Mathematics for Joint Entrance Examination JEE (Advanced): Calculus 9353500826, 9789353500825

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Contents

ix

Preface 1.

1.1-1.82

Relations and Functions

Real Nunmber System and Inequalities Intervals as Subset of Set R Inequalities

Sign Scheme Method

to Solve

Inequalities

Subtraction

Modulus of Addition and 1.1 Application Exercise

Concept Relation

Number of Relations

1.13

and Range of Function Domain, Codomain Relation between

Independent and

Dependent Variables

Vertical Line Test Some Elementary Functions

Identical Functions Piecewise Functions Algebra of Functions Exercise

1.3

Concept Application Functions (Mappings) Different Types of Functions O n e - O n e and Many-One and Into

Functions

Concept Application

Exercise

1.4

Algebraic Functions

Exercise

Exercise

1.6

Inverse Trigonometric Functions

Concept Application Exponential

Exercise

and Logarithmic

1.7 Functions

Exponential Function Logarithmic Function

Concept Application

Exercise 1.8

Some M o r e Functions Greatest Integer Function Fractional P a r t Function

min. {g1x). g2(x), Exercise 1.9 Application Concept

8,r)} orfx)

=

...

8,(x)}

Even and Odd Functions

Even Function Odd Function Function Even and Odd Extension of Exercise 1.10

Concept Application Periodic Functions

Functions Period of Some Elementary Functions Period of Transformed Functions Period of Composite from Algebraic Period of Functions Resulting Operations of Functions Exercise 1.11

Concept Application

Concept Application

1.15 1.16

Inverse Functions

.20 20 1.21 24

1.24 1.26 1.29 1.29 1.32 1.32 1.32

1.33 1.36 1.36 1.36 1.39

1.43 1.43 1.43 1.44 1.44 1.45 1. 5

1.45 1.46 1.47 1.48 1.49

Injective and Surjective

1.14

1.18 1.18 1.19 1.19

1.41

1.42

Composite Functions

Nature of Composite

Function

1.25 1.5

Trigonometric Functions

Concept Application

1.13

1.24

Quadratic Functions

Concept Application

.10

1..13

Exercise 1.2

Function (Mapping)

Onto

1..10

1.11

Types of Relation

as a

1.9 1.9 1..10

Relation and Codomain of

Function

14

.10

Domain of Relation

Concept Application

1.1 1.2 1.6

Modulus of Real Number

Range

1.1

1.41

Signum Function Functions of the Form fx) = max. {g,(x), g2(x), ...

Exercise 1.12

Inverse Properties of

Concept Application

Functional Equations Problems Based

Function

Exercise 1.13

on

Value of Finding the

Point Function at Some Problems Based

on

Problems Based

on

Concept Application

Finding Function Function Properties of

Exercise 1.14

Transformation of Graphs Vertical Shift Horizontal Shift

Related to Modulus from the Known

Graph ofy (l)] the Graph of y =/x), where [] Represents The

=

1.52 1.54 1.54

1.54 1.55

1.56 1.57 1.58

1.59 1.59

1.59

Horizontal Stretch or Squeeze Vertical Stretch or Squeeze Vertical and Horizontal Flip Transformation

1.49

1.51

Greatest Integer Function

Concept Application

Exercise 1.15

1.60 1 .60 60 1.60

1.62 1.63

Solved Examples

1.63

Exercises Single Correct Answer Type

1.67

Multiple Correct Answers Type

1.67 1.74

iv Contents

76 78 79

Linked Comprehension Type Matrix ateh Tipe Numerical Value 7vpe

1.80 Archives

1.82

Answers Key 2.

2.1 2.44

2.1 2.1

Limit of a Function Number Neighbourhood (NBD) of Real Value of a Function Existence of Limit of a Function at

x =a

Concept Application Exercise 2.1 Rules to Find Limit Algebra of Limits

Finding Limit Using Important Concepts

Direct Substitution

Concept Application

Exercise 3.4

Differentiation Using Logarithm

Concept Application

Exercise 3.5

Indeterminate Forms

Using

Rationalization Method

2.5 2.5 2.6

Concept Application Exercise Solved Examples

2.8 2.8 2.9

Standard Formula for Limits

2.13 2.15

2.3 Concept Application Exercise

Limits Using Expansion Concept Application Exercise 2.4 Trigonometric Limits Application Exercise 2.5

Concept

the form l

Concept Application

Indeterminate Forms 0 and o Concept Application Exercise 2.8 Soved Examples

Linked Comprehension Type Matrix Match Type Numerical Value Type

Archives Answers Key

as

Concept Application

Slope

Tangent

to

a

Derivative of Composite Function (Chain Product Rule for Differentiation Quotient Rule for Differentiation

Concept Application

Exercise 3.2

Differentiation of Implicit Functions

4.14.45

4.1 4.1

Continuity

Obtained by Algebraic Given Functions

2.35 2.35 2.38 2.40 2.41 2.42

Operations

on

4.3

4.5 4.5

Twvo

Concept Application Exercise 4. 1 of Few Specific Functions

Continuity

4.8 4.9 4.10

Continuity of Function Involving Greatest Integer

Function Function Continuity of Function Involving Signum

Continuity of Function Involving Limit at Infinity Continuity of Function Differently Defined for

4.10 4.12

4.13

Rational and Irrational Values

4.14

Continuity of Composite Function Concept Application Exercise 4.2

4.15 4.15

2.43

Intermediate Value Theorem (IVT)

4.16

2.44

Concept Application Exercise 4.3

4.17

Differentiability: Definition Differentiability and Continuity Reasons for Non-Differentiability Differentiability in an Interval

4t.17

3.1

Rule)

3.5 3.5 3.6 3.9 3.9

Rules of Differentiation

3.38

Answers Key

.26 2.27

Curve

Exercise 3.1

3.36

Archives

2. 25

3.1

Differentiation: Definition

Matrix Match Type Numerical Value Type

Discontinuity and its Types Directional Continuity Continuity in Interval Continuity and Discontinuity of Functions

3.1-3.38

Differentiation

Single Correct Answer Type Multiple Correct Answers Type Linked Comprehension Type

Continuity of Function

2.30

Exercises ingle Correct Answer Type Multiple Correct Answer's Type

3.29 .29 .33 .34 3.35 3.36

Exercises

2.17 2.17 2.22

2.27 2.29 2.30

L'Hospital's Rule

3.9

Continuity and Differentiability

2.25

Limit of indeterminate Form 1

Higher Order Derivatives Application Exercise 3.8

2.15

2.23

Exponential and Logarithmic Limits Concept Application Exercise 2.66

of a

Differentiation of Determinant

Concept

2.10

of

Exercise 3.6

Differentiation of Functional Relations

Finding Limits at Infinity

Easier Method to Find Limit Exercise 2.7

Concept Application

2.5

.8

Concept Application Exercise 2.22 Limit of Indeterminate Forms Limits

3.16 3.17 3.17 3.18 3.18 3.20 3.21 3.23 3.24

Function

2.2 2.4

2.7

Sandwich Rule

Differentiation

3.13 .13 3.14 .14 3.16

Concept Application Exercise 3.7

Limiting

3.

Exercise 3.3

Differentiation of One Function w.r.t. Another

Limits

Finding

Concept Application

Differentiation of Functions In Parametric Form

3.11 3.11

Concept Application

Exercise 4.4

4.18 4.18

4.18 4.20

Examining Differentiability Using Differentiation and Graph of the Function

Differentiability of Functions Obtained by Algebraic Operations on Two Given Functions Continuity of Derivative

4.20 4.25 4.26

Contentsv 6.4

Concept Application Evercise 4.5

4.27

Sohved Examples

4.28

Exercises

4.33 4.33

Concept Application Exercise 6.1

6.7

4.37

Applications of Monotonicity

6.8

Single Correct Answer Type Multiple Correct Answers Type Linked Comprehension 71pe Matrix Match Type Numerical Value 7ype

Nature of the Composite Functions Monotonic Function Non-Monotonic Function

Finding Range of Roots of the Equation Using Monotonicity Proving Inequalities Using Monotonicity

4.41 4.42 4.43

Answers Key

6.5

of the Function and Number

4.40

Archives

6.5

4.45

6.8 6.9

Concept Application Exercise 6.2

6.11

Concavity of Curve and Point of Inflection

6.11

6.12 6.13

Incqualities Using Concavity 5.

Application of Derivatives Tangent and Normal

Tangent and Normal at a Point on the Curve Tangent and Normal from External Point Condition for which Given Line is Tangent or Normal to Given Curve Concept Application Exercise 5.1 Angle between Curves

Orthogonal Intersection of Curves Concept Application Exercise 5.2

Tangent, Normal, Subtangent and Subnormal Concept Application Exercise 5.3

6.14 6.15

Maxima and Minima of Function

Identifying Maxima and Minima by Checking

5.4

Value of Function in the Neighbourhood of Point of Extrema

5.5

6.16

Second Derivative Test for Maxima and Minima of

5.6

5.6 5.7 5.8

Differentiable Function First Derivative Test for Maxima/Minima of

6.18

Differentiable Function

6.19

Tests for Maxima and Minima of Discontinuous and

5.8 5.9

Non-Differentiable Functions Global (Absolute) Maximum and Minimum Concept Application Exercise 6.4

5.9

6.20

6.23 6.24

Applications of Tangent and Normal Concept Application Exercise 5.4

5.12

Derivative as Rate Measure Concept Application Exercise 5.5

5.12 5.15

Approximation Using Derivative Concept Application Exercise 5.6

5.16

Applications of Extrema in Optimization.

5.17

Plane Geometry and Coordinate Geometry

6.30

Rolle's Theorem

5.17 5.17

Concept Application Exercise 6.6

6.33

Applications of Extrema in Solid Geometry Concept Application Exercise 6.7

6.33

Solved Examples

6.35

Exercises

6.43 6.43 6.48

Graphical Interpretation of Rolle"'s Theorem Application of Rolle's Theorem and Selection of

Concept Application

Applications of Extrema in Curve Tracing and Analysing Roots of Equation Concept Application Exercise 6.5

5.18

Function Exercise 5.7

Mean Value Theorems Lagrange's Mean Value Theorem (LMVT) Cauchy's Mean Value Theorem Concept Application Exercise 5.8

5.19 5.19

Single Correct Answer Type Multiple Correct Answers Type

5.19 5.22

6.25 6.29

6.35

Linked Comprehension Type Matrix Match Type

6.50

5.23

Solved Examples

5.23

Numerical Value Type

6.54

Exercises Single Correct Answer Type

5.28

Multiple Correct Answers Type

Linked Comprehension Type Matrix Match Type Numerical Value Type

6.

Point of Inflection Concept Application Exercise 6.3

5.1-5.36 5.1 5.1

28

5.31 5.32 5.33 5.34

Archives

5.34

Answers Key

5.35

7.

6.53

Archives

6.55

Answers Key

6.58

Indefinite Integration Integration as Reverse Process of Ditferentiation

Integration of Commonly Used Functions Basic Properties of Indefinite Integration

7.1-7.46 7.1 7.1 7.1

Concept Application Exercise 7.1

7.2

Monotonicity and Maxima-Minima of

Integration of Function Aax + b)

7.3

6.1-6.59 Functions 6.1 Introduction 6.1 Classification of Functions Based on Monotonicity

Concept Application Evercise 7.2 Integration Using Substitution

7.4

Increasing Function Strictly Increasing Function Decreasing Function

6.1 6.1 6.1

Strictly Decreasing Function

6.2

Integration of tan, cot, sec and cosec Functions Some Standard Trigonometric Substitutions

Concept Aplication Exercise 7.3 Integration of Functionf(glr))g' («)

Concept Application Exercise 7.4

4 7.6 8

7.9 7.10 7.13

vi Contents Integration of Function

which is Reciprocal

Concept Application

of Quadratic Integration of Function in which Linear Function is Divided by Quadratic Integration of Expressions Involving Biquadratic

Concept Application 7.15

Concept Application Exercise 7.6

Integration of Functions of Some Standard Forms

Concept Application Exercise 7.7 Integration by Partial Fractions Concept 4pplication Exercise 7.8

Matrix Match Type Numerical Value Type

Linked Comprehension Type

7.19 7.22

7.22 7.25

7.31

Exercises

7.36

. .28 .28 .29

30

36 7.42 7.43

Single Correct Anser Type Multiple Correct Answers Type Linked Comprehension Type Matrix Match Type Numerical Value Type

7.44 7.45

8.54

Answers Key

8.58

Area

9.1-9.29 9.1

Area Bounded by Curve and Axis

Area Bounded by Curve and Axis when Graph of 9.5 9.5 9.6

Function Intersects x-axis

Area Using Integration along y-Axis

Concept Application Exercise 9.1

9. 6

Area Bounded By Two Curves Area Bounded by Curves in Given Interval Area Bounded by Two Curves when Curves Intersect at Two Points Area Bounded by Two Curves when Curves

9.6 9.7

9.10

Intersect at more than Two Points

9.12

9.12 9.12

8.1

Area Bounded By Miscellaneous Curves Concept of Variable Area Area Bounded by Inequalities Area Bounded by Inverse of Given Function without Getting Inverse Function

8.2

Concept Application Exercise 9.3

9.16

Solved Examples

9.16

Exercises Single Correct Answer Type

9.21

7.45

Answers Key

7.46

8.1-8.59

Definite Integration as Limit of Sum Definite Integration as Area Function Exercise 8.1

Concept Application

8.4

Theorems of Definite Integration

8.4

Definite Integration by Parts

8.6

Sum of Infinite Series Using Integration Concept Application Exercise 8.2

9.

Archives

Concept Application Exercise 9.2

Archives

Definite Integration

8.41 8.41 8.47 8.49 8.51 8.53

Exercises

7.19 7.19

Solved Examples

8.8 8.8

Inequalities in Definite Integration Concept Application Exercise 8.3

8.9 8.11

Elementary Properties of Definite Integration Concept Application Exercise 8.4

8.11 8.15

9.13 9.15

9.21

Multiple Correct Answers Type

9.23 9.24

Linked Comprehension Type Matrix Match Type

9.26 9.26

Numerical Value Type Archives

9.27 9.29

Answers Key

Replacing Dummy Variable with Sum of the Limits Minus Dummy Variable Concept Application Exercise 8.5

8.15 8.19

Halving The Upper Limit

8.19

Important Result

8.21 8.22

Exercise 8.6

Definite Integration of Odd and Even Functions Exercise 8.7

Definite Integration of Periodic Functions

8.22 8.24 8.25

Concept Application Exercise 8.8

8.27

Concept Application

Leibniz's Rule (Differentiation Under Integral Leibniz's Rule when Integrand is flx, 1)

Solved Examples

7.17

7.25

8.33 8.34 8.35

Exercise 8.11

Single Correct Answer Type Multiple Correct Answers Type

Integration by Parts Integration of e" sin bx and e cos bx Integration of Square Root of Quadratic Integration by Cancellation Concept Application Exercise 7.9

Concept Application

Concept Application

7.17

Integration of Function which is Reciprocal of Square Root of Quadratic Integration of Function in which Linear Function is Divided by Square Root of Quadratic

8.30 8.32

Exercise 8. 10

Definite Integration by Reduction Formula

7.15

Function

Concept Application Exercise 7.5

8.

7.14

8.30

Exercise 8.9

Definite Integration by Typical Substitution

Sign)

8.27 8.29

10. Differential Equations

10.1-10.38

Differential Equation Order and Degree of Differential Equation Concept Application Exercise 10.1 Formation of Differential Equation

Concept Application Exercise 10.2

10.1 10.1 10.2 10.2

104

Solving Differential Equation of Variable

Separable Type Differential Equations Reducible to the Variable

Separable Type

Concept Application Exercise 10.3

10.4 10.6

10.6

Contents v

Homogencous Ditlerential Equation

Concept Application Evercise 10.4

10.6 10.9

Solving Differential Equation Using Gieneral Form of Vaniable

Separation

10.10 10.11

Linear Difterential Equations

10.11 10.13

Concept Application Exenise 10.5 Conept Application Exercise 10.6 Diterential Equations Reducible To Lincar Ditterential Equations

Conept Application Exercise 10.7 Application of Differential Equation in Curve

Isogonal Trajectory

10.13 10.14 10.1 .17 10.1

Concept Appication Exercise 10.8 Miscellaneous Applications of Differential Equation 10.18

Matrix Match Type

Numerical Valhue Type Archives

10.36

Answers Key

10.3

Solutions Chapter 1

S.I S.38

Chapter 3 Chapter 4

S.63 S.87

Chapter 5

S.117

Chapter 6

140

Chapter 7

S.181 S.208 S.257

10.20

Chapter8

Sohed Examples

10.20

Exercises

10.27 10.27 10.32

Chapter 9 Chapter 10

10.33

S.1-5.319

Chapter 2

Concept 4pplication Exercise 10.9

Single Correct Answer Type Multiple Correct Answers Type Linked Comprehension Type

10.3 35

S.289

Appendix 1:

Chapterwise Solved January 2019 JEE Main

Questions (AlI Sets)

A.1-A.6