) and (2, 3) are any two points on this line, and 3 –1 2 = = ∞ . That is, the 2– 2 0 y-coordinate could vary indefinite without any change in the x-coordinate. The implicit relationships between zero slope and infinite slope may be noted. Information embedded in slope in a function : The asymptote of slope in being a defying features of functions cannot be over emphasised. Hence, here is a detailed listing of the specific information embedded in the nature of slope of functions. the slope of which is given by m =

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• Q uantitatively, it is a measure of how steep a function is, i.e., how rapidly the outcome of function change with unit change in its input(s).

In above graph, we see that line B has a greater slope than line A. It is also clear that line B is steeper than line A. Thus, a greater slope describes a steeper function. • A positive slope describes an increasing direction of change, and a negative slope describes a decreasing direction of change. In the above graph, line A and B have a positive slope whereas line C has a negative slope. • For a function y = f(x), the slope describes the degree of sensitivity of the dependent variable (y) on the independent variable x, that is, the quantum of change in the former due to a small change in the latter. For example, a slope of 4 at a point means that the y-axis will grow 4 times the (small) change in x-axis. • It is quick means of comparing any set of functions to know if they are parallel, perpendicular, or converging, and the rate of convergence. Parallel lines have the same slope.

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Perpendicular lines are such that the slope of one is the negative reciprocal of the slope of the other. That is, if the slope of the first line is 2, then the slope of the line −1 perpendicular to it will be (graphically presented ahead). 2

• S lope is widely used to find the maximum and minimum value of a function – local (within a limited range of the domain) or global (over the entire set of possible values of the domain). Finding maxima and minima is of great practical and scientific value, and it is the value of the function at which the slope is zero (and turns positive or negative thereafter). Geometrically, it is about the tangent drawn at the maxima and minima being parallel to the horizontal axis. We can use derivative, calculus, to find those points – a purely mathematical way of knowing where the slope is 0! In general, the slope of the tangent at a point is the same as the slope of the curve at the point – a critical part of knowing more about or visualising, a point on a curve.

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In the above graph, we see that the slope at A is 0, that being a horizontal line; hence, A is the point where the maximum of the function occurs. • Using the slope, we can find whether the function increases or decreases after the location of the point of maxima or the minima. This can be found by once again finding the slopes at the points where we want to figure out the nature of the function. Thus, the slope, or the rate of change, can be further scrutinised in terms of ‘rate of change of the rate of change’. The sign of this term tells us whether the slope of the tangent line to the function is increasing or decreasing. In the left part of the graph, we observe that the slope is increasing as we move from left to right. This means that the ‘rate of change of rate of change’ is positive. Whereas, in the part of the graph on the right, the slope is decreasing as we move from left to right, thus, the ‘rate of change of rate of change’ is negative. • Interestingly, wherever the ‘second slope’ (or double differentiation) is 0, there are flattening of the curve, but without those being maxima and minima. There are points of plateau in a curve which are neither maxima nor minima. These are called saddle points (unlike the names of the extremes – maxima or minima).

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In the above graph, we see that the slope remains constant throughout. Thus, there is no rate of change in the slope, that is, the ‘second slope’ = 0. We also observe that for such graphs, there is no point where the function is maximum or minimum, except when we define this function on an interval. It must be emphasised that understanding the minima and maxima and the saddle points helps know function better. When things are thrown/fired up straight, we can explain the path of travel – the maximum height too. But it is not that straight when the motion is curvilinear. The concept and computation of instantaneous rate of change of functions, i.e., the slope of their curve, is tailor made for finding maximum and minimum values of functions. • Instantaneous value of (constantly) changing quantities can only be found through the knowledge of the rate of change at all instants; many quantities around us are continuous ones, even in domains such as economics (e.g., stock markets), biology (e.g., population growth), fluid dynamics, etc. And the instantaneous values matter. They are real, e.g., the impact of speed in an accident is directly affected by the speed at that instant! It is very different from the values we

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use – ‘average’ (it is good when change is minimal, or some amount of change does not matter). • The rate of change is particularly important for things in motion – slope is the way to get two pieces of information – velocity and acceleration of motion; in fact, force is the rate of change of momentum. It may also be added that calculus was born under the shadow of the need to know more about the motion (especially speed and acceleration) of earth and other heavenly bodies (which were moving at varying speeds, and instantaneous speed was the only relevant speed). This need was made pressing with Kepler’s laws around elliptical orbits of heavenly bodies, discovered earlier in the same century as Newton’s work – the seventeenth century. Of course, the direction of motion was also a varying feature – in a curvilinear motion, it continuously varies. • L ast but not the least, finding the distance travelled (a length) in a curvilinear path (to find the exact position of the earth at any instant) or finding the volume of the earth (not a perfect sphere at the poles) makes the knowledge of the changing slope of the shape very important. Indeed, the most important characteristics of non-linear functions are their slope, the continuously changing slope, to be precise.

Function

This chapter must be read after the first chapter is read in its entirety, and in the flow. The first chapter inventively presents a unique story of the foundation of calculus, and seamlessly integrates all its basic building blocks – derivative, anti-derivative, differential equations, limits, and continuity. Yet, for the fraction of the readers landing straight on this chapter we must offer a summary context of function, and that is what follows. Welcome to the reality of variation Variation is the law of nature. Not many kinds of things remain as they are forever. Our body keeps renewing itself by every second. The skin cells are in regular birth and death cycle, and so are the cells in all other tissues and organs. Trees grow, animals grow/move/age, and even mountains reshape due to tectonic shifts; earth is in constant state of change. The man-made world also has most things changing their position, shape, composition, structural strength, pressure, stress, etc. with the passage of time. For example, no two car tyres, of the same car, would visibly wear off the same way, recognisable difference in the treads of tyres is the norm. Ongoing variations in the different features of things, or situations, is a reality; for a car in motion, strictly speaking, the speed varies all along the duration of its motion.

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The need of instantaneous values The only way to know the exact nature of variation is to know the instantaneous values of the varying quantity at all points of interest; for example, to know the variation in the speed of a moving thing, we need to know many instantaneous values of speed along the motion, or several instantaneous values of speed during the specific time period of interest. A particular instantaneous value – Rate of change (Variation) For simplicity, let us start this conversation with the example of motion, nothing varies like motion! Recall, the change in position is how we most apparently see, experience, measure, and express motion. Pertinently, the change in position can be measured and expressed in a few ways – the quantum of change in position, the pace (quickness or slowness) of the change, and the steadiness or changing nature of the pace of the change. There is a world of difference in the computation of the quantum of change in position due to motion, and the computation of the pace of change (and the pace of pace of change). The former is a simple distance between two chosen points on a surface, but the latter is a challenge, it is the rate of change of distance at any instant of motion. The rate of change of distance has instantaneous values (magnitude, and/or direction); it needs to be specifically computed, it is not obvious like the way the change in distance traversed during a motion is calculated. The challenge in computing instantaneous values The challenge in finding an instantaneous rate is the measurability of changes in ‘an instant, or a particular point/condition (for example, measuring distance travelled at an instant, i.e., measuring the distance covered for a duration that is nearly zero) – the divisor in the such computation of rate is nearly zero (We call these nearly zero quantities as ‘infinitesimal’, which means infinitely small). Mathematically, such quantities/numbers

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can be visualised but any attempt to measure such changes is rarely possible; imagine measuring the distance travelled by car in 0.001 seconds (taken to represent ‘an instant’). Thus, computation of rate at any instant gets near impossible. Recall, the only way to find the rate of change of anything is to use the mathematical concept of rate (which is the quotient of two changing quantities). Turning rate into a measure of instantaneous values of changing quantities requires reducing the divisor to infinitesimal magnitude, and there lies the problem. Avoiding measurements of instantaneous situations The difficulty in measurement of instantaneous values of changing quantities has led us to invent another concept, and method, of calculating instantaneous values. It avoids actual physical measurement of quantities and involves an indirect way of computing instantaneous values that does not compute them out of any instantaneous measurements. How do we exactly avoid actual physical measurements? We use a kind of mathematical ‘formula/equation’ to represent the nature of change, mathematically relating the various variables which change in a given situation. To be precise, the mathematical ‘relationship’ that comprehensively imitates the behaviour of the varying quantities (variables) of a particular situation, is a function. For example, interest = principal × rate of interest × time is a function, just as distance = speed × time, y = sin x, area = length × breadth (for a square or rectangular figure), or y = 3x + 2; where (x, y) are coordinates of a line; are functions, representing a certain kind of relationship between changing quantities. Function – Capturing realities in mathematical expressions Functions are mathematically-expressed relationships of real-world situations, and they are such that for each change in any of the variables in the relationship, howsoever small, a

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change is observed in some other variable of the relationship. Functions model the behaviour of some reality, using the language of mathematics. Let us briefly recall the algebraic function that relates the area of a square and the length of its sides, this function displays the behaviour of area which changes as the length of the sides of the square changes. The area function can be written as f(x) = x 2 where x is the length of each side of a square. For each value of the length of the side (x), there would be a unique value of the area function f(x). For example, When x = 3 units ⇒ f(x) = 9 sq. units When x = 4 units ⇒ f(x) = 16 sq. units The same is true the other way round as well, and the (length) function can also be represented in terms of the area of the square. In that case, f(x) represents the length of the side of a square and x is the area of the square. This can be expressed as f(x) = x . Here, again, for each value of the area, there is a unique value of length of a side of a square. The notation f(x) and y of a function is used interchangeably. Input and output of functions Let us start by thinking about how should we express the function that relates the area and length of the sides of a square. Recall that the function can represent both area and length of the side of a square depending on what is given and what needs to be found out. For example, to know the length of a side of a square when the area of a square is known, the function can be expressed as f(x) = x , where x is the area. And, if the area of a square is required when the length is known, the function can be expressed as f(x) = x2, where x is the length of a side of a square. Thus, a function is expressed mathematically based on the requirement of what is to be calculated. An interesting feature of a function is that there is a corresponding value for every value of the independent

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variable. For example, in f(x) = x2, where x is the length of a side of a square, for every value of x, there is a corresponding value of f(x). Pictorially, f(x) = x2, where x is the length of a side of a square can be represented as: Length

Area

1

1

2

4

3

16

4

9

when length, x = 1 unit, area, f(x) = 1 sq. units when length, x = 2, area, f(x) = 4 sq. units and so on. Similarly, the function f(x) = x , where x is the area of the square can be seen as: Length Area 1

1

4

2

16

3

9

4

when area, x = 1 sq. unit, length, f(x) = 1 unit when area, x = 4 sq. units, length, f(x) = 2 units and so on. Thus, for any value of input, there is a definite value of an output. Input is a quantity that is usually entered into a function. The quantity should be such that it is valid for the function and it returns a solution for the same. For example, the function f(x) = x is only valid for positive values of x. When the input has been processed as defined in the function, we finally get the solution. This solution is termed

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as the output of the function. Thus, the output is what we get when the relationship is applied and the calculations are done on the input value. The processing of input to output is done by a function; therefore a function is a translator between input and output. As discussed, there is a relation between the independent and the dependent variables. This property can also be expressed in terms of a set, i.e., ordered pairs. Every function generates a set that can be written as an ordered pair. For example, the area function f(x) = x2 generates an ordered pair represented by (side length, area) and given by (1, 1), (2, 4), (3, 9) and so on. Here, length = x is the independent variable and area = f(x) is the dependent variable, since its values depend on the values of the length (= x). Similarly, the length function f(x) = x also generates an ordered pair represented by (area, side length) and given by (1, 1), (4, 2), (9, 3), and so on. Here, area = x is the independent variable and length = f(x) is the dependent variable, since its values depend on the values of the area (= x). Analogous to this, when we press any key on the keyboard while using a computer, it processes the data by checking the typed letters and storing the same in its memory and finally shows the typed document on the screen. This is a specific example of Input – Processing – Output unit. Functions also act in the same way and are, thus, the Input – Processing (Relationship) – Output expressions, specific to situations. Function as processor Once the input value is entered into the function, the next step is that it applies the relationship, as defined, to this input value. For example, f(x) = x 2 implies that the input x will be processed using the relationship ‘f(x) = x2’. Similarly, the input for the function f(x) = x will be processed using the relationship ‘f(x) = x ’, but only for the positive values, since the negative values as input are not valid for

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this function. This step is, thus, the processing of the input. The above-mentioned steps can also be shown through a table for f(x) = x . Input

Process

Output

x=2

2

1.414

x=4

4

2

x=5

5

2.236

Similarly, for f(x) = x3, the above steps can be explained as: Input x=2 x = –3 x=4

Process 2×2×2 –3 × –3 × –3 4×4×4

Output 8 –27 64

The nature of input and output There is a world of things and situations that are quantified through the act of counting, and a world that is quantified through the act of measurement. The marks obtained by a student in an exam is quantified by the evaluator by first counting the marks obtained in individual questions and then adding them all up, whereas the weight of students in a class is quantified by measuring the weight of individual student on a weighing scale. Recall, functions take some kind of quantities as inputs and produce some other, or similar kind of quantities as output, which could be obtained by counting, or by measurement. The nature of quantities obtained by counting is what we call as discrete. For example, marks obtained by a grade X student in all subjects in a school have discrete values, graphically seen as follows:

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The nature of quantities obtained by measurement/computation is what we call continuous, expressed using real numbers. For example, the temperature of a person with fever tracked after every hour gives a range of values, graphically shown as follows:

5

The temperature takes every value between 100°F and 101°F between 1 pm and 2 pm. Thus, the nature (discrete or continuous) of quantities determines the nature of functions dependent on these quantities. Discrete or continuous function Expectedly, the nature of output could also be discrete or continuous, depending upon the mode of quantification – counting or measurement.

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Let us take a function f(x) = x2 and take different values to know how this function behaves. When input values are taken as discrete, say, –2, –1, 0, 1, ..., the nature of output is also discrete and it is 4, 1, 0, 1, …, corresponding to the input values in the given function. (2, 4)

(– 2, 4)

(– 1, 1)

f(x) = x2

(1, 1)

Domain: Integers

Discrete Function

On the other hand, when input values are taken as interval, say [–2, 2], it takes every value between –2 and 2. The output from the given function is also an interval [0, 4], which is continuous.

f(x) = x2

Domain: Real numbers

Continuous Function

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Thus, for a different domain of input values, the same function f(x) = x2 will have a different range of the output values. The first of the above graphs have discrete points (integers) which can be easily counted, thus representing a discrete function while the second graph has a range of values (interval) making it a continuous function. Graphing functions for fuller understanding Let us consider the trigonometric function sin x to understand why graphs are important and necessary to understand the true nature of functions. We will explore the values (behaviour) of the curve of the function over a cycle with a period of 2π. Let us input few angles (x) in the function f(x) = sin x to plot the graph of the outputs of the function, when π x = 0, f(x) = 0; x = , f(x) = 1; 2 3π x = π, f(x) = 0; x= , f(x) = –1; 2 x = 2π, f(x) = 0

-1

The above graph does not depict the nature of the function, because the function’s values between the points could be anything; the graph leaves a lot of ‘holes’. Let us add some more angles as input to the graph, and observe if the graph becomes more specific.

98

Function

Apart from the points in the first graph of the function, we have few more points as: π 3π 1 1 When x = , f(x) = ; x= , f(x) = ; 4 4 2 2 x=

5π 1 , f(x) = – ; 4 2

x=

7π 1 , f(x) = – 4 2

-1

Clearly, the graph is still not specific, there are still many holes that could give any shape to the curve. It may now be obvious that we need many points to be plotted to know the true nature of a function; in fact, we need infinite points. And, if we have infinite points we will get a curve like the one below for the function, sin x. Briefly, this is why graphs uniquely help in capturing the comprehensive behaviour of functions. Graphical representation of f(x) = sin x

f(x) = sin x

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99

This graphical tool also enables us to know how the function behaves when any relation is to be explained. For example, the relation cos x = sin (90° – x) states that the sine curve when shifted by an angle of 90° gives us the graph of cos x. Graphical representation of f(x) = cos x f(x) = cos x

y

=

x

Similarly, a linear equation of the type y = x when multiplied with a constant gives no understanding of how a function would behave or look like. But, when graphically shown, it gives a better understanding. Graphical representation of the function y = x

100

Function

y = 5x

Graphical representation of the function y = 5x

In the graph y = x, the line makes an angle of 45° with the x-axis while a change in the function, say when x is multiplied by 5 gives a new function y = 5x and shows a steeper line. Therefore, graphs are particularly good tools to represent and study changing quantities, in general, and in visualising functions. Infinite realities call for infinite functions Recall, functions represent reality. One of the most amazing realisations of a better understanding of the real world using the language of math is that the infinitely colorful and diverse things and their interactions can be accurately represented by (mathematical) functions. Interestingly, infinite realities call for infinite functions. Any reality which is dependent on certain factors can be expressed mathematically. So, as many are the realities of life, as many are the functions to express them.

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The magic of a few Fortunately, there are just a few tens of functions that can define the world of infinite variations in functions. For example, a linear function y = f(x) = ax + b, where (x, y) are the coordinates in the xy-plane is a line and can become horizontal, steeper, flatter or vertical with a little modification in the same function f(x) = ax + b. This can be seen in the following graph where the line y = x can be shifted 1 unit up to obtain the function y = x + 1 or it can be reflected about the y-axis to obtain the function y = –x. Graphical representation of f(x) = x, f(x) = –x, and f(x) = x + 1

)=

f(x

)=

f(x

x

+

1

–x

= x)

x

f(

Similarly, a curved function f(x) = ax 2 + b, can be translated vertically or horizontally with simple modifications, graphically a parabola, in the function.

102

Function

Graphical representation of f(x) = x2 and f(x) = –x2

f(x) = x2

f(x) = –x2

Indeed, the world of functions can be identified to be a part of a limited family/set of functions; these distinguishable set of functions are called ‘the parent functions’. Indeed, there are a set of disparate functions which act as the source/base of many other functions. In real life also, we are all witness to the similar genes, characteristics, and behaviour within individual families. The same similarity is observed even at a social level. For example, the various series that we watch on the OTT platform are distinguishable along clear genres, but episodes/stories in each genre looks similar. For example, each of the episodes in, say, a comedy genre looks similar. The same characters perform with little variation and usually, the endings are on the same note. All the later episodes are derived from the broad pattern set in the first episode and are connected to it. Such connection between parents and siblings, and between the first and the latter episodes is what we call as ‘parent functions’.

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Similarly, a family of functions share similar properties with minor variations in the graphs of the specific functions within a family. For example, y = 2x 2 + 1 and y = x 2 are different functions but the base/family of the two functions is the same, i.e., x 2. The parent functions Just like our human families are characterised by parents, a family of functions also has a characteristic parent function. However, this parent function has to be in the simplest form of expression so that other (complex) functions in that family can be easily recognised/graphed bearing the marked features of the parent function. A parent function is a function in its most basic form and shows the relationship between the independent and dependent variables in its simplest form. Studying just the parent (form of the) function would immediately give us an idea of how any function from the same family would look like. All the functions within a family of functions can be derived from the parent function by taking the parent function’s graph through various shifts, flips or stretches. There are infinite possibilities of creating unique function from a given parent function. For example, y = x, y = –x, and y = x + 1 all represent a family of straight lines as can be seen from their graphs as follows:

104

Function

)=

f(x

)=

f(x

x

+

1

–x

) f(x

=

x

If we observe the graph carefully, we will notice that the graph of y = x + 1 is shifted up by 1 unit up, from y = x though both have the same shape of the graph. Similarly, y = –x is a reflection of y = x about the y-axis. However, both these functions y = x + 1 and y = –x look similar to y = x. These transformations are in no way changing the shape of the graph, so all of these transformed functions in a family have the same shape, look similar and follow the basic characteristics of the parent function. Thus, these functions are transformations of the basic parent functions, which are a way to create new functions belonging to the same family. The properties that are essentially shared among the same family of functions along with the basic parent function are: • The shape of their graphs (we need to keep in mind that the family of functions only have similar shape of the graph. Their slopes may vary depending upon the ratio of rise to run or change in y to change in x). • The degree or the highest power of the independent variable x. • The number of roots or solutions of the function.

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Let us specifically study the graphs of the family of the (parent) function y = x. The graph of y = x – 1, can be drawn by identifying the following coordinates (0, –1), (1, 0), and (2, 1). Graphical representation of the function y = x – 1 (2, 1)

y

(1, 0)

=

x

–

1

(0, –1)

y

=

x

C omparatively, the graph of y = x for coordinates (–1, –1), (0, 0), and (1, 1) would be:

(1, 1) (0, 0) (–1, –1)

Now, the imposition of the above two functions is shown graphically as:

106

Function

y

=

x y

=

x–

1

We see that y = x – 1 is a straight line passing through (0, –1), (1, 0), and (2, 1), and it is parallel to the straight line y = x. It has the same degree 1 as that of y = x and can be obtained by shifting y = x, by 1 unit down. Similarly, many other graphs can be obtained from y = x by simply transforming it; for example, y = x + 1, y = x – 5, y = x – 1000, y = 2x, y = 8x, y = 400x, y = 0.25x + 2 belong to the same family of functions with the parent function as y = x. Following are some graphs of family of functions of parent function y = x to show the similarity in the shape and properties between them. Graphical representation of the function y = x + 1

(-1, 0)

y

=

x

+

1

(0, 1)

Calculus For Professionals

Graphical representation of the function y = x – 5

y

=

x

–

5

(5, 0)

(–0, –5)

107

108

Function

Graphical representation of the function y = x – 1000

y

=

x

–

10

00

(1000, 0)

(0, –1000)

Graphical representation of the function y = 0.25x + 2

5x + 2

y = 0.2 (–8, 0)

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2x

y

=

x

y=

y = 8x

Graphical representation of the functions y = x, y = 2x, and y = 8x

An another example, let us draw the graph of y = (x + 1)2. Following the standard graphing technique, we create the following table of coordinates to plot the points on the graph.

x y

–2 1

–1 0

0 1

Graphical representation of the function y = (x + 1)2 (1, 4)

y = (x + 1)2

(0, 1)

(-2, 1)

(-1, 0)

1 4

110

Function

Let us also create the graph for y = x2 using the same graphing technique and making a table of corresponding values.

(2, 4)

(–2, 4)

y = x2 (0, 0)

The two graphs imposed on a single graph would look like as follows:

y = x2

y = (x + 1)

2

(-1, 0) (0, 0)

If we observe carefully, y = (x + 1) 2 is a parabola with its vertex at (–1, 0). Its graph is similar to that of y = x 2, a parabola, with vertex at (0, 0). Thus, knowing the graph and properties of y = x2 can help us to know the graph and properties of the function y = (x + 1) 2 as well. Hence, y = x 2 is called as the parent function of all other degree 2 polynomials, such as y = (x – 1)2

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and y = (3x – 4)2. The graphs of both these functions are same, however the vertex in case of y = (x – 1) 2 is (1, 0) and that for 4 y = (3x – 4)2 is ( , 0). 3 Graphical representation of the function y = (x – 1)2

y = (x – 1)2

(1, 0)

Graphical representation of the function y = (3x – 4)2

y = (3x – 4)2

( 4 ,0 3

(

We can see that each parent function gives us many other new functions which can be derived through transformation of the parent function.

112

Function

Given the scope of this book, we must explore the derivative and anti-derivative of the parent function. We will study a set of parent functions that are the means of understanding and knowing the properties of all the other functions/graphs. Let us start with the simplest one, y = x; we start with this only because there is more to be discussed even further. Let us consider f(x) = x

f(x

(1, 1)

)=

x

This function represents relationship between two quantities that are linearly proportional to each other. Graphically, it is a straight line and passes through (0, 0), (1, 1), and (–1, –1). Graphical representation of f(x) = x

(0, 0)

(– 1, – 1)

It is the parent function of all linear functions. One of the most beneficial tools for future predictions is to apply linear functions. For example, if an employee who is paid hourly at 500, if works for 2 hours will be paid 1000, for 3 hours will be paid 1500, and so on. Thus, the linear function which describes this situation is y = 500x, where x is the number of working hours employed. Being a linear function, it has the similar graph and characteristics as the parent function y = x.

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Graphical representation of f(x) = 500x

=5

1000

f(x)

Hourly wages

00x



1500

500

Number of hours

Let us look at some more such linear functions and find their properties from its parent function. Consider f(x) = 2x which represents a linear function that passes through (0, 0), (1, 2), (–1, –2), where the value of the function y is twice the value of x. This function can be obtained by dilating the parent function y = x, as the values in this function is twice (>1) as that in case of its parent function value.

114

Function

f(x)

=2 x

Graphical representation of f(x) = 2x

(1, 2)

(0, 0)

(–1, –2)

x which represents a linear function that looks 4 1 1 like the parent function, passing through (0, 0), (1,1, ), (2, 2, ) and 4 2 the function’s value y is one-fourth the value of x. This function can be obtained by contracting the parent function y = x, as the value is quarter ( 0 and decreasing for x < 0. This suggest the following graphs y = x, y = x3, and y = x 5. Graphical representation of g(x) = x3 and g(x) = x g(x) = x3 g(x) = x

Graphical representation of g(x) = x5 and g(x) = x g(x) = x5 g(x) = x

128

Function

2x )= g(x

g(x) = 5

x

However, we also observe a gradual decrease/increase in the graph. No sudden dips/rise in the values of the slope are seen from the tangents considered in the graph of the function f(x) = x 2. This suggests that the values of the derivative function would not be high for a small value of the input. That is, the derivative function cannot be a curve such as x3 or x5, where for a small value of x we have a large value of the function. This suggests a straight line as a derivative for x2. There can be many functions with this possibility, as shown in the graph:

x = x) 2

g(

The takeaway from this derivative graph is that the derivative of all quadratic functions is a linear function. We have logically derived the nature of the derivative function of a quadratic function. In the next chapter we will explore which of the above graphs is the actual derivative graph of the given function. It is the graph of y = 2x. Thus, the derivative of x2 is 2x. Hence, we can generalise that the derivative of all quadratic functions is a linear function.

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Anti-derivative of x2 We aim to find a function whose derivative is x2. Similar to the discussion in the linear parent function, the anti-derivative function is one whose range values are same as the various values of the area under the curve y = x2 in some interval. The graph of x2 in the interval [0, 2] can be obtained by making a table of the various points of x and the corresponding points of y. x

0

1 2

1

2

y = x2

0

1 4

1

4

And, it can be shown as follows: y (2, 4)

f(x) = x2

(1, 1) ( 1 , 1) 4 2 (0, 0)

x

Arithmetically, it is tough to find the exact area of the region that is curved. However, finding its approximate area is always possible and that basically serves our current purpose of broadly finding the nature of the function that is the anti-derivative of x2. Logically, breaking the intervals [0, 2] into small sub-intervals would make better sense.

130

Function

(2, 4)

f(x) = x2 (

3 9 , ) 2 4

(1, 1)

(1 , 1) 2 4 (0, 0)

1 in the interval [0, 0.5] 2 3 1  2 2  2 1  11  11 = × × =   ∗  =   2  22  216 22

Area corresponding to the point ≈

1 1 1 11 1 1 × b≈ × h∗ ×=b ∗ ×h = × ∗ = 2 2 2 2 24 2

Area corresponding to the point 1 in the interval [0, 1] ≈

13 1 1 1 2 × b × h = × 1 × (1) = = 2 2 2 2

3 in the interval [0, 1.5] 2 3 3 2 2 2 1 1 3 1 9 3 1 3 3  27 3 1   ≈ × ≈b × h∗ b= ∗ h× = × ∗ = ∗ ×  ×= ==   2 2 2 2 4 2 2 2 2  22  2 22

Area corresponding to the point

Area corresponding to the point 2 in the interval [0, 2] ≈

3 1 1 1 1 2 1 223 2 2 )4 = × b ×≈ h = ∗ b ∗× h2 = × 4 =∗ 2 ∗× (22 )× (= 2 2 2 2 2 22

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From the above calculations, we can deduce the following: • The approximate area under the curve y = x2 corresponding to any interval [0, x] is

x3

. 2 • We see that as the value of x increases, the approximate area corresponding to the point also increases and at a very fast rate.

3

3

1 3 3 2  2  2 1 ( ) 1 1 1 1  1 1 31 2 3  21  27   ,... on the • ≈ The points , , = × 1∗= × (∗1)= = = ∗ bfunction ∗≈h ≈=×1 b∗that ∗×b h∗∗has h =the = =  2 2 22 2  2 2 216  2  2 2 2 2 1 3 y-axis corresponding to the points , 1, , ... on the x-axis is an 2 2 approximate graph of the anti-derivative function of x2, i.e., x3 2

. This being not exact (but approximate) would just vary

in the coefficient of x3. That is, instead of the anti-derivative being

x3

x3

, it can be , where c can be any real number. c 2 • At this point of conceptual exploration, it is fair to ignore the constant ‘c’. The graph of the anti-derivative function h(x) = x3 is as follows:

132

Function

h(x) = x3

-0.5

Thus, the anti-derivative of any quadratic function of the type ax2 + bx + c would be a three degree function. We will see in the chapters to follow that the anti-derivative x3 of x 2 is . But for now, just observe that the anti-derivative 3 corresponds to the curve which comprises of the point on the x-axis and area as its value on the y-axis. On combining the result for the linear and quadratic functions, and from the properties which we will study in the chapters to come, the derivative of y = x 2 + 2x is the sum of the derivatives of x2 and 2x. The derivative of x 2 is 2x and that of 2x is 2. Thus, the derivative of y = x2 + 2x is 2x + 2. The anti-derivative of y = x 2 + 2x is the sum of the antix3 2 2 derivative of x and 2x. The anti-derivative of x is and that 3 3 x of 2x is x 2 . Thus, the anti-derivative of y = x2 + 2x is + x2. 3 The graph of the function y = x2 + 2x, its derivative x3 2x + 2 and its anti-derivative + x2 is as follows: 3

133

g(x

f(x) = x2 + 2x

)=

2x +

2

Calculus For Professionals

3 h(x) = x + x2 3

Let us consider f(x) = x3 Volume of three-dimensional figures such as a sphere or a cube is an immediate use of studying cubic functions. f(x) = x3 is the parent function of all the cubic functions of the type f(x) = ax3 + bx2 + cx + d. The graph of the parent function f(x) = x3 is as follows:

f(x) = x3

(0, 0)

The graph of other cubic functions such as f(x) = (x – 1)3 [obtained by shifting the vertex of the parent cubic function to (1, 0)] is as follows:

134

Function

y = (x – 1)3 (1, 0)

Function f(x) = 3x 3 is obtained by dilation of the parent function. The value of the function is three times the value of the parent function, that is, at x = 2, the value of the parent function x 3 is 2 3 = 8 and that of the given function 3x 3 is 3 times 8, which is 24. The graph of f(x) = 3x3 is as follows:

f(x) = 3x3

(0, 0)

Now, consider the function f(x) = x 3 – 3x 2 + 3x + 3. Its vertex can be obtained by making it a perfect cube as follows: f(x) = y = x3 – 3x2 + 3x + (4 – 1) ⇒ y = (x3 – 3x2 + 3x – 1) + 4 ⇒ y – 4 = (x – 1)3 ⇒ Y = X3 (Let us assume that Y = y – 4 and X = x – 1) Substituting x – 1 = 0 and y – 4 = 0 gives the vertex of the function as (1, 4).

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The graph of the given function is as follows:

(1, 4)

f(x) = x3 – 3x2 + 3x + 3

The general properties of a cubic function which follows from its parent function are: • The graph shape is a unique ‘standing wiggle’. • The highest degree of the independent variable is three and hence the number of solutions of the function is three. • The domain and range of the cubic functions are all real numbers. Derivative of x3 To find the derivative of the function, let us start with the graph of the given function, along with three tangents drawn at different points to know the nature of slope of the function.

136

Function

Graphical representation of f(x) = x3

f(x) = x3 f(x) = x3

We can observe the following in the graph: • For negative and positive values of x, the slope of the given function is positive as it is making an acute angle with the horizontal in the anti-clockwise direction or the tangent at each point on the curve is tending upwards when moving from left to right. • It is zero at x = 0, where the tangent is horizontal to the x-axis, in fact it coincides with the x-axis. Based on these observations, two imperatives emerge, for the derivative function of the given function, y = x3: • The graph of the derivative function, g(x) has positive values for all values of x (either x > 0 or x < 0) except at x = 0, where it is zero. • The derivative function passes through the origin. Such a description of the derivative of x3 holds for functions where the values are always positive for any value of x and it is zero for x = 0. These functions could be y = x 2 or y = x 4 or y = x 6.

137

g(x) = x4

g(x) = x6

Calculus For Professionals

g(x) = x2

However, the nature of the values of the slope suggests that though the values are increasing but the increment is not very high, which points out to y = x 2 as the derivative function. Thus, the graph of the derivative function of x3 is as follows:

g(x) = x2

However, in the chapters to follow, we will see that the derivative of x3 is 3x2. Similarly, the derivative of 3x 3 is also a quadratic function and we will see that it is 9x 2. The derivative of (x + 1) 3 is given by 3(x + 1) 2.

138

Function

Thus, all cubic functions have quadratic functions as their derivatives. Anti-derivative of x3 The anti-derivative of x 3 is a function whose derivative is x 3. As discussed for the previous parent functions, we would seek a function whose values in range are same as the area under the curve y = x 3. Its graph in the chosen interval [0, 2] is as follows: (2, 8)

f(x) = x3

(

3 27 , 2 8 )

(1, 1) 1 1 ( , ) 2 8

Following the reasons similar to the function y = x2, we would try to approximate the area under the curve y = x3 in the interval [0, 2]. However, for a better approximation, it is always better to split the interval [0, 2] into smaller sub-intervals. 1 Now, area corresponding to the point in the interval [0, 0.5] 2 44 11 33    11  44 ≈≈ 1 × b × h = 1 ×≈11 ×∗ b1∗= h =1 1× ∗11 × ∗   ==   2 2 22 8 2 2 22  22 2

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Area corresponding to the point 1 in the interval [0, 1] 1 1 3 31 1144 1 1 × b≈× h∗=b ∗ h×= 1 × ∗(11)∗ (=1) = = 2 2 2 2 2 2 3 Area corresponding to the point in the interval [0, 1.5] 2 44 33 33   1 1 31 27 11 3 33 22 ≈≈ × b × h = ×≈ ∗× b ∗ h== ×∗ ∗×  = =   2 2 2 8 22 2 22 2 ≈≈

Area corresponding to the point 2 in the interval [0, 2] ≈≈

1 1 1 11 2244 × b × h = ≈ × 2∗ b× ∗8h= = ×∗22 ×∗ (2) (2)33 == 2 2 2 2 22

From the above calculations, we can infer the following: • As x increases, the approximate area corresponding to it also increases. Thus, the graph of the anti-derivative of x3 in the interval [0, 2] is increasing. • The graph is drawn out of the ordered pairs (coordinate) where x-coordinate is same as the function’s x-coordinate while the y-coordinate is the coordinate of the area under the curve at that corresponding point on x-axis. This graph shows the approximate graph of the anti-derivative function x4 3 of x , i.e., . c • The ordered pairs of the anti-derivative function are ,   1144  1  4    1344              2     (1) 4  2         42 0.5,  2 , etc. 0.5,   1, 2   0.5, 0.5, 1.5,        22    22   22               • It is fair to ignore the coefficient of x 4 at this point of conceptual exploration. The graph of x4 is as follows:

140

Function

h(x) = x4

Thus, the anti-derivative of any cubic function of the type ax3 + bx2 + cx + d is a 4-degree function. (In the chapters to follow, we will learn that the anti-derivative x4 .) 4 Let us consider f(x) = |x| of x3 is

This is known as the modulus function and it actually means the following piecewise function or sub-functions over different intervals: − x, when x < 0  f(x) = x =    x, when x ≥ 0 

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141

Graphical representation of f(x) = |x|

f(x) = |x|

We see that for any value of x, be it positive or negative, the function takes the positive value only. Hence, such a function is also known as the absolute function. Let us look at some more such functions, after learning about a couple of everyday application of this function. We usually use an absolute function to highlight that a function’s value must always be positive, its most common use is to measure distance between two points. We find distance between any two positions by subtracting the initial position from the final position. For example, on a number line, the distance between 1 and 3 is 3 – 1 = 2. The distance between 3 and –1 is –1 – 3 = –4. Here, we take its absolute value, that is, |–4| = 4, since distance can never be negative. Thus, distance between a and b is always defined as |a – b|.

142

Function

Distance = 4 units –3

–2

–1

0

1

2

3

Distance = 2 units

Another situation where we make use of absolute value is when we are in debt, that is, we do not say that we will get (–50), instead we are in debt of 50. Some modulus functions are: • Function f(x) = |2x| (obtained by dilation from the parent function, that is, the value of this function is twice as the value of the parent function |x|), graphically drawn as:

f(x) = |2x|

It looks exactly like the parent function |x|, but every value of the function is twice as much as a corresponding value of x in |x|. 1 • The function f(x) = | x | can be obtained from the parent 2 function |x| by contraction, that is, the value of the function is 1 the value of |x|, as can be seen graphically: 2

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1 f(x) = 2 x

Thus, the absolute functions of the type |ax| got either by contraction or dilation of |x|, graphically look similar to the parent function |x|. • Similarly, the graph of the function f(x) = |x + 1| is as follows:

f(x) = |x + 1|

(–1, 0)

The graph looks like the parent absolute (modulus) function, with its vertex shifted to (–1, 0). The vertex is obtained by 1 taking 2x x +−11 = = 0 ⇒ x == –1. and y = 0, 2

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• The function f(x) = |2x – 1| is another modulus function 1 11 with vertex given by 2x andyy==0,0, i.e., ( , 0). 2x –− 1 == 0 ⇒ xx == and 22 2 Its graph is given as follows:

f(x) = |2x – 1|

(1 ,0) 2

Thus, all functions of the type f(x) = |ax + b| derive their properties from the parent function f(x) = |x|. The general properties of the absolute function derived from parent function are • It has a V-shaped graph. • The domain of the function are all real numbers, and its range is [0, ∞). Derivative of |x| To find the derivative of the function f(x) = |x|, let us explore its graph:

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f(x) = |x|

Unique tangent does not exist at x = 0

From the graph of y = |x|, and by drawing tangents at different points on the curve, we observe the following: • For negative values of x, the slope of the given function is negative as the tangent at any point is tending downwards from left to right on the graph. • For positive value of x, the slope is positive as the tangent at any point is tending upwards from left to right on the graph. • Unique tangent cannot be drawn at x = 0, thus the derivative of |x| does not exist at x = 0. • Recall, the slope of a linear function is always a constant value. Thus, the slope for x > 0 and x < 0 is a constant value. From the above inferences, we can conclude that • For x > 0, the slope is constant and positive. As x increases by 1 unit, y also increases by 1 unit. Thus, the slope is 1 for x > 0. • For x < 0, the slope is constant and negative. As x increases by 1 unit, the value of y decreases by 1 unit.

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Function

Thus, the derivative of |x|, for x ≠ 0, i.e., g(x) is a step function  1, x > 0 given by  and its derivative does not exist at x = 0 as  −1, x < 0 shown in the graph:

1

–1

For the function,  x + 1, x + 1 ≥ 0  x + 1, x ≥ −1 f(x) = x + 1 =  =   − (x + 1), x + 1 < 0  − (x + 1), x < −1 with its graph as shown as follow, the slope is 1 for values of x > –1 and it is –1 for values of x < –1.

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)=

x

+

147

1

f(x

(–1, 0)

 1, x > −1 Its derivative is given by the step function =  and  −1, x < −1 the derivative does not exist at x = –1. Thus, absolute functions are of the type   b    ax + b, ax + b ≥ 0  ax + b, x ≥ − a |ax + b| =  =   − (ax + b), ax + b < 0    −(ax + b), x < − b   a  Their derivative is given by the step function   a, x > − b  a , whereas the derivative does not exist at – b .  a   −a, x < − b  a

Anti-derivative of |x|

 x, x ≥ 0 For the function, f(x) = |x| =  .  − x, x < 0

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Function

This being a linear function has the same anti-derivative as discussed for linear function, that is a parabola. Thus, for x > 0, the anti-derivative of |x| is a quadratic function of the type x 2 (an upward parabola) and for x < 0, the antiderivative is a function of the type –x2 (a downward parabola).   Thus, the anti-derivative function of |x|, h(x)=  − where c is any constant.  Graphically, it can be shown as follows:

x2 , x≥0 c x2 , x – and a downward parabola of the type c   a  x2  b parabola of the type –  + x  for x < – . a  c 

1 x The reciprocal function is not defined at x = 0 as the function is not having any value in its range. Thus, the graph of this function sees a vertical asymptote at x = 0. The presence of a vertical asymptote at a point signifies that the value of the function is not defined at this point. 1 Graphical representation of f(x) = x Let us consider f(x) =

1 f(x) = x

This function is used in modelling situations where if the value of one variable increases, the value of the other variable decreases, that is there is inverse proportionality between the two variables. For example, the frequency of a pendulum’s swinging motion can be found out by taking the reciprocal of time period.

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Function

1 , 1, ( x + 1) 2x 2 can be drawn by transforming the parent reciprocal function 1 . x x 1 The function is not defined for x = –1, thus a vertical x +1 asymptote can be seen be seen at x = –1 in the graph of this 1 2 function. Both the function and are not defined at x = 0, 2x x thus a vertical asymptote exists at x = 0. All these graphs can be seen as follows: 1 Graphical representation of f(x) = 2x

The graphs of other reciprocal functions like

f(x) =

1 2x

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1 Graphical representation of f(x) = x +1

f(x) =

1 x+1

151

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Function

Graphical representation of f(x) =

2 x

2 f(x) = x

Some of the characteristics of the reciprocal functions are: • Their graph has horizontal and vertical asymptotes. This is because the fraction is not defined when denominator is 0. 1 That is, is not defined at x = 0, thus the y-axis acts as the x 1 vertical asymptote. Similarly, y = is also not defined for x y = 0, thus x-axis is the horizontal asymptote. • The domain and range of the parent function are all real numbers except 0. Derivative and Anti-derivative of The graph of the function f(x) =

1 x

1 is as follows: x

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f(x) = 1x

1 can be learnt by drawing tangents on a x curve which shows nature of slopes at different points on the graph. Graphically, the curve with tangents is shown as follows: The derivative of

1 f(x) = x

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Function

We see that the slope is always negative for any value of x. This is because the tangent at any point on the curve is tending downwards when moving from left to right. This brings us to the conclusion that • The graph of the derivative function (which is obtained by plotting various input points of the given function and the slope corresponding to that point) is always negative for any value of the independent variable (or the input value). • As the value of x tends towards 0 from both the sides of the function, the tangent gets steeper and almost near to being vertical. This verticalness of the tangents shows the slope is tending to infinity and thus cannot be defined. This suggest a vertical asymptote at x = 0 in the derivative function graph. • For x > 0, slope is negative. As x tends to 0, the tangent gets near to vertical, thereby showing value of slope is increasing. Here, the slope is undefined due to verticalness of tangent (any value which is undefined would either be + ∞ or – ∞ ). Since, the slope is negative for f(x) = 1 , it will tend to – ∞ . x As the value of x increases, the tangent gets flatter and for a large value of x, the tangent will be horizontal. Thus, the slope gets closer to 0. Hence, the value of the slope for x > 0 increases from – ∞ to 0. • For x < 0, the value of slope is decreasing. As the value of x decreases, the tangent is getting flatter (and the slope is getting closer to 0). Thus, the value of slope is decreasing (as x increases, the value of the slope goes from 0 to – ∞ ). From the above conclusions, the graph of derivative function 1 of can be drawn as follows: x

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g(x) = – 12 x

1 Similarly, the anti-derivative of can be interpreted by the x area under it. Let us draw some inferences. 1 Consider f(x) = in the interval [0, 2]. Since at x = 0, the x function is not defined, thus we can consider taking its antiderivative in the interval (0, 2]. The area corresponding to the sub-interval [0.1, 0.5] is less than the area corresponding to 1 [0.1, 1]. Thus, the curve of the anti-derivative function of x is increasing. Although, the area corresponding to [0.1, 2] is more than that corresponding to the interval [0.1, 1], but the difference is not much. This corresponds to the curve of the anti-derivative function that slowly increasing after a certain point as can be seen from the graph:

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Function

f(x) = 1 x

Such a graph of the anti-derivative function corresponds to the graph of log x can be seen as follows:

f(x) = log x

We will see in the chapters to come that the anti-derivative of 1 is log x. x

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Let us consider f(x) =

157

x

The famous Pythagoras theorem for right-angled triangle involves the square root function. Architects and carpenters use extensively right angles due to the inherent stretched strength of things joined at right angles and the Pythagoras theorem to build large buildings. Statisticians use this square root function to analyse how diverse are the data points from the average, while finding the standard deviation of the given data. The graph of f(x) =

x is as follows:

f(x) = √x

Based on this parent function, there are other square root xx ,and , , x –x 1, – 1, x +1 x +1 and 4x4x , which have 22 similar graphs and properties as that of the parent function x x . It can be shown graphically as functions of and 4x 2 are both defined for x > 0, thus they pass through the origin.

functions such as

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Function

f(x) =

x

√2

x –1 is defined for x – 1 > 0, i.e., x > 1. For the same reason, the graph of x +1 starts from x = –1 after which it is positive and the square root would be defined. However,

f(x) = √x – 1

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f(x) = √x + 1

f(x) = √4x

Thus, square root functions are of the type f(x) = ax + b , b which are defined for ax + b ≥ 0 or x ≥ – . a The general properties of a square root function are: • It is defined for only those values where ax + b > 0 (to make the square root function be actionable on real numbers), i.e., b its domain is x > – . a

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Function

• The range is also x > –

b . a

Derivative and Anti-derivative of

x

The derivative and anti-derivative of x can be studied in the same fashion as previous parent functions. For studying the derivative of the function x , we draw tangents at various points and consider the slope of these tangents at the point of consideration.

f(x) = √x

We infer the following from the graph of the given function and the slopes at different points: • There is a vertical tangent at x = 0, that is, the slope is not defined here. This shows that the graph of the derivative function also has a vertical asymptote at x = 0. • The slope is always positive, since the tangent at any point on the curve is tending upwards when move from left to right. • As the value of x increases, the tangent tends to become horizontal to the x-axis, which means the slope is decreasing.

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These inferences point out to the fact that the derivative function of the given function x is decreasing. It is defined for positive values of x only and gives the output as a positive value only. This suggests a reciprocal function as the derivative function. The description can be seen in the following graph:

g(x) =

1 2√x

The graph corresponds to that of

1 2 x

, which we will learn in

the chapters to follow which is the derivative of x . We must mention that the anti-derivative of the function f(x) = x can be studied by studying the area under the curve in any given interval. The anti-derivative function would then be having a graph consisting of the ordered pairs (point on the x-axis of the given function, area corresponding to this point).

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Function

x in the interval [0, 4].

Consider the graph of f(x) =

f(x) = √x

(1, 1)

(2, √2)

(4, 2)

(3, √3)

We observe the following from the above graph: • The area under the curve in the interval [0, 2] is lesser as compared to that in the interval [0, 4]. Thus, the anti-derivative function is an increasing one. • We would try to approximate the area under the curve y = x in the interval [0, 3]. For a better approximation, we consider its sub-intervals. Area corresponding to the point 1 in the interval [0, 1] 3

1 1 1 12 ≈ × b × h = ×1× 1 = = 2 2 2 2

Area corresponding to the point 2 in the interval [0, 2] 3

22 1 1 ≈ ×b×h= ×2 × 2 = 2 2 2

Area corresponding to the point 3 in the interval [0, 3] 3

32 1 1 ≈ × b×h = ×3× 3 = 2 2 2

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Area corresponding to the point 4 in the interval [0, 4] ≈

3 2

4 1 1 ×b×h= ×4× 4 = 2 2 2

Thus, the area corresponding to any point x in the interval 3 2

x 2 This description enables us to plot the approximate graph

[0, x] ≈

3

of the anti-derivative function of learn later in the chapters to follow.

h(x) =

x2 x , i.e.,≈ , which we 2

2 32 x 3

Let us consider f(x) = 2x Such type of functions are the exponential functions, where the value of the function increases mani-folds. For example, for the input 5, the value of the function is 25 = 32.

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Function

The graph of the above stated function is as follows:

f(x) = 2x

The most common exponential function is e x, where e is an irrational number with value 2.718, just like the irrational number Pi (π), its graphical representation is shown as follows:

f(x) = ex

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The exponential functions model – population growth of micro-organisms, such as bacteria in a culture shows enormous growth (that is, a very large number of something and each are multiplying rapidly). These functions are also used by financers while finding the amount received when the interest is compounded continuously (that is, the amounting periods are extremely large). The properties of exponential functions are: • They have a horizontal asymptote (the x-axis as can be seen from the graph). As we keep substituting negative values of x in the function, we will get smaller values of the function. The more the negative value of x, the lesser is the value of the function. However, nowhere the function will be 0. For example, for x = –1000 (say), 1 f(x) = 2x = 2–1000 = 1000 , which is a very small value ≠ 0. 2 • The domain of the function is the set of all real numbers. The range is (0, ∞). As discussed above, there is no value of x which will give the output as a negative real number or a zero. The function f(x) = e x is the parent function for all other exponential functions like 3 –x, 2 x+1, 2 1–x, whose graphs are drawn as follows:

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Function

Graphical representation of f(x) = 3–x

f(x) = 3–x

Graphical representation of f(x) = 2x+1

f(x) = 2 x+1

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Graphical representation of f(x) = 21-x

f(x) = 2 1–x

Merging the above shown graphs, we get

f(x) = 2 x+1

f(x) = 3–x

f(x) = 2 1–x

The nature of the graph of the function 2 x + 1 is similar to that of 2 x, where with the increasing values of x, the function increases rapidly. For the function 21–x, we can see a decreasing graph. This is because, the more the negative value of x, the more is the

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Function

value of the function. This is also true for the other function 3–x, where the curve is decreasing. Derivative and Anti-derivative of ex The derivative of ex can be studied from the nature of the slopes of its curve.

f(x) = ex

We draw the following inferences from the graph of the function and the tangents at different points: • We see that the slope is always positive, since the tangent at any point on the curve is tending upwards when moving from left to right. • We also see that the values of the slopes are increasing with increasing value of x. This depicts an increasing derivative function. • The values of the slopes are increasing rapidly with an increase in the value of x. This suggests that the graph of the derivative function is similar to the function itself. In fact, we will study later that the derivative of ex is ex. Recall, the anti-derivative of any function can be interpreted from the area under the curve of the function ex in a specific

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interval. Thus, to study the anti-derivative of ex, we need to study the area under the curve of ex in the interval, say, [–1, 3].

f(x) = ex

Since this chapter focuses on the conceptual understanding of the functions, without getting into its technical definition, we intend to draw few inferences. As the interval in which the area is considered goes from [–1, 0] to [–1, 1] and to [–1, 3], the area is increasing and it increases rapidly. This claims that the anti-derivative is an increasing function and can be described similar to ex. We see in the later part of the book that the antiderivative of ex is ex. Let us consider f(x) = log x Logarithmic functions are reverse of exponential functions and are used to model the magnitude of the earthquakes, among

170

Function

many diverse application of the functions. We usually express the magnitude of earthquakes on the Richter scale where a difference of 1 point on the scale equates to a 10-fold increase in the strength of the earthquake. For example, hypothetically an earthquake expressed to be 2.59 on the Richter scale would indicate a strength 390 (there are a few measures of magnitude of earthquake). Even the unit of measurement of sound, decibel is on the logarithmic scale. The log function can be graphically seen to be:

f(x) = log x

Here, the value of the function does not exist for x = 0, which acts as the asymptote (the y-axis does not ever meet the log x function graph). x The other logarithmic functions such as log(x + 1) + 2, log   2 can be transformed accordingly from the graph of the parent log function, log x. Here is the graph of a function made out of the parent function log x.

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f(x) = log

171

x 2

x x is not defined for = 0 or x = 0 and has the similar 2 2 shape as that of log x. Similarly, log (x + 1) + 2 is not defined for x + 1 = 0, that is, for x = –1. The nature of its graph is similar to that of log x and the asymptote can be seen at x = –1. log

f(x) = log (x + 1) + 2

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Function

The characteristics of the logarithmic function log x are: • The domain of the function is (0, ∞), that is, log function is defined only for positive value of x. Its range is the set of all real numbers. log 0 is not defined and log 1 = 0. Between 0 and 1, log function takes all negative values. Beyond 1, it is positive. • log x is not defined for x = 0, thus having a vertical asymptote at x = 0. Derivative and Anti-derivative of log x One way to discuss the derivative of log x is to find a function 1 whose anti-derivative is log x. Since the anti-derivative of is x 1 log x, thus the derivative of log x is , graphically shown as: x

1 g(x) = x

We can also draw inferences about the derivative of log x through the usual way of studying the slope of the curve of log x at various points.

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The following graph shows several tangents on the log x curve, each tangent offers the slope of the curve at the point of tangency.

f(x) = log x

We observe the following from the graph: • The tangent at each point of the curve makes an acute angle with the horizontal or the tangent at each point on the curve is tending upwards from left to right. It means the slope is always positive.. Thus, the derivative function will always take positive values (for x > 0, because log x is defined only for x > 0). • As the value of x increases, the slope decreases rapidly till x = 1 (the curve is steeper near x = 1) and then decreases slowly (the curve is flatter beyond x = 1 when tending towards + ∞ ). Thus, the derivative function is decreasing rapidly till x = 1 and then decreases slowly thereafter. These two graphical characteristics point to a graph such as this one:

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Function

1 g(x) = x

1 The above graph is that of g(x) = for x > 0, which is the x derivative of log x. The anti-derivative of log x can be studied from the area under its curve in the interval [0, 2] (Recall, we need to define an interval for finding anti-derivative). Since, log x is not defined at x = 0, we can consider the interval to be (0, 2]. The graph depicting the area under the curve in the chosen interval follows.

f(x) = log x

Since log x is not defined for x = 0, a vertical asymptote occurs at x = 0 for the anti-derivative function as well.

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We must also take the following points into consideration: • We cannot ignore the fact that area is always a positive quantity, and that anti-derivative can be obtained by signifying the area under the curve (in any interval). • We must also consider the values of y while considering the area. That is, if the area under the curve to be considered lies above the x-axis, it takes positive values of y, and thus a positive value of the area can be corresponded to. However, if the area to be considered lies below the x-axis, it then takes negative values of y, and area can be taken as negative. We must note that a negative area corresponds to the curve lying below the x-axis. It has got nothing to do with the magnitude being negative. Keeping in mind the above two points, we make the following inferences: • The area below the x-axis can be taken as negative and that above x-axis can be taken as positive. • The shaded area below the x-axis in the interval (0, 0.5] is less than the area in the interval (0, 1]. Since this shaded area carries a negative sign, thus it can be restated that – the shaded area below the x-axis in the interval (0, 0.5] is less (negative) than the area in the interval (0, 1]. This confirms that the anti-derivative function curve decreases till x = 1. • Beyond x = 1, the area is positive. The area under the curve corresponding to the interval [1, 2] is obviously more as compared to that in the interval [1, 1.5]. Thus, the antiderivative function increases after x = 1.

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Function

The description of such a graph is as follows:

h(x) = (x – 1) log x

We will discover that the anti-derivative of log x is given by (x – 1) log x represented by the above graph. Trigonometric functions Trigonometry is a widely applied domain of math. It is used in finding the height and width of buildings, by measuring the horizontal distance from the base of the building to the view point and using a secant to find the angle of elevation to its top. It is because of this branch that finding the height of mountains has become an easy task. Aviation industry uses trigonometry to calculate the speed and direction of airplanes while landing and taking off. Sine and cosine functions have great applications in CAT and MRI scanning, in detecting tumour or in laser treatment. Interestingly, the sinusoidal waves represent sound waves or musical notes and hence, sine function is used to develop computer music, stereo or any musical tones.

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In trigonometry, the parent functions are sin x, cos x, and tan x. Other trigonometric ratios are derived out of these three. 1 1 1 cot x = sec x = ; ; tan x cos x sin x The graphs of the parent functions are given below. Along with the parent function are some of their transformations. cosesc x =

f(x) = sin 2x

f(x) = sin x

Above are the graphs of the parent function sin x and one of its transformation function sin 2x. We observe that sin 2x has double the amplitude as that of sin x. f(x) = cos(1 + x)

f(x) = cos x

The above is the graph of the parent function cos x and one of its transformation function cos (1 + x). cos (1 + x) is the cosine curve which is 1 unit behind cos x.

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Function

x

x 2

f(x) = tan

f(x) =

tan x 2

Graphical representation of f(x) = tan x and f(x) = tan

Above is the graph of the parent function tan x and its x x transformation function tan ( ). tan has a similar graph as 2 2 tan x and has half the amplitude as that of tan x. These parent functions can be transformed to create and sketch interesting trigonometric graphs of sin 2x, cos (x + 1), x tan , and many more. 2 The properties of the parent function sin x are: • Its graph is a sinusoidal curve. • Its domain is the set of all real numbers and its range is [−1, 1]. The properties of the parent function cos x are: • Its graph is a sinusoidal curve (having the form of a sine curve). • Its domain is the set of all real numbers and its range is [−1, 1]. The difference between the sine and cosine function is the value which they take in their domain. Sine curve passes through the origin whereas cosine curve takes the value 1 at

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π x = 0. Sine takes the value 1 at x = . Thus, the cosine and the 2 sine function differ by π . This can be seen in the graphs of the two functions as well. 2 f(x) = sin x

f(x) = cos x

The properties of the parent function tan x are: • The graph of tan x is symmetric with respect to the origin. π • The tangent function is not defined at odd multiples of , 2 sin x π since tan x = and cos x = 0 at odd multiples of . 2 cos x • The range of the tangent function includes all real numbers. Derivative of sin x The derivative of sin x can be explored by studying the values of the slopes at various points of the curve. For that, consider the sine curve in the interval [0, 2π] and the tangents at various points of the sine curve.

180

Function

f(x) = sin x

From the graph and the tangents, we draw the following inferences: • The slope of the sine function is positive between x = 0 to π as the tangent at any point on the curve in this interval is 2 tending upwards when moving from left to right. π At x = , the slope is zero as the tangent at this point is 2 horizontal to the x-axis. π This imples that the derivative graph decreases till and 2 π at x = it touches the x-axis. 2 • Further till x = π, the slope is negative as the tangent at any point on the curve in this interval is tending downwards when moving from left to right. This depicts that the derivative graph takes negative values till π.

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The above description suggests the following graph: f(x) = cos x

This is the graph of the derivative of sin x and is that of cos x. Thus, the derivative of sin x is cos x. Similarly, the derivative of cos x can be figured out from its curve and the tangents drawn at different points on it. • We observe that the tangent at x = 0 is horizontal to the x-axis, which means the slope is 0 at x = 0. This suggests that the derivative function passes through the point (0, 0). • We also discover that the slope is negative throughout in the interval [0, π], since everywhere in this interval the tangent at any point on the curve is tending downwards when moving from left to right. Thus, the derivative function is decreasing in this interval [0, π]. • Further at x = π, the slope is 0. Thus, the derivative curve touches (π, 0). This suggests that the following graph is that of –sin x. Thus, the derivative of cos x is –sin x.

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Function

Graphical representation of g(x) = –sin x

(π ,0)

(0,0)

g(x) = – sin x

The graph of tan x in the interval [0, π] is given by

f(x) = tan x

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We can also draw some inferences of the derivative of tan x. For this, let us consider the graph of tan x in the interval [0, π]. We observe the following from the graph and the tangents: • The slope of the tan curve is always positive. Thus, the derivative function always takes the positive value. • The slope increases rapidly till

π indicating a rapidly 2

π . 2 π • At x = , there is a vertical asymptote showing that the 2 derivative function is not defined here. increasing function till

π , though the slope is positive, it decreases till π, 2 which indicates a decreasing curve in this region.

• After

The following is the graph of the derivative function, which is also that of sec2 x.

g(x) = sec2 x

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Function

Anti-derivative of sin x Recall, the anti-derivative of a function can be studied from the area under the curve of the given function. Thus, to discuss the anti-derivative of sin x, we consider the  π π function in the interval (say)  − ,  and consider the area  2 2 under this curve.

f(x) = sin x

π π Let us consider the intervals  − , 0  and 0,  :  2   2  • Approximate area of the shaded region in the interval   π  with base = π and height = –1  sin  − π  =   − 1   − 2 , 0  2  2   1 π π ≈ × × − 1 = − = − 0.8 ≈ − 1. 2 2 4 Though area cannot be negative, but just for representation, area can be taken with a negative sign suggesting that the curve is below the x-axis (or has negative values of the function). Similarly, area above x-axis with positive values of the function can be taken as positive.

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The negative area suggests a decreasing anti-derivative  π  function below the x-axis in the interval  − , 0  .  2  Also, the value –1 corresponds to the point 0 on the x-axis. Thus, the anti-derivative function passes through (0, –1). • We observe that the same area is occupied by the curve on the other side of the y-axis. Thus, if –1 is the value of the area on the left side, +1 is the value of the area on the right π side of y-axis (in the interval 0,  ).  2  π Approximate area of the shaded region in the interval 0,   2  π  π  with base = and height = 1  sin   = 1 2 2   ≈

1 π π × ×1 = = 0.8 ≈ 1 . 2 2 4

• Approximate area of the shaded region in the interval  π π  − 2 , 2  = (Approximate area of the shaded region in the interval  π   − 2 , 0  ) + (Approximate area of the shaded region in the π interval 0,  ) = –1 + 1 = 0.  2  π Thus, the value 0 (of area) corresponds to the point x = 2 on the x-axis. • For the same reason as above, on considering the sine curve  −3π −π  in the interval  , , the value 0 would correspond to 2   2 −π the point as well. 2

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Function

We thus sketch the following graph based on the above description.

h(x) = – cos x

 π π The above graph is that of –cos x in the interval  − ,  .  2 2 Thus, the anti-derivative of sin x is –cos x. Similar inferences can be drawn while studying the antiderivative of the function cos x, which is sin x. For considering the anti-derivative of cos x, we consider its graph in the interval [0, π].

f(x) = cos x

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π π  Let us consider the intervals 0,  and  , π  :  2  2  π • Approximate area of the shaded region in the interval 0,   2  π with base = and height = 1 ( cos 0 = 1) 2 1 π π ≈ × ×1 = = 0.8 ≈ 1 . 2 2 4 The positive area suggests an increasing anti-derivative π function above the x-axis in the interval 0,  .  2  π • Also, the value 1 corresponds to the point on the x-axis. 2 π Thus, the anti-derivative function passes through ( , 1). 2 • We observe that the same area is occupied by the curve for the remaining portion of the curve as well, but below the x-axis. Thus, if 1 is the value of the area above the x-axis, –1 is the value of the area below the x-axis (in the interval π   2 , π  ). Approximate area of the shaded region in the interval π π  , π with base = and height = –1 [ cos (–π) = –1]  2  2 1 π π × × − 1 = − = − 0.8 ≈ − 1 . 2 2 4 • Approximate area of the shaded region in the interval ≈

[0, π] = (Approximate area of the shaded region in the interval  π ) 0, 2  interval

+ (Approximate area of the shaded region in the π   2 , π  ) = 1 + (–1) = 0.

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Function

Thus, the value 0 corresponds to the point x = π on the x-axis. • For the same reason as above, on considering the cosine curve in the interval [–π, 0], the value 0 would correspond to the point x = –π as well. We thus sketch the following graph based on the above description: h(x) = sin x

The above graph is that of sin x in the interval [0, π]. Thus, the anti-derivative of cos x is sin x. These are precisely the few parent functions which gives way to study the infinite functions that we encounter. Just by understanding the basic concepts of calculus applied to these parent functions, these concepts can be applied to the infinite functions as well and can be understood with ease. However, there can be other situations which we deal with in our daily lives. The coming section talks about these situations and how to model them. Crafting functions We hope that the choice of the functions presented till now does not pass on an impression that functions are all standard, and created by mathematicians. The reality canot be farther than truth – function are used to mathematise everyday-life situations, and we all can create mathematical equivalent of

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everyday situations. Here are just a few examples of how we create functions. Consider the following situations: • Consider a water-bottling plant. It has two key operations – filling and capping the bottles (of course, there are other steps, such as cleaning, labeling too). If the filling of water is considered as one function (relating time and quantity of filled water), say f, and capping the bottle as another function, say g, then the quantity of water bottles produced in the plant is a result of these two steps. In terms of functions, the quantity of water bottles produced in a factory and the time taken to fill water bottles can be considered as a combination function as f + g, where + is a replacement of ‘and’, mathematically. • Imagine a warehouse that creates invoices the sales in tonnes, but the weight is measured physically and thus, the unit of measurement is kilogram (electronic weigh bridges can measure much larger weight and directly in tonnes). Expectedly, for invoicing/billing, the weight is first converted to tonnes. Now, assume f be the function which represents the price of a thing for one tonne, and g be the function which converts weight in kilogram to tonnes. Since, 1 kilogram is 0.001 tonne and suppose the price of the item per tonne is 200000, then the first function becomes f(x) = 200, and second function becomes g(x) = 0.001x. For instance, if an item weighs 100 kg then the invoice price of the item can be calculated by applying first the function g to it, i.e., g(100) = 0.001 × 100 = 0.1. The price of the item f(x) × g(10) = 200000 × 0.1 = 20000. • Think of a commonplace vending machine at a food court; users put in money, press a specific button and a particular item drops into the delivery slot. In many older version of the machines, the number of times a button is pressed represents the required quantity of the item. Let f(x) be considered

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Function

to be the function which represents the price of the item where x is the number of items and g(y) the function which represents the quantity of the items, where y is the number of times the button is pressed. Then, g(y) = y. A user who wants 2 bottles of coke, will press the coke button twice, i.e., g(2) = 2. If the cost of one coke is 20, then f(x) = 20x. The total cost of the two coke bottles is then given by the function f(g(2)) = f(2) = 20 × 2 = 40. If the user has entered more money (say 50) than required by the function rule, the change ( 10 here) is also delivered along with the required item. • We work for 20 hours in a week at some store and receive 5000 as the weekly salary, plus a 5% commission on sales over 1000. Suppose we sell enough this week to get the commission. How can we describe the commission through a function? Assume a function f(x) = 0.05x and k(x) = x – 1000. We take the sales x, subtract off 1000 on which we do not get the commission, and then multiply by 5%. This is equivalent to 0.05(x – 1000) = f(x – 1000) = f(k(x)) • Consider a deflated football which is being inflated by pumping air (one that is still almost spherical, just shrunken). Let us say that the pumping volume is such that till the football is completely inflated, the radius of the ball is changing at the rate of 2 cm/sec. Then, the radius can be considered a function of time and taken to be given by the function r(t) = 2t. How can we find the function that represents the rate at which the volume of the football 4 3 π r is the volume of the is changing? We know, V(r)= 3 football, and then the rate at which the volume changes is 4 4 8 3 3 3 given by V(t) = V(r(t))= π(r(t)) = π(2t) = π(t) . Thus, if 3 3 3 the radius is changing at the rate of 2 cm/sec, the volume will change at the rate of 8 cm3/sec.

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These are all examples of composition of functions, which we encounter in various situations in our day-to-day life. Indeed all functions are some unique combinations of parent functions. Graphical representation of combination functions We have used graphs for finding the derivative and antiderivative of the parent functions. We also used the first principle – the most basic definition of derivative and antiderivative – to find the same for the parent functions. We need not do the same for non-parent functions, the functions made out of ‘combining’ parent functions (mathematically relating, using operations); there are infinite number of such functions. The next chapter is all about how we find derivative of functions, and that would be the right place to find the derivative of non-parent, combined functions. However, the next chapter will not consider graphs of functions. In this chapter, thus, we will explore the graphs of some combined functions, to get a feel of how we must visualise combined functions (enabling visualisation of the derivative and anti-derivative of the graph of the combined functions). To get a better understanding of how some of the functions look, let us combine a few parent functions and explore their properties: 1. x + log x The function is the sum of a linear and a logarithmic function.

Function

)=

x

Graphical representation of f(x) = x

f(x

192

Graphical representation of f(x) = log x f(x) = log x

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Combining both the functions, we have the following graph for f(x) = x + log x

f(x) = x + log x

We infer the following from the above graph: • The function log x is only defined for x > 0, thus the function f(x) = x + log x is also not defined for negative values of x. Hence, an asymptote can be seen at x = 0. • A linear function dominates the log function, that is, x > log x. This can be verified by substituting different values of x. For example, for x = 1, 1 > log 1 (= 0) and for x = 2, 2 > log 2. The graph of the combined function x + log x, thus, takes the properties and graph of both the linear and the log function. For x = 1, f(x) = 1 + log 1 = 1 For x = 2, f(x) = 2 + log 2 = 2 + a very small value For x = 3, f(x) = 3 + log 3 = 3 + a very small value… Thus, the shape of its graph is very similar to that of the linear function, since the logarithmic values are not very large.

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Function

f(x )=

x

2. x + sin x We now study the function which is the sum of a linear and a sine function. Graphical representation of f(x) = x

Graphical representation of f(x) = sin x

f(x) = sin x

The following graph can be observed if both the above functions are combined as f(x) = x + sin x:

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f(x) = x + sin x

The above graph is an outcome of the fact that sine takes only values between –1 and 1. Thus when added to the linear function either decreases it or increases it by maximum a unit. This small oscillation can be seen in the graph as well.

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Function

f(x )=

x

3. x + ex We now study the function which is the sum of a linear and a exponential function. Graphical representation of f(x) = x

Graphical representation of f(x) = ex

f(x) = ex

When both the above functions are combined, the graph of the function to follow takes the properties from its parent functions. The graphical representation of the combined function f(x) = x + ex can be seen as:

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f(x) = x + ex

We draw the following inferences from the graph: • The values of the exponential function is always greater than that of the linear function, that is, e x > x. This can be verified by substituting various values of x. For example, for x = 0, e 0 (=1) > 0; for x = 1; e1 > 1; and so on. • When the exponential function is added to the linear function, it results in a value which is much greater than the value of the linear function. For example, For x = 0, f(x) = 0 + e0 = 1 For x = 1, f(x) = 1 + e1 = 1 + a large value For x = 2, f(x) = 2 + e2 = 2 + a large value For larger value of x, f(x) is very large. Thus, the shape of the curve of the combined function is more than that of the exponential curve, that is, very large values of f(x) for larger values of x.

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Function

All the below given combination functions (which takes the characteristics of both the parent functions) can be obtained in the similar way as the above functions. 4. sin x + log x We now study the function which is the sum of a linear and a log function. Graphical representation of f(x) = sin x

f(x) = sin x

Graphical representation of f(x) = log x

f(x) = log x

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Graphical representation of the combined function f(x) = sin x + log x is given by:

f(x) = sin x + log x

At x = 0, sin 0 = 0 and log is not defined. Thus, in the graph of the combined function, there is a vertical asymptote at x = 0. For x > 0, the occurrence of the wave is because of the sine function as the logarithmic values are not very large. Thus, the addition of the logarithmic values (which are very small) and the sine values (which are between –1 and 1) results in the above graph. 5. ex + log x This combined function is the sum of the exponential function and the logarithmic function.

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Function

Graphical representation of f(x) = ex

f(x) = ex

Graphical representation of f(x) = log x

f(x) = log x

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Graphical representation of the combined function f(x) = ex + log x is given by:

f(x) = ex + log x

We draw the following inferences from the graph: • The combined function exists only for x > 0 as the log function can be defined only for x > 0. • A vertical asymptote can be seen at x = 0 as log 0 is not defined. • Thereafter, for x > 0, e x lets the function increase exponentially, but the log function keeps the values small. 6. sin x + ex Such a function is the sum of the sine function and the exponential function.

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Function

Graphical representation of f(x) = sin x

f(x) = sin x

Graphical representation of f(x) = ex

f(x) = ex

Calculus For Professionals

Graphical representation f(x) = sin x + ex is given by:

of

the

combined

203

function

f(x) = sin x + ex

We draw the following inferences from the graph: • The sine function always takes the values between –1 and 1. However, exponential function takes a much larger value as compared to the input values of x. • The exponential function dominates the sine function for x > 0. Thus, for x > 0, more of an exponential curve can be seen. • The combined function shows the effect of both the sine function (seen as waves for x < 0) and the exponential curve (for x > 0). For x < 0, sine function dominates e x. This is because ex takes small values for negative values of x.

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Function

1 x Given function is the sum of the sine function and the reciprocal function. Graphical representation of f(x) = sin x

7. sin x +

f(x) = sin x

Graphical representation of f(x) =

1 x

f(x) =

1 x

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The sum of the above two functions result in a new function 1 f(x) = sin x + that is given by: x

f(x) = sin x +

1 x

We infer the following from the functions and the graph: • The function is not defined for x = 0, thus a vertical asymptote can be seen. This is because of the presence of the reciprocal function, which is not defined for x = 0. • Sine values are between –1 and 1. The presence of the sine function gives the combined function some waves. 1 decreases. Thus, the graph of x the combined function is much steeper for smaller values of x.

• As the value of x increases,

8. log x +

1 x

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Function

Graphical representation of f(x) = log x

f(x) = log x

Graphical representation of f(x) =

1 x

f(x) =

1 x

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Thus, combining the functions results in the following graph of 1 f(x) = log x + is: x

f(x) = log x +

1 x

We draw the following inferences form the graph and the individual functions: • The function is not defined for x < 0 because of the presence of the log function. • A vertical asymptote can be seen at x = 0 because 1 both log x and are not defined for x = 0. x • For increasing values of x, the reciprocal function is decreasing. Also, the log values are anyway small as compared to the input values. Thus, a gradual and small increase in the graph of the combined function can be observed (due to the log value). 9.

ex +

1 x

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Function

Graphical representation of f(x) = ex

f(x) = ex

Graphical representation of f(x) =

1 x

f(x) =

1 x

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Thus, the graphical representation of the combined function 1 f(x) = ex + obtained by adding both the functions are: x

f(x) = ex +

1 x

The following inferences can be drawn from the graphs: • For x < 0, ex has very small values, thus the graph of the 1 combined function is similar to that of for x < 0. x • At x = 0, the combined function is not defined, since the reciprocal function is also not defined for x = 0. • For x > 0, ex increases by a large value as compared to the values of x. Thus, an exponential increase can be seen for x > 0.

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Function

10. (sin x)2 Graphical representation of f(x) = (sin x)2 f(x) = (sin x)2 1

The function (sin x)2 can be obtained by squaring the sine function, whose graph is as follows: f(x) = sin x

We know that the square of any function is positive. Thus, squaring the sine curve results in positive values of the new function for every point in the domain. This can be seen in the graph as the function takes only positive values for any value of x. Functions and the mathematical operations that calculus works on have expanded our ability to understand our world beyond what we can learn out of purely personal experiences, way beyond the physical touch and feel of the ways of the world. We have abstracted the real world into functions and operations on them. The mathematical expression of functions in place of actual situations involving physical actions and objects not only saves costs and make the study of changes simple, and it also makes the study of many situations feasible; function has made calculus possible.

Differentiation

Beyond change – The rate of change There are infinite changes continuously occurring around us, such as the changing atmospheric temperature during day and night, speed of an airplane when it is in motion, the pressure exerted by the water/any fluid on the walls of the pipes in any pipeline system (for fluids). The natural as well as the man-made world is full of situations and things that constantly evolve, morph their magnitude, direction, shape, place, etc. More pertinently, even the way the change happens is almost never really constant; for example, the revolution of the planet around the sun varies by the minute, both in terms of speed and direction. The volume of a football or a basket ball varies with the rate at which the air is pumped into them. The rate of the change itself varies for most changes around us; it is indeed more common that the amount of change of something varies with unit time (the change could also be in terms of any other independent variable such as change in pressure due to change in temperature, or the change in volume due to change in the radius). Clearly, to precisely and comprehensively know the nature of all such continuously changing things around us, we need to specifically compute the rate of change at all the times, and situations of interest. The need for functions Functions are used to express all kinds of situations which have quantifiable changing conditions, and events. Functions 211

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Differentiation

are the mathematical equivalent of situations expressed in natural languages. We functionalise situations in terms of variables and their relationships. Rate of change as a proxy of differentiation Consider an object in motion. To express the situation mathematically, we functionalise it to know the length of distance travelled in a given duration. To find out how fast or slow the object is moving, in other word to find its rate of change, the process of differentiation is used. Differentiation numericalises how much has the dependent variable changed with a unit change in the independent variable. For example, the consumption of a bundle of goods determines the level of utility derived from it. As consumption increases, the level of utility increases. Thus, the utility function is dependent on the consumption of bundles of goods. Broadly, the marginal utility of a good is the rate of change of utility function with respect to the consumption of goods. If x is the amount of the goods, U(x) is the utility function, d and represents a change in the utility function with a small dx change in the amount of goods, then mathematically, marginal d U(x) . utility is given by MU = dx Similarly, in a conductor, as the flow of electric charges increase, the electric current increases (keeping other factors constant). The magnitude of the current is the rate of change in the flow of electric charges through a cross-section of the conductor. Broadly, the current is the derivative of the quantum/amount of charge per unit time. If I is the electric current, Q is the d electric charge and is the change in the electric charge dt

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function with a small change in time is mathematically, given by, dQ . I= dt d d Let us elaborate our understanding for or . dx dt The concept of differentiation Differentiation is a mathematical process where we find the instantaneous rate of change of a function with respect to a variable. The following ‘logical steps’ may be visualised through an object in motion: 1. Something is continuously changing (slow, fast, regular or irregular, …); for example, a car in motion is continuously changing its position. It can be fast when on a highway or slow when in a traffic jam. Whatever the case is, it is continuously changing its position. 2. The change in position is measured by a (physically measurable) quantity. Distance is that quantity which measures the change in position of any object in motion. 3. That quantity reflects the change, i.e., motion. Distance reflects the change which is happening for any moving object. 4. The rate of change of that quantity could also be changing. The change in distance with respect to time could also be a variable and could be changing. 5. The instantaneous value of the rate of change represents ‘something else’. The rate of change in distance with respect to time represents speed, which also changes. 6. The value of ‘something else’ is the value derived out of another quantity. Speed, which is the rate of change of distance with respect to time, is derived out of distance. 7. The derived quantity is, thus, called a derivative. Speed is the derivative of distance. 8. The derived quantity is not a primary measurable observation. Speed cannot be directly measured. It can be measured only when there is a change in the position, or

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motion. When there is no change in position, there is no motion, and thus, no speed. 9. The derived quantity changes in tandem with the primary quantity. As the distance (which is the primary quantity here) varies, the speed (which is the derived quantity out of distance) also varies. The derived quantity does not describe the nature of the derivative. It takes an intensive understanding of the thing that is changing to describe and define the derived quantity. To appreciate the nature of speed, we must command an understanding of the real-world motion. In general, derived quantity is a new, special function derived from a primary function. It is a new ‘fact, detail, or knowledge’ about a changing situation. For example, current is the derived quantity from electric charge. Current, thus, adds to the knowledge of electric charge in motion. The derived quantity is the value of the (primary) function (that is changing) at an instant. 10. Theoretically, a derived quantity is also a function which can change with respect to any other quantity. It must be possible for the derived quantities to derive another quantity. Rate of change of velocity/speed is acceleration. Acceleration is thus is derived out of velocity, which is also derived quantity out of the changing distance. Given the conceptual understanding of ‘derivatives’ as ‘rate of change’, let us formulate it mathematically and technically for an in-depth understanding of one of the most basic concepts of calculus. Welcome to derivatives We have already learnt in Chapter 1, Introduction that for a straight line, the rate of change remains constant, whereas it keeps changing in case of a curve.

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Graphically,

y 3 2

y uniform rate y1

of change

non-uniform rate of change

y0

1 0

1

2

3

x

0

x0 x1

2

x

This is the catch – the rate of change is rarely constant in the real world. So, how do we define the rate of change when it is changing? Rate of change is the ratio of the change in value of the function of x, i.e., y = f(x) by change in value of x. Mathematically, Rate of change =

(y – y ) ∆y = 1 0 ∆x ( x1 – x 0 )

where (x0, y0) and (x1, y1) are coordinates of graph. =

f(x1 ) – f(x 0 )

[Substituting y = f(x)] x1 – x 0 In case of a curve, where the rate of change is not uniform, the points x0 and x1 can be taken very close to each other to know the immediate rate of change. Thus, x 1 can be taken as x 0 + h, where h is a very small quantity. These points can be shown on the graph as:

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Differentiation

y

f(x0)

x0 x0 + h

0

Thus, rate of change = =

f(x0 + h)

x

f(x1 ) − f(x 0 ) f(x 0 + h) − f(x 0 ) = x1 − x 0 (x 0 + h) − x 0

f(x1 ) − f(x 0 ) f(x 0 + h) − f(x 0 ) = x1 − x 0 (x 0 + h) − x 0



[Substituting f(x1) = f(x0 + h)]

=

[ f(x 0 + h) − f(x 0 )]

h where h is a very small quantity. The ‘rate of change’ can also be termed as ‘slope’. This new notation of slope in terms of the function shows that slope is a function itself and is useful for advancing from the idea of the slope of a line to a more general  concept for slope of functions/curves. We aim to find the slope of a curve at a point, for which we need to make the change in x as small as possible, , x0+h is very close to x0. This can be achieved when h is approaching 0.

 f ( x 0 + h ) – f(x 0 ) Slope = lim  h →0 h The notation of ‘h’ when approaching to 0 is denoted as lim . h →0

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This formula is derived from the function. This formula for slope can be termed technically as ‘derivative’. Thus, derivative of a function gives a formula to find the slope of a function at a given value of x, and is referred to as the instantaneous rate of change of a quantity with respect to the other quantity. For a car in motion, the derivative at each point of its motion gives the velocity with which it is travelling. Precisely, ‘How fast is the car travelling?’ is answered by finding the derivative. Derivatives in real life Derivatives are a way to quantify the change in something (a quantity) with respect to a change in something else (another quantity). For example, the change in distance with respect to change in time gives the speed. Questions such as below are all about derivatives: 1. By how much the demand of a product increases/ decreases when the price decreases/increases, keeping other things constant? Demand of a product is dependent on its price. Higher the price, lower would be its demand. As the price of the product increases with time, its demand decreases. Thus, demand is the rate of change of the price with respect to change in time.

y 4

f(p)

3 2 1 0

1

2

3

Price (p)

4

x

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Differentiation

2. By how much should be the dose of a medicine be increased/decreased to lower a person’s cholesterol level? The amount of medicines to be administered to the patient depends on its effectiveness. If the rate at which it is lowering the cholesterol level of the patient is high, the dosage can be low. Thus, the rate of change of the cholesterol level with respect to change in time gives the dosage of the medicine. 3. By how much the cell count of a colony of bacteria increase with initial increase in temperature? Bacterial growth is dependent on temperature. As temperature increases initially, the cell count of bacteria growing in a culture increases. Thus, bacterial growth is the rate of change of number of bacteria with respect to change in temperature. Growth of bacteria

y

Temperature

x

4. How does the stock market fluctuate with time? Stock prices are dependent on the news released on earnings and profits, and future estimated earnings of the company. As the earnings and profit increases, share price also increases. Therefore, measure of stock market fluctuation is the rate of change of stock prices with respect to earnings and profits of the company. 5. How do the number of cases of the affected people in a pandemic increase or decrease in a country?

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The number of people affected by the pandemic are dependent upon the precautions and dosage of vaccine taken. Higher the precautions and dosage, lesser the number of casualties. 6. By how much has the literacy rate increased/decreased in the rural area? Literacy rate depends upon the government schemes in the rural schools and awareness campaign. Higher the impact of awareness and government schemes, higher the literacy rate. 7. By how much has the consumption of petrol/diesel increased/decreased with the hike in their prices? Higher the price, lesser the consumption. Therefore, petrol consumption is dependent upon the rate of change of the petrol prices with respect to time. These are some examples where derivatives play an important role. How are derivatives represented? By the terminology of derivatives, derivative at a point is measured by the slope at that point. For any curve, y = f(x), the derivative at  f ( x + h ) − f(x 0 ) any point (x0, f(x0)) is given by lim  0 . However, we h →0 h need a simplified way to represent/denote this instantaneous rate of change. Derivatives are, thus, represented by

dy or dx

df ∆y or f ′ (x). It is the same as the ordinary rate of change, , dx ∆x

except that both the changes ∆y and ∆x are extremely small. Thus, dy and dx are imagined to be infinitesimally small and are those infinitesimal bits in which the problem is chopped and solved.

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Differentiation

1 1 Consider the function y = x2 and any point  ,  on it. Since 2 4 derivatives are represented by slope, dy is an infinitesimal small change in y and can be thus considered as an infinitesimal rise and dx is an infinitesimal small change in x and can be thus considered as an infinitesimal run. Therefore, infinitesimal rise dy = slope = infinitesimal run dx Geometrically, as shown in case of the curve y = x 2 below, the rise over run is characterised by the trigonometric formula tan θ, where θ is the angle which the secant joining the points P and Q and the horizontal line drawn through P. Since, slope is all about infinitesimal changes, thus it is logical to consider tangent at a point rather than considering a secant passing through two points. Thus, slope at a point can be represented by the tan of the angle at which the tangent is inclined at that point.

Points (x, y) and (x + dx, y + dy) are located on the parabola y = f(x) = x2, then f(x) = y + dy = (x + dx)2 = x2 + 2x(dx) + (dx)2

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The term (dx) 2 being very small can be discarded. Thus, we have, y + dy = x2 + 2x(dx). dy Since, y = x 2, we are left with dy = 2x(dx), or = 2x. This dx gives the slope of the curve at any point on the curve. 1 This means that an infinitesimal change in x near gets

1

2

converted to an infinitesimal change in y near and is given 4 dy by = 2x. dx Thus, the derivative of x2 is 2x, which means that the slope of the curve/function is 2x at all points. Since the value of x changes at every point, so is the change in the value of its slope.

1 2

At x = , the slope is given by 2 ×

1 = 1. 2

y = x2

Using the fact that slope = tan θ, where θ is the angle made by the tangent with the horizontal in the anti-clockwise direction, we observe from the graph that when x < 0, the slope is negative. Also, from the ‘derivative of x 2, which is 2x,’ we see that for a negative x, the term 2x is also negative, which signifies a negative slope.

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Differentiation

That means for an increase in the value of x (for x < 0), the value of y is decreasing at double the rate. Similarly, for x > 0, the derivative as well as from the graph of 2 x , it is clear that the slope is positive and hence signifies that as the value of x increases (for x > 0) the value of y also increases at double the rate. However, applying this technique to complicated functions such as trigonometric or higher order polynomials is tedious. Thus, we have to ‘explore’ another approach to find the derivatives. More about derivatives Let us recall, derivative is denoted as:  f ( x 0 + h ) − f(x 0 ) slope = lim  h →0 h Let us understand derivatives in detail using the above mathematical expression. To get around the difficulty of finding the slope of a curve, the following process is used. Considering the above discussed expression for the parabola y = x2, its derivative/slope at any point x0=

1 is given by first 2

 f ( x 0 + h ) − f(x 0 ) calculating the ratio  , which is as follows: h 2 2  1  1  1 2 + h −      +h +h −   2    4  2 = h h

1  4

(h =

2

+ h) h

=

h(h +1) =h+1 h

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This ratio approaches 1 as h → 0. Graphical representation of parabola showing slope: y

f(x0 + h) f(x0)

x0 x0 + h

x

df (x0), dx  f(x + h) − f(x 0 ) or Df(x0), is defined as lim  0 if this limit exists. h →0 h It can also be expressed as: f ( x + ∆x ) − f ( x ) df = lim →0 dx ∆x∆ ∆x y = lim . ∆x →0 ∆x is a small change along x-axis, this quantity can also (Since be taken as h). f (x + h) − f (x ) df = lim dx h→0 h df ∆y or, = lim dx ∆x →0 ∆x Derivatives, thus, use the concept of limits for its evaluation. Thus, the derivative  of  f(x) at  x0, written as  f(x0),

Example 1 Differentiate f(x) =

x , using limits.

Solution Using the formula of derivatives in terms of limits, we have, x+h – df f(x + h) – f(x) = lim = lim h →0 dx h →0 h h

x

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Differentiation

= lim

x+h – x

Substituting,f(x) = x   

h

h →0

Dividing numerator and denominator by ( x + h + x ), = lim

(

x+h– x h

h →0

= lim h →0

= lim h →0

h

h

= lim h →0

=

∴

(

x+h+ x

x+h+ x

( x + h) – x x

)

x+h+ x

)

x+h+

h

)

)

1 x+h+ x

1 x+0+

1 x + 1 = 2 x =

(

(

)(

x

df 1 = dx 2 x

x

( h → 0 )

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Graphical representation of f(x) =

x and

225

df 1 = dx 2 x

y = √x

df 1 = dx 2√x

Example 2 Differentiate f(x) = Solution

1 3 x – , using limits. 2 5

Using the formula of derivatives in terms of limits, we have df f(x + h) – f(x) = lim dx h→0 h 1 3 1 3 (x + h) – –  x –  1 3 2 5 2 5  = lim Substituting f(x) = x –   h →0 h 2 5  1 1 3 1 3 x+ h– – x+ 2 5 2 5 = lim 2 h →0 h

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Differentiation

1 h = lim 2 h →0 h 1 = 2 df 1 ∴ = dx 2 Graphical representation of f(x) =

df 1 = dx 2

1 3 df 1 x – and = 2 5 dx 2 3 – 5 1 x )= 2

f(x

1 . 2 This means that y increases half a unit for every unit increment in x.

This being a linear function increases at a constant rate of

Example 3 Differentiate f(x) = 5x2 – 3x + 7, using limits. Solution Using the formula of derivatives in terms of limits, we have

df f(x + h) – f(x) = lim h → 0 dx h

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df {5(x + h)2 – 3(x + h) + 7} – {5x 2 – 3x + 7} = lim dx h→0 h [Substituting f(x) = 5x2 – 3x + 7] = lim h →0

5x 2 + 5h 2 + 10xh – 3x – 3h + 7 – 5x 2 + 3x – 7 h

df 5h 2 + 10xh – 3h = lim dx h→0 h

df h(5h + 10 x – 3) = lim dx h →0 h df = lim (5h 5h + 10 3 10xx ––3) dx h →0 = 10x + 3

df = 10x – 3 dx

(∵ h  0)



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Differentiation

Graphical representation of f(x) = 5x 2 – 3x + 7 and

df = 10x − 3 dx

f(x) = 5x2 – 3x + 7

df = 10x – 3 dx

Example 4 Differentiate f(x) = 4 – x + 3 , using limits. Solution Using the formula of derivatives in terms of limits, we have =

f(x + h) − f(x) df = lim h → 0 dx h

= lim h →0

(4 –

) (

x +h+3 – 4 – x +3 h

)

(Substituting(x) = 4 – x + 3 )

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df dx df dx

x+3 − x+h + 3

= lim

(Simplifying)

h

h →0

x+3 − x+h+3

= lim

h

h →0

229

x+3 + x+h+3

×

x+3 + x+h+3

(Multiplying and dividing by the conjugate of the numerator)

( x+3) – ( x+h+3)

= lim

h

h →0

df dx

df dx

= lim

h→ 0

= lim

h →0

= =

(

h

(

x+3 + x+h+3

)

−h x+3 + x+h+3

[By using (a – b) (a + b) = a2 – b2]

)

−1 x+3 + x+h+3

–1 x +3 + x + 3 –1

2 x +3 −1 df = dx 2 x +3

(∵ h  0)



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Differentiation

Graphical representation of f(x) = 4 – x + 3 and

df −1 = dx 2 x+3

f(x) = 4 – x + 3

df −1 = dx 2 x+3

Example 5 Differentiate f(x) = Solution

x +1 , using limits. 2−x

Using the formula of derivatives in terms of limits, we have

df dx

= lim

h →0

f(x+h) − f(x) h (x+h)+1

df dx

= lim

h →0

2 − (x + h) h

−

(Substituting f(x) =

x +1 2−x

)

x+1 2−x

x +1   Substituting f(x) =  2 – x 

As the function is not defined when the denominator = 0, i.e., f(x) is not defined for x = 2. Thus, we intend to find the derivative of the given function for x ≠ 2.

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= lim

(x+h+1)(2 – x) – (x+1) (2 – x – h)

df dx

2

= lim

x →0

=

df dx

2

2x+2h +2 − x − xh − x − (2x − x − xh + 2 − x − h)

(2 − x − h ) (2 − x ) h

h→0

= lim

3h

(2 – x –h) (2 – x ) h 3

( 2– x ) ( 2 – x ) =

(Simplifying)

h 2 –(x + h) ( 2 – x )

h →0

3

( 2 – x )2

(∵ h → 0)

231



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Differentiation

Graphical representation of f(x) =

x +1 df 3 and = 2−x dx (2 − x)2

df 3 = dx (2 − x)2

f(x) =

x +1 2− x

Example 6 Differentiate f(x) = sin x, using limits. Solution Using the formula of derivatives in terms of limits, we have df f(x + h) − f(x) = lim dx h →0 h df sin (x + h) − sin x = lim h → 0 dx h

[Substituting f(x) = sin x]

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233

df sin x cos h + cos x sin h − sin x = lim h →0 dx h [∵ sin (A + B) = sin A cos B + cos A sin B]

df sin x (cos h − 1)+ cos x sin h = lim h →0 dx h df sin x (cos h − 1)+ cos x sin h = lim h →0 dx h

(Taking sin x common )

 cos h − 1  = lim sin x   + cos h→ 0 h  

 sin h  x   h  df cos h − 1 sin h = lim (sin sin xx ×× 0 + cos xx ××1) 1 ( lim =0 and, lim =1, h →0 h →0 h →0 dx h h cos h − 1 sin h   = 0 and, lim = 1  lim h→ 0 h → 0 h h  

= lim cos x h →0

= cos x ∴

df = cos x dx

Graphical representation of f(x) = sin x and

f(x) = sin x

df dx

= cos x

df = cos x dx

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Differentiation

Example 7 Differentiate f(x) = cos 3x, using limits. Solution Using the formula of derivatives in terms of limits, we have df f(x + h) − f(x) = lim h → 0 dx h df cos 3(x + h) − cos 3x = lim [Substituting f(x) = cos 3x] h → 0 dx h df cos 3x cos 3h − sin 3x sin 3h − cos 3x = lim h → 0 dx h [ cos (A + B) = cos A cos B – sin A sin B]

df cos 3x (cos 3h − 1) − sin 3x sin 3h = lim (Taking cos 3x common) h → 0 dx h cos 3h − 1 sin 3h  df  = lim  cos 3x − sin 3x h → 0 dx h h   cos 3h − 1 sin 3h  df  = lim  3cos 3x − 3sin 3x dx h →0  3h 3h  (Multiplying and dividing by 3 in both the terms)

 cos 3h − 1  sin 3h  df cos h − 1 si  − 3 sin 3x  lim = 3 cos 3x  lim (lim = 0 and lim   h→0 dx 3h h   h → 0 3h  h → 0  h→0 ( lim h →0

df = 3 cos 3x(0) − 3 sin 3x (1) dx df = −3 sin 3x dx

cos h – 1 sin h = 0 and lim = 1) h →0 h h



Calculus For Professionals

Graphical representation of f(x) = cos 3x and

f(x) = cos 3x

df dx

235

df = –3 sin 3x dx

= – 3 sin 3x

Steps to find the slope of a function at a point Derivatives give the function for the slope of a function at various points on it. This relationship gives way to the steps of finding the slope of a function at a given point x0. • Find the derivative of the function. There are many different ways to do this, depending on the function. One way has been shown as above. (We will study other ways to find derivatives of various functions in the coming sections in detail.) • Plug the x-value, x0, of the point into the derivative. This is the slope of the function at the required point. Ways to find the derivative of functions As mentioned in the beginning of the chapter, there are just a few parent functions. The rest of the functions can be derived from these parent functions. Thus, to find the derivatives/slopes of functions, it is enough to find for just the parent functions and then find the derivatives of the other related functions using those for the parent functions.

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Differentiation

Rules for finding the derivative There are certain basic rules that are used at different levels to find the derivative of the functions. These are as follows: Basic derivative formula rules 1. Constant Rule: The derivative of a constant function is zero, i.e.,

d (˘ ) dx

2. Constant Multiple Rule: 3. Power Rule:

d d cf ( x ) = c f ( x ) dx dx

d n x = nx n-1 dx

( )

4. Sum Rule: The derivative of the sum of functions is the sum of the derivative of the functions, i.e.,

d d d [ f(x)+ g(x)] = f(x)+ g(x) dx dx dx

5. Difference Rule: The derivative of the difference of functions is the difference of the derivative of the functions,

d d d [ f(x) − g(x)] = f(x) − g(x) dx dx dx d d d 6. Product Rule: [ f(x)g(x)] = f(x) g(x)+ g(x) f(x) dx dx dx d d − g(x) f(x) f(x) g(x) d  f(x)  dx dx 7. Quotient Rule:  = 2 dx  g(x)  [g(x)] d d d 8. Chain Rule: f [ g(x)] = f [ g(x)]. g(x) dx dx dx i.e.,

Some Derivative Formulas Listed below are some important derivative formulas. 1. Elementary Functions (a) d x n = n.x n-1

dx

( )

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(b) (c) (d) (e) (f )

d ( ˘ ) , where k is a constant dx d x x ˘ dx d x a = a x log e a, where a > 0, a ≠ 1 dx d 1 log x ) = , x > 0 ( dx x d 1 x = dx 2 x

( ) ( )

( )

2. Trigonometric Functions

d ( sin x ) = cos x , –∞ < x < ∞ dx d (b) ( cos x ) = − sin x , –∞ < x < ∞ dx (a)

(c)

( 2n +1) π d tan x ) = sec2 x, x ≠ , n∈i ( dx 2

(d)

d − cosec2 x, x ≠ nπ , n ∈ i ( cot x ) = dx

(e)

d ( 2n +1) π , n ∈ i sec x ) = sec x tan x, x ≠ ( dx 2

(f )

d ( cosec x ) = –cosec x cot x, x ≠ nπ, n ∈ i dx

3. Hyperbolic Functions

d coshxx ) ( sinh x ) = (cosh dx d (b) ( cosh x ) = (sinh sinhxx ) dx (a)

237

238

Differentiation

d tanh x ) = sech 2 x ( dx d 2 (d) ( coth x ) = –cosech x dx d (e) ( sech x ) = –sech x tanh x dx d (f ) ( cosech x ) = –cosech x coth x dx (c)

4. Inverse Trigonometric Functions

d 1 sin–1 x ) = , – 1< x 1 ( dx x x2 –1

5. Inverse Hyperbolic Functions (a)

d 1 sin h–1 x ) = ( dx 1 + x2

(b)

d –1 cos h–1 x ) = 2 , x >1 ( dx x –1

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239

d 1 tan h–1 x ) = ,|x|1 (d) ( cot h x ) = dx 1– x2 (c)

d –1 cosec h–1 x ) = ,x≠0 ( dx | x | 1– x 2 d –1 sec h–1 x ) = (f ) ,0 0 for every x in that interval. • Decreasing in an interval if f ′ (x) < 0 for every x in that interval. • Constant in an interval if f ′ (x) = 0 for every x in that interval.

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To find the interval where the function is increasing/decreasing, we need to first evaluate the point where the slope = 0. In the above figure, we see that the slope is zero at x = 1, i.e., the tangent to the curve at the point x = 1 is parallel to the x-axis. Such a point where the derivative [written as

df dx

at x = c or f ′ (c)] is 0 is known as the critical point. The term ‘critical’ is used in the sense of a point where a situation changes. Primarily at these points, the slope changes from positive to negative or vice-versa. Example 1 Find all critical points of f(x) = x4 – 8x2. Solution To find the critical points, we first evaluate f ′ (x), and then substitute f ′ (x) = 0. We then find all the points x which satisfy f ′ (x) = 0. Using the differentiation rules, we have d n n–1  f ′ (x) = 4x3 – 16x  dx ( x ) = nx  For critical point, f ′ (x) = 0 ⇒ 4x3 – 16x = 0 ⇒ 4x(x2 – 4) = 0 ⇒ 4x(x + 2)(x – 2) = 0 ⇒ x = 0 or x = –2 or x = 2 Let us evaluate the value of the function at all these points and the corresponding critical points. f(–2) = (–2)4 – 8(–2)2 = –16 ⇒ (–2, –16) 4 2 f(0) = (0) – 8(0) = 0 ⇒ (0, 0) 4 2 f(2) = (2) – 8(2) = –16 ⇒ (2, –16) Hence, the critical points of  f(x)  are (−2, −16), (0, 0), and (2, −16).

254

Differentiation

We can verify these critical points from the graph of the function as well. At these points, the derivative is 0. Graphical representation of f(x) = x4 – 8x2

f(x) = x4 - 8x2

Extending the use of critical points for checking the intervals where the function is increasing/decreasing, we see that: • • • •

slope < 0 for x < –2 slope > 0 for –2 < x < 0 slope < 0 for 0 < x < 2 slope > 0 for x > 2

Verifying from the definition for f ′ (x) = 4x3 – 16x. • For x < –2, choose the point x = –3; f ′ (–3) = 4(–3)3 – 16(–3) = –60 < 0 • For –2 < x < 0, choose x = –1; f ′ (–1) = 4(–1)3 – 16(–1) = 12 > 0 • For 0 < x < 2, choose x = 1; f ′ (1) = 4(1)3–16(1) = –12 < 0 • For x > 2, choose x = 3; f ′ (3) = 4(3)3 – 16(3) = 60 > 0

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Therefore, the function is increasing in the interval –2 < x < 0 and for x > 2. And, the function is decreasing for x < –2 and in the interval 0 < x < 2. Example 2 2

Determine all the critical points for the function f ′ (x) = xe x . Solution In this case, the derivative is given as follows: 2 dx d d x2 f(x) = x e + ex dx dx dx

( ) 2

= (2x 2 )e x + e x 2

2

2

(2x)+ e x = e x (1+ 2x 2 ) 2

 e x > 0 and 1 + 2x2 > 0 for all real values of x. 2

∴ e x (1 + 2x2) > 0 for all real values of x. This function will never be zero for any real value of x, as the exponential function is never zero and the polynomial (1 + 2x 2) is zero if x is complex and recall that we only want real values of x for critical points. Thus, this function will not have any critical points. Hence, it is not necessary that all functions have critical points. Example 3 Two cars start out 500 miles apart. Car A is to the west of Car B and starts driving to the east (i.e. towards Car B) at 35 mph and at the same time Car B starts driving south at 50 mph. After 3 hours of driving, at what rate is the distance between the two cars changing? Is it increasing or decreasing?

256

Differentiation

Solution Distance between car A and car B = 500 miles Speed of car A = 35 mph Speed of car B = 50 mph We know, Distance = Speed × Time ∴ After 3 hours of driving, Distance covered by car A = 35 × 3 = 105 miles Distance covered by car B = 50 × 3 = 150 miles ∴ Final position of car A = x = 500 – 105 = 395 miles Final position of car B, y = 150 miles Using the Pythagoras theorem to find z at a given time: z2 = x2 + y2 (∵ x = 395 miles and y = 150 miles) = z2

( 395 ) + (150 ) 2

2

z 2 = 178525

= z

= 178525 422.522

Now, to find whether the distance between the cars is increasing or decreasing, we differentiate the equation with respect to time:

z2 = x2 + y 2

⇒ 2z dz = 2x dx + 2y dy

dt dt dt dz dx dy ⇒ z = x +y dt dt dt

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Substituting the values of the variable, we get

dz dx dy z=(422.522) x + y = 395(–35) + 150(50) dt dt dt dz dx dy z= ≈ x–14.9696 +y dt dt dt So, after three hours the distance between the cars is decreasing at the rate of 14.9696 mph. 3. Linear approximation We are aware that the square of 3 is 9. However, we would definitely need a calculator to find the square of 3.2. We would guess it to be little more than 9 but we would not be able to get the exact value. Let us try and resolve it. Graphical representation of the function y = x2

If we zoom in on the point (3, 9) on the graph, we notice that the graph now looks very similar to a line, which is the tangent at this point. We can use this tangent line, which is very close to the curve around this point, to approximate other values along the curve (as long as we stay near this point).Linear

258

Differentiation

approximation method or linearisation is used to find the approximate value of a function at a particular point. The equation of the tangent through the point (x 0 , y 0 ) = (a, f(a))  with slope  m = f ′ (a)  has its equation in the point-slope form given by y – y0 = m(x – x0) ⇒ y – f(a) = f ′ (a)(x − a) {Here, m = f ′ (a) and (x, y)= [a, f(a)]} which can also be expressed as y = f ′ (a)(x − a) + f(a). Thus, the equation of the tangent line is given by L(x) = f(a) + f ′ (a)(x − a). Since, the value of the function at a point is approximately equal to the value of the tangent line at the same point, the value of the function is given by f (x) ≈ L(x). Graphical representation of f (x) ≈ L(x).

This fact that every curve when zoomed in small enough will always look like a line, enables us to approximate another point on the curve that is close to the zoomed-in point. To find the approximate value of the function at a point, we can directly substitute the value in the equation of the tangent. This process is known as linear approximation. Let us understand this through an example.

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259

Example 1 Find the linearisation of the function f(x) = x2 at a = 3 and use it to approximate f(3.2). Solution We first find the point by substituting into the function to find f(a), i.e., f(3) = (3)2 = 9. Thus, the point under consideration is (3, 9). Then, we find the derivative f ′ (x) which is given by f ′ (x) = 2x ⇒ f ′ (3) = 2 × 3 = 6 The equation of the tangent line is given by L(x) = f ′ (3)(x − 3) + f(3) = 6(x − 3) + 9 = 6x − 18 + 9 = 6x − 9 Since, = 6 × 3.2 = 19.2 9 ≈ 10.2 f (3.2) ≈ L (3.2)=6(3.2) −9 =− 919.2 −9 = −10.2. Thus, the approximate value of (3.2) 2 is 10.2, whereas on calculating it using the basic arithmetic rules, we get its exact value as 10.24. So, we are really close. Example 2 Estimate the value of

9.1..

Solution Here the function can be taken as f(x) = y = x . Using the value of 9 and linear approximation, we can find the value of f(x) = x ⇒ f ′ (x) =

9.1.

1 2 x

Now, the equation of tangent is given by L(x) = f(a) + f ′ (a)(x − a). Substituting a = 9 and x = 9.1 in L(x) = f(a) + f ′ (a)(x − a), we get L(9.1) = f(9) + f ′ (9) (9.1 – 9)

1 ⇒ L(9.1) = 3 + (9.1 − 9) 6

1 1   f ′ ( 9 ) = 2 9 = 6 

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Differentiation

⇒ L(9.1) = 3.0166206 This value is very close to the actual value of 9.1. (= 3.01662). Hence, by using derivatives, we can find the linear approximation of function to get the value near to the function. We have observed that linear  approximations come to the rescue while approximating the value of the functions. These can also be used to estimate the amount a function value changes as a result of a small change in the input. To put this more formally, we define the concept of differentials, which provide us with a way of estimating the amount a function changes as a result of a small change in input values. Differentials

dy , dy and dx are infinitesimal dx small changes in the values of x and y. Given dx and dy are small changes in the values of x and y, if a non zero real number is assigned to dx, then dy can be defined as dy = f ′ (x)dx, and these expressions dx and dy are called differentials. With the notation of derivative as

Example 1 Find the differential dy when x = 3 and dx = 0.1 for the function y = x 2. Solution The derivative of the function is given by f ′ (x) = 2x. It can also be rewritten as dy = f ′ (x) dx ⇒ dy = 2x dx [∵ f ′ (x) = 2x] ⇒ dy = 2 × 3 × 0.1 = 0.6 (Substituting x = 3 and dx = 0.1) Thus, with a changes of 0.1 units in the input variable (x), the value of the function changes by 0.6 units ‘approximately’. To get the ‘exact value’ of the change in the value of the function is given by,

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261

dy = f (3.1) – f(3) dy = (3.1)2 – 32 = 9.61 – 9 = 0.61 which is very close to the differential dy, the error being 0.61 – 0.6 = 0.01. Differentials are useful while estimating the error in finding the volume of a box if the measurements are made with a certain amount of accuracy. 4. Tangent and normal to a Curve Ever sat on a merry-go-round! We experience certain forces which move our body towards the centre of the swing. There are also forces which keep our body tilted. What we actually experience is normal force on our body which pushes our body towards the centre of the swing and the speed of the swing acts in the tangential direction which keeps our body tilted. While travelling in a car, if we drive over something slippery (like oil or water), the car continues to slide in some direction. That direction is tangent to the curve of motion of the car. Similarly, when an object is in motion on a circular track, there are two types of forces acting on it – one that makes us go around the circular track (such a force is directed towards the centre and is thus acting in the normal direction) and the other is in the tangential direction. The forces are shown in the figure.

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Differentiation

Let us define these special lines to any curve for reference. A tangent at a point to a curve is a line that touches the given curve at that point only. It is the best approximation to a curve near that point. As discussed, the slope of the tangent line is the derivative of the function (which defines the curve) at that point. If m is the slope of the tangent line, then the equation of the tangent line touching the curve of the function y = f(x) at the point (a,f(a)) can be obtained by using the point-slope form (y − f(a)) m = f ′f'(a) (a) = (x − a) ⇒ y = f(a) + f ′ (a) (x – a) A normal line to a curve is the line perpendicular to the tangent –1 −1 = . Hence, the f ′(a) m equation of a normal line to a curve of a function y = f(x) at a point [a, f(a)] can be obtained by using the equation  – 1 (y − f(a)) f'(a) = m (x − a) line. Thus, the slope of a normal line is

⇒

y – f(a) –1 = f ′(a) x–a

 m = f ′(a)

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⇒

–(x – a) = y – f(a) f ′(a)

⇒ y = f(a) –

(x – a) f ′(a)

Graphical representation of tangent and normal line

Tangent

P Normal line

Example 1 Consider the curve given by y = f(x) = x3 – 2x + 5.

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Graphical representation of f(x) = x3 – 2x + 5

f(x) = x3 – 2x + 5

(a) Find the equation of the line tangent to the curve at the point (1, 4). (b) Find the equation of the line normal (perpendicular) to the curve at the point (1, 4). Solution (a) To find the equations of the tangent and the normal line, we need to first find the slope at the point x = 1, i.e., f′(1). f(x) = x3 – 2x + 5 ⇒ f(1) = 13– 2(1) + 5 = 4 (Substituting x = 1) 2 f ′ (x) = 3x – 2 [On differentiating f(x)] 2 ⇒ f ′ (1) = 3(1) – 2 = 1 (Substituting x = 1)

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Thus, the equation of the tangent line is given by L(x) = f(a) + f ′ (a) (x – a) Putting a = 1, y = f(1) + f ′ (1) (x – 1) ⇒ y = 4 + 1(x – 1) [Substituting f(1) = 4 and f ′ (1) = 1] ⇒y=x+3 Thus, y = x + 3 is the equation of the tangent line. −1 (b) Now, the slope of the normal line is given by = −1 . f ′(1) Thus, the equation of the normal line is given by (x – a) y = f(a) – f ′(a) ⇒ y = f(1) – ⇒ y=

(x – 1) f ′(1)

[Put a = 1]

4 – ( x – 1) 1

[Substituting the value of f(1) =4 and f ′ (1)=1]

⇒y=5–x Thus, y = 5 – x is the equation of the line normal to the curve. 5. Maxima, minima and point of inflection Any manufacturing business would aim to maximise its profit. The maximum profit a company can make gives an idea of the salaries that can be offered to its employees.  The engineers would aim to determine the maximum possible speed of an airplane. This can help them choose the  materials  that would be strong enough to withstand the pressure due to such high speeds. If a function can be determined which describes the cholesterol level in the blood, its maximum and the minimum values can help doctors to determine the dosage they need to prescribe to the patients. Thus, finding the maximum or the minimum

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value is of utmost importance in every field. Derivatives are helpful in finding these points of maxima, minima, and inflection. To visualise, maxima and minima are just like the peaks and valleys in a curve. Together they are referred to as the extrema. Suppose you are working with a company for over 5 years. You face a situation in your life where you are bound to leave the present corporate job and go on for your further studies. This situation took your life in a different direction and altered your corporate life. These situations are termed as inflections. Mathematically also, the point of inflection is the part of the curve where the curve changes its nature (from concave down to concave up or vice versa). Graphical representation of maxima, minima and inflection point

Nature of curve Consider the time-distance graph of a travelling car, the slope of which represents velocity. In case of normal traffic, the car is speeding up. As the driver sees some traffic, brakes are applied and the speed starts decreasing gradually. Thus, the point where the brakes are applied leads to a change in the speed of the car. Now, an increasing speed characterises an increasing slope and a decreasing speed characterises a decreasing slope. Consider the graph below where the slope is increasing till the origin (0, 0) and then starts

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decreasing. Thus, (0, 0) is the point where nature of the curve changes. This point is known as the point of inflection. When the slope is increasing, the curve is termed as concave up whereas when the slope is decreasing, it is termed as concave down. Graphical representation of concave up and concave down

How do we discover these points of extrema and inflection? We see from the graph that at the points of extrema, the tangent is horizontal to the x-axis, i.e., the slope is zero. Thus, the derivative of the function at these points is zero. Once we check the value of the derivative function at the points lying to the left and right of the curve, we can then determine the nature of this point. This forms the basis of the first derivative test, where we equate the derivative of the function at a point c to 0, i.e., f ′ (c) = 0 and then check the values of f ′ (x) at the points lying to the left and the right of the curve. The point is then the point of • Maxima if to the left of c, f ′ (x) > 0 and to the right of c, f ′ (x) < 0. f(c) is the maximum value. • Minima if to the left of c, f ′ (x) < 0 and to the right of c, f ′ (x) > 0. f(c) is the minimum value. • Inflection when the sign of f ′ (x) does not change as we move near to c.

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Now there can be many local grocery shops in an area but only one supermarket in the same area. Similarly, there can be many local extremas but only one global extrema. Look at the graph below. Graphical representation of absolute/local maximum/minimum

We, thus define the following: 1. f(x)  has an  absolute (or global) maximum  at  x = c,  f(x) ≤ f(c) for every x in the given domain. 2. f(x)  has a  relative (or local) maximum  at  x = c,  f(x) ≤ f(c) for every x in some open interval around x = c. 3. f(x)  has an  absolute (or global) minimum  at  x = c,  f(x) ≥ f(c) for every x in the given domain. 4. f(x)  has a  relative (or local) minimum  at  x = c,  f(x) ≥ f(c) for every x in some open interval around x = c.

if if  if  if

Example 1 Identify the absolute and relative extrema for the function f(x) = x2 on [−2, 3].  Solution To find the absolute and relative extrema for the function, f(x) = x2 on [–2, 3].

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Graphical representation of f(x) = x2 on [–2, 3].

f(x) = x2

From the graph, it shows that both the relative minimum and an absolute minimum exist at x = 0. However, at the point x = 3, f(x) ≤ f(3) for every point in the domain [−2, 3]. Thus, x = 3 is the point of absolute maxima. We observe that x = −2 is not a point of relative maxima because we cannot find any open interval about x = −2 where f(x) ≤ f(−2). Thus, no relative maxima exists for this function. Example 2 Identify the absolute and relative extrema for the function f(x) = sin x.

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Solution Graphical representation of f(x) = sin x f(x) = sin x

In the graph of function f(x) = sin x, there is no domain on which the extrema are to be found. The sine curve has both relative and absolute maximums of 1 at: −7 π −3π π 5π 9π x = ... , , , , ,... 2 2 2 2 2 It has both relative and absolute minimums of −1 at: −9π −5π 3π 7 π , , , ,... 2 2 2 2 We notice that all these functions are continuous. The above examples and the observations from them lead us to the following theorem. x = ...

Extreme value theorem It states that if a function f(x)  is continuous on any closed interval [a, b], then there are two values c and d in the interval [a, b] with a ≤ c and, d ≤ b  such that  an absolute maxima is attained at c and f(c) is an absolute maximum for the function; an absolute minima is attained at d and  f(d)  is an absolute minimum for the function, i.e., the function always attains the absolute extrema. As in the graph below, we see that c and d are points in the interval [a, b] where the maxima and the minima are attained.

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Graphical representation of function f(x) on close [a, b].

This theorem has many applications in the real world, especially in business, where to maximise the profit, the right decision in terms of setting the maximum and the minimum prices of the goods and services is to be made. It is also useful in pharmacology, where a pharmacist has to figure out the smallest dosage to be administered for effectiveness or administering the largest dosage without causing harm. Example 1 Find the maximum and minimum values of f(x) = 3x 3 − 2x + 1 on [0, 2]. Solution f(x) = 3x3 – 2x + 1 f ′ (x) = 9x2 – 2

(given)

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Differentiation

Graphical representation of f ′ (x) = 9x2 – 2

f '(x) = 9x2 – 2

The function is continuous on [0, 2]. Its derivative is f ′ (x) = 9x2 −2. For critical points, we know f ′ (x) = 0 ⇒ 9x 2 – 2 = 0 ⇒ x =± − x = ± ⇒

2 3

−22 does not lie in the interval [0, 2] 3

2 ∴ The critical point occurs x= . ⇒ at x= ± 3

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3

 2  2  2 2 2 2 2 ⇒ f – + 1 = 0.37  = 3  – 2 = 3 3 9 3 3       At end points, f(0) = 3(0)3 – 2(0) + 1 = 1 f(2) = 3(2)3 – 2(2) + 1 = 21 ∴ Maximum function value = 21 at x = 2 2 Minimum function value = 0.37 at x = 3 Another test to verify whether the function has an extremum at the critical points is the second derivative test, according to which, if x = c is a critical point of f(x) such that  f ′ (c) = 0 and that  f ′′ (x) is continuous in a region around x = c. Then, • If  f ′′ (c) < 0, then x = c is a relative maximum. • If  f ′′ (c) > 0, then x = c is a relative minimum. • If f ′′ (c) = 0, then the test is inconclusive and x = c can be a relative maximum, relative minimum or neither. Let us verify the following function for the relative extrema using the second derivative test. Consider the function f(x) = x2. From the graph, it has a relative minimum at x = 0. Now, using the test, we first find the critical point c which satisfies f ′ (c) = 0, i.e., 2c = 0 or c = 0. f ′′ (x) = 2 f ′′ (c) = f ′′ (0) = 2 (∵ c = 0 is the critical point) ∵ f ′′ (0) = 2 > 0 ∴ x = 0 is a relative minimum.

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Graphical representation of point of relative minima at x = 0

The graph of  the function f(x) = −x2  has a relative maximum at  x = 0. It can be verified using the second derivative test also. For this, we first find the critical point c using f ′ (c) = 0, i.e.,− 2c = 0 or c = 0. Now, f ′′ (x) = −2, or f ′′ (0) = −2 < 0. Thus, x = 0 is a relative maximum. Graphical representation of point of relative maximum at x = 0

The function f(x) = x3 seems to have neither a relative minimum nor a relative maximum at x = 0. It has a point of inflection at this point.

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Graphical representation of point of inflection at x = 0

Let us use the second derivative test to verify. f ′ (c) = 0 ⇒ 3c2 = 0 or c = 0 is the critical point. Now, f ′′ (x) = 6x, i.e., f ′′ (0) = 0. Thus, nothing can be said about the nature of the critical point, c = 0 from the test. Example 2 Classify the critical points of the function f(x) = –2x3 + 6x2 + 1 Solution f(x) = –2x3 + 6x2 + 1 For second derivative test, first find the critical points. f ′ (x) = –6x2 + 12x f ′ (c) = 0 ∴ –6c2 + 12c = 0 ⇒ –6c (c – 2) = 0 Thus, c = 0 and c = 2 are the critical points. Then, find the second derivative. f ′′ (x) = –12x + 12 Further, evaluate the sign of f ′′ (x) at the critical points. f ′′ (0) = 12 > 0, which shows that c = 0 is the point of minima. and f ′′ (2) = –12 < 0, which implies that c = 2 is the point of maxima. The minimum/maximum value of the function is at the critical point where the minima/maxima occurs.

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Differentiation

Thus, minimum value of the function is given by f(0) = –2(0)3 + 6(0)2 + 1 = 1 (At c = 0) And the maximum value of the function is given by f(2) = –2(2)3 + 6(2)2 + 1 = 9 The extremas can be seen from the graph of the function as well. Point of maxima

f(x) = –2x3 + 6x2 + 1

Point of minima

The second derivative test also helps in determining the point of inflection. We know that if f ′′ (x) > 0, then the function is concave up and if f ′′ (x) < 0, then the function is concave down. Thus, we look for that point x = c where the second derivative of the function changes its sign from positive to negative, or from negative to positive. This concludes that at that point, the second derivative should be zero, i.e., f ′′ (c) = 0. Example 3 Determine the inflection f(x) = –2x3 + 6x2 + 1.

point

for

the

function

Solution The second derivative test also helps in determining the point of inflection. We know that if f ′′ (x) > 0, then the function is concave up and if f ′′ (x) < 0, then the function is concave down.

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Graphically,

f(x) = –2x3 + 6x2 + 1 P (1, 5)

Point of inflection

1

f(x) = –2x3 + 6x2 + 1 f ′ (x) = –6x2 + 12x (On differentiating) f ′′ (x) = –12x + 12 (On differentiating) For critical points, put f ′ (x) = 0 ⇒ –6x2 + 12x = 0 ⇒ x(–6x + 12) = 0 ⇒ x = 0 or x = 2 Put f ′′ (x) = 0, we get –12x + 12 = 0 ⇒x=1 To check for the nature of the function around x = 1, we substitute x = 0 and 2 in f ′′ (x). So, f ′′ (0) = –12(0) + 12 = 12 > 0 f ′′ (2) = –12(2) + 12 = –12 < 0 Thus, the function changes its nature from concave up to concave down while crossing x = 1. Now, f(1) = –2(1)3 + 6(1)2 + 1 = –2 + 6 + 1 = 5. Thus, (1, 5) is the point of inflection of the function.

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Differentiation

6. Optimisation Optimisation problems are problems to find the extrema of a function subject to some restrictions/constraints. For example, a company would always want to maximise its profit. However, this comes with a set of restrictions such as the number of employees working in a company, the cost of marketing the goods, the salaries of the employees, the working hours, etc. The company’s aim would still be to maximise the profit, but now subject to these restrictions. Let us understand its working through an example. Example 1 A farmer wants to build a rectangular fence that will enclose 120 square feet of area. The two long sides of the fence are to be made of steel at a cost of ₹5 per ft. The two shorter sides are to be made of wire at a cost of ₹6 per ft. What are the dimensions of the fence that will minimise cost? Solution Let the length of rectangular fence be l and the breadth of the rectangular fence be b. We know that, Perimeter of the rectangular fence is given by, P = 2(l + b) According to the question, Cost of length = ₹5 per ft. Cost of breadth = ₹6 per ft. ∴ Cost of the fence = 2 [(5 × l) +(6 × b)] = 2 (5l + 6b) = 10l + 12b The constraint in this problem is the area to be covered, which is, A = lb = 120. However, we cannot take the first derivative of the cost equation because it is not in terms of one variable; therefore, our next step is to get the cost equation in terms of one variable. For this, we

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use the constraint equation to solve for one of the variables in the objective equation. 120 120 Since lb = 120, therefore, b = and by substituting for b in l l the cost equation, we get

 120  C(= l ) 10l + 12    l  1440 = 10l + l = 10l + 1440l −1 ∴ C′(l) = 10 + (–1440)–2

(On differentiating)

Setting the first derivative of the cost equation C′(l) = 0 and solve for l, we get 0 = 10 – 1440l–2 ⇒ 1440l–2 = 10 1440 ⇒ 2 10,or1440 l 2 , or 144 l 2 . = = 10= l ⇒ l2 = 144 ⇒ l = ±12 Since length cannot be negative, thus l = 12.

120 120 = 10. ,⇒ b = 12 l Thus, the cost would be minimum when the dimensions are l = 12 ft and b = 10 ft. Now, b =

Example 2 An open top box is to be made from 24 inches by 36 inches piece of cardboard by cutting a square from each corner of the box and folding up the flaps on each side. What size a square should be cut out of each corner to get a box with the maximum volume?

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Differentiation

Solution Let x be the side length of the square to be removed from each corner such that, the remaining four flaps can be folded to form an open top box.

Let V be the volume of the resulting box. Volume of box, V = l.b.h where l, b, and h are the length, breadth, and height of the box respectively. From the figure, we see that the height of the box is x inches, the length is (36 – 2x) inches, and the breadth is (24 – 2x) inches, therefore, the volume of the box is given by V(x) = (36 – 2x) (24 – 2x)x = 4x3 – 120 x2 + 864x Now, we can understand that the side length of the square cannot be greater than or equal to half the length of the short side, 24 inches; otherwise, one of the flaps would be completely cut off. Since V is continuous at the closed interval [0, 12], V will have an absolute maximum in the closed interval. V must have an absolute maximum (and an absolute minimum) since V(x) = 0 at the end point and V (x) > 0 and 0 < x < 12. The maximum must occur at the critical point. ∴ V′(x) = 12x2 – 240x + 864 ⇒ 12x2 – 240x + 864 = 0 [Put V′(x) = 0] ⇒ x2 – 20x + 72 = 0 (On simplifying) ⇒ x=

 ca4 − b2b± ±bb2 − 4ac  20 ± (−20)2 − 4(1)72 = x ,alumrof citardau ∵ x =  u sin g quadratic formula,   a 2 2a 2  

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20 ± 112 2 ⇒ x =10 ± 2 7 ⇒x =

Since 10 + 2 7 is not in the domain of consideration, therefore the only critical point we need to consider is 10 − 2 7 . ∴ Maximum volume (V) =4(10 − 2 7 )3 − 120(10 − 2 7 )2 + 864(10 − 2 7 ) = 640 + 448 7

  1825 inches3



inches, Graphical representationMaximum of V(x) volume = 4x3 –1825 120x2cubic + 864x when x = 4.7 inches

V(x) = 4x3 – 120x2 + 864x

7. Business In the business world, we usually encounter terms such as cost, profit, revenue, etc, which are generally dependent on the number of units of the goods. Thus, if x is the number of units of the item, then C(x) is the cost function, p(x) is the demand/price function. The revenue function is determined by the number of units of the item and its price. Thus, R(x) = x × p(x)

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Differentiation

Profit P(x) is the difference of the revenue earned and the cost of the item. Thus, P(x) = R(x) - C(x) The cost to produce an additional item is called the  marginal cost  which is approximated by the rate of change of the  cost function,  C(x). Thus, the  marginal cost function  is the derivative of the cost function or, C′(x). Similarly, the marginal revenue function  is  given by R′(x)  and the  marginal profit function is given by P′(x) and these represent the revenue and profit respectively if one more unit is sold. Example 1 The weekly cost to produce x widgets is given by C(x) = 75000 + 100x − 0.03x 2 + 0.000004x 3, 0 ≤ x ≤ 10000 and the demand function for the widgets is given by, p(x) = 200 − 0.005x, 0 ≤ x ≤ 10000. Determine the marginal cost, marginal revenue, and marginal profit, when 2500 widgets and 7500 widgets are sold. Assume that the company sells exactly what they produce. Solution Cost Function, C(x) = 75000 + 100x − 0.03x2 + 0.000004x3 Demand function, p(x) = 200 – 0.005x Revenue function, R(x) = x × p(x) = x(200 – 0.005x) = 200x – 0.005x2 Profit function, P(x) = R(x) – C(x) = 200x – 0.005x2 – (75000 + 100x – 0.03x2 + 0.000004x3) = –75000 + 100x + 0.025x2 – 0.000004x3 Differentiate the functions to get marginal functions. Now, all the marginal functions are, C′(x) = 100 − 0.06x + 0.000012x2 R′(x) = 200 − 0.01x P′(x) = 100 + 0.05x − 0.000012x2 At 2500 widgets or x = 2500, marginal function are as follows:

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C′(2500) = 100 − (0.06)(2500) + (0.00012)(2500)2 = 25 R′(2500) = 200 − (0.01)(2500) = 175 P′(2500) = 100 + (0.05)(2500) − (0.00012)(2500)2 = 150 Similarly, at 7500 widgets or x = 7500, marginal function are as follows: C′(7500) = 325; R′(7500) = 125; P′(7500) = −200 On the other hand, when the company produces 7501st widget, it will cost ₹325 while it will earn ₹125 as revenue and incur a loss of ₹200. 8. Mean value theorem

You are driving on a highway on which the speed limit is 100 km/hr. You cross two toll gates 20 kms apart in 10 minutes. At the first toll gate, your speed was recorded as 60 km/hr and at the other, it was recorded as 80 km/hr. On crossing the second toll gate, you were issued a speeding violation ticket. How is that possible, when it clearly seems that you have not crossed the speed limit ever? Consider another case. A few strains of bacteria are allowed to multiply in a culture to determine its characteristics. The difference in the number of bacteria is calculated in an hour to find how fast they have multiplied. But what does this tell? Both the above situations can be justified using the mean value theorem which guarantees that the car’s instantaneous speed is its average speed at least once in a given time interval. Travelling 20 kms in 10 mins gives the average speed as 120 km/hr. Thus, the instantaneous speed was 120 km/hr at least once in the 20 km stretch of highway. No matter how slow the traffic was or how fast you zoomed in some lane, at some point along your way you were going at a speed of exactly 120 km/hr. Applying the mean value theorem to the bacteria culture would allow you to find the exact time where the bacteria multiplied at the same rate as the average speed.

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Differentiation

Mean-value theorem: Suppose f(x) is a function that satisfies the following: • f(x) is continuous on a closed interval [a, b]. • f(x) is differentiable on the open interval (a, b). Then, there exists a number c such that a < c < b and f ( b ) − f (a ) f ′(c) = b−a Geometrically,

f ( b ) − f (a ) is the slope of a line joining points b−a

A (a, f(a)) & B (b, f(b)), and for a tangent to f(x) at x = c, slope is  f ′ (c).

But, why do we talk of continuity in a closed interval and differentiability in the open interval? To define the secant, the value of the function on the endpoints is to be defined and thus, the function has to be continuous on the closed interval. However, in this case, differentiability at the endpoints is not required. We do not even need to assume the derivative is well defined on them. Saying that a function is differentiable in an interval means derivatives exist at all points of the interval, or there exist unique tangents  at all points  in that interval. Differentiability in a closed interval implies existence of unique tangents at the endpoints as well. But the function can follow

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infinitely many paths outside the interval, for which case there would be infinitely many tangents (and thus infinitely many derivatives) just at the end points. Thus, talking about the differentiability at the endpoints is not required. Since, the behaviour of the function inside the interval gives no information about its behaviour outside  the interval; hence, giving no information about the slope of the tangent at the end points.

This theorem leads us to two facts: • If  f ′ (x) = 0 for all x in an interval (a, b), then f(x) is constant on (a, b). • If  f ′ (x) = g′ (x)  for all  x  in an interval  (a, b),  then in this interval we have f(x) = g(x) + c where c is a constant. Example 1 A ball is dropped from a height of 100 ft. Its position  at t  seconds, before hitting the ground is given by the function  s(t) = −16t2 + 100. (a) How long does it take before the ball hits the ground? (b) Find the average velocity  of the ball, when the ball is released and hits the ground. (c) Find the time  t  guaranteed by the Mean value theorem, when the instantaneous velocity of the ball is  its average velocity as well.

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Differentiation

Solution (a) When the ball hits the ground, its position is s(t) = 0 ⇒ −16t2 + 100 = 0 ⇒ 16t2 = 100 10 = ± 2.5 sec 4 Since time cannot be negative, therefore t = 2.5 sec Hence, the ball hits the ground 2.5 sec after it is dropped. t=±

(b) Distance at t = 2.5, s(2.5) = −16(2.5)2 + 100 = 0 Distance at t = 0, s(0) = −16(0)2 + 100 = 100 ft ∴ The average velocity is given by Vavg = =

s(t = 2.5) − s(t = 0) t(final) − t(initial) 0 − 100 2.5 − 0

= −40 ft/sec (c) By the Mean value theorem, where the function s(t) is continuous in the interval [0, 2.5] and differentiable in (0, 2.5). We need to find a time t in the interval (0, 2.5) for which the instantaneous velocity = average velocity. If the instantaneous velocity is given by s′(t), Vavg = s′(t) And, Vavg = −40 ft/sec ∴ s′(t) = −40 ft/sec

[From solution (b)] ...(i)

Taking the derivative of the distance function s(t), s(t) = −16t2 + 100 ⇒ s′(t) = −32t ...(ii)

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Therefore, the equation reduces to −32t = −40 ⇒ t = 1.25

287

[From eq. (i) and (ii)]

∴ During its free fall, the instantaneous velocity equals to the average velocity of the rock at t = 1.25 sec.

A particular case of the Mean value theorem is the Rolle’s theorem. Rolle’s theorem Suppose f(x) is a function that satisfies the following: • f(x) is continuous on a closed interval [a, b]. • f(x) is differentiable on the open interval (a, b). • f(a) = f(b) Then, there exist a number c such that a < c < b and f′(c) = 0. The function in such circumstances always has a critical point.

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Differentiation

Geometrically,

Example 1 Show that the function  f(x) = sin x  has a critical point in the interval (0, π). Solution The function sin x is differentiable (and hence also continuous) on the interval (−∞, ∞) and since sin 0 = sin π = 0, we can apply Rolle’s Theorem to  sin x  on the interval  [0, π]. The theorem says that there exists a number  c  between  0  and  π  such that  f ′ (c) = 0. Thus,  f(x) = sin x  has a critical point in the

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interval (0, π). Moreover, we can find c, since  f ′ (x) = cos x and π the equation cos x = 0 has x = as a solution in the interval. 2 9. Newton Raphson method In all the above applications, we are either finding the rate of change of the function or the solutions to the equations. However, we may come across equations such as cos x = x where we need to find an x in the interval [0, 2] which satisfies the equation. Such equations are difficult to solve manually or by any of the abovementioned techniques. We may not be able to find the exact solution in this case, but we may look for an approximate solution. To find approximate solutions, we make use of the famous Newton Raphson method which is as follows: • We aim to find an approximate value of the function f(x) = 0 at some point in a given interval. For that, we start with a guess x 0, which acts as the initial approximation. This guess can be a mid-point of the interval so that reaching to the solution of the interval which may lie to the left or to the right of the mid-point is with equal probabilities. • Corresponding to the point (x 0, f(x 0)), we draw a tangent there, with equation as y = f(x0) + f′(x0) (x−x0).

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Differentiation

This can be understood graphically as:

Consider a tangent line at  x0. Our aim is to find the value where the curve touches the x-axis. We see that the tangent line crosses the  x-axis much closer to the actual solution to the equation than  x 0  does. Let us call this point x 1. Now, this point x 1 becomes our new approximation. We also see that (x1, 0) is a point lying on the tangent line passing through x 0 and it thus satisfies the equation of the tangent line. Thus, = 0 f(x 0 ) + f ′(x 0 ) (x1 – x 0 ) f(x 0 ) ⇒ (x1 − x 0 ) = − f '(x 0 ) f(x 0 ) ⇒ x1 = x 0 − f '(x 0 ) This is defined only when f ′ (x0) ≠ 0. The same process can be repeated to find an even better approximation. We draw the tangent line to  f(x) at  x1  and call  x2 the point where the tangent intersects with the x-axis as a new approximation to

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the

solution. We f(x1 ) x 2 = x1 − f '(x1 )

then

have

the

following

291

formula:

This point can also be seen on the graph above. On continuing this process, we get a sequence of numbers that are getting very close to the actual solution. This process is termed as Newton Raphson method. Thus, if f(xn)  is an approximation of the solution of f(x) = 0 and if f ′ (xn) ≠ 0 then, the next approximation is given by x n+1 = x n −

f(x n ) . f '(x n )

Example 1 Determine an approximation to the solution to cos x = x in the interval [0, 2]. Solution We start with an initial approximation, say x0 = 1. Rewriting the equation as: cos x − x = 0 Where, f(x) = cos x − x And, f ′ (x) = −sin x − 1. Using the Newton Raphson method to find the approximations, we have f(x n ) x n+1 = x n − f '(x n ) cos x n − x n −sin x n − 1 cos x n − x n = xn + sin x n + 1

⇒ x n+1 = x n − ⇒ xn+1

On substituting n = 0, we have

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Differentiation

x1 = x 0 +

cos x 0 − x 0 sin x 0 + 1

⇒ x1 = 1 +

cos1 − 1 = =0.75036 0.75036 sin1 + 1

(∵ x0 = 1)

Similarly, when n = 1, we get cos x1 −1 x 2 = x1 + sin x1 + 1 =0.75036 +

cos(0.75036) − 0.75036 sin(0.75036)+1

= 0.73911. Continuing this process, we get: x3 = 0.73908 and x 4 = 0.73908, which are both same and hence, we can stop here. Thus, the approximate solution to the equation is x = 0.73908. It is thus a quick way to find approximate solution of some complicated equations. 10. L’Hospital’s Rule and Indeterminate forms Consider  lim x →0

sin x . x

On substituting the value x = 0, we get 0 .

0

ex Similarly, on considering lim 2 and substituting infinity we x →∞ x ∞ get  .

∞

Both these limits are not defined and are hence called the indeterminate forms. There are other types of indeterminate forms as well, such as the cases where we may end up after finding the limits as, 0 (±∞) 1∞ 00 ∞0 ∞ ±∞ To handle such cases of limits, we have the L’Hospital’s rule which is as follows:

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293

If we have

f(x) 0 = x → a g(x) 0

• lim

f(x) ±∞ = x → a g(x) ±∞

• lim

where  ‘a’  can be any real number or infinity or negative infinity, then according to the L’Hospital’s rule, we have f(x) f ′(x) lim = lim x →a g(x) x →a g ′(x) Thus, for indeterminate forms of the type

0 ±∞ or , all we ±∞ 0

need to do is differentiate the numerator and the denominator separately and then take the limit. Example 1 Evaluate: lim x →0

sin x x

Solution On substituting the value x = 0 in the numerator and denominator, we get the 0 indeterminate form. Thus, applying 0

the L’Hospital’s rule, we get = lim x→ 0

sinx x

cosx x→ 0 1

= lim

=

1 = 1 1

(On differentiating sin x and x) (∵ x  0)

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Differentiation

Example 2

ex x →∞ x 2

Evaluate: lim Solution

This is the ∞ form. On applying the L’Hospital’s rule and ∞

differentiating the numerator and denominator, we have

ex ex lim 2 = lim x →∞ 2x   x →∞ x The limit is again of the form ∞ . Thus, on again applying ∞ the L’Hospital’s rule, we have

ex ex lim = x →∞ x 2 x →∞ 2x

lim

ex (On differentiating ex and 2x) x →∞ 2 =∞ (∵ x  ∞) Thus, in such cases, we need to apply L’Hospital’s rule more than once. = lim

Example 3

1  1 Evaluate: lim  − x →0 x sin x   Solution 1  1 = ∞−∞ lim  − x →0 x sin x  

(As x  0) 0 This is an indeterminate form. We try converting it into = a or 0 ∞ form, and then apply the L’Hospital’s rule. =∞−∞

∞

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1  1 lim  −  x →0  x sin x  = lim x →0

=

sin x − x x.sin x

0 0

( x  0)

Therefore, again applying L’Hospital’s rule sin x − x lim x →0 x.sin x cos x − 1 = lim x →0 sin x + x.cos x (On differentiating numerator and denominator) 0 = ( x  0) 0 Again, applying L’Hospital’s rule, cos x − 1 lim x →0 sin x + x.cos x − sin x x →0 cos x + cos − x.sin x (On differentiating numerator and denominator) =0 ( x  0) = lim

Example 4 Evaluate: lim ( tan x − 1) sec 2x x→

Solution

π 4

limπ ( tan x − 1) sec 2x x→

4

Such cases can be again converted to

0 ∞ or (whichever is 0 ∞

easier to evaluate) form for applying the L’Hospital’s rule. We

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Differentiation

see that sec 2x can be easily expressed as  tan x − 1  0 = limπ   cos 2x  0 x→  4

1 . Thus, we have cos 2x

Applying L’Hospital’s rule,  sec 2 x  limπ   −2 sin 2x  x→  4 2 = = −1 −2 Example 5

(On differentiating numerator and denominator) π   x →  4 

Evaluate: lim ( cot x ) Solution

sin x

x →0

On substituting the value x = 0, we get the limit as ∞0, which is indeterminate. In such cases, the following approach is to be followed. Let L = lim ( cot x )

sin x

x →0

Taking log on both sides, sin x log L = lim  log ( cot x )  x →0   = limsin x(log cot x) x →0

= lim x →0

(On differentiating the function)

log cot x cosec x

Applying L’Hospital’s rule = lim x →0

1 (−cosec2 x) cot x −cosec x cot x

(On differentiating numerator and denominator)

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= lim x →0

297

cosec x cot 2 x

Again, applying L’Hospital’s rule, −cosec x cot x = lim x →0 −2cot x cosec 2 x (On differentiating numerator and denominator) 1 x →0 −2cosec x

= lim

Thus, on taking the limit as x  0, 1 ⇒ log L = =0 ∞ ⇒ log L = 0 ∴ L = e0 = 1 Example 6 1

x −2 Evaluate: lim  x  x →2 2  

Solution On substituting the value x = 0, we get the limit as 1∞, which is indeterminate. In such cases, the following approach is to be followed. Indeterminate forms of the type 1∞ can be solved by following the same approach as in the case of ∞0. 1

 x  x −2 lim   = 1∞ x →2  2  1

 x  x −2 Let A =   2

(Given)

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Differentiation

ln A =

1 x ln   x −2  2 

⇒ A=e

1 x ln  x −2  2 

⇒ lim A = e

lim

 1

x → 2  x − 2

(Taking log on both sides)

(Taking exponential function on both sides) x  In    2 

(Taking lim on both sides) x →2

x →2

⇒ lim A = e

  x   ln 2   lim     x →2  x −2   

x →2

⇒ lim A = e

1 lim 2 x →2 x 2

x →2

(On differentiating numerator and denominator of power) 1

( x  0)

⇒ A = e2 ⇒A = e Example 7 Evaluate: lim x sinx x →0

Solution

lim x sinx = 00 x →0

The same approach is followed as above. Let A = x sinx Taking log on both sides, ln A = sin x ln x sin x ln x ⇒ A=e

(Taking exponential function on both sides)

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lim (sin x ln x )

⇒ lim A = e x →0 x →0

⇒ lim A = e

299

(Taking lim on both sides) x →0

ln x lim x →0 1 sin x

x →0

⇒ lim A = e

1 x lim − cos x x →0 sin2 x

x →0

(On differentiating numerator and denominator of power) ⇒ lim A = e

− sin2 x lim x → 0 x cos x

x →0

⇒ lim A = e

−2 sin x cos x lim x → 0 cos x − x sin x

x →0

(On differentiating numerator and denominator of power) ⇒ A = e0

( x  0)

⇒A=1 Some real-life applications The applications of derivatives (discussed earlier) can be put into practice through these examples. Example 1  Suppose that the amount of water in a holding tank at t minutes is given by V(t) = 2t2 −16t + 35. Determine each of the following: (a) Is the volume of water in the tank increasing or decreasing at t = 1 minute? (b) Is the volume of water in the tank increasing or decreasing at t = 5 minutes? (c) Is the volume of water in the tank changing faster at  t = 1 or t = 5 minutes? (d) Is the volume of water in the tank ever not changing? If so, when?

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Differentiation

Solution To know the volume of water at specific time, we need to find the rate of change. For this, we will differentiate the given function at any time t. V(t) = 2t2 – 16t + 35 The derivative of the function is: V′(t) = 4t – 16 ⇒

dV = 4t – 16 dt

If the rate of change is positive, then the quantity is increasing and if the rate of change is negative, then the quantity is decreasing. We can now work on the problem. (a) To know whether the volume of water is increasing or decreasing, we will find the rate of change, i.e., derivative of the given volume function at t = 1 minute. At t = 1, V′(1) = 4(1) – 16  dV  ⇒   = − 12  dt  t =1 Since,  the rate of change is negative therefore the volume must be decreasing at t = 1 minute. (b) Again, we will need the rate of change of the volume function at t = 5 minute. V′(5) = 4(5) – 16  dV  or, ⇒   =4  dt  t = 5 In this case, the rate of change is positive and so the volume must be increasing at t = 5 minutes. (c) To answer this question, we look at the size of the rate of change. The larger the number, the faster the rate of change. So, in the first case, we observed that at t = 1, the rate of change was –12 and at t = 5, the rate of change is 4.Therefore, the volume of water in the tank is changing faster at t = 1 minute.

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(d) The volume will not be changing if rate of change is zero. In order to have a rate of change zero, the derivative must be zero. We, thus, solve for V′(t) = 0 ⇒ 4t – 16 = 0 ⇒ 4t = 16 ⇒ t = 4 sec Thus, at t = 4 minutes, the volume is not changing. Example 2  Suppose that the position of an object is given by s(t)= − tet Does the object ever stop moving? Solution If we assume that the object will stop at some point of time, then the velocity at that point would be zero. i.e., s′(t) = 0. Now, s′(t)= et + tet = (1 + t)et We need to solve for a ‘t’ for which (1 + t)et = 0. Now, we know that exponential functions are never zero and so this will only be zero at t = −1. , which is not possible. Thus, there would be no time for which the object would ever stop. Example 3  Air is being pumped into a spherical balloon at a rate of 5 cm 3 /min. Determine the rate at which the radius of the balloon is increasing when the diameter of the balloon is 20 cm. Solution  We see that both the volume and the radius the balloon will vary with time and hence can be treated as function of time i.e. V(t) and r(t).

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Differentiation

Since, the air is being pumped into the balloon at a rate of 5 cm3/min. This is the rate at which the volume is increasing, which is given by V′(t) = 5 We want to determine the rate at which the radius is changing,

d 10 = = 10 cm. 2 2 The volume of the sphere is given by 4 3 3  4 π r ( t )  V(t)=  V = π  3   3 when r(t) =

Differentiating both sides with respect to t, we get

V′(t)

2 4  = π 3 ( r ( t ) )  r′(t) 4 πr(t)2 r′(t)  3 

Substituting the given values, we have 5 = 4π (102) r′(t) 1 ⇒ r′(t) = π cm/min 180 Thus, the rate at which the radius of balloon is increasing is π cm / min . 80 In addition to  first-order derivatives (which provide us with information such as the instantaneous rate of change of a function) and the second-order derivatives (which can measure the acceleration of a moving object or help find the extrema of a function), the higher-order derivatives can also be equally useful.  Higher Order Derivatives Consider the function y = f(x) = 4x4 − 3x2 + 10x + 2. Its derivative is given by

dy = f ′(x)= 16x 3 − 6x + 10 dx

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303

This is called as the first derivative. It being a function can again be differentiated using the rules for differentiation and we get

d 2 y d  dy  = ′′(x) [f ′(x)]′ = 16x 3 − 6x + 10 ) ′ = 48x 2 − 6 ( =  f= 2 dx dx  dx  This is called as the second derivative. On differentiating this again, we get the third derivative as well, written as, d3y = f= ′′′(x) [f ′′(x)]′ = 96x dx 3 Continuing like above, we can differentiate again and get the fourth derivative. We can keep adding on primes, but that will get cumbersome after a while. Hence, the notation changes as: f(4)(x) = [ f ′′′ (x)] = 96. And the fifth derivative comes out to be zero, i.e., f(5)(x) = 0. Collectively the second, third, fourth,  etc. derivatives are called higher order derivatives. Higher order derivatives are used to model real-life phenomena in many transportation devices such as cars, roller-coasters, and trampolines. We know, the first derivative describes how fast a function is changing with respect to time (i.e., a car’s velocity) and the second derivative describes how fast the velocity is changing with respect to time (i.e., a car’s acceleration). The third derivative is how fast the acceleration changes and can be physically felt by our body when a car’s driver suddenly accelerates or quickly applies break. Another important application of higher-order derivatives is its use in the construction of the Taylor polynomial which is a way to approximate continuous functions. Let us see how the functions are approximated.

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Differentiation

Linear approximations For a function as complex as f(x) = cos x (which is otherwise hard to evaluate manually), if approximated by a linear function (which is much easier to graph and evaluate) would make calculations easier. Linear functions are of the form, y = mx + b = g(x) where m is the slope of the line and b is the y-intercept.

y

=

m

e op sl (x, y)

(0, b)

x

What could be the possible values of m and b for the linear function which best approximates cos x? To initiate the process of finding the linear function, we can start with g(0) = f(0) ⇒ g(0) = m(0) + b = 1 (∵ cos 0 = 1) ⇒b=1 Thus, g(x) = mx + 1 Since m represents slope of the linear function g(x) and f ′ (x) at x = 0 also represents the slope of the function f(x), thus m = f ′ (0) = − sin(0) = 0. This gives us g(x) = 1 as our linear approximation.

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Graphical representation of g(x) = 1 and f(x) = cos x

g(x) = 1 f(x) = cos x

We see from the graph that for x close to 0, g(x) = 1 is a reasonable approximation of cos x. Thus, the value of the function cos x for very small values of x (when x is near to 0) is close to 1. But this approximation seems inappropriate for larger values of x. Thus, for better approximations, we move to quadratic approximation. Quadratic approximations The linear approximation g(x) = 1 for f(x) = cos x did not take into account any of the curviness of the graph of cos x. To capture the curviness of the function yet keeping it simple, it is best to approximate it using polynomials, to start with, consider a quadratic polynomial of the form g(x) = a + bx + cx2 which is a parabola with ‘a’ representing the y-intercept, ‘b’ representing the slope at the y-intercept and c opening the parabola up or down. Below is the graph of such a quadratic function, g(x) = 2 + 3x + x2.

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Differentiation

Graphical representation of g(x) = 2 + 3x + x2

g(x) = 2 + 3x + x2

Just as in the case of linear approximation, we would still want g(0) = f(0). ∵ g(x) = a + bx + cx2 ⇒ g(0) = a + b(0) + c(0)2 =a And, f(x) = cos x Now, f(0) = cos 0 = 1 ∵ f(0) = g(0) ∴a=1 To capture the slope of the tangent line near x = 0, we want f ′ (0) = g′ (0) Now, g′ (x) = b + 2cx ⇒ g′ (0) = b And, f ′ (x) = −sin x ⇒ f ′ (0) = −sin 0 = 0 Therefore, b = 0. Since the first derivatives are coincident, for a better approximation, let us ensure that the second derivatives of

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307

g(x) and f(x) also coincide at x = 0. The second derivative determines how the first derivative changes, and so making the second derivatives coincide will roughly give us the graphs of f(x) = cos x and of the quadratic approximation g(x) changing at the same rate around x = 0. Now, g′′ (x) = 2c ⇒ g′′ (0) = 2c And, f ′′ (x) = − cos x ⇒ f ′′ (0) = − cos 0 = −1 ∵ g′′ (0) = f ′′ (0) ⇒ 2c = −1 ⇒ c = −1 2 Hence, the quadratic approximation is g(x) = 1 + 0.x −

x2 x2 =1− 2 2

Graphical representation of g(x) = 1 −

x2 and f(x) = cos x 2 f(x) = cos x

g(x) = 1 −

x2 2

The quadratic approximation is indeed a better approximation of the original function near x = 0. Thus, polynomials of higher degree would be able to better approximate the given function f(x) = cos x.

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Differentiation

Higher order approximations (Taylor series) Let the nth order approximation be given by, g(x) = a0 + a1x1 + a2x2 + … + an–1xn–1 + anxn where a0, a1,..., an are constants. Matching the values of the first n derivatives of f(x) = cos x and g(x) at x = 0, we get f(0) = g(0) ⇒ cos 0 = a0 ⇒ a0 = 1 f ′ (0) = g′ (0) ⇒ − sin0 = a1 ⇒ a 1 = 0 (∵ sin 0 = 0) 1 f ′′ (0) = g′′ (0) ⇒ − cos0 = 2a2 ⇒ a2 = − (∵ cos 0 = 1) 2 f ′′′ (0) = g′′′ (0) ⇒ 0 = (3 × 2) a3 ⇒ a3 = 0 Similarly, f(m) (0) = g(m)(0) = m.(m – 1). (m – 2) …1 am = m! am f (m) (0) m! Below are the graphs of various polynomial approximations of the function:

⇒ f(m)(0) = m! am or a m =

f(x) = cos x x2 g(x)=1 − 2 x2 x4 h(x)=1 − + 2 4! x2 x 4 x6 j(x)=1 − + − 2 4! 6! Graphical representation of x2 + 2 x2 j(x) = 1 − + 2 x2 g(x) = 1 − 2 h(x) = 1 −

x4 4! x4 x6 − 4! 6!

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h(x) = 1 −

x2 x4 + 2 4!

g(x) = 1 −

x2 2

f(x) = cos x

j(x) = 1 −

309

x2 x4 x6 + − 2 4! 6!

Continue this process indefinitely, we get the Taylor series approximation as g(x) = a0 + a1x + a2x2 + … where, m! am = f(m)(0)

f (m) (0) m! Thus, any function can be approximated around x = 0 using

Or, a m =

the Taylor series approximation (when the centre is x = 0, Taylor series is called as the Maclaurin series expansion) given by: f(0)+

f' (0) f'' (0) 2 f''' (0) 3 f (n) (0) n x+ x + x +...+ x +... 1! 2! 3! n!

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Differentiation

In case of approximation of the function f(x) near any point x = a where ‘a’ may not be zero, as discussed in all cases above we make the derivatives of f(x) and g(x) coincide at x = a. The Taylor series approximation is, then, given by: f (n) (0) 3 n f (n) (0) f' (a) f'' (a) f''' (a) a) + ... (x − n)n +... (x − a)+ (x − a)2 + ...+ (x − (x a) −+...+ 1! 2! 3! n! n! Thus, to approximate any complicated function numerically and get a nice polynomial expression, Taylor series expansion is one of the most powerful tools. The more the number of terms of the polynomial, the better the approximation of the function. Some of the common Taylor Series expansion is given as: f(a)+

1.

x2 x3 e  = 1 + x +   +   + ... 2! 3!

2.

sin x = x − 

3.

x2 x4 cos x = 1 −   +   − ... 2! 4

4.

1 = 1 + x + x 2 + x 3 + ... for x < 1 = 1− x

x

x3 x5  +  − ... 3! 5!

x2 x3 x4 5. ln (1+x) = x – + – +... 2 3 4

6. (1 + x)p = 1 + px +

=

xn ∑ n=0 n!

=

(−1)n 2n+1 x ∑ n=0 (2n+1)!

=

(−1)n 2n x ∑ n=0 (2n)!

∞

=

∞

∞

∞

∑x n=0

∞

=

n

∑ (–1)

n =1

n–1

xn n

p(p − 1) 2 p(p − 1)(p − 2) 3 x + x + ... 2! 3!

p(p – 1) ... (p – n + 1) n x n! n =1 But, one needs to know ‘How good is the approximation?’  The Taylor series approximation of any function f(x) around a point x = a is given by =1+

∞

∑

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f(x) = f(a)+

311

(n) (n) f' (a) f''f' (a) f'''(a) (a) (a)f (a) 3 (a) 2 f'' n f 2 3 f''' ... (x a)+ (x a) + (x a) +...+ (x a) +... − − − − f(x) = f(a)+ a)+ (x − a) + (x − a) +...+ (x − a)n + 1! 2! 1! 2!3! 3! n! n!

which can be written as f(x) = Pn(x) + Rn(x) Where, (n) f' (a) f''f' (a) (a) (a) f''f'''(a) f (n) (a) 3 f''' (a)f ... (x − a)(x3 +...+ Pn (x) = f(a)+ Pn (x) (x=−f(a)+ a)+ (x − a)+ a)2 + − a)n (x(x−−a)a)2 ++...+ (x − a)n 1! 2! 3! n! 1! 2! 3! n! is the Taylor polynomial which approximates the function, and f (n +1) (c) R n (x)= (x − a)n +1 (n+1)! for some c between a and x, is the remainder or error term for the approximation of the function by the Taylor polynomial A good approximation means the function value to be very close to its approximate value obtained from the Taylor polynomial. Thus, the smaller the error/remainder the better approximation it is. Till now, we have discussed function with two variables (one independent x and another dependent y) both explicit and implicit forms. However, practically we deal with functions having two or more variables. For example, the amount of petrol to be filled in a car depends on the type of the car (sedan or SUV) and the distance it has to travel. The cost of the paint depends on the paint chosen and the area to be painted. The time taken by a student to finish the course of any particular subject depends on his understanding of the subject, the surrounding where he is studying, his state of mind, the time of the day, and many more. In such cases, we have to deal with the rate of change or derivatives in a different manner. Partial Differentiation Consider the function z = f(x, y), where z is dependent on x and y. Graphically such functions are surfaces in three-dimension such as a hemisphere with centre (0, 0, 0) and radius ‘a’ has

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Differentiation

equation given by x2 + y2 + z2 = a2.

Similarly, a paraboloid with vertex at (0, 0, 0) has the equation z2 = x2 + y2 graphically seen as:

Looking back at the example where the time taken by a student to finish his course depends on various factors such as his state of mind or the time of the day or his surroundings, there might be a change in the time taken to finish when only his surroundings changed. Thus, for any function which is changing, we need to mention the variable in which the change has occurred due to which the dependent variable has changed. Similarly, for z = f(x,y), discussing the rate of change of the function z, we need to mention the variable in which respect

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we need the change. We either talk of the rate of change of z with respect to x or the rate of change of z with respect to y. While finding the rate of change of z with respect to x, we keep the other variable y as constant. Similarly, for finding the rate of change of z with respect to y, we keep x as constant. ∂z ∂z These can be denoted by or . These are termed as partial ∂y ∂x derivatives. Similar to derivatives, partial derivatives are geometrically viewed as slopes to the surface. Example 1 Consider the function z = x2 + y 2 to find partial differentiation. Solution Given function is z = x2 + y2. Then, its partial derivative with respect to x is given by ∂z ∂(x 2 + y 2 ) = ∂x ∂x and can be found by differentiating (x2 + y2) with respect to x and keeping y constant. Thus, we get ∂z = 2x ∂x Similarly, the partial derivative of z with respect to y, keeping x constant is given by ∂z = 2y. ∂y

314

Differentiation

Above is the graph of the paraboloid z = x 2 + y 2 and one of the partial derivatives 2x. We can see that 2x is a plane and represents the slope (called as tangent plane) to the surface. Example 2 Consider the function x 2 + y 2 + z 2 = 1 to find partial differentiation. Solution Given function is x2 + y2 + z2 = 1 To find

∂z ∂z and , we use the chain rule. ∂x ∂y

Partially differentiating with respect to x (keeping y constant and z as a function of x and y), we have 2x + 0 + 2z ⇒ 2x + 2z ⇒

∂z x =− ∂x z

Similarly,

∂z =0 =0 ∂x

∂z =0 =0 ∂x

y ∂z = − . z ∂y

APPENDIX

Limit and Continuity

The role of limit and continuity The following text is reproduced here from the first chapter to facilitate the best recall of the role of limit and continuity. The challenges in computing instantaneous speed in a distance-time graph using the idea of rate of change are obvious, as under: • Physically, and in most everyday situations, it is impossible to accurately record time which is less than a few seconds and to correctly record the distance traversed in very small durations (by accurately pinpointing the start and end points of motion in that duration); any error in computation will get greatly pronounced because the measured quantities can only be small in magnitude for such small measurement windows. • Mathematically, rate involves a division operation by almost zero (at an instant) when computing instantaneous values. The solution to this division by almost zero evaded centuries till a new mathematical foundation for infinitesimal quantities was invented. The advantage of 315

316

Limit And Continuity

math is that it knows no limit on the magnitude/quantum and these infinitesimal quantities, however small, can be quantified/derived. Real numbers are just for that; math can indefinitely create smaller and smaller intervals to give us the most accurate measure of the instantaneous value of changing quantities. • Conceptually, what is ‘near zero’, what number may be nearly zero, is it 1 second, 0.1 second, 0.01 second, or even less, when measuring the time period of changing motion; how small is really as small as possible? There is also the issue of the ‘real’ quantitative difference between any infinitesimal and zero. For example, how does one really visualise the difference and distinction between 0.0001 and zero? But let us not forget that smaller the infinitesimal, the more accurate the computed instantaneous value. • Importantly, the idea of infinitesimal also implies infinite instantaneous values within the smallest time intervals (or any other changing quantity). This understanding will be required to appreciate certain transformations truly. And then infintesimal would also vary with situations with the kind of changes being studied. • These challenges were specifically addressed. For example, the physical challenge (to find the distance travelled in small time periods, given an object in motion) was addressed using the idea of derivative that avoided such difficult measurements the idea of limit addressed the mathematical challenge (the mathematical formulation of finding the distance traversed in a small time interval) and the conceptual challenges (that such small quantities do exist) were addressed using the idea of continuity. The idea of limit and continuity are expectedly very mathematical (and intensely conceptually, and imaginatively created).

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Welcome to limit Limit is at the foundation of conceptualising and computing derivatives and anti-derivatives of functions; where functions are the mathematisation of specific situations in terms of the variables of the situations. At the heart of derivatives (and antiderivatives) is the rate of change of a function, at a point (or over a duration, or range of some variable). The instantaneous rate of change of a function at a point is the slope of the tangent at that point. The conception of limit helps us find the slope at a point in a way that we get the slope of the function at a point from the slope of the tangent (a straight line whose slope is easy to compute). Concept of Limit The exact way limit helps us find slope of the function is by making the slope of the tangent become the best approximate value of the slope of the function at the point of interest. This is achieved by making the tangent infinitesimal small around the point of interest and thereby making the curve of the function almost coincide with the tangent. The coinciding of the two is achieved by taking two points on the curve of the function that is infinitesimally close to the point of interest (but not the same as the point).

318

Limit And Continuity

PR tangent at point Q on the curve shows the best approximate value of the slope of the curve as it nearly coincides with the slope of line PR. Limit enables the computation of the slope despite the involvement of infinitely small quantities; we know how very small denominator quantities are undesirable in division situations (such as the use of the ‘average formula’). So, this undesirability can be reduced by making the denominator as small as possible such that it is non-zero but approaches zero. Thus, the non-zero approach calls for the importance of the notion of a limit. Limit in real life Calculus backs on the concept of limits. However, the motivation to study the same in depth is because of its usage in our daily lives.

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319

• When a chemical reaction takes place between two chemicals, a new compound is formed as time passes. Thus, the new compound is the limit as time period becomes larger. • Tossing a coin gives a head or tail. To know the probability of the outcome, we may flip a coin a large number of times. As the number of trials increase, the time period becomes almost larger in magnitude and hence, the number of heads becomes almost equal to the number of tails. So, on tossing the coin large number of times, we get the probability of getting an equal number of heads or tails. What are these quantities precisely? Our aim is to measure these quantity in an ‘instant’, and we define this ‘instant’ by a limit. What is limit? The word ‘limit’ is the best descriptor of what the value of a quantity, say A is when there is the smallest change in the value of some other quantity on which A is dependent. Functions are a way to mathematise real-life situations dependent on different quantities. This understanding of limits applied to functions gives us the mathematical formulation of the former, described as follows. Mathematically, written as limf(x)=t , this expression is read x →c as ‘limit of the function f as x approaches c is t.’ It can also be phrased as ‘As x approaches c, the value of the function f gets arbitrarily close to t.’ That is, f(x) → t as x → c. This can be numerically and graphically explained through an example. Example 1 Consider a function f(x) = x + 3. What is the value of this function as x approaches 2? What are the ways to approach 2? The integer 2 can be approached either from points on its left (that is, taking the values very close to 2 lying on its side and are less than 2, say 1.8, 1.9) or from points on its right (that is, taking the values

320

Limit And Continuity

close to 2 lying on its right and are greater than 2, say 2.1, 2.2). The values of the given function as x takes different values near 2 can be mathematically studied as: Points on the left of 2 Points on the right of 2 At x = 1.8, f(x) = 1.8 + 3 = 4.8 At x = 2.1, f(x) = 5.1 At x = 1.9, f(x) = 4.9 At x = 2.2, f(x) = 5.2 The values can be seen on the graph as follows:

)



f(x

f(x)5+

+ =x

3



f(x)5–

x2–





x2+

This closeness of x to c and f(x) to t is the essential concept of limit. But the way it has been formulated in the above example is not very appropriate. Conceptually, we have been relating words such as ‘approaching’, ‘sufficiently close to’ or ‘near to’ with limit. But is there a way to be precise in defining the limit of a function, which should be consistent with the methods which we will study in the later part of the chapter? Before we head to defining limits formally, it will be unfair if we do not mention the man behind this formalisation, Bolzano introduced this technique to precisely and algebraically define limits and continuous functions (also covered as a concept in this chapter).

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The Epsilon-Delta technique for limit Limit of a function exists if and only if for any (epsilon) ε > 0, there exists (delta) δ > 0, such that if 0 < |x – c| 0, such that as x approaches c, then f(x) approaches t, that is, when x → c (for δ > 0) then f(x) → t (for ε > 0) ⇒ 0 < |x – c| < δ then 0 < |f(x) – t| < ε As found above the limit of f(x) as x approaches 1 is 3. Let us assert the same using the definition. Let us also note from the definition the order in which ε and δ are given. We are given ε first and then we have to verify that the limit exist only if we can find a δ that works for this ε. ε > 0, say ε = 0.01 We aim to find a corresponding δ, such that whenever 0 < |x – 1| < δ , 0 < |f(x) – 3| < 0.01. Example 1 Consider |f(x) – 3| = |(2x + 1) – 3| = |2x – 2| = 2 |x – 1| The above inequality reduces to, 0 < 2 |x – 1| < 0.01 ⇒ 0 < |x – 1| < 0.005 We can choose δ = 0.005 And, δ < ε [ ε = 0.01] However, for 0 < |x – 1| < 0.005 to be true, δ can take any value less than 0.005 as well, that is, δ < 0.005. Example 2 What is the limit of the function f(x) = 5x + 7 when x approaches – 1? Graphically, we see that as x approaches – 1 (from both left and right), the functions f(x) approaches 2.

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Limit And Continuity

f(x) = 5x + 7

Let us verify the same by taking any arbitrary ε . choose ε > 0 We need to show the existence of a δ such that whenever 0 < |x – (–1) < |δ and 0 < |f(x) – 2| < ε Consider f(x) − 2 = =

5x + 7 − 2 5x + 5

= 5 x +1

The inequality reduces to 0 < 5 |x + 1| < ε or 0 < |x + 1| < Thus, corresponding to any ε, we can choose δ
0, there exists a δ > 0 such that 0 < |x − c| < δ implies |f(x) − L| < ε.  Suppose the function f(x) has a different right-hand limit (say R) and left-hand limit (say L). Applying the same definition to these two different limits L and R, gives the following: For the right-hand limit R (when x approaches c from the right), there exists a δ1 such that: as, x → c, then f(x) → R 0 < |x – c| < δ1, then |f(x) – R| 0, there exist a real number N < 0 such that, |f(x) – L| < ε ∀ |x| < N. We thus need to find the value of the function to which it will approach when x becomes extremely small. Example 1 Consider the function f(x) =

1 . x

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Limit And Continuity

1 y= x

We see that as x becomes larger, the value of the function 1 approaches 0. Thus, lim = 0 x →∞+ x Example 2 For f(x) = 5 –

2 , evaluate lim f(x). x →+∞ x2

Solution

2  lim f(x) = lim  5 − 2  x → +∞ x → +∞ x   2 == lim 5 − lim 2 x →+∞ x →+∞ x 1   5 − 2  lim 2  == xlim →+∞  x →+∞ x 

= 5 – (2 × 0) =5

(∵ x  +∞)

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347

Example 3 Evaluate: f(x) = 4 − Solution

2 at lim f(x). x 3 x →−∞

2  lim f(x) = lim  4 − 3  x → –∞ x  

x → –∞

 2 = lim 4 − lim  3  x → –∞ x → –∞ x   1  = lim 4 − 2  lim 3  x → –∞ x → –∞ x  

= 4 – (2 × 0) (∵ x  –∞) =4 As x approaches to ∞, function f(x) can also approach ∞. Example 4 Evaluate: lim f(x) for f(x) = –5x3. x →∞

We see that as x approaches ∞, the function approaches –∞. Example 5 Evaluate: lim 3x + 2x 2 . x → −2

Solution lim 3x + 2x 2

x → −2

= 3 lim x + 2 lim x 2 x → –2

x → –2

= 3 (–2) + 2(–2)2



 lim [ f(x)+ g(x)] = lim f(x)+ lim g(x) x→c x→c  x → c  (∵ x  –2)

= –6 + 8 = 2

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Limit And Continuity

f(x)



Graphical representation of f(x) = 3x + 2x2 f(x)2+



f(x)2-

f(x) = 3x + 2x2





x2- x2+

This can be verified from the graph as well that as x approaches −2, the function approaches the value 2. Example 6

x 2 − 11x + 28 x→7 x −7

Evaluate: lim Solution

 limf(x) f(x1 )  f(x)1 ) lim x x→→ cc  lim f(x   ==  xx →→cc f(x 2 )  g(x) lim g(x) lim ) f(x 2 x→ x→ cc    On substituting the value where x approaches 7, we see that the denominator vanishes leaving the function undefined. In such cases where denominator becomes zero on direct substitution of the value of limit, we factorise the numerator and then evaluate it. x2 – 11x + 28 = (x – 7) (x – 4) Therefore, x 2 − 11x + 28 lim x →7 x−7

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= lim x →7

= lim

349

x 2 − 11x + 28 x−7

( x − 7 )( x − 4 )

x →7

(x − 7)

= lim (x – 4) x →7



=7–4 (∵ x  7) =3 From the graph of f(x) = x – 4, we verify that as x approaches 7, the function approaches 3. f(x)3+



f(x)3-





x7- x7+

) f(x

Example 7  16 − h − 4  lim   h→0 h  

=

x

–

4

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Limit And Continuity

Solution On substituting h = 0 in the function, we see that both the numerator and the denominator vanishes. In such cases, where square root is present, we take the conjugate of the terms and then evaluate.  (16 − h ) − 4    = lim h→0 h   Multiplying the numerator and denominator with the conjugate of

(

(16 – h ) – 4 )

(16 − h ) − 4

= lim

h

h→0

= lim h →0 = lim h →0 =

lim h →0

=

(16 − h ) + 4 (16 − h ) + 4

×

(16 − h ) − 16 h ( (16 − h ) + 4 ) h

(

(

−h

(16 − h ) + 4 ) −1

(16 − h ) + 4 )



−1 −1 = = − 0.125. 4+4 8

(∵ h  0)

−1 −1 = = − 0.125. 4+4 8 This can be verified from the graph as well that as h approaches 0, the straight line is intersecting the vertical axis at −0.125.

 lim Graphical representation of f(h) = h→0 

(16 − h ) − 4  h

 

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351

Example 8

x 2 + 4x – 12 Evaluate: lim x → 2 x 2 – 2x Solution

We observe that on substituting x = 2, both the numerator and denominator becomes 0. Therefore, in such cases, we factorise numerator and denominator and then evaluate further.

x 2 + 4x – 12 x →2 x 2 – 2x (x – 2) (x + 6) = lim x →2 x(x – 2) lim

= lim

x →2

=

x+6 x

8 =4 2

(x → 2)

f(x)4–

f(x) =



f(x)4+



Limit And Continuity



x2 + 4x – 12 x2 – 2x



352

x2– x2+

Example 9 Evaluate: lim

t – 3t + 4 4–t

t → 4

Solution On substituting t = 4, both numerator and denominator becomes 0. Therefore, on rationalising numerator and denominator because of the presence of square root, we get

lim

t – 3t + 4

t → 4

4–t

= lim

t → 4

(t –

) (t + (4 – t) (t + 3t + 4

) 3t + 4 ) 3t + 4

Multiplying the numerator and denominator with the conjugate of t – 3t + 4 .

= lim

t → 4

(t –

) × (t + (4 – t) (t + 3t + 4

) 3t + 4 ) 3t + 4

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= lim

t → 4

t 2 – ( 3t + 4 )

(4 – t) (t +

3t + 4

353

)

t 2 – 3t – 4

= lim

( 4 – t ) ( t + 3t + 4 ) ( t – 4 ) ( t + 1) = lim ( Factorising numerator ) ( 4 – t ) ( t + 3t + 4 ) ( t – 4 ) ( t + 1) = lim t → 4

t→ 4

(

–(t – 4) t + 3t + 4

t → 4

= lim

t → 4

=

(

( t + 1)

– t + 3t + 4

–5 = − 0.625 8

= –0.625

)

)

( t = 4 )

(∵ t = 4)

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Limit And Continuity

t − 3t + 4

lim Graphical representation of f(t) =

4–t

t→4

f(t) =

Example 10

t – 3t + 4 4–t

 1    x

Evaluate: lim x 2 cos  Solution

x → 0

We know that, –1 ≤ cos x ≤ 1 , thus for any value of x

 ∴ –1 ≤ cos  

1  ≤1 x   1 –x 2 ≤ x 2 cos   ≤ x 2  x

(Multiplying throughout by x2)

( )

Now, lim x 2 = 0 and lim –x 2 = 0 x → 0

x → 0

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 ∴ lim x 2 cos  x →0 

1  1 =0 x  x

355

(Using Squeeze theorem)

The above calculations show that as x tends to 0, the limit of g(x) = x 2 and h(x) = –x 2 approaches 0. Therefore, by Sandwich 1 theorem, x2 cos   would also approach 0. x

cos ( 1 x) f(x) = x 2

g(x)

= x2

Graphically, it can be represented as:

h(x) 2

= –x

To summarise, the limit of a function at any given point is  unique and exists if and only if the left hand and the right hand limits are same. The idea of limits lies in the fact that if the values of x are close enough, then the values of the function f(x) will also be close enough to the limit. But, ‘How close is the function to the limit?’ is a question to be addressed. While evaluating the limit of a function as x approaches c, we simply calculate the value of the function f(x) at c. This gives us a sense of continuity of the function.

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Limit And Continuity

Welcome to Continuity The idea of continuity is the other concept required to find the derivative of a function that would be valid at the point of interest. Let us explore this through an example. Suppose a rocket is launched. The distance travelled by the rocket is governed by some motion equations which depend on the position and velocity of the rocket with which it is launched. Using mathematical tools, these motion equations along with the initial conditions can be solved, and thus, the position and velocity of the rocket at any time t can be found out. The three equations of motion are: v = u + at v² = u² + 2as 1 s = ut + at² 2 where, u = initial velocity v = final velocity t = time taken a = acceleration s = displacement Now, changing the initial conditions for the position and velocity would bring a change in the distance the rocket has travelled and its velocity. We, thus, say that the final position and velocity is ‘continuously’ dependent on the initial conditions. Similarly, the water flow in ocean and sea is ‘continuous’, we are ‘continuously’ aging with time describes the behaviour of how the water flow is and how we age respectively. Understanding continuity Consider the paths on the following two graphs:

357

f(x )=

x

Calculus For Professionals

Graph (i)

f(x) =

1 x

Graph (ii) If we start walking on these paths from the starting point –1 till the end point 1, there is a smooth ride in case of path (i) whereas in the graph (ii), it is impossible to get to the second part of the path because of the jump from negative y on the left side to a positive y on the right side. We thus say that path (i) is continuous in nature whereas path (ii) is not.

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Limit And Continuity

f(x) =

3x

Following inferences can be drawn on considering the path followed by the graphs of the function: • Continuity, in the above sense, describes how a graph behaves. Since we sketch graphs of functions thus, continuity precisely describes the behaviour of a function. • Continuity can be best described as ‘for a small change in the input values, there will be a small change in the output values as well’. Studying continuity of functions fundamentally, makes the functions more intuitive for visualising and prediction. For example, in the above path (i), a small change along the x-axis results in a small change in the values along y-axis, whereas in path (ii), while crossing the origin (0, 0) from negative to positive value of x, the value of y jumps from −∞ to +∞ . Thus, a small change in value of x results in a huge change in the value of the function. • Graphically and conceptually, we can see that continuous curves/functions are single and unbroken. For example, the function f(x) = 3x shown below is continuous at x = 2.

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359

Similarly, the function f(x) = x2 is continuous at x = 0.

f(x) = x2

However, the greatest integer function f(x) = [x] is not continuous at any of the integer values. However, if we consider the function in the interval, say (0,1), it is continuous, because there are no jumps while following its path.

f(x) = [x]

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Limit And Continuity

This is how continuous functions can be graphically observed and conceptually understood. However, for a rigorous understanding of mathematics, a more technical definition is required. This is where limit helps, using which, we will learn a better and far more precise way of defining and understanding continuity. Fortunately, finding the continuity at a point can be universalised in terms of certain conditions of the behavior of the function around that point. Some Conditions for Continuity For the function f(x) to be continuous at a point x = c in its domain, the following three conditions should be satisfied: 1. f(c) exists. The value of the function at x = c exists and is finite. 2. lim f(x) exists. x →c

For the limit to exist, the left-hand limit = right-hand limit, that is, lim f(x) = lim+ f(x) (Both are finite).

x →c −

x→ c

3. lim f(x) = f(c) f(a) x →c

The limit of the function as x approaches c coincides with the value of the function at x = c. The function f(x) is said to be continuous on the real line if the above three  mentioned conditions are satisfied for every point on the real line.

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361

Graphically, the above conditions can be understood as follows: 1. f(c) exists

2. lim f(x) exists x →c

362

Limit And Continuity

3. lim f(x) = f(a) f(c) x →c

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363

More about the conditions For the function to be continuous, all three conditions should hold. What if one of the conditions is not true? Does that make the function discontinuous? Let us get into the details. 1. The first condition for continuity states that f(c) should be defined. We work with the case that f(c) is undefined. Consider the graph as given below. The graph has a hole at c, which means the value of the function at c is not defined. Thus, there is a gap in the graph while crossing the point x = c, making the function not continuous at x = c.

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Limit And Continuity

Consider a function f(x) =

x2 − 9 x−3

x 2 –- 9 f(x) = x –- 3

f(3) =

32 − 9 0 = i.e., f(3) is not defined. We also see a hole 3−3 0

at x = 3 in the graph which verifies that the function is not continuous at x = 3. Example 1

 x + 3 if x > 0  if x ≤ 0  −x

Consider a function f(x) =  Solution

 x + 3 if x > 0  if x ≤ 0  −x

Graphical representation of f(x) = 

We check the continuity of this function at x = 0.

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365

Now, f(0) = 0. That means the first condition holds. Graphically, we can see that there is a jump in the graph around x = 0. Let us move to the second condition, LHL = lim− f(x) = lim− (− x) = 0 x →0

x →0

[For x < 0, f(x) = –x]

RHL = lim+ f(x) = lim+ (x + 3) = 0 + 3 = 3 [For x > 0, f(x) = x + 3] x →0

x →0

Since both the left-hand and the right-hand limits are different, hence the limit does not exist. Thus, the second condition for continuity is violated. 2. The second condition states that the limit should exist, that is, lim f(x) exists. x →c

In the above example, we see that the limit does not exist and the function is not continuous at x = 0. The violation of the second condition of continuity can be graphically shown as:

Limit And Continuity

 x a–



366

x a+

The above graph though is defined at c, but has a jump at x = c, thus it is not continuous. 3. The third condition for continuity states that the limit and value of the function at the point to be discussed for continuity should coincide. Consider the case when lim f(x) and f(c) are not same. x →c

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367

Though both conditions (i) and (ii) are true in the graph, that is, f(c) is defined and lim f(x) exists, but both the values x →a are not same.  sin x , x ≠0  For example, consider the function f(x) =  x  0 , x = 0 Graphically we see a gap which accounts for the function being not continuous at x = 0.

Now, f(0) = 0. The first condition holds. sin x sin x = 1 and lim+ =1 x →0 x →0 x x sin x ⇒ lim = 1, the second condition also holds. x →0 x sin x = 1 ≠ 0 = f(0). The third condition is However, lim x →0 x violated making the function not continuous at x = 0. Thus, if any of the three conditions do not hold at a certain point, then the function is not continuous at the mentioned point. Let us quote few more examples for a better understanding. lim−

Example 2 Consider the function f(x) = 3x. Graphically, we see that it is continuous at x = 2.

Limit And Continuity

f(x) =

3x

368

Applying the above conditions, we see that, LHL RHL lim f(x) = 3x lim f(x) = 3x x →2 x → 2− lim f(x) = 3 (2) lim −=f(x) = 3 (2) f(x) 3(2) xf(x) → 2 = 3(2) x→ 2 f(x) = 6 f(x) = 6 =6 lim f(x) lim − f(x) = 6 x →2 2 x → We also see that f(2) = 3 × 2 = 6. +

+

+

lim f(x) f(x) = f(2). Thus, lim f(x) = lim x → −2

x → x→ +22 +

It is thus verified that the function is continuous at x = 2. It can be verified that this function is continuous at all points of the real line, thus proving that f(x) = 3x is continuous on the real line. Example 3 Consider the function f(x) = x2

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369

f(x)

= x2

We see that f(x) = x2 is graphically continuous at x = 0.

Also, lim− f(x) = 0 and lim+ f(x) = 0 . And, f(0) = 0. x →0

x →0

Thus, lim− f(x) = lim f(x) = f(0) . x →0

x → 0+

Thus, the three conditions in the definition are sufficient to check if the function is continuous or not. These conditions are a consequence of the graphical representation of the functions. Since, it is difficult to visualise the graph of complex functions, these conditions come to the rescue. Example 4 Polynomial functions of the type axn + bxn–1 +…+ c = 0 are continuous on the real line.

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1. Consider a polynomial function f(x) = x 2 – 3x and its graphical representation.

f(x) = x2 – 3x





x1+



x1–

f(x)2+



f(x)2–

f(x)

Graphically, it seems to be continuous at x = 1. Let us check for continuity at x = 1 using the three conditions. LHL RHL lim f(x) = x 2 – 3x

x →1–

f(1) = (1)2 – 3(1) =1– 3 = –2

lim f(x) = x 2 – 3x

x →1+

f(1) = 1 – 3(1) =1– 3 = –2

Thus, lim f(x) = −2 =f(1) , showing that the function is x →1

continuous at x = 1. 2. Consider the polynomial function, f(x) = x 6 + 3x 2 + 2 and its graphical representation.

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

f(x) = x6 + 3x2 + 2

f(x) 6+



f(x) 6–



 x 1–

x 1+

Let us check the continuity at x = 1 using the three conditions. LHL RHL 6 2 lim+ f(x) = (1)6 + 3(1)2 + 2 lim− f(x) = (1) + 3(1) + 2 x →1

x →1

lim f(x) = 1+ 3 + 2

x → 1−

lim f(x) = 6

x → 1−

lim f(x) = 1+ 3+ 2

x → 1+

lim f(x) = 6

x → 1+

Also, f(1) = 6 Thus, lim f(x) = 6 = f(1), showing that the function is x →1

continuous at x = 1. It can be similarly shown that all polynomials are continuous at every point on the real line. Example 5

p(x) , q(x) ≠ 0 , where p(x) q(x) and q(x) are polynomial functions. Rational functions are also continuous everywhere except where q(x) = 0, since for q(x) = 0, the rational function is not defined. Rational functions are of the type

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(x 2 − 3x) For example, the function defined by f(x) = is (x − 2) continuous at x = 1.

f(x) =

(x2 – 3x) (x – 2)

We see that there is no break in the graph until x = 2 and is thus continuous at x = 1. Actually, the function is continuous at all points x < 2. We also see graphically that the function is continuous at all points x > 2. The only point of concern is x = 2, where the function is not defined and a vertical asymptote can be seen in the graph. Let us now check by applying the conditions at x = 1.

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LHL 2 3x xx2 ––3x lim f(x) = lim lim –f(x) = lim –  x x→→ 1–1 x x→→ 1–1  xx––22    – x x 3   ( ) lim–x ( x – 3 )  ==lim x x→→ 1–1  xx––22   11((11––33)) == 1 – 2   1 – 2  ––22 == –1==22 –1

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RHL xx22––3x 3x lim lim+ +f(x) f(x)==lim lim  + + xx→→ 11 xx→→ 11  xx––22 

xx((xx––33)) == lim lim+ +  xx→→ 11  xx––22  11((11––33)) ==   11––22  ––22 == ==22 ––11



(12 − 3 × 1) Also, f(1) = =2 (1 − 2) Since, all three conditions are satisfied, thus the function is continuous at x = 1. On the similar lines, we can show that the function is continuous for all other values of x (not equal to 2). However, at x = 2, f(2) = defined.

(22 − 3 × 2) −2 = , which is not (2 − 2) 0

Since the first condition, that is, f(2) should be defined is violated, the function is not continuous at x = 2. Thus, rational functions are continuous everywhere except where the denominator is zero. Example 6 Trigonometric functions such as sin x, cos x are continuous everywhere in the domain of x as can be seen in the graph below. Their continuity can also be verified by applying the three conditions.

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f(x) = sin x

f(x) = cos x

For example, to check the continuity at x = 0, f(0) = sin 0 = 0 and lim− f(x) = 0 = lim+ f(x), thus sine function is continuous, x →0

x →0

so is the cosine function. Consider tangent function. θ tan θ

− 3π 2 ∞

– –1

−π 2 ∞

0 0

π

2 ∞

f(x) = tan x

 1

33 π 22 ∞

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We

see that there is a break in the graph at −π − 3π π 3π x = and so on. We also know , , , 2 2 2 2  π     = ∞ = tan 33π  , the  , function is not defined that tan     2 2   2  at these points, making it discontinuous. However, if we just  −π π  take the interval say  ,  , the tan function is continuous  4 4 at every point of this interval, thus making it a continuous function in that domain. Thus, continuity of a function, in a way, is concerned with the interval in which it is defined. Example 7 Consider exponential functions that are continuous on the real line. Graphical representation of f(x) = ex

f(x) = ex

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Limit And Continuity

Graphical representation of f(x) = 2x

f(x) = 2x

Graphically or by proving all three conditions, we can see that exponential functions are continuous on the real line. Example 8 Consider some of the logarithmic functions such as log x and log (x – 1). Graphical representation of f(x) = log x

f(x) = log x

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Graphical representation of f(x) = log (x – 1)

f(x) = log (x – 1)

log x is defined for x > 0 because log 0 = –∞. Thus, log x is continuous for every value x > 0. Similarly, log (x – 1) is defined for x – 1 > 0, that is, x > 1. Thus, log (x – 1) is continuous for every value x > 1. Example 9 The modulus or the absolute function of the type |x| is continuous throughout the real line. Below are the graphs of |x| and |x – 2|, which are continuous on the real line. This can be proved both graphically and using the conditions of continuity.

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Limit And Continuity

f(x)

= |x |

Graphical representation of f(x) = |x|

Graphical representation of f(x) = |x – 2|

f(x) = |x - 2|

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Example 10 Consider the square root function

x.

As square root is defined for positive values, it is thus continuous for all positive real numbers. Graphical representation of f(x) =

f(x) =

x

√x

We are now aware of the conditions which can be used to evaluate whether the function is continuous or not at values in the domain of the function. Let us now discuss when and how some complex functions can be continuous as well. Properties of continuous functions Suppose the two functions f(x) and g(x) are continuous at say x = c, then the following are true: 1. f(x) + g(x), f(x) – g(x), and f(x).g(x) are also continuous at x = c. Consider the functions f(x) = x2 and g(x) = log x which are continuous at x = 1.

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Limit And Continuity

Graphical representation of f(x) = x2

f(x) = x2

Graphical representation of g(x) = log x

g(x) = log x

Let us examine the following functions for their continuity at x = 1.

Calculus For Professionals

(i) f(x) + g(x) = x2 + log x Graphical representation of the function y = x2 + log x

y = x2 + log x

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Limit And Continuity

(ii) f(x) – g(x) = x2 – log x Graphical representation of the function y = x2 – log x

y = x2 – log x

(iii) f(x).g(x) = x2 log x Graphical representation of the function y = x2 log x

y = x2 log x

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We see graphically that the function f + g is continuous at x = 1 and so are f – g and fg. Their continuity can also be verified by applying the three conditions. 2.

f is also continuous at x = c provided g(c) ≠ 0. g Considering the same functions f(x) and g(x) as above, we

f(x) x2 = is not defined g(x) log x for g(1) = 0. We also have seen in the previous examples that rational functions of the type p(x) are continuous q(x) see that g(1) = log 1 = 0. Thus,

f ( x0 ) is not defined for g(x) g(1) = 0, hence it is not continuous at x = 1, as can be seen from the graph as show. everywhere they are defined. Since

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Limit And Continuity

f(x) x2 = Graphical representation of g(x) log x

f(x) x2 = g(x) logx

1

However, it can be seen that

f(x) is continuous at x = 2 g(x)

(say), where both the functions f(x) and g(x) are continuous. 3. If f is continuous at g(a), then the composition function (f o g) is also continuous at x = a. Consider the same functions f(x) and g(x) as above. The function f(x) = x2 is continuous at 0 = g(1) = log 1. Now, consider the composition function f o g(x) = f(g(x)) = f(log x) = (log x)2

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Graphical representation of f[g(x)] = (log x)2

f(g(x)) = (log x)2

As can be seen from the graph, f o g = (log x) 2 is continuous at x = 1. This can also be verified using the three conditions for continuity. This is all about evaluating the functions for continuity, which can be intuitively verified using graphs (no jumps/breaks in the graph of the function) and the continuity conditions. So if we are at a point where continuity holds, then with a sufficiently small amount of change around this point, the function value will change arbitrarily little. So f(x) is approximately equal to f(c), for x in the vicinity of c. Thus the value f(c) may be useful to approximate the value f(x). However, a formalisation of this feature of continuity (that a sufficiently small change in the argument causes arbitrarily small change in the function value) is definitely

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Limit And Continuity

required to understand the mathematical rigour. This can be understood in a similar fashion as defining the limit of a function using the epsilon-delta (ε-δ) technique. Formal definition of continuity A function f is continuous at a point a ∈ R if given ε > 0 there exists a δ > 0 such that if |x – a| 0. We must show that whenever |x – 1| 0 2. Consider the function f(x) =  – x if x ≤ 0 The graph has a jump at x = 0, hence it is discontinuous.

We have already discovered in the previous sections that the limit doesn’t exist as x  0.

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Limit And Continuity

This is because both the left-hand limit and the right-hand limits are different. lim f(x) = 0 and lim+ f(x) = 3 .

x → 0–

x →0

Since, the value of f(x) ‘jumps’ as x approaches 0 – and 0+, hence such type of discontinuity is termed as jump discontinuity. Such a discontinuity happens when the  limit  of the function as it approaches  c –  and  c+  are different and are usually observed in functions which are defined differently for different domain on R, that is in case of piecewise functions. As the limit does not exist, there is no way that this discontinuity can be removed. Thus, it is also termed as a non-removable discontinuity. 3. f(x) =

1 is discontinuous at x = 0 x

f(x) =

1 x

In this case, the limit does not exist when x approaches 0. Also, the function is not defined at x = 0, due to which

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a vertical asymptote can be seen in the graph of the function. Thus, all the three conditions are not being satisfied. Due to the presence of vertical asymptotes at a particular value of x, such type of discontinuities is termed as infinite/asymptotic discontinuity. Another example of asymptotic discontinuity can be seen in f(x) = tan x, where asymptotes can be seen at and −π π 3π , , , so on. 2 2 2 Graphically, f(x) = tan x can be represented as:

f(x) = tan x

In this case, the limit does not exist when x approaches

π . 2

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Limit And Continuity

π . Therefore, all the 2 three conditions are not satisfied in this example too, making it a function with asymptotic/infinite discontinuity. Hope the placement of this chapter in the Appendix of the book, and out of the way of the (linear) plan of learning differentiation (the first of the calculus operation), removes one major obstacle in gaining confidence in working with calculus. Differentiation becomes ‘more arithmetical/operational’ without the mechanics of limit, and continuity, as the gateway to it. Yet, the concept of limit may be seen as the critical solution to working with all kinds of ‘irregular (real world) situations, events, surfaces, shapes, scientific relationships, etc.’; the idea of limit allows us ‘regularise’ the irregular quantities by slicing them into a series of ‘near-zero regular quantities’. And that is why limit formed the ‘arithmetical/operational’ foundation of differentiation.

Also, the function is not defined at x =

Acknowledgment

When God wants to give you something, he does not ask for anything but just bestows! With the countless blessings and best wishes of people known and unknown to me, I have been fortunate enough to be on the receiving end of everything good 😊. And this book is one of the most prized opportunities I have received, for which I would like to thank Sandeep. He has been my mentor whose love and guidance enabled me to decide on charting the path few have taken, home-schooling my children, and now giving me an opportunity to make a difference in the lives of countless children through this book. I would also like to extend my love and gratitude to my children – Avyana and Redan, my husband – Mausam, and the entire family who has been a constant support during the writing of this book and everywhere else. Their love and understanding made herculean task like writing this book feel easy than it was. Obviously, there is an extremely talented and committed team behind the scene! I must specifically acknowledge all the members who I have directly been with for weeks to give a life to this book. Nikita Gupta may come on top of the list, for she anchored the brass tracks since day one of the book writing and editing. She is currently majoring in Mathematics at an undergraduate level and aspires to be an analyst as well as to establish her own analytics company. Nikita is one of the most inquisitive mathematics students I have come across especially in the domain of calculus and analysis. 393

Saloni Srivastava, Sandeep’s wife, is polymathic when it comes to editing, to say the least; she has the keenest eye as an editor and straddles across domain validation, design, language, as well as continuously helps the team to develop into its higher level of perfection. The reading experience of the book is owed to her immense contribution. Dipti Chauhan, the backbone of the content and consulting team at IYCWorld Softinfratucture Pvt. Ltd., who relentlessly and gracefully organised, handled, and directed the entire group, and kept their spirits high at all times. She has given a concrete shape to my ideas and thoughts in the form of this book and has been there right from its first rough draft to its final publishing. Nikita Todarwal and Nidhi Gandhi, the professional editors with excellent academic credentials. Nikita has an eagle’s eye when it comes to proofreading; nothing can go unnoticed when she diligently takes on her onus of producing a fool-proof book. Nidhi is an amazing person who puts all her heart and soul into her work. She is filled with enormous determination, dedication, and devotion; leaving no stone unturned in bringing out the best turnouts. Manish Kumar, Tanu Gaur, and Karan Anand are responsible for designing and fabricating the entire book. Manish is a quick and zealous learner who can travel to boundaries with little guidance. He has been extremely energetic and earnest in his ways of working throughout the book tour. Tanu is one of the sincerest persons I have come across in years. She is focused and mindful of her responsibilities; a person who does not move from her chair till she puts the completed work on the table. Karan is a jovial person, serving us with his grin at all times even while working under pressure. His solemn, punctual, and altruistic attitude are what has kept this book just right on its tracks. 394

Despite this being the first work in publishing a Calculus book, all three have admirably wrapped their head around it in no time to make the book what it is today. Mention must also be made to Sandeep’s longstanding colleagues at IYCWorld Softinfrastructure Pvt. Ltd., Krishan Tikoo, Rakesh Mani Tripathi, Virendra Agrawal, and Abhishek Pandey. They all have been the prime motivators and the drivers of the entire team; filling us with the necessary resources and all the means we required at every step of our journey. And then there are some who I do not know, or cannot know, who have also played their part in helping me become the source through which this book got manifested. Dr. Garima V Arora

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