Note Taking Guide for Stewart/Redlin/Watson Precalculus: Mathematics for Calculus, 7th Edition

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Name ___________________________________________________________ Date ____________

Chapter 1 1.1

Fundamentals

Real Numbers

I. Real Numbers Give examples or descriptions of the types of numbers that make up the real number system. Natural numbers:

aa aa aa aa a a a a a a a aaa aaa a aa aa aa aa aa aa a a a

.

aaaaaaaaaaa aa aaaaaa aaa aaaaaa aa aaaaaaaa

.

Integers: Rational numbers: Irrational numbers:

.

aaaa aaaaaaa aaaa aaaaaa aa aaaaaaaaa aa a aaaaa aa aaaaaaaa

The set of all real numbers is usually denoted by the symbol

.

.

The corresponding decimal representation of a rational number is

aaaaaaaaa

corresponding decimal representation of an irrational number is

. The

aaaaaaaaaaaa

.

II. Properties of Real Numbers Let a, b, and c be any real numbers. Use a, b, and c to write an example of each of the following properties of real numbers. Commutative Property of Addition:

aaaaaaa

Commutative Property of Multiplication:

aa a aa

Associative Property of Addition:

.

aaaaa a aaaaa aaa a aa a aa a aa aa a aaa a aa a aa

Example 1:

.

aa a aa a a a a a aa a aa

Associative Property of Multiplication: Distributive Properties:

.

.

. .

Use the properties of real numbers to write 4(q + r ) without parentheses.

III. Addition and Subtraction The additive identity is number a has a

a

because, for any real number a,

aaaaaaaa

, −a, that satisfies a  (a) 

To subtract one number from another, simply

Note Taking Guide for Stewart/Redlin/Wats on

Precalculus:

Copyright © C engage Lear ning. All rights res er ved.

aaaaa a

. Every real

.

aaa aaa aaaaaaaa aa aaaa aaaaaa

.

th

Mathematics for Calculus, 7 Edition

1

2

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CHAPTER 1

Fundamentals

Complete the following Properties of Negatives. 1.

(1)a 

aa

2.

(a) 

a

3.

(a)b 

aaaaa a aaaaa

4.

(a)(b) 

5.

(a  b) 

aa a a

.

6.

(a  b) 

aaa

.

. . .

aa

.

Use the properties of real numbers to write 3(2a  5b) without parentheses.

Example 2:

IV. Multiplication and Division The multiplicative identity is

a

nonzero real number a has an

because, for any real number a, , 1/a, that satisfies a  (1/ a) 

aaaaaaa

To divide by a number, simply

aaaaa

aaaaaaaa aa aaa aaaaaaa aa aaaa aaaaaa

a

. Every . .

Complete the following Properties of Fractions. 1.

a c   b d

2.

a c   b d

aa a aa a aa a aa

3.

a b   c c

aa a aa a a

4.

a c   b d

aaa a aaa a aa

5.

ac  bc

6.

Example 3:

If

aaaa a aaaa

aaa

a c  , then b d

Evaluate:

.

.

.

.

.

aa a aa

.

4 19  9 30

aaaaa

Note Taking Guide for Stewart/Redlin/Wats on

Precalculus: Mathematics for Calculus, 7th Edition Copyright © Cengage Learning. All rights reserved.

SECTION 1.1

|

3

Real Numbers

V. The Real Line On the real number line shown below, the point corresponding to the real number 0 is called the aaaaa

. Given any convenient unit of measurement, each positive number x is represented by . Each negative number −x

aaa aa aaa aaaa a aaaaaaaa aa a aaaaa aa aaa aaaaa aa aaa aaaaaa is represented by

aaa aaaaa a aaaaa aa aaa aaaa aa aaa aaaaaa

.

0

The real numbers are ordered, meaning that a is less than b, written . The symbol a  b is read as

aaaaaa

aaa

a a aaaaaa

, if

a aa aaaa aaaa aa aaaaa aa a

.

VI. Sets and Intervals A set is a

aaaaaaaaaa aaaaaaa

the set. The symbol  means aa aaaaaaa aa

, and these objects are called the

aaaaaaaa

, and the symbol  means

aa aa aaaaaaa aa

of aa aaa

.

Name two ways that can be used to describe a set. aaaa aaaa aaa aa aaaaaaaaa aa aaaaaaa aaaaa aaaaaaaa aaaaaa aaaaaaa aaaaaaa aaa aa aaaaaaaa a aaa aa aa aaa aaaaaaaaaaa aaaaaaaaa

The union of two sets S and T is the set S  T that consists of

. The intersection of S and T is the set S  T that consists of

a aa aa aaaa

. The symbol ∅ represents

aaaaaaaa aaaa aaa aa aaaa a aaa a aaaa aaaa aaa aaa aaa aaaa aaaaaaaa aa aaaaaaa Example 4:

aaa aaaaaaaa aaaa aaa aa a aa aaa aaa aaaaa

.

If A = {2, 4,6,8,10} , B  {4, 8,12,16}, and C = {3,5,7} , find the sets (a) A  B (b) A  B (c) B  C aaa aaa aa aa aa aaa aaa aaa aaa aaa aa aaa a

If a < b , then the open interval from a to b consists of is denoted

aaa aa

is denoted

aaa aa

aaa aaaaaaa aaaaaaa a aaa a

. The closed interval from a to b includes

aaaaaaaaaa

.

Note Taking Guide for Stewart/Redlin/Wats on Precalculus: Copyright © Cengage Learning. All rights reserved.

Mathematics for Calculus, 7th Edition

and and

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CHAPTER 1

Fundamentals

VII. Absolute Value and Distance The absolute value of a number a, denoted by aaaa

aaa

. Distance is always

, is

aaa aaaaa aa a aa aaa aaaa aaaaaa

aaaaaaaa aa aaaa

, so we have a  0 for

every number a. If a is a real number, then the absolute value of a is

 a   Example 5:

Evaluate. (a) 12  8 (b) 9  15 (c)

77

aaa a aaa a aaa a

Complete the following descriptions of properties of absolute value. 1.

The absolute value of a number is always

aaaaaaaa aa aaaa

.

2.

A number and its negative have the same

aaaaaaaa aaaaa

.

3.

The absolute value of a product is

aaa aaaaaaa aa aaa aaaaaaaa aaaaaa

4.

The absolute value of a quotient is

aaa aaaaaaaa aa aaa aaaaaaaa aaaaaa

. .

If a and b are real numbers, then the distance between the points a and b on the real line is

d (a, b)  Example 6:

a aa a a

.

Find the distance between the numbers −16 and 7. aa

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Precalculus: Mathematics for Calculus, 7th Edition Copyright © Cengage Learning. All rights reserved.

SECTION 1.2

|

5

Exponents and Radicals

Name ___________________________________________________________ Date ____________

1.2

Exponents and Radicals

I. Integer Exponents If a is any real number and n is a positive integer, then the

is a n  a  a 

aaa aaaaa aa a

a .

n factors

The number a is called the

aaaa

, and n is called the

If a  0 is any real number and n is a positive integer, then a 0  Example 1:

a

Evaluate.

1 (b)   9

(a) (2)5

0

(c) 42

aaaa aaaaaaaa aaaaaa aaaa

II. Rules for Working with Exponents Complete the following Laws of Exponents. 1. 2.

am an = am a

n

aaaa

.

aa aa

=

.

3.

(a m ) n =

a

4.

(ab)n =

a a

5.

a b   

6.

a b  

aa

.

a a

.

n

7.

an bm

Example 2:

aa a aa

.

n



a a a a aa



.

aa a aa

.

Evaluate. 6 8 (a) y y aa

5 3 (b) ( w ) aa

b (c)   2

3

a

aaa a aaaaa a aaaaa aaa

Note Taking Guide for Stewart/Redlin/Wats on Precalculus: Copyright © Cengage Learning. All rights reserved.

Mathematics for Calculus, 7th

aaaaaaaa and a  n 

.

aaa

a

.

6

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CHAPTER 1

Fundamentals

III. Scientific Notation Scientists use exponential notation as a compact way of writing aaaaaaa

aaaa aaaaa aaaaaaa aaa aaaa aaaaa

.

A positive number x is said to be written in

if it is expressed as x  a 10n ,

aaaaaaaaaa aaaaaaaa

where 1  a  10 and n is an integer. Example 3:

Write each number in scientific notation. (a) 1,750,000 (b) 0.0000000429 a

a

aaa aaaa a aa aaaa aaaa a aaa

IV. Radicals The symbol

means

aaaa aaaaaaaa aaaaaa aaaa aaa

.

If n is any positive integer, then the principal nth root of a is defined as follows: n

a =b

aa a a

means

If n is even, we must have

.

a a a aaa a a a

.

Complete the Properties of nth Roots. 1.

n

ab =

2.

n

a = b

3.

mn

4. 5.

Example 4:

a

. .

a =

n

an =

n

an =

. a

aa a aa aaa

.

a a a aa a aa aaaa

Evaluate:

4

.

64w5 y8

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Precalculus: Mathematics for Calculus, 7th Edition Copyright © Cengage Learning. All rights reserved.

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SECTION 1.2

Exponents and Radicals

7

V. Rational Exponents For any rational exponent m/n in lowest terms, where m and n are integers and n > 0, we define

am/ n =

or equivalently

If n is even, then we require that Example 5:

aaa

am/ n =

n

am

.

Evaluate. a) b7/8b9/8 b)

4

x 2 ( x5 ) 2

aa aa aa aa

VI. Rationalizing the Denominator; Standard Form Rationalizing the denominator is the procedure in which

a aaaaaaa aa a aaaaaaaaa aa aaaaaaaaaa aa

aaaaaaaaaaa aaaa aaaaaaaaa aaa aaaaaaaaaaa aa aa aaaaaaaaaaa aaaaaaaaaa

.

Describe a strategy for rationalizing a denominator.

A fractional expression whose denominator contains no radicals is said to be in Example 6:

Rationalize the denominator:

aaaaaaaa aaaa

x 3y

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Mathematics for Calculus, 7th Edition

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CHAPTER 1

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Fundamentals

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SECTION 1.3

|

9

Algebraic Expressions

Name ___________________________________________________________ Date ____________

1.3

Algebraic Expressions

A variable is

a aaaaaa aaaa aaa aaaaaaaa aaa aaaaaa aaaa a aaaaa aaa aa aaaaaaa

An algebraic expression is

.

aaa aaaaaaaaa aa aaaaaaaa aaaa aa aa aa aaa aa aaa aaaa aaaa

aaaaaaa aaaaa aaaaaaaaa aaaaaaaaaaaa aaaaaaaaaaaaaaa aaaaaaaaa aaaa a aaaaa aa aaaaaaaaaa aa aaa aaaa aaa a aaaaa a aa a aaaa aaaaaa aaa a aa

A monomial is a aaaaaaaaaaa aaaaaaa

.

A binomial is

a aaa aa aaa aaaaaaaaa

A trinomial is

a aaa aa aaaaa aaaaaaaaa

. .

an xn  an1 xn1 

A polynomial in the variable x is an expression of the form where a0 , a1 ,

.

 a1x  a 0

,

, an are real numbers, and n is a nonnegative integer. If an  0 , then the polynomial has

degree

. The monomials ak x k that make up the polynomial are called the

a aaaaa

of the polynomial.

The degree of a polynomial is aaa aaaaaaaaaa

aaa aaaaaaa aaaaa aa aaa aaaaaaaa aaaa aaaaaaa aa .

I. Adding and Subtracting Polynomials We add and subtract polynomials by aa aaaaaaa aaa

aaaaa aaa aaaaaaaaaa aa aaaa aaaaaaa aaaa aaaa aaaaaaaaa . The idea is to combine

with the same variables raised to the same powers, using the

aaaa aaaaa

, which are terms

aaaaaaaaaa aaaaaaaa

When subtracting polynomials, remember that if a minus sign precedes an expression in parentheses, then

. aaa

aaaa aa aaaaa aaaa aaaaaa aaa aaaaaaaaaaa aa aaaaaaa aaaa aa aaaaaa aaa aaaaaaaaaaa

II. Multiplying Algebraic Expressions Explain how to find the product of polynomials or other algebraic expressions. aaa aaa aaaaaaaaaaaa aaaaaaaa aaaaaaaaaaa aa aaaaaaaaaaa aaaaa aa aaaaa aaaaa aa aaa aaaaaaa aa aaa aaaaaaaaaa aa aaa a aaaa aaaa aaaa aa aaaaaaaa aaa aaa aaaaaaa aa aaaaaaaaaaa aaaa aaaa aa aaa aaaaaa aa aaaa aaaa aa aaa aaaaa aaaaaa aaa aaaaaa aaaaa aaaaaaaaa

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Mathematics for Calculus, 7th Edition

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CHAPTER 1

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Fundamentals

Explain the acronym FOIL. aaa aaaaaaa aaaa aaaaa aa aaaaaaaa aaaa aaa aaaaaaa aa aaa aaaaaaaaa aa aaa aaa aa aaa aaaaaaaa aa aaa aaaaa aaaaaa aaa aaaaa aaaaaa aaa aaaaa aaaaaa aaa aaa aaaa aaaaaaa

Example 1:

Multiply: ( x  5)(3x  7)

III. Special Product Formulas Complete the following Special Product Formulas. Sum and Difference of Same Terms (A + B)(A  B) =

a

a aa

a

a

Square of a Sum and Difference (A + B)2 = (A  B) = 2

aa a aaa a aa a

a a aaa a a

a

a

a

Cube of a Sum and Difference (A + B)3 =

aa a aaaa a aaaa a aa

a

(A  B)3 =

aa a aaaa a aaaa a aa

a

The key idea in using these formulas is the Principle of Substitution, which says that aaa aaaaaaaaa aaaaaaaaaa aaa aaa aaaaaa aa a aaaaaaa

Example 2:

aa aaa aaaaaaaaaa .

2 Find the product: (2 y  5) .

IV. Factoring Common Factors Factoring an expression means

aaaaa aaa aaaaaaaaaaaa aaaaaaaa aa aaaaaaa aaa aaaaaaa aa aaaaaaaaa

aaaaaaaaa aaaaaaaaaaa aaaa a aaaaaaa aa aaaaaaa aaaaaaaaaaa

Note Taking Guide for Stewart/Redlin/Wats on

.

Precalculus: Mathematics for Calculus, 7th Edition Copyright © Cengage Learning. All rights reserved.

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SECTION 1.3

Example 3:

Algebraic Expressions

Factor: 14 x3  2 x2

V. Factoring Trinomials To factor a trinomial of the form x2  bx  c , we note that ( x  r )( x  s)  x 2  (r  s) x  rs so we need to choose numbers r and s so that

a a a a a aaa aa a a

.

To factor a trinomial of the form ax2 + bx + c with a  1 , we look for factors of the form px + r and qx + s :

ax2  bx  c  ( px  r )(qx  s)  pqx2  ( ps  qr ) x  rs . Therefore, we try to find numbers p, q, r, and s such that aa a aa Example 4:

aa a aa

aa a aa a a

.

Factor: 6 x2  7 x  3

VI. Special Factoring Formulas Complete the following Special Factoring Formulas. Difference of Squares A B = 2

2

aa a aa aa a aa

a

Perfect Squares A2 + 2AB + B2 =

aa a aaa

a

A2  2AB + B2 =

aa a aaa

a

Difference and Sum of Cubes A B = 3

3

3

3

A +B =

a

a

a

a

a

a

aa a aaaa a aa a a a aa a aaaa a aa a a a

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Mathematics for Calculus, 7th Edition

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CHAPTER 1

Example 5:

|

Fundamentals

Factor: 2 (a) 36  25x

2

2

(b) 49x + 28xy + 4y

aaa aa a aaaaa a aaa

aaa aaa a aaaa

Describe how to recognize a perfect square trinomial. a

a

a

a

a aaaaaaaaa aa a aaaaaaa aaaaaa aa aa aa aa aaa aaaa a a aaa a a aa a a aaa a a a aa aa aaaaaaaaa a aaaaaaa aaaaaa aa aaa aaaaaa aaaa aaaa aa aaaaa aa aaaa aa aaaaa aaaaa aaa aaaaaaa aa aaa aaaaaa aaaaa aa aaa aaaaa aaa aaaaaa

VII. Factoring by Grouping Terms Polynomials with at least four terms can sometimes be factored by

aaaaaaaa aaaaa

.

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SECTION 1.4

|

Rational Expressions

13

Name ___________________________________________________________ Date ____________

1.4

Rational Expressions

A fractional expression is

a aaaaaaaa aa aaa aaaaaaaa aaaaaaaaaaa

A rational expression is

a aaaaaaaaaa aaaaaaaaaa aa aaaaa aaaa aaa aaaaaaaaa

aaa aaaaaaaaaaa aaa aaaaaaaaaaa

.

I. The Domain of an Algebraic Expression The domain of an algebraic expression is

aaa aaa aaa aaaaaaa aaaa aaa aaaaaaaa aa aaaaaaaaa

aa aaaa

Example 1:

.

Find the domain of the expression

2x 2

x + 6x + 5

.

aaa aaaaaa aa a

II. Simplifying Rational Expressions Explain how to simplify rational expressions. aa aaaaaaaa aaaaaaaa aaaaaaaaaaaa aa aaaaaa aaaa aaaaaaaaa aaa aaaaaaaaaaa aaa aaa aaa aaaaaaaaa aaaaaaaa aa aaaaaaaaaa a aaaaa aaaaaa aa aa aaaaaa aaaaaa aaaaaaa aaaa aaa aaaaaaaaa aaa aaaaaaaaaaaa a

Example 2:

Simplify:

2x + 2 2

x + 6x + 5

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Mathematics for Calculus, 7th Edition

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CHAPTER 1

Fundamentals

III. Multiplying and Dividing Rational Expressions To multiply rational expressions, use the following property of fractions:

A C   B D

aaaaa

This says that to multiply two fractions, we

aaaaaaaa aaaaa aaaaaaaaaa aaa aaaaaaaa aaaaa

aaaaaaaaaaaa

.

To divide rational expressions, use the following property of fractions:

A C   B D This says that to divide a fraction by another fraction, we aaaaaaaa

Example 3:

aaaaaa aaa aaaaaaa aaa

.

Perform the indicated operation and simplify:

x2  1 x2  2 x  1  . x3 x2  9

IV. Adding and Subtracting Rational Expressions To add or subtract rational expressions, we first find a common denominator and then use the following property of fractions:

A B + = C C It is best to use the least common denominator (LCD), which is found by

aaaaaaaaa aaaa

aaaaaaaaaaa aaa aaaaaa aaa aaaaaaa aa aaa aaaaaaaa aaaaaaaa aaaaa aaa aaaaaaa aaaaa aaaa aaaaaaa aa aaa aa aaa aaaaaaa

Example 4:

.

Perform the indicated operation and simplify:

x  1 x2  2x  1  x3 x2  9

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Precalculus: Mathematics for Calculus, 7th Edition Copyright © Cengage Learning. All rights reserved.

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SECTION 1.4

Rational Expressions

15

V. Compound Fractions A compound fraction is

a aaaaaaaa aa aaaaa aaa aaaaaaaaaa aaa aaaaaaaaaaaa aa aaaa

aaa aaaaaaaaaa aaaaaaaaaa aaaaaaaaaaa

.

Describe two different approaches to simplifying a compound fraction. aaa aaa aa aaaaaaaa a aaaaaaaa aaaaaaaa aa aa aaaaaaa aaa aaaaa aa aaa aaaaaaaaa aaaa a aaaaaa aaaaaaaa aaa aaaaaaa aaa aaaaa aa aaa aaaaaaaaaaa aaaa a aaaaaa aaaaaaaaa aaaa aaaaaa aaa aaaaaaaaa aaaaaaa aaa aaaaaaaa aa aa aaa aaaaaaaaaaaaa aaaaaa aaa aa aaaaaaaa a aaaaaaaa aaaaaaaa aa aa aaaa aaa aaa aa aaa aaa aaaaaaaaa aa aaa aaaaaaaaaaa aaaa aaaaaaaa aaaa aaa aaaaaaaaa aaa aaa aaaaaaaaaaa aa aaaa aaaa aaaaaaaa aaa aaaaaaa

VI. Rationalizing the Denominator or the Numerator If a fraction has a denominator of the form A  B C , describe how to rationalize the denominator. aa aaaaaaaaaaa aaa aaaaaaaaaaaa aaaaaaaa aaa aaaaaaaaa aaa aaaaaaaaaaa aa aaa aaaaaaaa aa aaa aaaaaaaaa aaaaaaa a

If a fraction has the numerator

3 y  2 , how would you go about rationalizing the numerator?

aa aaaaaaaaaaa aaa aaaaaaaaaa aaaaaaaa aaa aaaaaaaaa aaa aaaaaaaaaaa aa aaa aaaaaaaa aa aaa aaaaaaaaa aaaaaaa a

VII. Avoiding Common Errors Identify the error in the following solution and show the correct solution.

3 2 3 2 5 1     2 y 3y 2 y  3y 5y y aaa aaaaa aaaaaa aa aaa aaaaa aaaaa aa aaa aaaaaaaa aaaaaaaaaaaa aa aaaa aaaaa aaaa a aaaaaa aaaaaaaaaaaa aaa aaaaa aaaa aa aaa aaaaaaaa aaaaaaaaaaa aaaaa aaa aaaaaaaaaaaa aaaaa aaaaaa aaa aaaaaaa aaaaaaaa aa

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Mathematics for Calculus, 7th Edition

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CHAPTER 1

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Fundamentals

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Precalculus: Mathematics for Calculus, 7th Edition Copyright © Cengage Learning. All rights reserved.

|

SECTION 1.5

17

Equations

Name ___________________________________________________________ Date ____________

1.5

Equations

An equation is

a aaaaaaaaa aaaa aaa aaaaaaaaaaaa aaaaaaaaaa aaa aaaaa

The solutions or roots of an equation are

.

aaa aaaaaa aa aaa aaaaaaaaaa aaaaaaa

aaaa aaaa aaa aaaaaaaa aaaa

. The process of finding these solutions is called

aaaaaaa aaa aaaaaaaa

.

Two equations with exactly the same solutions are called

aaaaaaaaa aaaaaaaaa

.

Describe how to solve an equation. aa aaaaa aa aaaaaaaaa aa aaa aaa aaaaaaaaaa aa aaaaaaaa aa aaa aa aaaa a aaaaaaaa aaaaaaaaaa aaaaaaaa aa aaaaa aaa aaaaaaaa aaaaaa aaaaa aa aaa aaaa aa aaa aaaaaaa aaaaa

Give a description of each property of equality. 1.

A  B  AC  B C aaaaaa aaa aaaa aaaaaaaa aa aaaa aaaaa aa aa aaaaaaaa aaa aa aaaaaaaaaa aaaaaaaa

. .

2.

A  B  CA  CB (C  0) aaaaaaaaaaa aaaa aaaaa aa aa aaaaaaaa aa aaa aaaa aaaaaaa aaaaaa aaaaa aa aaaaaaaaaa aaaaaaaa

. .

I. Solving Linear Equations The simplest type of equation is a linear equation, or an

aaaaaaaaaa aaaaaaaa

aaaaaaaa aa aaaaa aaaa aaaa aa aaaaaa a aaaaaaaa aa a aaaaaaa aaaaaaaa

aa aaa aaaaaaaa

.

A linear equation in one variable is an equation equivalent to one of the form

aa a a a a

a and b are real numbers and x is the variable. Example 1:

, which is

Solve the equation 5x  8  2 x  7 . a a aa

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Mathematics for Calculus, 7th Edition

, where

18

CHAPTER 1

Example 2:

|

Fundamentals

 ab . Solve for the variable b in the equation A  h    2 

II. Solving Q uadratic Equations Quadratic equations are

aaaaaa

degree equations. aaa a aa a a a a

A quadratic equation is an equation of the form

, where a, b,

and c are real numbers with a ≠ 0. The Zero-Product Property says that

aa a a aa aaa aaaa aa a a a aa a a a

.

This means that if we can factor the left-hand side of a quadratic (or other) equation, then we can solve it by aaaaaaa aaaa aaaaaa aaaaa aa a aa aaaa

Example 3:

.

2 Find all real solutions of the equation x  7 x  44 .

a a aa aaa a a aa

2 The solutions of the simple quadratic equation x  c are

.

If a quadratic equation does not factor readily, then we can solve it using the technique of aaa aaaaaa

.

2 In this technique, to make x  bx a perfect square, add

aaaaaaa aaa aaaaaa aa aaa aaaaaaaaaaa 2

aa a

aaaaaaaaaa

. This gives the perfect square

Note Taking Guide for Stewart/Redlin/Wats on

b b  x 2  bx      x   2 2 

2

.

Precalculus: Mathematics for Calculus, 7th Edition Copyright © Cengage Learning. All rights reserved.

SECTION 1.5

|

Equations

19

State the Q uadratic Formula. aaa aaaaa aa aaa aaaaaaaaa aaaaaaaa a aaaaa a aaa a

Example 4:

Find all real solutions of the equation 2 x2  3x  6  0 .

2 The discriminant of the general quadratic equation ax  bx  c  0, (a  0), is

a a aa a aaa

.

1.

If D > 0, then

aaa aaaaaaaa aaa aaa aaaaa aaaa aaaaaaaaa

.

2.

If D = 0, then

aaa aaaaaaaa aaa aaaaaaaa aaaa aaaaaaaa

.

3.

If D < 0, then

aaa aaaaaaaa aaa aa aaaa aaaaaa

.

Example 5:

2 Use the discriminant to determine how many real solutions the equation 5x  16 x  4  0 has.

aaa aaaaaaaa aaaa aaaaaaaaa

III. O ther Types of Equations When you solve an equation that involves fractional expressions or radicals, you must be especially careful to

aaaaa aaaa aaaaa aaaaaaa

. When we solve an equation, we may end up with

one or more extraneous solutions, which are aaa aaaaaaaa aaaaaaaa

Note Taking Guide for Stewart/Redlin/Wats on Precalculus: Copyright © Cengage Learning. All rights reserved.

aaaaaaaaa aaaaaaaaa aaaa aa aaa aaaaaaa .

Mathematics for Calculus, 7th Edition

20

|

CHAPTER 1

Fundamentals aaa a aa a a a aa aaaaa a aa

An equation of quadratic type is an equation of the form aa aaaaaaaaa aaaaaaaaaa

Example 6:

.

Find all solutions of the equation x4  52 x2  576  0 . a a aaa aa

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SECTION 1.6

Complex Numbers

21

Name ___________________________________________________________ Date ____________

1.6

Complex Numbers

A complex number is an expression of the form 2

numbers and i =

aa

imaginary part is

a

a a aa

, where a and b are both real

. The real part of this complex number is

a

and the

. Two complex numbers are equal if and only if

aaa aaaa aaaaa

aaa aaaaa aaa aaaaa aaaaaaaaa aaaaa aaa aaaaa

.

A complex number which has real part 0 is called a(n)

aaaa aaaaaaaaa aaaaaa

.

I. Arithmetic O perations on Complex Numbers To add complex numbers, To subtract complex numbers,

aaa aaa aaaa aaaaa aaa aaa aaaaaaa aaaaa

.

aaaaaaaa aaa aaaa aaaaa aaa aaaaaaaaa aaaaa

.

aaaaaaaa aaaa aaaaaaaaa aaaaa aa a a a

To multiply complex numbers,

.

Complete each of the following definitions. Addition: (a + bi) + (c + di) =

aa a aa a aa a aaa

Subtraction: (a + bi) − (c + di) =

aa a aa a aa a aaa

Multiplication: (a + bi)  (c + di) = Example 1:

.

aaa a aaa a aaa a aaaa

.

Express the following in the form a + bi. (a) (b) (c)

1  3i    2  5i  1  3i    2  5i  1  3i  2  5i 

aaa a a aa

aaa a a a aa

aaa aaa a aaa

Division of complex numbers is much like aaaaaaaaaa

z

.

aaaaaaaaaa aaa aaaaaaaaaaa aa a aaaaaaa

. For the complex number z = a + bi we define its complex conjugate to be a a aa

. Note that z  z 

a  bi , c  di aaaaaaaaa aa aaa aaaaaaaaaaa To simplify the quotient

a

.

aaaaaaaa aaa aaaaaaaaa aaa aaa aaaaaaaaaaa aa aaa aaaaaaa

Note Taking Guide for Stewart/Redlin/Wats on Precalculus: Copyright © Cengage Learning. All rights reserved.

.

Mathematics for Calculus, 7th Edition

22

|

CHAPTER 1

Example 2:

Fundamentals

Express the following in the form a + bi:

6  5i 2i

II. Square Roots of Negative Numbers If −r is negative, then the principal square root of −r is roots of −r are

a

and

a a

. The two square

.

III. Complex Solutions of Q uadratic Equations For the quadratic equation ax2  bx  c  0 , where a  0 , we have already seen that if b2  4ac then the equation has no real solution. But in the complex number system this equation will always aaaaaaaaa Example 3:

a

0, aaaa

. Solve the equation: x2  6 x  10

If a quadratic equation with real coefficients has complex solutions, then these solutions are aaaaaaaaaa aa aaaa aaaaa

aaaaaaa

.

Homework Assignment Page(s) Exercises

Note Taking Guide for Stewart/Redlin/Wats on

Precalculus: Mathematics for Calculus, 7th Edition Copyright © Cengage Learning. All rights reserved.

SECTION 1.7

|

Modeling w ith Equations

23

Name ___________________________________________________________ Date ____________

1.7

Modeling with Equations

I. Making and Using Models List and explain the Guidelines for Modeling with Equations. 1.

aaaaaaaa aaa aaaaaaaaa aaaaaaaa aaa aaaaaaaa aaaa aaa aaaaaaa aaaa aaa aa aaaaa aaaa aaaaaaaa aaa aaaaaaa aa aaaaaaaaaa aa a aaaaaaa aaaaaaa aa aaa aaaaaaaa aaaa aa aaaaa aa aaa aaa aa aaa aaaaaaaa aaaa aaaaaaaaa aaaaaaaa aaa aaa aaaaaaaaaaaaa aa a aa aaaa aaaaa aaaaaaa

2.

aaaaaaaaa aaaa aaaaa aa aaaaaaaa aaaa aaaa aaaaaaaa aa aaa aaaaaaa aaaaaa aaa aaaaaaa aaa aaa aaaaaaaaaa aaaaaaaaa aa aaa aaaaaaa aa aaaaa aa aaa aaaaaaaa aaa aaaaaaa aa aaaa aa aa aaaaaaaa aaaa aaaaaaaaaaaa aa aa aaaaaaaaa aaaaaaa aa aaaa a aaaaaaa aa aaaa a aaaaaa

3.

aaa aa aaa aaaaaa aaaa aaa aaaaaaa aaaa aa aaa aaaaaaa aaaa aaaaa a aaaaaaaaaaaa aaaaaaa aaa aaaaaaaaaaa aaa aaaaaa aa aaaa aa aaa aa aa aaaaaaaa aaa aaaaaa aaaa aaaaaaaaa aaaa aaaaaaaaaaaaa

4.

aaaaa aaa aaaaaaaa aaa aaaaa aaaa aaaaaaa aaaaa aaa aaaaaaaaa aaaaa aaaa aaaaaaa aaa aaaaaaa aa aa a aaaaaaaa aaaa aaaaaaa aaa aaaaaaaa aaaaa aa aaa aaaaaaaa

II. Problems About Interest What is interest? aaaa aaa aaaaaa aaaaa aaaa a aaaa aa aaaa a aaaa aaaaaaaaa aaaa aaaaa aa aaaaaaa aa aaa aaaa aa a aaaaaaa aaaaaaaa aaa aaaaaaaa aa aaaa aaaa aaaa aaa aaa aaa aaaaaaaaa aa aaaaa aaa aaaaaa aaa aaa aaaa aa aaaa aa aaaaaa aaaaaaaaa

The most basic type of interest is

aaaaaa aaaaaa

aaa aaaaa aaaaaa aaaaaaaa aa aaaaaaaaa Note Taking Guide for Stewart/Redlin/Wats on Precalculus: Copyright © Cengage Learning. All rights reserved.

, which is just an annual percentage of .

Mathematics for Calculus, 7th Edition

24

CHAPTER 1

|

Fundamentals

The amount of a loan or deposit is called the use of this money is the

aaaaaaaaa a

aaaaaa aaaa a

aaaaaa aa aaaaa aaaa aaa aaaa aa aa aaaaaaa aaaaaaaa aaaaaa

. The annual percentage paid for the . The variable t stands for

aaa

, and the variable I stands for the

aaaaa

.

The simple interest formula gives the amount of interest I earned when a principal P is deposited for t years at an interest rate r and is given by to convert r from a percentage to Example 1:

a a aaa a aaaaaaa

. When using this formula, remember .

Consider the following situation: Oliver deposits $22,000 at a simple interest rate of 4.25%. How much interest will he earn after 8 years? In this situation, identify the value of each variable in the simple interest formula and indicate which variable is unknown. a a aaaaaaa a a aaaaaaa aaa a a aa a aa aaaaaaaa

III. Problems About Area or Length Give formulas for the (1) area A and (2) perimeter P of a rectangle having length l and width w. aa aaaaa

a a aaaaaaa aaaaaaaaaa

a a aa a aa

Give formulas for the (1) area A and (2) perimeter P of a square having sides of length s. aa aaaaa

a a aaaaaaa aaaaaaaaaa

a a aa

Give formulas for the (1) area A and (2) perimeter P of the given triangle.

a

aa aaaaa aaa aaaaaaaaaa

c

a a a aa

h

aa a aa aa b

Note Taking Guide for Stewart/Redlin/Wats on

Precalculus: Mathematics for Calculus, 7th Edition Copyright © Cengage Learning. All rights reserved.

SECTION 1.7

|

Modeling w ith Equations

25

IV. Problems About Mixtures Problems involving mixtures and concentrations make use of the fact that if an amount x of a substance is dissolved in a solution with volume V, then the concentration C of the substance is given by a a aaa

.

Solving a mixture problem usually requires analyzing aaaaaaaa

aaa aaaaaa a aa aaa aaaaaaa aaaa aa aa aaa

.

V. Problems About the Time Needed to Do a Job When solving a problem that involves determining how long it takes several workers to complete a job, use the fact that if a person or machine takes H time units to complete the task, then in one time unit the fraction of the task that has been completed is

aaa

.

If it takes Faith 80 minutes to complete a task, what fraction of the task does she complete in one hour? aaa

VI. Problems About Distance, Rate, and Time Give the formula that relates the distance traveled by an object traveling at either a constant or average speed in a given amount of time. aaaaaaaa a aaaa a aaaa

Give an example of an application problem that requires this formula. aaaaaaa aaaa aaaaa

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Mathematics for Calculus, 7th Edition

26

CHAPTER 1

|

Fundamentals

Additional notes

Homework Assignment Page(s) Exercises

Note Taking Guide for Stewart/Redlin/Wats on

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SECTION 1.8

|

Inequalities

27

Name ___________________________________________________________ Date ____________

1.8

Inequalities

An inequality looks

aaaa aaaa aa aaaaaaaaa aaaaaa aaaa aa aaa aaaaa aa aaa aaaaa aaaa aa aaa

aa aaa aaaaaaa aa aa aa aa a

.

To solve an inequality that contains a variable means

aa aaaa aaa aaaaa aa aaa

aaaaaaaa aaaa aaaa aaa aaaaaaaaaa aaaa generally has

. Unlike an equation, an inequality

aaaaaaaaaa aaaa

solutions, which form

aaaaa aa aaaaaaaaa aa aaa aaaa aaaa

aa aaaaaaaa aa a

.

Describe how to solve an inequality. aa aaaaa aaaaaaaaaaaaa aa aaa aaa aaaaa aaa aaa aaaaaaaa aa aaa aaaa aa aaa aaaaaaaaaa aaaaa

Give a description of each property of inequality. 1.

A  B  AC  B C aaaaaa aaa aaaa aaaaaaaa aa aaaa aaaa aa aa aaaaaaa aaaaa aa aaaaaaaaaa aaaaaaaaaa

. .

2.

A  B  AC  B C aaaaaaaaaaa aaa aaaa aaaaaaaa aaaa aaaa aaaa aa aa aaaaaa aaaaa aa aaaaaaaaaa aaaaaaaaaa

. .

3.

If C > 0, then A  B  CA  CB aaaaaaaaaaa aaaa aaaa aa aa aaaaaaaaaa aa aaa aaaa aaaaaaaaaaaaa aaaaa aa aaaaaaaaaa aaaaaaaaaa

. .

4.

If C < 0, then A  B  CA  CB aaaaaaaaaaa aaaa aaaa aa aa aaaaaaaaaa aa aaa aaaa aaaaaaaaaaaaaa aaaaaaaa aaa aaaaaaaaa aa aaa aaaaaaaaaa

5.

If A > 0 and B > 0, then A  B

. .



1 1  A B

aaaaaa aaaaaaaaaaa aa aaaa aaaa aa aa aaaaaaaaaa aaaaaaaaaaa aaaaaaaaaa aaaaaaaa aaa aaaaaaaaa aa aaaaaaaaaaa

Note Taking Guide for Stewart/Redlin/Wats on Precalculus: Copyright © Cengage Learning. All rights reserved.

. .

Mathematics for Calculus, 7th Edition

28

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CHAPTER 1

6.

Fundamentals

If A  B and C  D , then A  C  B  D aaaaaaaaaaa aaa aa aaaaa

. .

7.

If A  B and B  C , then A  C aaaaaaaaaaa

. .

I. Solving Linear Inequalities An inequality is linear if aaaaaaaa

aaaa aaaa aa aaaaaaaa aa a aaaaaa aa aaa . To solve a linear inequality,

aaa aaaaaaaa aa aaa aaaa aa aaa aaaaaaaaaa aaaa Example 1:

a aaaaaa

.

Solve the inequality 5x  9  2 x  9 . a a aa

II. Solving Nonlinear Inequalities If a product or a quotient has an even number of negative factors, then its value is

aaaaaaa

.

If a product or a quotient has an odd number of negative factors, then its value is

aaaaaaa

.

State the Guidelines for Solving Nonlinear Inequalities. 1.

aaaa aaa aaaaa aa aaa aaaaa aa aaaaaaaaaa aaaaaaa aaa aaaaaaaaaa aa aaaa aaa aaaaaaa aaaaa aaaaaa aa aaa aaaa aa aaa aaaaaaaaaa aaaaa aa aaa aaaaaaa aaaa aa aaa aaaaaaaaaa aaaaaaaa aaaaaaaaaa aaaaa aaaa aa a aaaaaa aaaaaaaaaaaa

2.

aaaaaaa aaaaaa aaa aaaaaaa aaaa aa aaa aaaaaaaaaaa

3.

aaaa aaa aaaaaaaaaa aaaaaaaaa aaa aaaaaa aaa aaaaa aaaa aaaaaa aa aaaaa aaaaa aaaaaaa aaaa aaaaaa aaa aaaa aaaa aaaa aaaaaaaaaa aaaa aaa aaaaaaaaa aaaa aaa aaaaaaaaaa aa aaaaa aaaaaaaa

Note Taking Guide for Stewart/Redlin/Wats on

Precalculus: Mathematics for Calculus, 7th Edition Copyright © Cengage Learning. All rights reserved.

SECTION 1.8

4.

|

Inequalities

29

aaaa a aaaaa aa aaaaaaaa aaa aaaa aaaaaa aa aaaa a aaaaa aa aaaaaaa aa aaa aaaaa aa aaaa aaaaaa aa aaaa aaaaaaaaa aa aaa aaaa aaa aa aaa aaaaa aaaaaaaaa aaa aaaa aa aaa aaaaaaa aaa aaaaaaaaa aa aaaaa aaaaaaaa

5.

aaaaaa aaa aaa aaaa aaaaa aa aaaa aaa aaaaaaaaa aa aaaaa aaa aaaaaaaaaa aa aaaaaaaaaa aaaaa aaaaaaa aaa aaaaaaaaa aa aaaaa aaaaaaaaa aaaaaaa aaa aaaaaaaaaaa aaaaa aaa aaaaaa aa aaa aaaaaaaaaa aaaaaaaa a aa aaa

Example 2:

Solve the inequality x2  7 x  44 . aaaa aaa

III. Absolute Value Inequalities For each absolute value inequality, write an equivalent form. 1.

x c 

aa a a a a

.

2.

x c 

aa a a a a

.

3.

x c 

a a a a aa a a a

.

4.

x c 

a a a a aa a a a

.

Example 3:

Solve the inequality 4 x  2  10 . aaaa aa

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Mathematics for Calculus, 7th Edition

30

CHAPTER 1

|

Fundamentals

IV. Modeling with Inequalities Give an example of a real-life problem that can be solved with inequalities. aaaaaaa aaaa aaaaa

Additional notes

Homework Assignment Page(s) Exercises Note Taking Guide for Stewart/Redlin/Wats on

Precalculus: Mathematics for Calculus, 7th Edition Copyright © Cengage Learning. All rights reserved.

|

SECTION 1.9

The Coordinate Plane; Graphs of Equations; Circles

31

Name ___________________________________________________________ Da te ____________

1.9

The Coordinate Plane; Graphs of Equations; Circles

I. The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of numbers to form the aaaaa

aaaaaaaa aaaaa

or

aaaaaaaaaa

.

Describe how the coordinate plane is constructed. Include a description of its major components: x-axis, y-axis, origin, and quadrants. aa aaaa aaa aaaaaaaaaa aaaaaa aaaa aaa aaaaaaaaaaaaa aaaa aaaaa aaaa aaaaaaaaa aa a aa aaaa aaaaa aaaaaaaa aaa aaaa aa aaaaaaaaaa aaaa aaaaaaaa aaaaaaaaa aa aaa aaaaa aaa aa aaaaaa aaa aaaaaaa aaa aaaaa aaaa aa aaaaaaaa aaaa aaaaaaaa aaaaaaaaa aaaaaa aaa aa aaaaaa aaa aaaaaaa aaa aaaaa aa aaaaaaaaaaaa aa aaa aaaaaa aaa aaa aaaaaa aa aaa aaaaaa aa aaa aaa aaa aaaa aaaaaa aaa aaaaa aaaa aaaa aaaaaaa aaaaaa aaaaaaaaaa

On the coordinate plane shown below, label the x-axis, the y-axis, the origin, and Quadrants I, II, III, and IV.

aaaaaa 5

aaaaaaaa aa

1 -5

-3

aaaaaaaa aaa

aaaaaaaa a

3

-1 -1

aaaaaa 1

-3

3

aaaaaa

5

aaaaaaaa aa

-5

Any point P in the coordinate plane can be located by a unique ordered pair of numbers (a, b), where the first number a is called the aaaaaaaaaaaa aa a

aaaaaaaaaaaa aa a

, and the second number b is called the

.

Note Taking Guide for Stewart/Redlin/Wats on Precalculus: Copyright © Cengage Learning. All rights reserved.

Mathematics for Calculus, 7th Edition

32

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CHAPTER 1

Fundamentals

II. The Distance and Midpoint Formulas The distance between the points A( x1 , y1 ) and B( x2 , y2 ) in the plane is given by

d ( A, B)  ( x2  x1 )2  ( y2  y1 )2 Example 1:

.

Find the distance between the points (−5, 6) and (3, −5).

The midpoint of the line segment from A( x1 , y1 ) to B( x2 , y2 ) is

 x1  x2 y1  y2   2 , 2    Example 2:

.

Find the midpoint of the line segment from (−5, 6) to (3, −5). aaaa aaaa

III. Graphs of Equations in Two Variables The graph of an equation in x and y is aaaa aaaaaaa aaa aaaaaaaa

aaa aaa aa aaa aaaaaa aaa aa aa aaa aaaaaaaaaa aaaaa .

Explain how to graph an equation. aaa aaaaa aa aa aaaaaaaa aa a aaaaaa aa aa aaaaa aa aaaaaaaaa aa aaaa aa aaaa aaaaaa aa aa aaaa aaaa aaaaaaa aaaa aa a aaaaaa aaaaaa

Note Taking Guide for Stewart/Redlin/Wats on

Precalculus: Mathematics for Calculus, 7th Edition Copyright © Cengage Learning. All rights reserved.

|

SECTION 1.9

Example 3:

The Coordinate Plane; Graphs of Equations; Circles

Sketch the graph of the equation

33

1 x+ y =3. 2

y 5

3

1 -5

-3

-1 -1

1

3

5

x

-3

-5

IV. Intercepts Define x-intercepts. aaa aaaaaaaaaaaa aa a aaaaa aaa aaa aaaaaaaaaaaaa aa aaaaaa aaaaa aaa aaaaa aa aa aaaaaaaaaaaaa aaa aaaaaaa

Explain how to find x-intercepts. aaa a a a aa aaa aaaaaaaa aa aaa aaaaa aaa aaaaa aaa aa

Define y-intercepts. aaa aaaaaaaaaaaa aa a aaaaa aaa aaa aaaaaaaaaaaaa aa aaaaaa aaaaa aaa aaaaa aa aa aaaaaaaaaaaaaa aaa aaaaaaa

Explain how to find y-intercepts. aaa a a a aa aaa aaaaaaaa aa aaa aaaaa aaa aaaaa aaa aa

Example 4:

Find the x- and y-intercepts of the graph of the equation y  4 x  36 . aaa aaaaaaaaaaa aa aa aaa aaaaaaaaaaa aa aaaa

Note Taking Guide for Stewart/Redlin/Wats on Precalculus: Copyright © Cengage Learning. All rights reserved.

Mathematics for Calculus, 7th Edition

34

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CHAPTER 1

Fundamentals

V. Circles ( x  h)2  ( y  k )2  r 2

An equation of the circle with center (h, k) and radius r is This is called the

aaaaaaaa aaaa

.

for the equation of the circle. If the center of the circle a

is the origin (0, 0), then the equation is

a

a

a aa aa

.

Graph the equation ( x  2)2  ( y  1)2  4 .

Example 5:

y 5

3

1 -5

-3

-1 -1

1

3

5

x

-3

-5

VI. Symmetry A graph is symmetric with respect to the y-axis if whenever the point (x, y) is on the graph, then so is aaa aa aa

. To test an equation for this type of symmetry, replace

a

by

. If the resulting equation is equivalent to the original one, then the graph is symmetric

with respect to the y-axis. A graph is symmetric with respect to the x-axis if, whenever the point (x, y) is on the graph, then so is aaaaa aa

. To test an equation for this type of symmetry, replace

a

by

. If the resulting equation is equivalent to the original one, then the graph is symmetric

with respect to the x-axis. A graph is symmetric with respect to the origin if, whenever the point (x, y) is on the graph, then so is aaa aaa aa

. To test an equation for this type of symmetry, replace and replace

a

by

aa

a

by

. If the resulting equation

is equivalent to the original one, then the graph is symmetric with respect to the origin.

Homework Assignment Page(s) Exercises

Note Taking Guide for Stewart/Redlin/Wats on

Precalculus: Mathematics for Calculus, 7th Edition Copyright © Cengage Learning. All rights reserved.

|

SECTION 1.10

35

Lines

Name ___________________________________________________________ Dat e ____________

1.10 Lines I. The Slope of a Line The “ steepness” of a line refers to how quickly it aaaaa

aaaaa aa aaa aa aa aaaa aaaa aaaa aa

. We define run to be

we define rise to be

aaa aaaaaaaa aa aaaa aa aa aaaaa

aaa aaaaaaaaaaaaa aaaaaaaa aaaa aaa aaaa aaaaa aaa a

slope of a line is

aaa aaaaa aa aaaa aa aaa

If a line lies in a coordinate plane, then the run is rise is

and . The

. aaa aaaaaa aa aaa aaaaaaaaaaaa

aaa aaaaaaaaaaaaa aaaaaa aa aaa aaaaaaaaaaaa

and the

between any two points on the line.

The slope m of a nonvertical line that passes through the points A( x1 , y1 ) and B( x2 , y2 ) is

m

rise y2  y1  run x2  x1

The slope of a vertical line is

.

aaa aaaaaaa

Lines with positive slope slant slant

. The slope of a horizontal line is

aaaaa aa aaa aaaaa

aaaaaaaa aa aaa aaaaa

. The steepest lines are those for which .

Find the slope of the line that passes through the points (1, 9) and (15, 2). a a aaaa

II. Point-Slope Form of the Equation of a Line The point-slope form of the equation of the line that passes through the point ( x1 , y1 ) and has slope m is

y  y1  m( x  x1 ) Example 2:

.

Find the equation of the line that passes through the points (1, 9) and (15, 2). a a aa a aa

Note Taking Guide for Stewart/Redlin/Wats on Precalculus: Copyright © Cengage Learning. All rights reserved.

.

. Lines with negative slope

aaaaaaaa aaaaa aa aaa aaaaa aa aaa aaaaaaa Example 1:

a

Mathematics for Calculus, 7th Edition

aaa

36

|

CHAPTER 1

Fundamentals

III. Slope -Intercept Form of the Equation of a Line The slope-intercept form of the equation of a line that has slope m and y-intercept b is a a aa a a

Example 3:

.

Find the equation of the line with slope −5 and y-intercept −3. a a aaa a a

IV. Vertical and Horizontal Lines The equation of the vertical line through (a, b) is line through (a, b) is

aaa

aaa

. The equation of the horizontal

.

V. General Equation of a Line The general equation of a line is given as

aa a aa a a a a

, where A and B are

. The graph of every linear equation Ax + By + C = 0 is a

aaaa a

Conversely, every line is the graph of

a aaaaaa aaaaaaaa

aaa

aaa

.

.

VI. Parallel and Perpendicular Lines Two nonvertical lines are parallel if and only if

aaaa aaaa aaa aaaa aaaaa

m1m2  1

Two lines with slopes m1 and m2 are perpendicular if and only if slopes are

, that is, their

1 . Also, a horizontal line, having slope 0, m1 aaaaaaaa aaaaa aaaaaa aa aaaaa .

aaaaaaaa aaaaaaaaaaa

is perpendicular to a Example 4:

.

: m2  

Find the equation of the line that is parallel to the line 2 x  y  4 and passes through the point (0, 5). a a aa a a

Homework Assignment Page(s) Exercises

Note Taking Guide for Stewart/Redlin/Wats on

Precalculus: Mathematics for Calculus, 7th Edition Copyright © Cengage Learning. All rights reserved.

SECTION 1.11

| Solv ing Equations and Inequalities Graphically

37

Name ___________________________________________________________ Date ____________

1.11 Solving Equations and Inequalities Graphically I. Solving Equations Graphically Describe how to solve an equation using the graphical method. aa aaaaa aa aaaaaaaa aaaaa aaa aaaaaaaaa aaaaaaa aaaa aaa aaaaa aa aaa aaaaaaaa aa aaa aaaa aaa aaa aaaaa aa aa aaaaa aaa aaaa aaa aaaaaa aa a aaa aaaaa a aa aaaaa aa aaaaa aaaa aaa aaaa aaa aaaaaaaaaaaa aa aaa aaaaaa

Describe advantages and disadvantages of solving equations by the algebraic method. aaa aaaaaaaaa aa aaa aaaaaaaaa aaaaaa aa aaaa aa aaaaa aaaaa aaaaaaaa aaaaa aaa aaaaaaa aa aaaaaaaaaa aaa aaaaaaaa aa aaaaaa aa aaa aaaaaa aaaaa aa aa aaaaaaaaaa aaa aaaaaaaaa aaaaaaaaa aa aaa aaaaaaaaa aa aaa aaaaa aaaaa aaa aaaa aaaaaaaaa aa aa aaaaaaaaa aa aaaaaaaaaa aa aaaaaaa aa

Describe advantages and disadvantages of solving equations by the graphical method. aaa aaaaaaaaa aaaaaa aaaaa aaaa a aaaaaaaaa aaaaaaaaaaaaa aa aaa aaaaaaa aaaaa aa aa aaaaaaaaa aaaa a aaaaaaaaa aaaaaa aa aaaaaaaa aaa a aaaaaaaaaaaa aa aa aaaaa aaaaaa aa aaaaaaaaa aaaaa aaaaaaaa aa aaaaaaaa aaaaa aa aa aaaaaaaaa aaa aaa aaaaaaaa aa aaaaaaa aa aaaaa aaaaaa aa aaa aaaaaaaaa

II. Solving Inequalities Graphically 2 Describe how to solve the inequality  x  4  0 graphically.

aaaaaa aaaa aaa aaaaa aa aaa aaaa aaaaa aaaaaa aa a aaa aaaaa a a aa aaaaa aaa aaaaaa aaa aaaaaaaa aaa aaaaa aaa aaaaa aaaa aa aa aaaaa aaa aaaaaaa

Note Taking Guide for Stewart/Redlin/Wats on Precalculus: Copyright © Cengage Learning. All rights reserved.

Mathematics for Calculus, 7th Edition

38

CHAPTER 1

|

Fundamentals

y

y

x

y

y

x

y

x

y

x

y

x

y

x

x

y

x

x

Homework Assignment Page(s) Exercises

Note Taking Guide for Stewart/Redlin/Wats on

Precalculus: Mathematics for Calculus, 7th Edition Copyright © Cengage Learning. All rights reserved.

SECTION 1.12

|

Modeling Variation

39

Name ___________________________________________________________ Date ____________

1.12 Modeling Variation I. Direct Variation If the quantities x and y are related by an equation

a a aa

k  0 , we say that y varies directly as x, or simply

Example 1:

a aa aaaaaaaa aaaaaaaaaa aa a

a aa aaaaaaaaaa aa a

aa aaaaaaaaaaaaaaa

for some constant

. The constant k is called the

, or aaaaaaa

.

Suppose that w is directly proportional to t. If w is 21 when t is 6, what is the value of w when t is 19? a a aaaa

II. Inverse Variation k for some constant k  0 , we say that y is x a aaaaaa aaaaaaaaa aa a .

If the quantities x and y are related by the equation y = aaaaaaaaaaaa aa a

Example 2:

or

aaaaaaaa

Suppose that w is inversely proportional to t. If w is 21 when t is 6, what is the value of w when t is 9? a a aa

Note Taking Guide for Stewart/Redlin/Wats on Precalculus: Copyright © Cengage Learning. All rights reserved.

Mathematics for Calculus, 7th Edition

40

CHAPTER 1

|

Fundamentals

III. Combining Different Types of Variation If the quantities x, y, and z are related by the equation z is

a a aaa

aaaaaaaaaa aa aaa aaaaaaa aa a aaa a

saying that z

aaaaaa aaaa

aaaaaaaaaaaa aa a aaa a

. We can also express this relationship by

as x and y, or that

a aa aaaaaaa

.

If the quantities x, y, and z are related by the equation z  k aaa aaaaaaaaa aaaaaaaaaaaa aa a

x , we say that z is y

or that

a aaa aaaaaaaaa aa a Example 3:

, then we say that

aaaaaaaaaaaa aa a a aaaaaa aaaaaaaa aa

.

Suppose that z is jointly proportional to x and y. If z is 45 when x  3 and y  5 , what is the value of z when x  6 and y 

1 ? 2

a a aa

y

y

x

y

x

x

Homework Assignment Page(s) Exercises

Note Taking Guide for Stewart/Redlin/Wats on

Precalculus: Mathematics for Calculus, 7th Edition Copyright © Cengage Learning. All rights reserved.

Name ___________________________________________________________ Date ____________

Chapter 2 2.1

Functions

Functions

I. Functions All Around Us Give a real-life example of a function. aaaaaaa aaaa aaaaa

II. Definition of Function A function f is a

aaaa aaaa aaaaaaa aa aaaaaaaaa a aa a aaa a aaaaaaa aaa aaaaaaaa

aaaaaa aaaaa aa a aaa a

.

The symbol f(x) is read or the

aa aa aa aa aa aa aa

aaaaa aa a aaaaa a

The set A is called the

and is called the

aa aa a aa a

,

.

aaaaaa

of the function, and the range of f is

aaa aaa aa

aaa aaaaaaaa aaaaaa aa aaaa aa a aaaaaa aaaaaaaaaa aaa aaaaaa

.

The symbol that represents an arbitrary number in the domain of a function f is called aaaaaaaa

. The symbol that represents a number in the range of f is called

aaaaaaaa

. If we write y  f ( x) , then

a

aa aaaaaaaaaaa

is the independent variable and

a aaaaaaaaa a

is

the dependent variable.

III. Evaluating a Function To evaluate a function f at a number, Example 1:

aaaaaaaaaa aaa aaaaaa aaa aaa aaaaaaaaaaa

.

2 If f ( x)  50  2 x , then evaluate f (5) . a

Note Taking Guide for Stewart/Redlin/Wats on

Precalculus:

Copyright © C engage Lear ning. All rights res er ved.

th

Mathematics for Calculus, 7 Edition

41

42

CHAPTER 2

|

Functions

A piecewise-defined function is

aaaaaaa aa aaaaaaaaa aaaaaaaa aa aaaaaaaaa aaaaa

aa aaa aaaaaa

.

IV. Domain of a Function The domain of a function is

aaa aaa aa aaa aaa aaa aaa aaaaaaaa

. If the function

is given by an algebraic expression and the domain is not stated explicitly, then by convention the domain of the function is

aaa aaaaaa aa aaa aaaaaaaaa aaaaaaaaaaaaaa aaa aaa aaa aa aaa aaaa aaaaaaa aaa aaaaa

aaa aaaaaaaaaa aa aaaaaaa aa a aaaa aaaaaa

Example 2:

.

Find the domain of the function g ( x)  x 2  16 . aaaa aaaaaaaaa

V. Four Ways to Represent a Function List and describe the four ways in which a specific function can be described. aaaaaaaa aaa a aaaaaaaaaaa aa aaaaaaaaaaaaaaaaaaaa aaa aa aaaaaaaa aaaaaaaaaaaaaaaaa aaa a aaaaaaaaaaaaaaaaaa aaa a aaaaa aa aaaaaaaa

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SECTION 2.2

|

Graphs of Functions

43

Name ___________________________________________________________ Date ____________

2.2

Graphs of Functions

I. Graphing Functions by Plotting Points To graph a function f,

aa aaaa aaa aaa aaa aaaaa aa a aaaaaaaaaa aaaaaa aa aaaaa aaaaaa aa

aaaa aaa aaaaaa aaa aa aaaaa aaaaaaaaaaaa aa aa aaaaa aaaaaa aaaaaaaaaaaa aa aaa aaaaaaaaaaaaa aaaaaa aa aaa aaaaaaaa

. aa aaaaa a a a aa

If f is a function with domain A, then the graph of f is the set of ordered pairs

plotted in a coordinate plane. In other words, the graph of f is the set of all points (x, y) such that that is, the graph of f is the graph of the equation

a a aaaa

A function f of the form f ( x) = mx + b is called a

aaaaaa aaaaaaaa

aaaaaaaa Its graph is

aaa aaaaaaaaaa aaaa a a a

a

and y-intercept

aaa aaaa aaaaaaa aaaaaaa a .

II. Graphing Functions with a Graphing Calculator 3 Describe how to use a graphing calculator to graph the function f ( x)  5x  2 x  2 .

aaaaaaa aaaa aaaaa

Note Taking Guide for Stewart/Redlin/Wats on Precalculus: Copyright © Cengage Learning. All rights reserved.

;

because its graph is

. The function f ( x) = b , where b is a given number, is called a because all its values are

a aaaa

.

the graph of the equation y = mx + b , which represents a line with slope a

,

Mathematics for Calculus, 7th Edition

aaaaaaa

. .

44

CHAPTER 2

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Functions

III. Graphing Piecewise Defined Functions  1 3  x, if x  0 Describe how to graph the piecewise-defined function f ( x)   5 . 2 x 2 , if x  0  aaaaaaa aaaa aaaaa

The greatest integer function is defined by [[ x ]] =

aaa aaaaaaaa aaaa aaaa aaaa aa aaaaa aa a

The greatest integer function is an example of a

aaaa aaaaaaaa

A function is called continuous if

.

.

aaa aaaaa aaa aa aaaaaaa aa aaaaaaa

.

IV. The Vertical Line Test: Which Graphs Represent Functions? The Vertical Line Test states that

a aaaaa aa aaa aaaaaaaaa aaaaa aa aaa aaaaa aa a aaaaaaaa aa

aaa aaaa aa aa aaaaaaaa aaaa aaaaaaaaaa aaa aaaaa aaaa aaaa aaaa Is the graph below the graph of a function? Explain.

.

aaa aa aaaa aaa aaaa aaa aaaaaaaa aaaa aaaaa

y

x

Note Taking Guide for Stewart/Redlin/Wats on

Precalculus: Mathematics for Calculus, 7th Edition Copyright © Cengage Learning. All rights reserved.

SECTION 2.2

|

Graphs of Functions

V. Which Equations Represent Functions? Any equation in the variables x and y defines a relationship between these variables. Does every equation in x and y define y as a function of x?

aa

.

Draw an example of the graph of each type of function. Linear Function

aaaaaa aaaa aaaaa

Power Function

y

y

x

Root Function

x

Reciprocal Function y

y

x

Absolute Value Function

x

Greatest Integer Function

y

y

x

Note Taking Guide for Stewart/Redlin/Wats on Precalculus: Copyright © Cengage Learning. All rights reserved.

x

Mathematics for Calculus, 7th Edition

45

46

CHAPTER 2

|

Functions

Additional notes

y

y

x

y

y

x

y

x

x

y

x

x

Homework Assignment Page(s) Exercises

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Precalculus: Mathematics for Calculus, 7th Edition Copyright © Cengage Learning. All rights reserved.

|

SECTION 2.3

47

Getting Information from the Graph of a Function

Name ___________________________________________________________ Date ____________

2.3

Getting Information from the Graph of a Function

I. Values of a Function; Domain and Range To analyze the graph of a function,

aaaa aa aaaa aaaa aaa aaaa aa aaa aaaaa aa aaa aaaaa aa

aaa aaaaaaaaa aa aa aaa aaaa aaa aaa aaaaaa aa a aaaaaaaa aaaa aaa aaaaa

.

Describe how to use the graph of a function to find the function’s domain and range. aaaaaaa aaaa aaaaa

II. Comparing Function Values: Solving Equations and Inequalities Graphically The solution(s) of the equation f(x) = g(x) are a aaa a aaaaaaa

aaa aaaaa aa a aaaaa aaa aaaaaa aa

. The solution(s) of the inequality f(x) < g(x) are

aa a aaaaa aaa aaaaa aa a aa aaaaaa aaaa aaa aaaaa aa a

aaa aaaaaa .

Describe how to solve an equation graphically. aaaaa aaaa aaa aaaaa aa aaa aaaa aa aaa aaaaaaaa aaa aaaa aaaaa aaa aaaaaaaa aaaa aaaaaaaaaaa aa aaa aaaaaaa aaaa aa aaa aaaaaaaaa aa aaaa aaaaa aaa aaaaaaaaa aa aaa aaaaaaaa aaa aaa aaaaaaaaaaaa aa aaa aaaaaaa

III. Increasing and Decreasing Functions A function f is said to be increasing when decreasing when

aaa aaaaa aaaaa

aaa aaaaa aaaaa

and is said to be .

According to the definition of increasing and decreasing functions, f is increasing on an interval I if aaaaa a aaaaa on an interval I if Example 1:

whenever aaaaa a aaaaa

aa a a a whenever

in I. Similarly, f is decreasing aa a aa

Use the graph to determine (a) the domain, (b) the range, (c) the intervals on which the function is increasing, and (d) the intervals on which the function is decreasing.

aaa aaa aaaaaa aa aa aaa aaa aaaaa aa aaaa aaa aaa aaaaaaaaaa aa aaaa aaaaaaa aaaaaaaaaa aa aaa aa

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Mathematics for Calculus, 7th Edition

in I.

48

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CHAPTER 2

Functions

IV. Local Maximum and Minimum Values of a Function The function value f(a) is a local maximum value of f if this case we say that f has

f ( a )  f ( x)

a aaaaa aaaaaaa aa a a a

The function value f(a) is a local minimum value of f if this case we say that f has

when x is near a. In

.

f ( a )  f ( x)

a aaaaa aaaaaaa aa a a a

when x is near a. In

.

Describe how to use a graphing calculator to find the local maximum and minimum values of a function. aaaaaaa aaaa aaaaa

y

y

x

y

y

x

y

x

x

y

x

x

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Precalculus: Mathematics for Calculus, 7th Edition Copyright © Cengage Learning. All rights reserved.

SECTION 2.4

|

Av erage Rate of Change of a Function

49

Name ___________________________________________________________ Date ____________

2.4

Average Rate of Change of a Function

I. Average Rate of Change The average rate of change of the function y = f ( x) between x = a and x = b is

The average rate of change is the slope of the graph of f, that is, the line that passes through

Example 1:

For the function

aaaaaa aaaa

between x = a and x = b on the

aaa aaaaa aaa aaa aaaaa

.

f ( x)  3x2  2 , find the average rate of change between x = 2 and x = 4 .

aa

II. Linear Functions Have Constant Rate of Change For a linear function f ( x) = mx + b , the average rate of change between any two points is aaaaaaaa a

. If a function f has constant average rate of change, then it must be

aaaaaa aaaaaaaa Example 2:

aaa aaaa

.

f ( x)  14  6 x , find the average rate of change between the following points. (a) x  10 and x  5 (b) x = 0 and x = 3 (c) x = 4 and x = 9 For the function

aaa aaa

aaa aaa

aaa aa

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Mathematics for Calculus, 7th Edition

a

50

CHAPTER 2

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Functions

Additional notes

y

y

x

y

y

x

y

x

x

y

x

x

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SECTION 2.5

|

51

Linear Functions and Models

Name ___________________________________________________________ Date ____________

2.5

Linear Functions and Models

I. Linear Functions A linear function is a function of the form function is

aaaaaaaaaaaaaaaa

. The graph of a linear

aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

.

II. Slope and Rate of Change For the linear function f ( x)  ax  b , the slope of the graph of f and the rate of change of f are both equal to aaaaaaaaaaaaaaaaaaaaaaa

.

III. Making and Using Linear Models When a linear function is used to model the relationship between two quantities, the slope of the graph of the function is

a aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

Give an example of a real-world situation that involves a constant rate of change.

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Mathematics for Calculus, 7th Edition TE

.

52

CHAPTER 2

|

Functions

Additional notes

y

y

x

y

y

x

y

x

x

y

x

x

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Precalculus: Mathematics for Calculus, 7th Edition TE Copyright © Cengage Learning. All rights reserved.

SECTION 2.6

|

Transformations of Functions

53

Name ___________________________________________________________ Date ____________

2.6

Transformations of Functions

I. Vertical Shifting Adding a constant to a function shifts its graph aaaaaaaa

aaaaaaaaaa

and downward if it is

: upward if the constant is aaaaaaaa

.

Consider vertical shifts of graphs. Suppose c > 0. To graph y = f ( x) + c , shift

aaa aaaa aa a a aaaa

. To graph y  f ( x)  c , shift

aaaaaa a aaaaa aaaaaaaa a aaaaa

aaa aaaa aa a a aaaa

.

II. Horizontal Shifting Consider horizontal shifts of graphs. Suppose c > 0. To graph y  f ( x  c) , shift

aaa aaaaa aa

. To graph y = f ( x + c) , shift

a a aaaa aa aaa aaaaa a aa aa a a aaaa aa aaa aaaa a aaaaa

aaa aaaaa

.

III. Reflecting Graphs To graph y   f ( x) , reflect the graph of y = f ( x) in the

aaaaaa

.

To graph y  f ( x) , reflect the graph of y = f ( x) in the

aaaaaa

.

IV. Vertical Stretching and Shrinking Multiplying the y-coordinates of the graph of y = f ( x) by c has the effect of

aaaaaaa aaaaaaaaaa aa

aaaaaaaaa aaa aaaaa aa a aaaaaa aa a

.

To graph y = cf ( x) : If c > 1, If 0 < c < 1,

aaaaaaa aaa aaaaa aa a a aaaa aaaaaaaaaa aaaaaa aa a aaaaaa aaa aaaaa aa a a aaaa aaaaaaaaaa

aaaaaa aa a

. .

V. Horizontal Stretching and Shrinking To graph y = f (cx) : If c > 1, If 0 < c < 1,

aaaaaa aaa aaaaa aa a a aaaa aaaaaaaaaaaa aa aaaa aa aaa aaaaaaa aaa aaaaa aa a a aaaa aaaaaaaaaaaa aaaaaa aa aaa

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Mathematics for Calculus, 7th Edition

. .

54

CHAPTER 2

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Functions

VI. Even and O dd Functions Let f be a function. Then f is even if

aaaaa a aaaa aaa aaa a aa aaaaaaa aa a

.

Then f is odd if

aaaaa a aaaaa aaa aaa a aa aaa aaaa aa a

.

The graph of an even function is symmetric with respect to

aaa aaaaaa

The graph of an odd function is symmetric with respect to

y

aaa aaaaaa

y

x

y

. .

y

x

y

x

x

y

x

x

Homework Assignment Page(s) Exercises

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|

SECTION 2.7

Combining Functions

55

Name ___________________________________________________________ Date ____________

2.7

Combining Functions

I. Sums, Differences, Products, and Q uotients Given two functions f and g, we define the new function f + g by

aa a aaaaa a aa a aaaa

The new function f + g is called

aaa aaa aa aaa aaaaaaaa a aaa a

x is

.

aaaa a aaaa

. . Its value at

Let f and g be functions with domains A and B. Then the functions f  g , f  g , fg , and f / g are defined as follows.

( f + g )( x) =

aaaa a aaaa

, Domain is

a

a

( f  g )( x) 

aaaa a aaaa

, Domain is

a

.

( fg )( x) =

aaaaaaaa

, Domain is

 f    ( x)  g

aaaa a aaaa

, Domain is

a

a

.

.

The graph of the function f + g can be obtained from the graphs of f and g by graphical addition, meaning that we

aaa aaaaaaaaaaaaa aaaaaaaaaaaaa

Example 2:

2 Let f ( x) = 3x + 1 and g ( x)  2 x  1 .

.

(a) Find the function g  f . (b) Find the function

f +g.

aaa aaa

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Mathematics for Calculus, 7th Edition

56

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CHAPTER 2

Functions

II. Composition of Functions Given two functions f and g, the composite function f a aaa a

aaa aaaaaaaaaaa aa

( f g )( x)  f ( g ( x))

) is defined by

The domain of f

g is

.

aaa aaa aa aaa a aa aaa aaa aa a aaaa aaaa aaaa aa aa . In other words, ( f

aaa aaaaa aa a aaa aaaaaaa aaa aaaaaaa Example 2:

g (also called

g )( x) is defined whenever

aaaa aaaa

.

Let f ( x) = 3x + 1 and g ( x)  2 x2  1 . (a) Find the function f (b) Find ( f

g.

g )(2) .

aaa aa

It is possible to take the composition of three or more functions. For instance, the composite function

( f g h)( x)  f ( g (h( x)))

is found by

y y

.

y y

x x

f g h

y y

x x

x x

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SECTION 2.8

|

57

One-to-One Functions and Their Inv erses

Name ___________________________________________________________ Date ____________

2.8

One-to-One Functions and Their Inverses

I. O ne -to-One Functions A function with domain A is called a one-to-one function if aaaa aaa

a

aaaaaaaa

aa aaa aaaaaaaa .

a

An equivalent way of writing the condition for a one-to-one function is this: aa aa aaaa a

The Horizontal Line Test states that

.

a aaaaaaaa aa aaaaaaaaaa aa aaa aaaa aa aa aaaaaaaaaa

aaaa aaaaaaaaaa aaa aaaaa aaaa aaaa aaaa

.

Every increasing function and every decreasing function is

aaaaaaaaaa

.

II. The Inverse of a Function Let f be a one-to-one function with domain A and range B. Then its 1 domain B and range A and is defined by f ( y)  x 

aaaaaa aaaaaaaa

f 1 has

f ( x)  y for any y in B.

1 Let f be a one-to-one function with domain A and range B. The inverse function f satisfies the following cancellation properties:

1) aaa aaaaa a aa a 2) aaa aaaaa a aa a Conversely, any function

f 1 satisfying these equations is

aaa aaaaaaa aa a

.

III. Finding the Inverse of a Function Describe how to find the inverse of a one-to-one function. aaaaaa aaaaa a a aaaaa aaaa aaaaa aaaa aaaaaaaa aaa a aa aaaaa aa a aaa aaaaaaaaaa aaaaaaaa aaaaaaaaaaa a aaa aa aaa aaaaaaaaa aaaaaaaa aa a a a a

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Mathematics for Calculus, 7th Edition

58

CHAPTER 2

Example 1:

|

Functions

Find the inverse of the function

f ( x)  9  2 x .

IV. Graphing the Inverse of a Function The graph of f 1 is obtained by

aaaaaaa aaa aaaaa aa a aa aaa aaaa a a a

y y

y y

x x

.

y y

x x

x x

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Name ___________________________________________________________ Date ____________

Chapter 3 3.1

Polynomial and Rational Functions

Quadratic Functions and Models P( x)  an xn  an1xn1 

A polynomial function of degree n is a function of the form A quadratic function is a polynomial function of degree

f ( x)  ax2  bx  c, a  0

the form

a

 a1x  a0

.

. A quadratic function is a function of

.

I. Graphing Q uadratic Functions Using the Standard Form f ( x) = ax2 + bx + c can be expressed in the standard form f ( x)  a( x  h)2  k by

A quadratic function

aaaaaaaaaa aaa aaaaaa

. The graph of f is a

aaa aa if

aaa

Example 1:

Let (a) (b) (c) aaa aaa aaa

aaaaaaaa

with vertex

aa a

or downward

. The parabola opens upward if .

f ( x)  3 x 2  6 x  1 . Express f in standard form. What is the vertex of the graph of f? Does the graph of f open upward or downward?

aaaaaa

II. Maximum and Minimum Values of Q uadratic Functions If a quadratic function has vertex (h, k), then the function has a minimum value at the vertex if its graph opens aaaaaa

and a maximum value at the vertex if its graph opens

aaaaaaaa

.

2 Let f be a quadratic function with standard form f ( x)  a( x  h)  k . The maximum or minimum value of f

occurs at

aaa

.

If a > 0, then the minimum value of f is

aaaa a a

.

If a < 0, then the maximum value of f is

aaaa a a

.

Note Taking Guide for Stewart/Redlin/Wats on

Precalculus:

Copyright © C engage Lear ning. All rights res er ved.

th

Mathematics for Calculus, 7 Edition

59

60

CHAPTER 3

Example 2:

|

Polynomial and Rational Functions

Consider the quadratic function f ( x)  2 x2  4 x  8 . (a) Express f in standard form. (b) Does f have a minimum value or a maximum value? Explain. (c) Find the minimum or maximum value of f.

The maximum or minimum value of a quadratic function f ( x) = ax2 + bx + c occurs at

If a > 0, then the

aaaaaaa

 b  value is f   .  2a 

If a < 0, then the

aaaaaaa

 b  value is f   .  2a 

Example 3:

x

b 2a

.

2 Find the maximum or minimum value of the quadratic function f ( x)  0.5x  5x  12 , and state whether it is the maximum or the minimum.

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SECTION 3.2

|

61

Polynomial Functions and Their Graphs

Name ___________________________________________________________ Date ____________

3.2

Polynomial Functions and Their Graphs

I. Polynomial Functions P( x)  an xn  an1xn1 

A polynomial function of degree n is a function of the form

 a1x  a0

where n is a nonnegative integer and an  0 . The numbers a0 , a1 , a2 , . . . , an are called the The number a0 is the

aaaaaaaaaaaa

aaaaaaaa aaaaaaaaaaa

or

The number an , the coeffici ent of the highest power, is the the term an x n is the

aaaaaaa aaaa

of the polynomial. aaaaaaaa aaaa

aaaaaaa aaaaaaaaaaa

. , and

.

II. Graphing Basic Polynomial Functions The graph of P( x) = x n has the same general shape as the graph of 3 the same general shape as the graph of y = x when

a a aa a aa aaa

degree n becomes larger, the graphs become aaa aaaaaaa aaaaaaaaa

when n is even and . However, as the aaaaaaa aaaaaa aaa aaaaaa

.

III. End Behavior and the Leading Term The graphs of polynomials of degree 0 or 1 are 2 are

aaaaaaaaa

aaaaa aaa aa

, and the graphs of polynomials of degree

. The greater the degree of a polynomial, the more . However, the graph of a polynomial function is

meaning that the graph has no function is a

aaaaa

aaaaaa aa aaaaa

aaaaaa

The end behavior of a polynomial is

aaaaaaaaaaa aaa aaaaaaaaaa

,

. Moreover, the graph of a polynomial

curve; that is, it has no

aaaaaaa aa aaaaa aaaaa aaaaaaa

.

a aaaaaaaaaa aa aaaa aaaaaaa aa a aaaaaaa aaaaa aa aaa aaaaaaaa

aa aaaaaaaa aaaaaaaaa aaaaaaa aaaaa aa aaa aaaaaaaa aaaaaaaaa aaa aaaaaaaa aaaaaaaaa

. To describe end behavior we use x   to mean and we use x   to mean

Note Taking Guide for Stewart/Redlin/Wats on Precalculus: Copyright © Cengage Learning. All rights reserved.

a aaaaaaa aaaaa aa

.

n n 1 The end behavior of the polynomial P( x)  an x  an1 x 

aaa aaa aaaa aa aaa aaaaaaa aaaaaaaaaa

a

 a1x  a0 is determined by .

Mathematics for Calculus, 7th Edition

aaa aaaaaa a

62

CHAPTER 3

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Polynomial and Rational Functions

In your own words, describe the end behavior of a polynomial P with the following characteristics: If P has odd degree and a positive leading coeffici ent, then aaa aaaaaaaaaa aa aaa aaaaa

aaa aaaaa aa a aa aaaaaaaaaa aa aaa aaaa

.

If P has odd degree and a negative leading coeffici ent, then aaa aaaaaaaaaa aa aaa aaaaa

aaa aaaaa aa a aa aaaaaaaaaa aa aaa aaaa

.

If P has even degree and a positive leading coeffici ent, then aaa aaaaaaaaaa aa aaa aaaaa

aaa aaaaa aa a aa aaaaaaaaaa aa aaa aaaa

.

If P has even degree and a negative leading coefficient, then aaa aaaaaaaaaa aa aaa aaaaa Example 1:

aaa aaaaa aa a aa aaaaaaaaa aa aaa aaaa

.

Determine the end behavior of the polynomial P( x)  5x6  2 x4  3x2  9 . aaaaaaa a aaa aaaa aaaaaa aaa a aaaaaaaa aaaaaaaaaaaa aa aaa aaa aaaaaaaaa aaa aaaaaaaaaa

IV. Using Zeros to Graph Polynomials If P is a polynomial function, then c is called a zeros of P are the solutions of

aaaa

of P if P(c) = 0. In other words, the

aaa aaaaaaaaaa aaaaaaaa aaaa a a

P(c) = 0, then the graph of P has an x-intercept at

aa a

. Note that if

, so the x-intercepts of the graph are

the zeros of the function. If P is a polynomial and c is a real number, then list four equivalent statements about the real zeros of P. aa a aa a aaaa aa aa aa a a a aa a aaaaaaaa aa aaa aaaaaaaa aaaa a aa aa a a a aa a aaaaaa aa aaaaa aa a aa aa aaaaaaaaaaa aa aaa aaaaa aa aa

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Precalculus: Mathematics for Calculus, 7th Edition Copyright © Cengage Learning. All rights reserved.

|

SECTION 3.2

Polynomial Functions and Their Graphs

The Intermediate Value Theorem for Polynomials states that

63

aa a aa a aaaaaaaaaa

aaaaaaaa aaa aaaa aaa aaaa aaaa aaaaaaaa aaaaaa aaaa aaaaa aaaaaa aa aaaaa aaa aaa

aa a aaaaaaa a aaa a aaa

aaaaa aaaa a a

.

List guidelines for graphing polynomial functions. aa aaaaaa aaaaaa aaa aaaaaaaaaa aa aaaa aaa aaa aaaa aaaaaa aaaaa aaa aaa aaaaaaaaaaaa aa aaa aaaaaa aa aaaa aaaaaaa aaaa a aaaaa aa aaaaaa aaa aaa aaaaaaaaaaa aaaaaaa aaaa aaaaaa aa aaaaaaaaa aaaaaaa aaa aaaaa aa aaa aaaaaaaaaa aaaa aaaaa aa aaaaa aaa aaaaaa aa aaa aaaaaaaaa aaaaaaaaaa aa aaa aaaaaa aaaaaaa aaa aaaaaaaaaaa aa aaa aaaaaaa aa aaa aaaaaaaaa aaaaaaaaa aaa aaa aaaaaaaa aa aaa aaaaaaaaaaa aa aaaaaa aaaa aaa aaaaaaaaaa aaa aaaaa aaaaaa aaa aaaaa aa aaa aaaaaa aaaaaa a aaaaaa aaaaa aaaa aaaaaa aaaaaaa aaaaa aaaaaa aaa aaaaaaaa aaa aaaaaaaa aaa aaaaaaaaa

V. Shape of the Graph Near a Zero If c is a zero of P, and the corresponding factor x − c occurs exactly m times in the factorization of P then we say that c is a

aaaa aa aaaaaaaaaaaa a

.

The graph crosses the x-axis at c if the multiplicity m is is Example 2:

aaaa

aaa

and does not cross the x-axis if m

.

2 3 How many times does the graph of P( x)  ( x  3) ( x  5)( x  9) cross the x-axis?

aaaaa

VI. Local Maxima and Minima of Polynomials If the point (a, f(a)) is the highest point on the graph of f within some viewing rectangle, then f(a) is a local maximum value of f and the point (a, f(a)) is a

aaaaa aaaaaaa aaaaa

on the graph. If the

point (b, f(b)) is the lowest point on the graph of f within some viewing rectangle, then f(b) is a local minimum value of f and the point (b, f(b)) is a

aaaaa aaaaaaa aaaaa

Note Taking Guide for Stewart/Redlin/Wats on Precalculus: Copyright © Cengage Learning. All rights reserved.

on the graph.

Mathematics for Calculus, 7th Edition

64

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CHAPTER 3

Polynomial and Rational Functions

The set of all local maximum and minimum points on the graph of a function is called its . If P( x)  an xn  an1 x n1 

aaaaa aaaaaaa the graph of P has at most

Example 3:

aaa

 a1x  a0 is a polynomial of degree n, then

local extrema.

How many local extrema does the polynomial P( x)   x4  x3  x2  4 x  2 have? aaa

y

y

x

y

y

x

y

x

x

y

x

x

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SECTION 3.3

|

Div iding Polynomials

65

Name ___________________________________________________________ Date ____________

3.3

Dividing Polynomials

I. Long Division of Polynomials Describe the Division Algorithm. aa aaaa aaa aaaa aaa aaaaaaaaaaaa aaaa aaaa a aa aaaa aaaaa aaaaa aaaaaa aaaaaaaaaaa aaaa aaa aaaaa aaaaa aaaa aa aaaaaa a aa aa aaaaaa aaaa aaaa aaa aaaaaa aa aaaaa aaaa aaaaaaaaaaa a aaaa a aaaa a aaaaaaaaaaa aaaa aa aaaaaa aaa aaaaaaaaa aaaa aa aaaaaa aaa aaaaaaaa aaaa aa aaa aaaaaaaaa aaa aaaa aa aaa aaaaaaaaaaa

The division process ends when Example 1:

aaa aaaa aaaa aa aa aaaaaa aaaa aaaa aaa aaaaaaa

.

Show how to set up the long division of 3x3  1 by 2 x + 5 .

II. Synthetic Division Synthetic division is

a aaaaa aaaaaa aa aaaaaaaa aaaaaaaaaaa

used when the divisor is of the form Example 2:

aaa

. It can be

.

3 Show how to set up the synthetic division of 4 x  7 x  21 by x + 2 .

III. The Remainder and Factor Theorems The Remainder Theorem states that if the polynomial P(x) is divided by x  c , then the remainder is the value

aaaa

.

The Factor Theorem states that c is a zero of P if and only if

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a a a aa a aaaaa aa aaaa

Mathematics for Calculus, 7th Edition

.

66

CHAPTER 3

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Polynomial and Rational Functions

Explain how to easily decide whether x  6 is a factor of the polynomial P( x)  x7  5x5  2 x4  x2  9 without performing long division. aaaaaa aaaaa aaa aaaaaaaa aaaaaaaaa aa aaa aaaa aa aaa aaaa aaaa aa aaa aaaaaaaaa aaaaaaaa aa aa aaaa a a a aa a aaaaaa aa aaaaa

y

y

x

y

y

x

y

x

x

y

x

x

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SECTION 3.4

Real Zeros of Polynomials

67

Name ___________________________________________________________ Date ____________

3.4

Real Zeros of Polynomials

I. Rational Zeros of Polynomials State the Rational Zeros Theorem. aa aaa aaaaaaaaaa a aaa aaaaaaa aaaaaaaaaaaa aaaaaa a aaa aaa aaaa aaaaa aaaaaaaa aaaa aa a aa aa aaa aaaa aaaaa a aaa a aaa aaaaaaaa aaa a aa a aaaaaa aa aaa aaaaaaaa aaaaaaaaaaa

a

a

aaa a aa a aaaaaa aa aaa aaaaaaa

aaaaaaaaaaa aa

We see from the Rational Zeros Theorem that if the leading coeffici ent is 1 or 1 , then the rational zeros must be

aaaaaaa aa aaa aaaaaaaa aaaa

.

List the steps for finding the rational zeros of a polynomial. aa aaaa aaaaaaaa aaaaaa aaaa aaa aaaaaaaa aaaaaaaa aaaaaa aaaaa aaa aaaaaaaa aaaaa aaaaaaaaaaa aaaaaaa aaa aaaaaaaaa aaaaaaaa aa aaaaaaaa aaa aaaaaaaaaa aa aaaa aa aaa aaaaaaaaaa aaa aaa aaaaaaaa aaaaa aaaa aaa aaaaa aa aaaa aa aaaa aaa aaaaaaaaa aa aa aaaa aaa aaaaaaaa aaa aaaa aaaaaaaaaaaa aaaaaaa aaaaaa aaaaa a aaa a aaa aaa aaaaaaaaa aaaa aaaa aaa aaaaa a aaaaaaaa aaaa aa aaaaaaaaa aa aaaaaaa aaaaaaa aaa aaa aaa aaaaaaaaa aaaaaaa aa aaaaaa aa aaaa aaa aaaaaaaaa aaaaaa

II. Descartes’ Rule of Signs If P(x) is a polynomial with real coeffici ents, written with descending powers of x (and omitting powers with coefficient 0), then a variation in sign occurs whenever aaaaaaaa aaaaa

aaaaaaaa aaaaaaaaaaaa aaaa

.

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Mathematics for Calculus, 7th Edition

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CHAPTER 3

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Polynomial and Rational Functions

State Descartes’ Rule of Signs. aaa a aa a aaaaaaaaaa aaaa aaaa aaaaaaaaaaaaa aa aaa aaaaaa aa aaaaaaaa aaaa aaaaa aa aaaa aaaaaa aa aaaaa aa aaa aaaaaa aa aaaaaaaaaa aa aaaa aa aaaa aa aa aaaa aaaa aaaa aa aa aaaa aaaaa aaaaaaa aa aaa aaaaaa aa aaaaaaaa aaaa aaaaa aa aaaa aaaaaa aa aaaaa aa aaa aaaaaa aa aaaaaaaaaa aa aaaa aa aaaaa aa aa aaaa aaaa aaaa aa aa aaaa aaaaa aaaaaaa

III. Upper and Lower Bounds for Roots We say that a is a lower bound and b is an upper bound for the zeros of a polynomial if aaaa a aa aaa aaaaaaaaaa aaaaaaaaa a a a a a

aaaaa aaaa .

State the Upper and Lower Bounds Theorem. aaa a aa a aaaaaaaaaa aaaa aaaa aaaaaaaaaaaaa aa aa aa aaaaaa aaaa aa a a a aaaaa a a aa aaaaa aaaaaaaaa aaaaaaaa aaa aa aaa aaa aaaa aaaaaaaa aaa aaaaaaaa aaa aaaaaaaaa aaa aa aaaaaaaa aaaaaa aaaa a aa aa aaaaa aaaaa aaa aaa aaaa aaaaa aa aa aa aa aa aaaaaa aaaa aa a a a aaaaa a a aa aaaaa aaaaaaaaa aaaaaaaa aaa aa aaa aaa aaaa aaaaaaaa aaa aaaaaaaa aaa aaaaaaaaa aaa aaaaaaa aaaa aaa aaaaaaaaaaa aaaaaaaaaaa aaa aaaaaaaaaaaa aaaa a aa a aaaaa aaaaa aaa aaa aaaa aaaaa aa aa

The phrase “ alternately nonpositive and nonnegative” simply means

aaaa aaa aaaaa aa aaa aaaaaaa

aaaaaaaaaa aaaa a aaaaaaaaaa aa aa aaaaaaaa aa aaaaaaaa aa aaaaaaaa

.

IV. Using Algebra and Graphing Devices to Solve Polynomial Equations Describe how to use the algebraic techniques from this section to select an appropriate viewing rectangle when solving a polynomial equation graphically. aaaaaaa aaaa aaaaa

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SECTION 3.5

|

69

Complex Zeros and the Fundamental Theorem of Algebra

Name ___________________________________________________________ Date ____________

3.5

Complex Zeros and the Fundamental Theorem of Algebra

I. The Fundamental Theorem of Algebra and Complete Factorization State the Fundamental Theorem of Algebra. aaaaa aaaaaaaaaa aa aa aaaa aaaaaaa aaaaaaaaaaaa aaa aa aaaaa aaa aaaaaaa aaaaa

State the Complete Factorization Theorem. aa aaaa aa a aaaaaaaaaa aa aaaaaa aa aaaa aaaaa aaaaa aaaaaaa aaaaaaa a aaaaa a a aa aaaa aaaa aa

To actually find the complex zeros of an nth-degree polynomial, we usually

aaaaa aaaaaa aa aaaa aa

aaaaaaaaa aaaa aaa aaa aaaaaaaaa aaaaaaa aa aaaaa aaaa aa aaaaa aaaaaa aaaaaaa Example 1:

.

Suppose the zeros of the fourth-degree polynomial P are 5, −2, 14i, and −14i. Write the complete factorization of P.

II. Zeros and Their Multiplicities If the factor x  c appears k times in the complete factorization of P(x), then we say that c is a zero of aaaaaaaaaaaa a

.

The Zeros Theorem states that every polynomial of degree n  1 has exactly a zero of multiplicity k is counted Example 2:

a aaaaa

a

zeros, provided that

.

Suppose the zeros of P are 5 with multiplicity 1, −2 with multiplicity 3, 14i with multiplicity 2, and −14i with multiplicity 2. Write the complete factorization of P.

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CHAPTER 3

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Polynomial and Rational Functions

III. Complex Zeros Come in Conjugate Pairs The Conjugate Zeros Theorem states that if the polynomial P has real coefficients and if the complex number z is a zero of P, then

aaa aaaaaaa aaaaaaaaa aa aaaa a aaaa aa a

.

IV. Linear and Q uadratic Factors A quadratic polynomial with no real zeros is called

aaaaaaaaaaa aaaa a aaaa aaaaaaa

Every polynomial with real coefficients can be factored into aaaaaaaaa aaaaaaa aaaa aaaa aaaaaaaaaaaa

.

a aaaaaa aa aaaaaa aaa aaaaaaaaaaa .

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SECTION 3.6

|

71

Rational Functions

Name ___________________________________________________________ Date ____________

3.6

Rational Functions

I. Rational Functions and Asymptotes r ( x) 

A rational function is a function of the form

P( x) Q( x)

where P(x) and Q(x) are polynomial

functions having no factors in common. The domain of a rational function consists of aaaaaaaaaaa aa aaaa

aaa aaaa aaaaaaa a aaaaaa aaaaa aaa aaaaa aaa

. When graphing a rational function, we must pay special attention

to the behavior of the graph near

aaaaa aaaaaaaa aaa aaaaa aaa aaaaaaaaaaa aa a

x  a  means

a aaaaaaaaaa a aaaa aaa aaaa

x  a  means

a aaaaaaaaaa a aaaa aaa aaaaa

.

. .

x   means

a aaaa aa aaaaaaaa aaaaaaaaa aaaa aaa a aaaaaaaaa aaaaaaa aaaaa

.

x   means

a aaaa aa aaaaaaaaa aaaa aaa a aaaaaaaaa aaaaaaa aaaaa

.

Informally speaking, an asymptote of a function is

a aaaa aa aaaaa aaa aaaaa aa aaa aaaaaaaa aaaa

aaaaaa aaa aaaaaa aa aaa aaaaaaa aaaaa aaaa aaaa

.

The line x = a is a vertical asymptote of the function y = f ( x) if a aaaa aaa aaaaa aa aaaa

a aaaaaaaaaa aa aa a aaaaaaaaaa .

Draw an example of a graph having a vertical asymptote. y

aaaaaaa aaaa aaaaa x

The line y = b is a horizontal asymptote of the function y = f ( x) if aaaaaaaaaa aa

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Polynomial and Rational Functions

Draw an example of a graph having a horizontal asymptote. y

aaaaaaa aaaa aaaaa x

A rational function has vertical asymptotes where

aaa aaaaaaaa aa aaaaaaaaaa aaaa aaa aaaaa

aaa aaaaaaaaaaa aa aaaa

.

II. Transformations of y = 1/x A rational function of the form r ( x) =

ax + b can be graphed by shifting, stretching, and/or reflecting cx + d

aaaaa aa aaaa a aaa aaaaa aaaaaa aaaaaaaaaa aaaaaaaaaaaaaaa

aa .

III. Asymptotes of Rational Functions Let r be the rational function r ( x) 

an x n  an 1 x n 1 

bm x m  bm1 x m1 

1. The vertical asymptotes of r are the lines 2.

 a1 x  a0  b1 x  b0

.

a a aa aaaaa a aa a aaaa aa aaa aaaaaaaaaa

(a) If n < m, then r has

aaaaaaaaaa aaaaaaaaa a a a

.

(b) If n = m, then r has

aaaaaaaaaa

.

(c) If n > m, then r has

________________

aa aaaaaaaaaa aaaaaaaaa

.

.

IV. Graphing Rational Functions List the guidelines for sketching graphs of rational functions. aa aaaaaaa aaaaaa aaa aaaaaaaaa aaa aaaaaaaaaaaa aa aaaaaaaaaaa aaaa aaa aaaaaaaaaaaa aa aaaaaaaaaaa aaa aaaaa aa aaa aaaaaaaaa aaa aaa aaaaaaaaaaa aaaa aaa aaaaa aa aaa aaaaaaaa aa a a aa aa aaaaaaaa aaaaaaaaaaa aaaa aaa aaaaaaaa aaaaaaaaaa aa aaaaaaaaaaa aaa aaaaa aa aaa aaaaaaaaaaaa aaa aaaa aaa aaaaaaa a a a aa a a aa aa aaaa aaaa aa aaaa aaaaaaaa aaaaaaaaa aa aaaaa aaaa aaaaaaa

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SECTION 3.6

|

Rational Functions

73

aa aaaaaaaaaa aaaaaaaaaa aaaa aaa aaaaaaaaaa aaaaaaaaa aaa aaaaa aaaaa aaa aaaaaaaaa aaaaaaaaa aaaaaaaa aa aaaaaa aaa aaaaaa aaaaa aaa aaaaaaaaaaa aaaaaaaa aa aaa aaaaa aaaa aaaaaa aaaa aaaa aa aaaa aaaaaaaaaa aaaaaa aa aaaaaa aa aaaa aa aaa aaaa aa aaa aaaaa aa aaa aaaaaaaaa

When graphing a rational function, describe how to check for the graph’s behavior near a vertical asymptote.

aaaaaaa aaaa aaaaa

Is it possible for a graph to cross a vertical asymptote?

aa

Is it possible for a graph to cross a horizontal asymptote?

aaa

V. Common Factors in Numerator and Denominator If s( x)  p( x) / q( x) and if p and q have a factor in common, then we may cancel that factor, but only for aaaaa aaaaaa aa a aaa aaaaa aaaa aaaaaa aa aaa aaaa aaaaaaaa aaaaaaaa aa aaaa aa aaa aaaaaaa

. Since s is not defined at those values of x, its graph has a

aaaaaa

at

those points.

VI. Slant Asymptotes and End Behavior If r ( x) = P( x) / Q( x) is a rational function in which the degree of the numerator is aaa aaaaaa aa aaa aaaaaaaaaaa form r ( x) = ax + b +

aaa aaaa aaaa

, we can use the Division Algorithm to express the function in the

R( x) , where the degree of R is less than the degree of Q and a  0 . For large values of Q( x)

x , the graph of y = r ( x) approaches the graph of say that y = ax + b is a

aaa aaa a a aa a a

aaaaa aaaaaaaaa aa aa aaaaaaa aaaaaaaaa

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Mathematics for Calculus, 7th Edition

. In this situation we .

74

CHAPTER 3

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Polynomial and Rational Functions

Describe how to graph a rational function which has a slant asymptote. aaaaaaa aaaa aaaaa

y

y

x

y

y

x

y

x

x

y

x

x

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SECTION 3.7

Polynomial and Rational Inequalities

75

Name ___________________________________________________________ Date ____________

3.7

Polynomial and Rational Inequalities

I. Polynomial Inequalities An important consequence of the Intermediate Value Theorem is that the values of a polynomial function P aa aaa aaaaaa aaaa aaaaaaa aaaaaaaaaa aaaaa between successive zeros are either

. In other words, the values of P

aaa aaaaaa aa aaa aaaaaaaa

this means that between successive x-intercepts, the graph of P is aaaaaaaa aaaaa aaa aaaaaa

. Graphically, aaaaaaaa aaaaa aa

. This property of polynomials allows us to solve

polynomial inequalities like P( x)  0 by

aaaaaaa aaa aaaaa aa aaa aaaaaaaaaa aaa

aaaaa aaaa aaaaaa aaaaaaa aaaaaaaaaa aaaaa aa aaaaaaaaa aaa aaaaaaaaa aaaaa aaa aaaaaaaaaa

.

List the guidelines for solving polynomial inequalities. aa aaaa aaa aaaaa aa aaa aaaaa aaaaaaa aaa aaaaaaaaaa aa aaaa aaa aaaaaaa aaaaa aaaaaa aa aaa aaaa aa aaa aaaaaaaaaa aaaaaaaaaaa aaaaaa aaa aaaaaaaaaaa aaaaaa aaa aaaaaaaaaa aaaa aaaaaaaaaaa aaaaaaa aaa aaaa aaa aaaa aaaaa aa aaa aaaaaaaaaaaaaaa aaaa aaa aaaaaaaaaa aaaa aaa aaaaaaaaa aaaaaaaaaa aa aaa aaaa aaaaaaaaaa aaaa a aaaaa aa aaaaaaaa aaa aaaa aaaaaa aa aaaa a aaaaa aa aaaaaaa aa aaa aaaaa aa aaaa aaaaaa aa aaaa aaaaaaaaa aa aaa aaaa aaa aa aaa aaaaa aaaaaaaaa aaa aaaa aa aaa aaaaaaaaaa aa aaaa aaaaaaaaa aaaa aaaaaa aaaaaaaaa aaa aaaaaaaaa aa aaa aaaaaaaaaa aaaa aaa aaaa aaa aa aaa aaaaaa aaaaa aaaaaaa aaa aaaaaaaaa aa aaaaa aaaaaaaaa aaaaaaa aaa aaaaaaaaaaa aaaaa aaa aaaaaa aa aaa aaaaaaaaaa aaaaaaaa a aa a aaa

Example 1:

3 2 Solve the inequality 12x  x  x .

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Mathematics for Calculus, 7th Edition

76

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CHAPTER 3

Polynomial and Rational Functions

II. Rational Inequalities Unlike polynomial functions, rational functions are not necessarily continuous—the vertical asymptotes of a rational function r

aaaaa aa aaa aaaaa aaaa aaaaaaaa aaaaaaaaaa

on which r does not change sign are determined by

. So the intervals aaa aaaaaaaa aaaaaaaaaa

aa aaaa aa aaa aaaaa aa a

.

If r  x   P  x  / Q  x  is a rational function, the cut points of r are aaaaaa aaaa a a aa aaaa a a

aaa aaaaaa aa a aa aaaaa

. In other words, the cut points of r are

aa aaa aaaaaaaaa aaa aaa aaaaa aa aaa aaaaaaaaaaa

r  x   0 , we use test points between

aaa aaaaa

. So to solve a rational inequality like aaaaaaaaaa aaa aaaaaa

to determine

the intervals that satisfy the inequality. List the guidelines for solving rational inequalities. aa aaaa aaa aaaaa aa aaa aaaaa aaaaaaa aaa aaaaaaaaaa aa aaaa aaa aaaaaaa aaaaa aaaaaa aa aaa aaaa aa aaa aaaaaaaaaa aaaaaaa aaaaa aaa aaaaaaaaa aa a aaaaaa aaaaaaaaaaaa aaaa aaaaaa aaaaaaaaa aaa aaaaaaaaaaaa aaaaaa aaa aaaaaaaaa aaa aaaaaaaaaaa aaaa aaaaaaaaaaa aaaaaaaa aaa aaaa aaaa aaa aaa aaaaaaa aaaa aaaa aaa aaaaaaaaaa aaaa aaa aaaaaaaaa aaaaaaaaaa aa aaa aaa aaaaaaaaaaa aaaa a aaaaa aa aaaaaaaa aaa aaaa aaaaaa aa aaaa a aaaaa aa aaaaaaa aa aaa aaaaa aa aaaa aaaaaa aa aaaa aaaaaaaaa aa aaa aaaa aaa aa aaa aaaaa aaaaaaaaa aaa aaaa aa aaa aaaaaaaa aaaaaaaa aa aaaa aaaaaaaaa aaaa aaaaaa aaaaaaaaa aaa aaaaaaaaa aa aaa aaaaaaaaaa aaaa aaa aaaa aaa aa aaa aaaaaa aaaaa aaaaaaa aaa aaaaaaaaa aa aaaaa aaaaaaaaa aaaaaaa aaa aaaaaaaaaaa aaaaa aaa aaaaaa aa aaa aaaaaaaaaa aaaaaaaa a aa aaaa

Example 2:

Solve the inequality

x 2  x  12 0. x 1

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Name ___________________________________________________________ Date ____________

Chapter 4 4.1

Exponential and Logarithmic Functions

Exponential Functions

I. Exponential Functions True or false? The Laws of Exponents are true when the exponents are real numbers.

aaaa a

The exponential function with base a is defined for all real numbers x by

aaaa a a

,

where a > 0 and a ≠ 1. Example 1:

Let f ( x)  6 x . Evaluate the following:

f (3) (b) f (2) (a)

f ( 3)

(c)

aaa aaa aaa aaaaaaaaa aaa aaaaaaaa

II. Graphs of Exponential Functions x The exponential function f ( x)  a , (a  0, a  1) has domain

The line y = 0 (the x-axis) is a

rapidly. If a > 1, then f

aaa aa

.

. If 0 < a < 1 , then f

aaaaaaaaaa aaaaaaaaa aa a

aaaaaaaaa Example 2:

and range

a

aaaaaaaaa

rapidly.

1 Sketch the graph of the function f ( x)  3 .

y 5

3

1 -5

-3

-1 -1

1

3

5

x

-3

-5

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Precalculus:

Copyright © C engage Lear ning. All rights res er ved.

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Mathematics for Calculus, 7 Edition

77

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CHAPTER 4

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Exponential and Logarithmic Functions

III. Compound Interest In terms of an investment earning compound interest, the

aaaaaaaaa

P is the amount of

money that is initially invested. Compound interest is calculated by the formula

where

A(t) =

aaaaaa aaaaa a aaaaa

P=

aaaaaaaaa

r=

aaaaaaaa aaaa aaa aaaa

n=

. . .

aaaaaa aa aaaaa aaaaaaaa aa aaaaaaaaaa aaa aaaa

t=

aaaaaa aa aaaaa

.

.

If an investment earns compound interest, then the annual percentage yield (APY) is the

aaaaaa

aaaaaaaa aaaa aaaa aaaaaa aaa aaaa aaaaaa aa aaa aaa aa aaa aaaa

y

.

y

x

y

x

x

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SECTION 4.2

|

The Natural Exponential Function

79

Name ___________________________________________________________ Date ____________

4.2

The Natural Exponential Function

I. The Number e  1 The number e is defined as the value that 1    n The approximate value of e is

n

aaaaaaaaaa aa a aaaaaaa aaaaa

aaaaaaaaaaaaaaaaaaaaaa

It can be shown that e is a(n)

.

.

aaaaaaaaaa aaaaaa

, so we cannot write its exact

value in decimal form.

II. The Natural Exponential Function a

The natural exponential function is the exponential function

aaaa a a

It is often referred to as the exponential function. Example 1:

Evaluate the expression correct to five decimal places: 4e0.25 aaaaaaa

III. Continuously Compounded Interest Continuously compounded interest is calculated by the formula where

A(t) =

aaaaaa aaaaa a aaaaa

P=

aaaaaaaaa

r=

aaaaaaaa aaaa aaa aaaa

t=

aaaaaa aa aaaaa

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. . . .

Mathematics for Calculus, 7th Edition

with base e.

80

|

CHAPTER 4

Example 2:

Exponential and Logarithmic Functions

Find the amount after 10 years if $5000 is invested at an interest rate of 8% per year, compounded continuously. aaaaaaaaaa

y

y

x

y

y

x

y

x

x

y

x

x

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SECTION 4.3

|

81

Logarithmic Functions

Name ___________________________________________________________ Date ____________

4.3

Logarithmic Functions

I. Logarithmic Functions Every exponential function f ( x) = a x , with a > 0 and a ≠ 1, is a the Horizontal Line Test and therefore has

aaaaaaaaa aaaaaaaa aa aaaaaaa aaaaaaaa

inverse function f 1 is called the

by . The

aaaaaaaaaaa aaaaaaaa aaaa aaaa a

and is

denoted by log a . For the definition of the logarithmic function, let a be a positive number with a ≠ 1. The logarithmic function with base a, denoted by log a , is defined by

So log a x is the

aaaaaaaa

to which the base a must be raised to give

a

.

Complete each of the following properties of logarithms. 1. log a 1 =

a

2. log a a =

.

a

x 3. log a a =

. a

log x 4. a a =

a

. .

II. Graphs of Logarithmic Functions 1 Recall that if a one-to-one function f has domain A and range B, then its inverse function f has domain

a domain

and range

x . Since the exponential function f ( x) = a with a ≠ 1 has

a

1 and range (0, ) , we conclude that its inverse function, f ( x)  log a x , has domain

aaa aa

and range

a

.

1 The graph of f ( x)  log a x is obtained by

aaa aaaa a a a

aaa

aaaa aaa aaaaa aa a aa

.

The x-intercept of the function y = loga x is

a

. The

vertical asymptote of y = loga x .

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Mathematics for Calculus, 7th Edition

aaaaaa

is a

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CHAPTER 4

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Exponential and Logarithmic Functions

y

Sketch the graph of the function f ( x) = log3 x .

Example 1:

5

3

1 -5

-3

-1 -1

1

3

5

x

-3

-5

III. Common Logarithms The logarithm with base 10 is called the a

aaaaaa aaaaaaaaa

and is denoted by

. Evaluate log 30 .

Example 2:

aaaaa

IV. Natural Logarithms The logarithm with base e is called the a

aaaaaaa aaaaaaaaa

and is denoted by

.

The natural logarithmic function y = ln x is the inverse function of aaaaaaaa a a aa

aaa aaaaaaa aaaaaaaaaaa

.

Complete each of the following properties of natural logarithms. 1. ln1 = x 3. ln e =

Example 3:

a

. a

.

2. ln e =

a

.

ln x 4. e =

a

.

Evaluate ln 30 . aaaaaa

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SECTION 4.4

83

Law s of Logarithms

Name ___________________________________________________________ Date ____________

4.4

Laws of Logarithms

I. Laws of Logarithms Let a be a positive number, with a  1 . Let A, B, and C be any real numbers with A > 0 and B > 0. Complete each of the following Laws of Logarithms and give a description of each law. 1. log a ( AB) = aaaaaaaaaa aa aaa aaaaaaaa

a

 A 2. log a    a B aaa aaaaaaaaaa aa aaa aaaaaaaa

..aaa aaaaaaaaa aa a aaaaaaa aa aaaaaaa aa aaa aaa aa aaa

. aaa aaaaaaaaa aa a aaaaaaaa aa aaaaaaa aa aaa aaaaaaaaaa aa

3. log a ( Ac ) = a aaaaa aaa aaaaaaaaa aa aaa aaaaaaa

. aaa aaaaaaaaa aa a aaaaa aa a aaaaaa aa aaa aaaaaaaa

II. Expanding and Combining Logarithmic Expressions The process of writing the logarithm of a product or a quotient as the sum or difference of logarithms is called aaaaaaaaa a aaaaaaaaaaa aaaaaaaaaa

.

The process of writing sums and differences of logarithms as a single logarithm is called aaaaaaaaa aaaaaaaaaaa aaaaaaaaaaa

.

Note that although the Laws of Logarithms tell us how to compute the logarithm of a product or quotient, there is no corresponding rule for the

Example 1:

aaaaaaaaa aa a aaa aa a aaaaaaaaaa

Expand the logarithmic expression ln

x4 y . 3

a aa a a aa a a aa a

Example 2:

Combine the logarithmic expression 2log w + 3log(2w +1) into a single logarithm. a a aaaaa aaa a aa a

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84

CHAPTER 4

|

Exponential and Logarithmic Functions

III. Change of Base Formula The Change of Base Formula is given by

logb x 

log a x log a b

.

Describe the advantage to using the Change of Base Formula. aa aaa aaa aaaaaaaa a aaaaaaaaa aa aaa aaaa aa aaaaa aaa aaaaaa aa aaaa aaaaaaa aa aaaaaaa aaa aaaaaaaaa aa aaaaa aa aaaaaa aaaaaaaaaa aa aaaaaaa aaaaaaaaaa aaa aaaa aaaaa a aaaaaaaaaaa

Explain how to use a calculator to evaluate log13 150 . aaaaa aaa aaaaaaaaaaaaaa aaaaaaaa aaaaaaaa aaaaaa aaaa aaaa a aaaa aaa aa aaa aaaa a aaa aaaa aaa aaaaaaa aaaa aa aaa aaaaa

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SECTION 4.5

|

Exponential and Logarithmic Equations

85

Name ___________________________________________________________ Date ____________

4.5

Exponential and Logarithmic Equations

I. Exponential Equations An exponential equation is one in which

aaa aaaaaaaa aaaaaa aa aaa aaaaaaaa

.

List the guidelines for solving exponential equations. aa aaaaaaa aaa aaaaaaaaaaa aaaaaaaaaa aa aaa aaaa aa aaa aaaaaaaaa aa aaaa aaa aaaaaaaaa aa aaaa aaaaa aaaa aaa aaa aaaa aa aaaaaaaaaa aa aaaaaa aaaa aaa aaaaaaaaaa aa aaaaa aaa aaa aaaaaaaaa

Example 1:

Find the solution of the equation 52 x1  20 , correct to six decimal places. aaaaaaaa

II. Logarithmic Equations A logarithmic equation is one in which

a aaaaaaaaa aa aaa aaaaaaaa aaaaaa

List the guidelines for solving logarithmic equations. aa aaaaaaa aaa aaaaaaaaaaa aaaa aa aaa aaaa aa aaa aaaaaaaaa aaa aaaaa aaaaa aaaa aa aaaaaaa aaa aaaaaaaaaaa aaaaaa aa aaaaa aaa aaaaaaaa aa aaaaaaaaaaa aaaa aaa aaaaa aaa aaaa aa aaaa aaaa aa aaa aaaaaaaaaa aa aaaaa aaa aaa aaaaaaaaa Example 2:

Solve 12ln( x  5)  2  22 for x, correct to six decimal places. aaaaaaaa

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III. Compound Interest Describe how to find the length of time rquired for an investment to double. aaaaaaa aaaa aaaaa

Example 3:

How long will it take for a $12,000 investment to double if it is invested at an interest rate of 6% per year and if the interest is compounded continuously. aaaaa aaaaa

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SECTION 4.6

|

Modeling w ith Exponential and Logarithmic Functions

87

Name ___________________________________________________________ Date ____________

4.6

Modeling with Exponential and Logarithmic Functions

I. Exponential Growth (Doubling Time) If the initial size of a population is n0 and the doubling time is a, then the size of the population at time t is given by

a

where a and t are measured in the same time units (minutes, hours,

days, years, and so on).

Example 1:

Suppose a community’s population doubles every 20 years. Initially there are 500 members in the community. (a) Find a model for the community’s population after t years. (b) How many community members are there after 30 years? (c) When will the population of the community reach 10,000?

aaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaa

II. Exponential Growth (Relative Growth Rate) A population’s relative growth rate r is the

aaaa aa aaaaaaaa aaaaaa aaaaaaaaa aa a aaaaaaaaaa aa

aaa aaaaaaaaaa aa aaa aaaa

.

A population that experiences exponential growth increases according to the model

a

.

where n(t) =

aaaaaaaaaa aa aaaa a

n0 =

aaaaaaa aaaa aa aaa aaaaaaaaaa

r= t=

, ,

aaaaaaaa aaaa aa aaaaaa aaaaaaaaaa aa a aaaaaaa aa aaa aaaaaaaaaaa aaaa

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Mathematics for Calculus, 7th Edition

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CHAPTER 4

Example 2:

|

Exponential and Logarithmic Functions

The initial population of a colony is 800. If the colony has a relative growth rate of 15% per year, find a function that models the population after t years.

III. Radioactive Decay The rate of radioactive decay is proportional to

aaa aaaa aa aaa aaaaaaaaa

Physicists express the rate of radioactive decay in terms of

aaaaaaa

. , the time it

takes for a sample of the substance to decay to half its original mass. The radioactive decay model states that if m0 is the initial mass of a radioactive substance with half-life h, then the mass remaining at time t is modeled by the function r=

aaa aaaa

a

, where

is the relative decay rate.

IV. Newton’s Law of Cooling Newton’s Law of Cooling states that the rate at which an object cools is

aaaaaaaaaaaa aa aaa

aaaaaaaaaaa aaaaaaaaaa aaaaaaa aaa aaaaaa aaa aaa aaaaaaaaaaaaa aaaaaaaa aaaa aaa aaaaaaaaaaa aaaaaaaaaa aa aaa aaa aaaaa

.

Newton’s Law of Cooling If D0 is the initial temperature difference between an object and its surroundings, and if its surroundings h ave temperature T s, then the temperature of the object at time t is modeled by the function

where k is a positive constant that depends on the type of object.

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SECTION 4.7

|

89

Logarithmic Scales

Name ___________________________________________________________ Date ____________

4.7

Logarithmic Scales

When a physical quantity varies over a very large range, it is often convenient to aaaaaaaaa

aaaa aaa

in order to have a more manageable set of numbers. On a logarithmic

scale, numbers are represented by

aaaaa aaaaaaaaaa

.

I. The pH Scale The acidity of a solution is given by its pH, defined as

+

a

, where [H ] is

the concentration of hydrogen ions measured in moles per liter (M).

Solutions with a pH of 7 are defined as

aaaaaa

and those with pH > 7 are

aaaaa

a aaaaaa aa aa

.

, those with pH < 7 are

aaaaa

,

. When the pH increases by one unit, [H+] decreases by

II. The Richter Scale The Richter Scale defines the magnitude M of an earthquake to be the

aaaaaaaa

a

, where I is

of the earthquake (measured by the amplitude of a seismograph reading taken

100 km from the epicenter of the earthquake) and S is the aaaaaaaaaa

aaaaaaaaa aa a aaaaaaaaaa

(whose amplitude is 1 micron =

The magnitude of a standard earthquake is

a

4 10 −

cm). .

III. The Decibel Scale According to the Decibel Scale, the decibel level B, measured in decibels (dB), is defined as

B  10log

I I0

12 where I0 is a reference intensity and I 0  10 W/m2 (watts per square meter).

The decibel level of the barely audible reference sound is The threshold of pain is about

aaa aa

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a aa

.

.

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Exponential and Logarithmic Functions

Additional notes

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Name ___________________________________________________________ Date ____________

Chapter 5 5.1

Trigonometric Functions: Unit Circle Approach

The Unit Circle

I. The Unit Circle The unit circle is equation is

Example 1:

aaa aaaaaa aa aaaaaa a aaaaaaaa aa aaa aaaaaa aa aaaaaaaa a

. Its

.

 2 2 Show that the point P  ,  2 2 

  is on the unit circle. 

II. Terminal Points on the Unit Circle Suppose t is a real number. Mark off a distance t along the unit circle, starting at the point (1, 0) and moving in a

aaaaaaaaaaaaaaa aaaaaaaaa

aaaaaaaaa aaaaa

if t is positive or in a

if t is negtaive. In this way we arrive at a point P(x, y), called the

aaaaaaaaa aaaaaaaa

determined by the real number t.

The circumference of the unit circle is C =

aa

unit circle, it travels a distance of circle, it travels a distance of

. To move a point halfway around the

a

. To move a quarter of the distance around the

aaa

.

List the terminal point determined by each given value of t. t 0

Terminal point determined by t aaa aa

 6



4

 3

 2

aaa aa

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Mathematics for Calculus, 7 Edition

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CHAPTER 5

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Trigonometric Functions: Unit Circle Approach

III. The Reference Number Let t be a real number. The reference number t associated with t is

aaa aaaaaaaa aaaaaaaa

aaaaa aaa aaaa aaaaaa aaaaaaa aaa aaaaaaaa aaaaa aaaaaaaaaa aa a aaa aaa aaaaaa To find the reference number t , it is helpful to know

.

aaa aaaaaaaaa aaaaa aaa aaaaaaaa aaaaa

aaaaaaaaaa aa a aaaa

. If the terminal point lies in quadrants I or IV,

where x is positive, we find t by

aaaaaa aaaaa aaaaaaa aa aaa aaaaaaaa aaaaaa

If it lies in quadrants II or III, where x is negative, we find t by aaaaaaaa aaaaaa

.

aaaaaa aaaaa aaa aaaaaa aa aaa

.

List the steps for finding the terminal point P determined by any value of t. aa aaaa aaa aaaaaaaaa aaaaaa a aa aaaa aaa aaaaaaaa aaaaa aaaa aa aaaaaaaaaa aa a aa aaa aaaaaaaa aaaaa aaaaaaaaaa aa a aa a aaaaa aaa aaaaa aaa aaaaaa aaaaaaaaa aa aaa aaaaaaaa aa aaaaa aaaa aaaaaaaa aaaaa aaaaa

Since the circumference of the unit circle is 2π, the terminal point determined by t is the same as that determined by

a a aa aa a a aa

. In general, we can add or subtract

aa

any

number of times without changing the terminal point determined by t. y

y

x

y

x

x

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SECTION 5.2

|

93

Trigonometric Functions of Real Numbers

Name ___________________________________________________________ Date ____________

5.2

Trigonometric Functions of Real Numbers

I. The Trigonometric Functions Let t be any real number and let P(x, y) be the terminal point on the unit circle determined by t. Complete the definitions of each trigonometric function.

sin t =

a

tan t =

a a a aa a aa

.

csct =

a a a aa a aa

.

sect =

a a a aa a aa

.

cot t =

a a a aa a aa

.

.

a

cost =

.

Because the trigonometric functions can be defined in terms of the unit circle, they are sometimes called the aaaaaaaa aaaaaaaaa

.

Complete the following table of special values of the trigonometric function. t 0

sin t a

cost a

tan t a

csct aa

sect a

cot t aa

 6



a

a

4  3

 2

a

a

The domain of the sine function is

aa aaa aaaa aaaaaaa

The domain of the cosine function is

a

aa

a

.

aaa aaaa aaaaaaa

.

The domain of the tangent function is

aaa aaaa aaaaa aaaaa aaaa aaa a aa aaa aaa aaaaaaa a

.

The domain of the secant function is

aaa aaaa aaa aaaaaaaa aaaa aaa a aa aaa aaa aaaaaaa a

.

The domain of the cotangent function is The domain of the cosecant function is

aaa aaaa aaaaaaa aaaaa aa aaa aaa aaaaaaa a aaa aaaa aaaaaaa aaaaa aa aaa aaa aaaaaaa a

. .

II. Values of the Trigonometric Functions To compute other values of the trigonometric functions, we first The signs of the trigonometric functions depend on aaaaa aa a aaaa

Note Taking Guide for Stewart/Redlin/Wats on Precalculus: Copyright © Cengage Learning. All rights reserved.

aaaaaaaa aaaaa aaaaa aaa aaaaaa

aa aa aaaaa aaa aaaaaaaa

.

Mathematics for Calculus, 7th Edition

.

94

|

CHAPTER 5

Trigonometric Functions: Unit Circle Approach

Complete the table listing the signs of the trigonometric functions. Quadrant I II III IV

Positive Functions aaa aaaa aaa aaaa aaa aaaa aaa

Negative Functions aaaa aaaa aaaa aaaa aaa aaaa aaaa aaaa aaa aaaa aaaa aaaa aaa

Since the trigonometric functions are defined in terms of the coordinates of terminal points, we can use the aaaaaaaaa aaaaaa

Example 1:

to find values of the trigonometric functions.

 7 Find the value of tan   4

 . 

aa

Complete each reciprocal relation. csct =

a

.

sect =

The odd trigonometric functions are

a

.

cot t =

a

.

aaaaa aaaaaaaaa aaaaaa aaa aaaaaaaaa

The even trigonometric functions are

.

aaaaaa aaa aaaaaa

.

III. Fundamental Identities Complete each of the following trigonometric identities.

1 = sin t

aaa a

.

1 = cos t

aaa a

.

sin t = cos t

aaa a

.

cos t = sin t

aaa a

.

sin 2 t + cos2 t =

1 + cot 2 t =

a

.

aaa a

.

a

tan 2 t + 1 =

1 = tan t

a

aaa a

aaa a

.

.

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SECTION 5.3

|

Trigonometric Graphs

95

Name ___________________________________________________________ Date ____________

5.3

Trigonometric Graphs

I. Graphs of Sine and Cosine A function f is periodic if there is a positive number p such that The least such positive number (if it exists) is the

aaaaaa

graph of f on any interval of length p is called The functions sine and cosine have period Example 1:

aaa a aa a aaaa aaaaa a

of f. If f has period p, then the

aaa aaaaaaaa aaaaaa aa a aa

.

.

.

Sketch the basic sine curve between 0 and 2π.

y

x

Example 2:

Sketch the basic cosine curve between 0 and 2π.

y

x

II. Graphs of Transformations of Sine and Cosine For the functions y = a sin x and y = a cos x , the number a is called the

aaaaaaaaa

is the largest value these functions attain. The sine and cosine curves y = a sin kx and y = a cos kx , (k > 0), have amplitude a and period aaaa

. An appropriate interval on which to graph one complete period is aaa aaaaa

.

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and

96

CHAPTER 5

|

Trigonometric Functions: Unit Circle Approach

The value of k has the effect of

aaaaaaaaa aaa aaaaa aaaaaaaaaaaa

aaaaaaaaaa aaa aaaaa aaaaaaaaaaaa

if k > 1 or the effect of

if k < 1.

The sine and cosine curves y  a sin k ( x  b) and y  a cos k ( x  b) , (k > 0), have amplitude a , period 2π/k, and horizontal shift

a

. An appropriate interval on which to graph one complete period is

aaa a a aaaaaaa

Example 3:

.

  Find the amplitude, period, and horizontal shift of y  2cos  x   . 6  aaaaaaaaa a aa aaaaaa a aaa aaaaaaaaaa aaaaa a aaa

III. Using Graphing Devices to Graph Trigonometric Functions When using a graphing device to graph a function, it is important to

aaaaaa aaa aaaaaaa aaaaaaaaa

aaaaaaaaa aa aaaaa aa aaaaaaa a aaaaaaaaaa aaaaa aa aaa aaaaaaaa

y

.

y

x

y

x

x

Homework Assignment Page(s) Exercises

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|

SECTION 5.4

97

More Trigonometric Graphs

Name ___________________________________________________________ Date ____________

5.4

More Trigonometric Graphs

I. Graphs of Tangent, Cotangent, Secant, and Cosecant The functions tangent and cotangent have period

a

The functions cosecant and secant have period

aa

The graph of y = tan x approaches the vertical lines x = aaaaaaaaaa

.

 2

.

and x  

 2

, so these lines are

aaaaaaaa

.

The graph of y = cot x is undefined for

a a aa

, with n an integer, so its graph has

vertical asymptotes at these values. To graph the cosecant and secant functions, we use the identities

csc x 

1 aaa sin x .

The graph of y = csc x has vertical asymptotes at

a a aaa aaa a aa aaaaaaa

The graph of y = sec x has vertical asymptotes at

a a aaaaa a aaa aaa a aa aaaaaaa

.

II. Graphs of Transformations of Tangent and Cotangent The functions y = a tan kx and y = a cot kx , (k > 0), have period

aaa

To graph one period of y = a tan kx , an appropriate interval is

To graph one period of y = a cot kx , an appropriate interval is

.

a

.

a

.

III. Graphs of Transformations of Cosecant and Secant The functions y = a csc kx and y = a sec kx , (k > 0), have period

aaaa

An appropriate interval on which to graph one complete period is

aaa aaaaa

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. .

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CHAPTER 5

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Trigonometric Functions: Unit Circle Approach

Additional notes

y

y

x

y

y

x

y

x

x

y

x

x

Homework Assignment Page(s) Exercises

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SECTION 5.5

|

99

Inv erse Trigonometric Functions and Their Graphs

Name ___________________________________________________________ Date ____________

5.5

Inverse Trigonometric Functions and Their Graphs

I. The Inverse Sine Function The inverse sine function is the function sin 1 with domain

aaaa aa

defined by sin 1 x  y

aaaaaa aaaa The inverse sine function is also called

aaaaaaa

and range

 sin y  x . , denoted by

y  sin 1 x is the number in the interval [ / 2,  / 2] whose sine is

aaaaaa

a

.

.

The inverse sine function has these cancellation properties:

sin(sin 1 x) 

a

for 1  x  1

sin 1 (sin x) 

a

for 

 2

x

 2

II. The Inverse Cosine Function The inverse cosine function is the function cos 1 with domain range

aaa aa

aaaa aa

1 defined by cos x  y

 cos y  x .

aaaaaaaa

, denoted by

The inverse cosine function is also called

y  cos1 x is the number in the interval [0,  ] whose cosine is

a

and

aaaaaa

.

.

The inverse cosine function has these cancellation properties:

cos(cos1 x) 

a

for 1  x  1

cos1 (cos x) 

a

for 0  x  

III. The Inverse Tangent Function 1 The inverse tangent function is the function tan with domain

aaaaaa aaaa

1 defined by tan x  y

The inverse tangent function is also called

aaaaaaaaa

  y  tan 1 x is the number in the interval   ,  whose tangent is  2 2

Note Taking Guide for Stewart/Redlin/Wats on Precalculus: Copyright © Cengage Learning. All rights reserved.

a

and range

 tan y  x . , denoted by a

Mathematics for Calculus, 7th Edition

aaaaaa .

.

100

CHAPTER 5

|

Trigonometric Functions: Unit Circle Approach

The inverse tangent function has these cancellation properties:

tan(tan 1 x) 

a

for x 

tan 1 (tan x) 

a

for 

 2

x

 2

IV. The Inverse Secant, Cosecant, and Cotangent Functions To define the inverse functions of the secant, cosecant, and cotangent functions, we restrict the domain of each function to

a aaa aa aaaaa aa aa aaaaaaaaaa aaa aa aaaaa aa aaaaaaa aaa

aaa aaaaaa

.

y

y

x

y

y

x

y

x

x

y

x

x

Homework Assignment Page(s) Exercises

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SECTION 5.6

Modeling Harmonic Motion

101

Name ___________________________________________________________ Date ____________

5.6

Modeling Harmonic Motion

Periodic behavior is

aaaaaaaa aaaa aaaaaaa aaaa aaa aaaa aaaaa

.

I. Simple Harmonic Motion A cycle is

aaa aaaaaaaa aaaaaaaaa aa aa aaaaaa

.

If the equation describing the displacement y of an object at time t is y = a sin t or y = a cos t , then the object is in

aaaaaa aaaaaaaa aaaaaa

.

In this case, the amplitude, which is the maximum displacement of the object, is given by The period, which is

2



aaa aaaa aaaaaaaa aa aaaaaaaa aaa aaaaa

. The frequency, which is

given by

aaa

.

, is given by

aaa aaaaaa aa aaaaaa aaa aaaa aa aaaa

, is

 . 2

The functions y = a sin 2 t or y = a cos 2 t have frequency

a

.

In general, the sine or cosine functions representing harmonic motion may be shifted horizontally or vertically. In this case, the equations take the form vertical shift b indicates that

a aa a

. The

aaa aaaaaaaaa aaaaaa aaaaaa aa aaaaaaa aaaaa a

The horizontal shift c indicates

.

aaa aaaaaaaa aa aaa aaaaaa aa a a a

.

II. Damped Harmonic Motion In a hypothetical frictionless environment, a spring will oscillate in such a way that its amplitude will not change. In the presence of friction, however, the motion of the spring eventually that is, the amplitude of the motion called

aaaaaaaaa aaaa aaaa

aaaaaa aaaaaaaa aaaaaa

aaaa aaaa . Motion of this type is

.

If the equation describing the displacement y of an object at time t is a the

2 /  is the

,

a

or

(c > 0), then the object is in damped harmonic motion. The constant c is

aaaaaaa aaaaaaaa aaaaaa

,

a

is the initial amplitude, and

.

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CHAPTER 5

Trigonometric Functions: Unit Circle Approach

Damped harmonic motion is simply harmonic motion for which the amplitude is governed by the function a

.

III. Phase and Phase Difference Any sine curve can be expressed in the following equivalent forms:

y  A sin  kt  b  , where the phase is



a

, and



y  A sin k t  bk , where the horizontal shift is The time b / k is called the

aaa

if b  0 (because P is behind, or lags, Q by

aaa aaaa

b / k time units) and is called the

.

if b  0 .

aaaa aaaa

It is often important to know if two waves with the same period (modeled by sine curves) are or

aaa

.

If the phase difference is a multiple of 2, the waves are aaa aa aaaaa

a

. For the curves y1  A sin(kt  b) and y2  A sin(kt  c) , the phase difference

aaa aa aaaaa

is

aa aaaa

aa aaaaa

, otherwise the waves are

. If two sine curves are in phase, then their graphs

y

y

x

aaaaaaaa

.

y

x

x

Homework Assignment Page(s) Exercises

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Name ___________________________________________________________ Date ____________

Chapter 6 6.1

Trigonometric Functions: Right Triangle Approach

Angle Measure

An angle consists of two rays with a common

aaaaaa

.

We often interpret an angle as a rotation of the ray R1 onto R2 . In this case, R1 is called the aaaa

, and R2 is called the

rotation is

aaaaaaaa aaaa

aaaaaaaaaaaaaaaa

clockwise, the angle is considered

aaaaaaa of the angle. If the

, the angle is considered positive, and if the rotation is aaaaaaaa

.

I. Angle Measure The measure of an angle is

aaa aaaaaa aa aaaaaaaa aaaaa aaa aaaaaa aaaaaaaa

aa aaaa a aaaa a

.

One unit of measurement for angles is the degree. An angle of measure 1 degree is formed by aaa aaaaaaa aaaa aaaaa aa a aaaaaaaa aaaaaaaaaa

aaaaaaaa

.

If a circle of radius 1 is drawn with the vertex of an angle at its center, then the measure of this angle in radians (abbreviated rad) is

aaa aaaaaa aa aaa aaa aaaa aaaaaaaa aaa aaaaa

For a circle of radius 1, a complete revolution has measure measure

a

aa

rad; a straight angle has

rad; and a right angle has measure

To convert degrees to radians, multiply by

aaaaa

To convert radians to degrees, multiply by

aaaaa

.

aaa

rad.

. .

II. Angles in Standard Position An angle is in standard position if it is drawn in the xy-plane with

aaa aaaaaa aa aaa aaaaaa

aaa aaa aaaaaaa aaaa aa aaa aaaaaaaa aaaaaa Two angles in standard position are coterminal if

. aaaaa aaaaa aaaaaa

.

III. Length of a Circular Arc In a circle of radius r, the length s of an arc that subtends a central angle of θ radians is or, solving for θ, we get

a a aaa

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a a aa

,

.

th

Mathematics for Calculus, 7 Edition

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104

CHAPTER 6

|

Trigonometric Functions: Right Triangle Approach

IV. Area of a Circular Sector

In a circle of radius r, the area A of a sector with a central angle of θ radians is

a

.

V. Circular Motion Linear speed is

aaa aaaa aa aaaaa aaa aaaaaaaa aaaaaaaa aa aaaaaa aa aaaaaa aaaaa aa aaa aaaaaaaa

aaaaaaaa aaaaaaa aa aaa aaaa aaaaaaa

.

aaa aaaa aa aaaaa aaa aaaaaaa aaaaa a aa

Angular speed is

aaaaaaaaa aa aaaaaaa aaaaa aa aaa

aaaaaa aa aaaaaaa aaaa aaaaa aaaaaaa aaaaaaa aa aaa aaaa aaaaaaa

.

Suppose a point moves along a circle of radius r and the ray from the center of the circle to the point traverses θ radians in time t. Let s = rθ be the distance the point travels in time t. Then the speed of the object is given by

Angular speed

a

.

Linear speed

a

.

If a point moves along a circle of radius r with angular speed ω, then its linear speed v is given by a

.

y

y

x

y

x

x

Homework Assignment Page(s) Exercises

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SECTION 6.2

|

105

Trigonometry of Right Triangles

Name ___________________________________________________________ Date ____________

6.2

Trigonometry of Right Triangles

I. Trigonometric Ratios Consider a right triangle with θ as one of its acute angles. Complete the following trigonometric ratios.

hypotenuse

opposite

θ adjacent

sin  

csc 

cos  

hypotenuse

sec 

hypotenuse

tan  

hypotenuse

cot  

hypotenuse

hypotenuse

hypotenuse

II. Special Triangles; Calculators Complete the following table of values of the trigonometric ratios for special angles.

𝜃 in degrees

30° 45°

60°

𝜃 in radians

sin 

cos

tan 

csc

sec

cot 

 6

 4  3

Calculators give the values of sine, cosine, and tangent; the other ratios can be easily calculated from these by using

aaaaaaaaaa aaaaaaaaaaaaa

.

III. Applications of Trigonometry of Right Triangles A triangle has six parts: means to

aaaaa aaaaaa aaa aaaaa aaaaa

. To solve a triangle

aaaaaaaaa aaa aa aaa aaaaa aaaa aaa aaaaaaaaaaa aaa aaaaa aaa aaaaaaaaa aaaa aaa aa aaaaaaaaa

aaa aaaaaaa aa aaa aaaaa aaaaa aaa aaa aaaaaaaa aa aaa aaaaa

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106

CHAPTER 6

Example 1:

|

Trigonometric Functions: Right Triangle Approach

Find the value of a in the given triangle. aa

28 60° a

If an observer is looking at an object, then the line from the eye of the observer to the object is called aaaa aa aaaa

aaa

. If the object being observed is above the horizontal, then the angle between

the line of sight and the horizontal is called

aaa aaaaa aa aaaaaaa

below the horizontal, then the angle between the line of sight and the horizontal is called aaaaaaaaaa

. If the object is aaa aaaaa aa

. If the line of sight follows a physical object, such as an inclined plane or

hillside, we use the term

aaaaa aa aaaaaaaaaaa

.

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SECTION 6.3

|

107

Trigonometric Functions of Angles

Name ___________________________________________________________ Date ____________

6.3

Trigonometric Functions of Angles

I. Trigonometric Functions of Angles P(x, y) r

y

θ O

x

Q

Let θ be an angle in standard position and let P(x, y) be a point on the terminal side. If r = x 2 + y 2 is the distance from the origin to the point P(x, y), then the definitions of the trigonometric functions are

sin  

cos  

tan  

csc  

sec  

cot  

The angles for which the trigonometric functions may be undefined are the angles for which either the x- or ycoordinate of a point on the terminal side of the angle is 0. These angles are called that is, angles that are

aaaaaaaaaa

aaaaaaaaa aaaaaa

,

with the coordinate axes.

It is a crucial fact that the values of the trigonometric functions

aa aaa aaaaaa

on the

choice of the point P(x, y).

II. Evaluating Trigonometric Functions at Any Angle Complete the following table to indicate which trigonometric functions are positive and which are negative in each quadrant. Q uadrant I II III IV

Positive Functions aaa aaaa aaa aaaa aaa aaaa aaa

Negative Functions aaaa aaaa aaaa aaaa aaa aaaa aaaa aaaa aaa aaaa aaaa aaaa aaa

Let θ be an angle in standard position. The reference angle  associated with θ is aaaaaa aa aaa aaaaaaaa aaaa aa a aaa aaa aaaaaa

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Mathematics for Calculus, 7th Edition

aaa aaaaa aaaaa

108

|

CHAPTER 6

Trigonometric Functions: Right Triangle Approach

List the steps for finding the values of the trigonometric functions for any angle θ. aa aaaa aaa aaaaaaaaa aaaaa a aaaaaaaaaa aaaa aaa aaaaa aa aa aaaaaaaaa aaa aaaa aa aaa aaaaaaaaaaaaa aaaaaaaa aa a aa aaaaaa aaa aaaaaaaa aa aaaaa a aaaaa aa aaa aaaaa aa aaa aaaaaaaaaaaaa aaaaaaaa aa a aa aaa aaaaa aaaaaa aaaaaaaa aaa aaaaa aa aaa aaaaa aa aaa aaaaaaaaaaaaa aaaaaaaa aa aa

III. Trigonometric Identities Complete each of the following fundamental trigonometric identities.

1 = sin 

aaa a

.

1 = cos 

aaa a

.

sin  = cos 

aaa a

.

cos  = sin 

aaa a

.

sin 2  + cos2  = 1 + cot 2  =

a a

aaa a

.

tan 2  + 1 =

1 = tan 

aaaa a

aaa a

.

.

.

IV. Areas of Triangles The area A of a triangle with sides of lengths a and b and with included angle θ is

A

1 ab sin  2

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SECTION 6.4

|

109

Inv erse Trigonometric Functions and Right Triangles

Name ___________________________________________________________ Date ____________

6.4

Inverse Trigonometric Functions and Right Triangles

I. The Inverse Sine, Inverse Cosine, and Inverse Tangent Functions The sine function, on the restricted domain the restricted domain domain

aaaaa aaaa

aaa aa

; the cosine function, on

; and the tangent function, on the restricted

aaaaaa aaaa

; are all one-to-one and so have inverses.

The inverse sine function is the function sin 1 with domain

aaaa aa

defined by sin 1 x  y

aaaaaa aaaa

 sin y  x .

The inverse cosine function is the function cos 1 with domain range

aaaa aa

defined by cos1 x  y

aaa aa

The inverse tangent function is the function tan 1 with domain defined by tan

aaaaaa aaaa

1

and range

and

 cos y  x .

a

and range

x  y  tan y  x .

The inverse sine function is also called

aaaaaaa

, denoted by

aaaaaa

.

The inverse cosine function is also called

aaaaaaaa

, denoted by

aaaaaa

.

The inverse tangent function is also called

aaaaaaaaa

, denoted by

aaaaaa

.

II. Solving for Angles in Right Triangles The

aaaaaaa aaaaaaaaaaaaa aaaaaaaaa

can be used to solve for angles in a right

triangle if the lengths of at least two sides are known. Which inverse trigonometric function should be used to find the measure of angle θ in the right triangle below?

12

θ 5 aaaaaaa aaaaaaa

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CHAPTER 6

|

Trigonometric Functions: Right Triangle Approach

III. Evaluating Expressions Involving Inverse Trigonometric Functions 12   Describe a possible solution strategy for evaluating an expression such as sin  tan 1  . 5  

aaaaaaaaaa aaaa aaaaa

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SECTION 6.5

|

The Law of Sines

111

Name ___________________________________________________________ Date ____________

6.5

The Law of Sines

T he trigonometric functions can be used to solve oblique triangles. List the four oblique triangle cases which may be solved. aaaa aa aaa aaaa aaa aaa aaaaaa aaa aaaaa aaaa aa aaaa aaaa aa aaa aaaaa aaa aaa aaaaa aaaaaaaa aaa aa aaaaa aaaaa aaaaa aaaa aa aaa aaaaa aaa aaa aaaaaaaa aaaaa aaaaa aaaa aa aaaaa aaaaa aaaaa

Which of these cases are solved using the Law of Sines? aaaaa a aaa a

I. The Law of Sines The Law of Sines says that

aa aaa aaaaaaaa aaa aaaaaaa aa aaa aaaaa aaa aaaaaaaaaaaa

aa aaa aaaaa aa aaa aaaaaaaaaaaaa aaaaaaaa aaaaaa

.

For a triangle ABC, state the Law of Sines.

II. The Ambiguous Case The triangle situation given in Case 2, in which two sides and the angle opposite one of those sides are known, is sometimes called the ambiguous case because

aaaaa aaa aa aaa aaaaaaaa aaa aaaaaaaaa aa aa

aaaaaaaa aaaa aaa aaaaa aaaaaaaaaa In general, if

aaa a a a

. , we must check the angle and its supplement as possibilities,

because any angle smaller than 180° can be in the triangle. To decide whether either possibility works, check to see whether the resulting sum of the angles exceeds

aaaa

possibilities are compatible with the given information. In that case, aaa aaaaaaaaa aa aaa aaaaaaa

Note Taking Guide for Stewart/Redlin/Wats on Precalculus: Copyright © Cengage Learning. All rights reserved.

. It can happen that both aaa aaaaaaaaa aaaaaaaaa

.

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CHAPTER 6

|

Trigonometric Functions: Right Triangle Approach

Additional notes

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SECTION 6.6

|

The Law of Cosines

113

Name _____________________________________________ ______________ Date ____________

6.6

The Law of Cosines

Which oblique triangle cases are solved using the Law of Cosines? aaaa a aaaaa aaa aaaa a aaaaa

I. The Law of Cosines The Law of Cosines says that in any triangle ABC, these three relationships exist among the angles A, B, and C and their opposite sides a, b, c:

In words, the Law of Cosines says that

aaa aaaaaa aa aaa aaaa aa a aaaaaa aa aaaaa aa aaa aaa

aa aaa aaaaaaa aa aaa aaaaa aaa aaaaaa aaaaa aaaaa aaa aaaaaaa aa aaaaa aaa aaaaa a aaa aaaaaa aa aaa aaaaaaaa aaaaa

.

II. Navigation: Heading and Bearing In navigation a direction is often given as a bearing, that is, as aaaaa aa aaa aaaaa

aa aaaaa aaaaa aaaaaaaa aaaa aaa

.

III. The Area of a Triangle Heron’s Formula for the area of a triangle is an application of

aaa aaa aa aaaaaaa

.

Heron’s Formula states that the area of a triangle ABC is given by

where s  aaaaaaaaa

1 ( a  b  c) 2

and is the semiperimeter of the triangle; that is, s is .

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Mathematics for Calculus, 7th Edition

aaaa aaa

114

CHAPTER 6

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Trigonometric Functions: Right Triangle Approach

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Name ___________________________________________________________ Date ____________

Chapter 7 7.1

Analytic Trigonometry

Trigonometric Identities

An identity is

aa aaaaaaaa aaaa aa aaaa aaa aaa aaaaaa aa aaa aaaaaa

A trigonometric identity is

.

aa aaaaaaaa aaaaaaaaa aaaaaaaaaaaaa aaaaaa

.

List the five trigonometric reciprocal identities.

List the three Pythagorean identities.

List three even-odd identities for trigonometry.

List the six trigonometric cofunction identities.

I. Simplifying Trigonometric Expressions Identities enable us to write the same expression

aa aaaaaaaaa aaaa

rewrite a complicated-looking expression as

. It is often possible to

a aaaa aaaaaaa aaa

trigonometric expressions, we use

. To simplify

aaaaaaaaaa aaaaaa aaaaaaaaaaaaa aaaaaaa aaaaaaa

aaaaaaaaa aaa aaa aaaaaaaaaaa aaaaaaaaaaaaa aaaaaaaaaa

.

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Mathematics for Calculus, 7 Edition

115

116

CHAPTER 7

|

Analytic Trigonometry

II. Proving Trigonometric Identities How can you tell if a given equation is not an identity? aaaa aaaa aa aaaaaa aaaa a aaaaa aaaaaaaa aa aaa aa aaaaaaaaa aaa aa aaaa aa aa aa aaaa aaaa aaa aaaaaaaa aaaa aaa aaaa aaa aaaa aaaaa aa aaa aaaaaaaa aa aaaaaaaaaa

List the guidelines for proving trigonometric identities. aa aaaaa aaaa aaa aaaaa aaaa aaa aaaa aa aaa aaaaaaaa aaa aaaaa aa aaaaa aaaa aaaa aa aa aaaaaaaaa aa aaaa aaa aaaaa aaaaa aaaa aaaaaaa aaaaaa aa aaaaa aaaa aaa aaaa aaaaaaaaaaa aaaaa aa aaa aaaaa aaaaaaaaaaa aaa aaaaaaa aaa aaa aaaaaaaaaa aaa aaaa aa aaaaaa aaa aaaa aaa aaaaaaa aaaaa aaaaa aaaaaaaaaa aaaaaaaaaaa aa a aaaaaa aaaaaaaaaaaa aaaaaaa aaa aaa aaa aaaaaaaaaaa aaaaaaaaaa aa aaaaaaaa aaaaaaaaaaaa aa aaaaaaa aa aaaaa aaa aaaaaaaa aa aaa aaa aaaaaa aaa aaa aaaa aa aaaaaaa aa aaaaaaa aaa aaaaaaaaa aa aaaaa aa aaaaa aaa aaaaaaaa

Here is another method for proving that an equation is an identity: if we can transform each side of the equation separately, by way of identities, to aaaaaaaa aa aa aaaaaaaa

aaaaaa aa aaa aaaa aaaa

, then

aaa

.

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|

SECTION 7.2

117

Addition and Subtraction Formulas

Name ___________________________________________________________ Date ______ ______

7.2

Addition and Subtraction Formulas

I. Addition and Subtraction Formulas List the addition and subtraction formulas for sine.

List the addition and subtraction formulas for cosine.

List the addition and subtraction formulas for tangent.

II. Evaluating Expressions Involving Inverse Trigonometric Functions When evaluating expressions involving inverse trigonometric functions, remember that an expression like

cos1 x represents a(n)

aaaaa

.

III. Expressions of the Form A sin x + B cos x We can write expressions of the form A sin x + B cos x in terms of a single trigonometric function using aaa aaaaaaaa aaaaaaa aaa aaaa

.

If A and B are real numbers, then A sin x + B cos x =

k = A2 + B2 , and  satisfies

cos  

a aaa aa a a

A A  B2

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2

aaa

Mathematics for Calculus, 7th Edition

, where .

118

CHAPTER 7

|

Analytic Trigonometry

Additional notes

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SECTION 7.3

|

Double-Angle, Half-Angle, and Product-Sum Formulas

119

Name ___________________________________________________________ Date ____________

7.3

Double-Angle, Half-Angle, and Product-Sum Formulas

The Double-Angle Formulas allow us to

aaaa aaa aaaaaa aa aaa aaaaaaaaaa aaaaaaaaa

aa aa aaaa aaaaa aaaaaa aa a

.

The Half-Angle Formulas relate

aaa aaaaaa aa aaa aaaaaaaaaaaaa aaaaaaaa aa aaaaaa

aa aaaaa aaaaaa aa a

.

The Product-Sum Formulas relate

aaaaaaaa aa aaaaa aaa aaaaa aa aaaa aa aaaaa

aaa aaaaaaa

.

I. Double -Angle Formulas List the Double-Angle Formula for sine.

List the Double-Angle Formula for cosine in all its forms.

List the Double-Angle Formula for tangent.

II. Half-Angle Formulas The Formulas for Lowering Powers allow us to write any trigonometric expression involving even powers of sine and cosine in terms of

aaa aaaaa aaaaa aa aaaaaa aaaa

List the three Formulas for Lowering Powers.

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120

CHAPTER 7

|

Analytic Trigonometry

List the three Half-Angle Formulas. Be sure to list both forms of the Half-Angle Formula for tangent.

The choice of the + or – sign depends on

aaa aaaaaaa aa aaaaa aaa aaaa

.

III. Product-Sum Formulas List the four Product-to-Sum Formulas.

List the four Sum-to-Product Formulas.

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SECTION 7.4

|

Basic Trigonometric Equations

121

Name _________________________________________________ __________ Date ____________

7.4

Basic Trigonometric Equations

A trigonometric equation is

aa aaaaaaaa aaaa aaaaaaaa aaaaaaaaaa aaaaaaaaa

.

I. Basic Trigonometric Equations Solving any trigonometric equation always reduces to solving a basic trigonometric equation, that is, an equation of the form

aaaa a aa aaaaa a aa a aaaaaaaaaaaaa aaaaaa aaa a aa a aaaaaaaa

To solve a basic trigonometric equation, first find the solutions in find all solutions of the equation by adding aaaaaaaaa

aaa aaaaaa

. , and then

aaaaaaa aaaaaa aa aaa aaaaaa aa aaaaa

.

Describe how to solve a basic trigonometric equation such as cos = 1 . aaaaaaa aaaaaa aaa aaaaaa aaa aaaaa aaaa aaa aaaaaaaaa aa aaa aaaaaaaa aa aaaaaa aa aaa aa aaaa aaaaa aaaaaaaaaa aaaa aaa aaaaaa aa aaa aaaa aaaaaa aaaa aaaa a aaaaaa aaaaa aa aa aaaaaaa aaa aaaaaa aaaaaaaa aaaaaaa aaa aaaaaa aaaaa aa aaaaaa aaaa aaa aaaaaaaaa aa aaa aaaaaaaa aa aaaaaa aaaaaaa aaaaaaaaa aa aa aa aaaaa aaaaaaaaaa

II. Solving Trigonometric Equations by Factoring Factoring is one of the most useful techniques for solving equations, including trigonometric equations. The idea is

aa aaaa aaa aaaaa aa aaa aaaa aa aaa aaaaaaaa aaaaaaa aaa aaaa aaa aaa aaaaaaaaaaaa

aaaaaaaa aa aaaaa

.

State the Zero-Product Property. aa aa a aa aaaa a a a aa a a aa

Give an example of a trigonometric equation of quadratic type. aaaaaaa aaaa aaaaa aaa aaaaaaa aa aa

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CHAPTER 7

|

Analytic Trigonometry

Additional notes

y

y

x

y

y

x

y

x

x

y

x

x

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SECTION 7.5

|

More Trigonometric Equations

123

Name ___________________________________________________________ Date ____________

7.5

More Trigonometric Equations

I. Solving Trigonometric Equations by Using Identities Describe the first step in solving a trigonometric equation such as cos 2  5cos  2  0 . aaa aaaaa aaaa aa a aaaaaaaa aaa aaa aaa aaaaaa aaaa aa a aaaaaaaa aa aa aa aa aaaaa aa aaaaa a aaaaaaaaaaaaa aaaaaaaa aa aaaaaaa aaa aaaaa aaaa aa a aaaaaaaa aa a aaaaa aaa aaaaaaaa aa aaaa aa aaa aaaaaaaaaaaa aaaaaaa aa

Describe a strategy for solving an equation such as cos = 3sin  +10 . aaa aaaaa aaaa aa aa aaaaaaaaa aaaa aaaaaaaa aaaa aaa aaaa aaaaaaaa aaaaaa aaaa aaaa aa aaaaaa aaaaa aa aa aaa aaaaaa aaaa aaaaa aa aaa aaaaaaaa aaa aaaa aaa aa aaaaaaaaaaa aaaaaaaaaaa aaaaaaaaa

Describe two methods of finding the values of x for which the graphs of two trigonometric functions intersect. aaa aaa aa aa aaa a aaaaaaaaa aaaaaaaaa aa aa aaa aaaaa aaaa aaaaaaaaa aa aaa aaaa aaaaaa aa a aaaaaaaa aaaaaaaaaa aaa aaa aaa aaaaa aa aaaaaaaaa aaaaaaaa aa aaaa aaa aaaaaaaaaaaaa aaaaaa aaaaa aaa aaaaaa aaaaaaaaaaaaa aaaaaa aaa aa aa aaa aa aaaaaaaaa aaaaaaaaa aa aaaa aaa aaaaa aaaaaaaaa aaa aaa aaa aaaaaaaaa aaaaa aaa aaaaa aaa aaaaaaaaa aaaaaaaa aaaaaaaaaaaaa aaaaa aaa aaaaaaa aa aaaa aaaaaaaa

II. Equations with Trigonome tric Functions of Multiple Angles When solving trigonometric equations that involve functions of multiples of angles, first solve for aaaaaaaa aa aaa aaaaaa aaaa aaaaaa aa aaaaa aaa aaa aaaaa

aaa .

Describe how to solve an equation such as cos5  1  0 aaaaa aa aaaaaaaaa aaa aa aaa aaaa aaaaaaa aaa aaa aaaaa aaa aaaaaaaa aa aaaaa aaa aa aaaaaa aaa aaaaaaaaa aa aa

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CHAPTER 7

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Analytic Trigonometry

Additional notes

y

y

x

y

y

x

y

x

x

y

x

x

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Name ___________________________________________________________ Date _ ___________

Chapter 8 8.1

Polar Coordinates and Parametric Equations

Polar Coordinates

I. Definition of Polar Coordinates The polar coordinate system uses distances and directions to a aaaaa aa aaa aaaaa plane called

aaaaaaa aaa aaaaaaaa aa

. To set up this system, we choose a fixed point O in the aaa aaaa aaa aaaaaaa

. Then each point P can be assigned polar coordinates P(r , )

aaaaa aaaa where r is

and draw from O a ray (half-line) called the

and θ is

aaa aaaaaaa aaaa a aa a

aaaaaaa aaa aaaaa aaaa aaa aaa aaaaaaa a

.

We use the convention that θ is positive if aaaaa aaaa

aaa aaaaa

aaaaaaaa aa a aaaaaaaaaaaaaaaa aaaaaaaaa aaaa aaa

or negative if

aaaaaaaa aa a aaaaaaa aaaaaaaaa

If r is negative, then P(r , ) is defined to be

.

aaa aaaaa aaaaa a a a aaaaa aaaa aaa aaaa aa

aaa aaaaaaaaa aaaaaaaa aa aaaa aaaaa aa a

.

Because the angles  + 2n (where n is any integer) all have the same terminal side as the angle θ, each point in the plane has

aaaaaaaaaa aaaa aaaaaaaaaaaaaaa aa aaaaaaaaaaaaaa

.

II. Relationship Between Polar and Rectangular Coordinates To change from polar to rectangular coordinates, use the formulas a a a aaa a aaaaaaa a a aaa a

To change from rectangular to polar coordinates, use the formulas aa

III. Polar Equations To convert an equation from rectangular to polar coordinates, simply a aaa aa aaa aaaa aaaaaaaa

Note Taking Guide for Stewart/Redlin/Wats on

aaaaaaa a aa a aaa a aaa a aa

.

Precalculus:

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th

Mathematics for Calculus, 7 Edition

125

126

CHAPTER 8

|

Polar Coordinates and Parametric Equations

Describe one or more strategies for converting a polar equation to rectangular form. aaaa aaa aaaa aa aaa aaa aaaaaaaaaa aaaaaaaaa

aaa a aa aaa aa aaaaaaaaa aa aaaaaaaa aaa aaaaa aaaaaaaa a aaaaaaa aa a aa a aaaaaaaaaaaaa aaaaaa aaaa aa aa

y

a

y

x

y

x

x

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SECTION 8.2

|

127

Graphs of Polar Equations

Name ___________________________________________________________ Date ____________

8.2

Graphs of Polar Equations

The graph of a polar equation r = f ( ) consists of

aaa aaaaaa a aaaaaa aa aaaaa aaa aaaaa

aaaaaaaaaaaaaa aaa aa aaaaa aaaaaaaaaaa aaaaaaa aaa aaaaaaaa

.

I. Graphing Polar Equations To plot points in polar coordinates, it is convenient to use a grid consisting of

aaaaaa aaaaaaaa

aa aaa aaaa aaa aaaa aaaaaaaaa aaaa aaa aaaa The graph of the equation r = a is

. a aaaaaa aa aaaaaa a a a aaaaaaaa aa aaaaaa

To sketch a polar curve whose graph isn’t obvious,

.

aaaa aaaaaa aaaaaaaa aaa aaaaaaaaaaaa

aaaa aaaaaa aa a aaa aaaa aaaa aaaa aa a aaaaaaaaaa aaaaa

.

The graphs of equations of the form r = 2a sin  and r = 2a cos are

aaaaaaa aaaa aaaaaa a

a a aaaaaaaa aa aaa aaaaaa aaaa aaaaa aaaaaaaaaaa aaa aaaa aaa aaa aaa aaaaaaaaaaa A cardioid has the shape of

a aaaaa

a

.

. The graph of any equation of the form

or

a

is a cardioid.

The graph of an equation of the form r = a cos n or r = a sin n is an n-leaved rose if a 2n-leaved rose if

a aa aaaa

a aaaaa

or

.

II. Symmetry In graphing a polar equation, it’s often helpful to take advantage of

aaaaaaaa

List three tests for symmetry. aa aa a aaaaa aaaaaaaa aa aaaaaaaaa aaaa aa aaaaaaa a aa aaa aaaa aaa aaaaa aa aaaaaaaaa aaaaa aaa aaaaa aaaaa aa aa aaa aaaaaaaa aa aaaaaaaaa aaaa aa aaaaaaa a aa aaa aaaa aaa aaaaa aa aaaaaaaaa aaaaa aaa aaaaa aa aa aaa aaaaaaaa aa aaaaaaaaa aaaa aa aaaaaaa a aa a a aa aaaa aaa aaaaa aa aaaaaaaaa aaaaa aaa aaaaaaaa aaaa a a aaa aaaa aaaaaaaa

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128

CHAPTER 8

|

Polar Coordinates and Parametric Equations

In polar coordinates, the zeros of the function r = f ( ) are the angles θ at which aaaaaaa aaa aaaa

aaa aaaaa

.

The graph of an equation of the form r = a ± b cos or r = a ± b sin  is a /2 y

aaaaaaa

/2 y



0 x



/2 y

x0



0 x

3/2

3/2

3/2

/2 y

/2 y

/2 y



0x



0 x



x0

3/2

3/2

3/2

/2 y

/2 y

/2 y



0 x

3/2



x0

3/2

.



0 x

3/2

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SECTION 8.3

|

Polar Form of Complex Numbers; De Moiv re’s Theorem

129

Name ___________________________________________________________ Date ____________

8.3

Polar Form of Complex Numbers; De Moivre’s Theorem

I. Graphing Complex Numbers How many axes are needed to graph a complex number? Explain. aaaa aaa aaa aaa aaaa aaaa aaa aaa aaa aaa aaaaaaaaa aaaaa The complex plane is determined by the

aaaa aaaa

and the

aaaaaaa aaaa

To graph the complex number a + bi in the complex plane, plot the ordered pair of numbers

.

aa aa

in

this plane. The modulus, or absolute value, of the complex number z = a + bi is

a

.

II. Polar Form of Complex Numbers A complex number z = a + bi has the polar form, or trigonometric form, where r = is the

and tan θ =

a aaaaaaa

a aaa

of z, and θ is an

. The number r

aaaaaaaa

of z.

The argument of z is not unique, but any two arguments of z differ by a When determining the argument, we must consider

aaaaaaaa aa aa

.

aaa aaaaaa aa aaaaa a aaaa

.

If the two complex numbers z1 and z2 have the polar forms z1 = r1 (cos 1+ i sin 1 ) and

z2 = r2 (cos 2 + i sin 2 ) , then the numbers are multiplied and divided as follows.

z1 z2 =

a

.

z1 = z2

a

a

.

This theorem says that to multiply two complex numbers, aaaaaaaaa

aaaaaaaa aaa aaaaaa aaa aaa aaa

. It also says that to divide complex numbers,

aaa aaaaaa aaa aaaaaaaa aaa aaaaaaaaa

aaaaaa

.

III. De Moivre’s Theore m De Moivre’s Theorem gives a useful formula for aaaaaaaa aaaaaaa a

aaaaaaa a aaaaaaa aaaaaa aa a aaaaa a aaa aaa

.

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|

Polar Coordinates and Parametric Equations

De Moivre’s Theorem states that if z = r (cos  + i sin  ) , then for any integer n,

zn =

a

.

Give an interpretation of De Moivre’s Theorem . aa aaaa aaa aaa aaaaa aa a aaaaaaa aaaaaaa aaaa aaa aaa aaaaa aa aaa aaaaaaa aaa aaaaaaaa aaa aaaaaaaa aa aa

IV. nth Roots of Complex Numbers aaa aaaaaaa aaaaaa a aaaa aaaa aa a a

An nth root of a complex number z is

.

If z = r (cos  + i sin  ) and n is a positive integer, then z has the n distinct nth roots

a

, for k = 0, 1, 2, . . . , n  1 .

When finding the nth roots of z = r (cos  + i sin  ) , notice that the modulus of each nth root is Also, the argument of the first root is

aaa

. Furthermore, we repeatedly add

a

.

aaaa

to

get the argument of each successive root. When graphed, the nth roots of z are spaced equally on

aaa aaaaaa aa aaaaaa a

.

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SECTION 8.4

|

Plane Curv es and Parametric Equations

131

Name ___________________________________________________________ Date ____________

8.4

Plane Curves and Parametric Equations

I. Plane Curves and Parametric Equations We can think of a curve as the path of a point moving in the plane; the x-coordinates and y-coordinates of the point are then

aaaaaaaaa aa aaaa

.

If f and g are functions defined on an interval I, then the set of points ( f (t ), g (t )) is a The equations x = f (t ) and y = g (t ) where t  I , are

aaaaa aaaaa

aaaaaaaaaa aaaaaaaaa

. for the

curve, with parameter t. A parametrization contains more information than just the shape of the curve; it also indicates aaa aaaaa aa aaaaa aaaaaa aaa

aaa

.

II. Eliminating the Parameter Often a curve given by parametric equations can also be represented by a single rectangular equation in x and y. The process of finding this equation is called to do this is

aaaaaaaaaa aaa aaaaaaaaa

. One way

aa aaaaa aaa a aa aaa aaaa aaaa aaaaaaaaaa aaaa aaa aaaaa

.

To identify the shape of a parametric curve,

aaaaaaa aaa aaaaaaaaa aaa aaaaaaa aaa aaaaaaaaa

aaaaaaaaaaa aaaaaaaa

.

III. Finding Parametric Equations for a Curve Describe how to find a set of parametric equations for a curve. aaa aaaaaaaaa aaaaaaaaaa aaaa aaaaaa aaa aaaaa aa aaaaa aaaaaaaaaa aaaaaaaaa aaa aaa aa aaaaaaa aaaa aaaaa aaaaaaaaa aaaa aaa aaaaaaa aaaaaa aaaaaaaaa aaa aaaaaaaaaa

A cycloid is

a aaaaa aaaa aa aaaaaa aaa aa a aaaaa aaaaa a aa aaa aaaaaaaaaaaaa aa a aaaaaa aa aaa

aaaaaa aaaaa aaaaa a aaaaaaaa aaaa

.

Name two interesting physical properties of the cycloid. aa aa aaa aaaaaa aa aaaaaaaa aaaaaaaa aa aaaa aa aaaaa aaa aaaaaa aa aaaaa aaaaaaaaa

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CHAPTER 8

Polar Coordinates and Parametric Equations

IV. Using Graphing Devices to Graph Parametric Curves A closed curve is

aaa aaaa aaaaa aaaaa aaaa aaa aaaa aaaaaaa aaa aaaaaa aaaaa

A Lissajous figure is the graph of a pair of parametric equations of the form

b

. a

and

where A, B, 1 , and 2 are real constants.

a

The graph of the polar equation r = f ( ) is the same as the graph of the parametric equations a

and

y

a

.

y

x

y

y

x

y

x

x

y

x

x

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Name ___________________________________________________________ Date ____________

Chapter 9 9.1

Vectors in Two and Three Dimensions

Vectors in Two Dimensions

A scalar is

a aaaaaaaa aaaaa aaa aa aaaaaaaaa aa aaaa aaa aaaaaa

.

Quantities such as displacement, velocity, acceleration, and force that involve magnitude as well as direction are called

aaaaaaaa aaaaaaaaaa

through the use of

. One way to represent such quantities mathematically is

aaaaaaa

.

I. Geometric Description of Vectors A vector in the plane is vector as

a aaaa aaaaaaa aaaa aa aaaaaaaa aaaaaaaaa

aa aaaaa aa aaaaaaa aaa aaaaaaaaa

denoted as AB , then point A is the aaaaa

. If a vector between points A and B is

aaaaa aaaaa

, and point B is the

. The length of the line segment AB is called the

the vector and is denoted by AB . We use

aaaaaaaa

Two vectors are considered equal if they have

. We sketch a

aaaaaaaa

aaaaaaaaa aa aaaaaa

of

letters to denote vectors.

aaaaa aaaaaaaaa aaa aaa aaaa aaaaaaaaa

.

If the displacement u = AB is followed by the displacement v = BC , then the resulting displacement is a as

. In other words, the single displacement represented by the vector AC has the same effect . We call the vector AC the

aaa aaaaa aaa aaaaaaaaaaaa aaaaaaaa

the vectors AB and BC , and we write represents

a

aa aaaaaaaaaaaa

aaa

of

. The zero vector, denoted by 0 ,

. Thus to find the sum of any two vectors u and v,

aa

aaaaaa aaaaaaa aaaaa aa a aaa a aaaa aaa aaaaaaa aaaaa aa aaa aaaaaaaa aaaaa aa aaa aaaaa If we draw u and v starting at the same point, then u + v is the vector that is aaaaaaaaaaaaa aaaaaa aa a aaa a

. aaa aaaaaaa aa aaa

.

Describe the process of multiplication of a vector by a scalar and the effect it has on the vector. aa a aa a aaaa aaaaaa aaa a aa a aaaaaaa aa aaaaaa a aaa aaaaaa aa aa aaaaaaaa aaa aaaaaa aa aaa aaaaaaaaa a a a a a a aaa aaa aaa aaaa aaaaaaaaa aa a aa a a a aaa aaa aaaaaaaa aaaaaaaaa aa a a aa aa a a aa aaaa aa a aa aaa aaaa aaaaaaa aaaaaaaaaaa a aaaaaa aa a aaaaaa aaa aaa aaaaaa aa aaaaaaaaaa aa aaaaaaaaa aaa aaaaaaa

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CHAPTER 9

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Vectors in Tw o and Three Dimensions

The difference of two vectors u and v is defined by

a

.

II. Vectors in the Coordinate Plane In the coordinate plane, we represent v as an ordered pair of real numbers v =

a1 is the

, where

and a2 is the

aaaaaaaaaa aaaaaaaa aa a

aaaaaaaaa aa a

a

aaaaaaaa

.

If a vector v is represented in the plane with initial point P( x1 , y1 ) and terminal point Q( x2 , y2 ) , then v=

a

.

The magnitude or length of a vector v  a1 , a2 is

a

.

If u  a1 , a2 and v  b1 , b2 , then a

u+v = uv  cu =

.

.

a a

.

The zero vector is the vector

a

.

List four properties of vector addition.

List the property for the length of a vector.

List six properties of multiplication by a scalar.

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SECTION 9.1

A vector of length 1 is called a defined by i =

a

|

135

Vectors in Tw o Dimensions

aaaa aaaaaa

. Two useful unit vectors are i and j,

and j =

a

.

The vector v  a1 , a2 can be expressed in terms of i and j by v =

a

Let v be a vector in the plane with its initial point at the origin. The direction of v is θ,

. aaa aaaaaaaa

aaaaaaaa aaaaa aa aaaaaaaa aaaaaaaa aaaaaa aa aaa aaaaaaaa aaaaaa aaa a

.

Horizontal and Vertical Components of a Vector Let v be a vector with magnitude | v | and direction θ. Then v  a1 , a2  a1i  a2 j , where

a1 =

a

, and a2 =

Thus we can express v as

a

.

a

.

III. Using Vectors to Model Velocity and Force The velocity of a moving object is modeled by

a aaaaaa aaaaa aaaaaaaaa aa aaa aaaaaaaaa aa

aaaaaa aaa aaaaa aaaaaaaaa aa aaa aaaaa

.

Force is also represented by a vector. We can think of force as aaaaaa

. Force is measured in

aaaaaaaaa a aaaa aa a aaaa aa aa aaaaaa

. If several forces are

acting on an object, the resultant force experienced by the object is aaaaa aaaaaa

.

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aaa aaaaaa aaa aa

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Vectors in Tw o and Three Dimensions

Additional notes

y

y

x

y

y

x

y

x

x

y

x

x

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SECTION 9.2

|

137

The Dot Product

Name ___________________________________________________________ Date ____________

9.2

The Dot Product

I. The Dot Product of Vectors If u  a1 , a2 and v  b1 , b2 are vectors, then their dot product, denoted by u v , is defined by u v = a

. Thus to find the dot product of u and v, we

aaaaaaaaaaaaa aaaaaaaaaa aaa aaa

.

The dot product of u and v is not a vector; it is a

aaaa aaaaaaa aa aaaaaa

aaaaaaaa

.

List four properties of the dot product.

Let u and v be vectors, and sketch them with initial points at the origin. We define the angle θ between u and v to be

aaa aaaaaaa aa aaa aaaaaa aaaaa aa aaaaa aaaaaaaaaaaaaaa aa a aaa a

,

thus 0     . The Dot Product Theorem states that if θ is the angle between two nonzero vectors u and v, then a

.

The Dot Product Theorem is useful because it allows us to find the angle between two vectors if we know the u v components of the vectors. If θ is the angle between two nonzero vectors u and v, then cos θ = u v

Two nonzero vectors u and v are called perpendicular, or orthogonal, if the angle between them is

aaa

We can determine whether two vectors are perpendicular by finding their dot product. Two nonzero vectors u and v are perpendicular if and only if

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.

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.

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Vectors in Tw o and Three Dimensions

II. The Component of u Along v The component of u along v (also called the component of u in the direction of v or the scalar projection of u onto v) is defined to be of u along v is the magnitude of

a

, where θ is the angle between u and v. Intuitively, the component

aaa aaaaaaa aa a aaaa aa aa aaa aaaaaaaaa aa a

.

comp v u 

The component of u along v (or the scalar projection of u onto v) is calculated as

u v v

.

III. The Projection of u onto v The projection of u onto v, denoted by proj v u, is aaaaaaaaa aa a aaaaa a projv u =

aaa aaaaaa aaaaaa aa a aaa aaaaa aaaaaa aa aaa . The projection of u onto v is given by

u v  v  v2   

u 2 is orthogonal to v, then u 1 =

. If the vector u is resolved into u 1 and u 2 , where u 1 is parallel to v and aaaaa a

and u 2 =

a a aaaaa a

.

IV. Work One use of the dot product occurs in calculating

aaaa

The work W done by a force F in moving along a vector D is

. a

.

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SECTION 9.3

139

Three-Dimensional Coordinate Geometry

Name ___________________________________________________________ Date ____________

9.3

Three-Dimensional Coordinate Geometry

I. The Three -Dimensional Rectangular Coordinate System To represent points in space, we first choose a fixed point O (the origin) and three directed lines through O that are perpendicular to each other, called the

aaaaaaaaaa aaaa

aaaaaaa aaaaaaa aaaaaa being horizontal and the

. We usually think of the

aaaaaa

aaaaaaaa aaaaaa

is the plane that contains the x- and y-axes. The aaaaaaa

aaaaaaaa

. The

aaaaaaaa

aaaaaaa

. aaaaaaa aaaaaa

first number a is

, the second number b is

aaa aaaaaaaaaaa aa a , and the third number c is

of real numbers (a, b, c). The

aaa aaaaaaaaaaaa aa a

ordered triples {( x, y, z) | x, y, z  } forms the

aaa aaaaaaaaaaaa . The set of all

aaaaaaaaaaaaaaaaa aaaaaaaaaaa aaaaaaaaaa

.

In three-dimensional geometry, an equation in x, y, and z represents a three-dimensional

aaaaaa

.

II. Distance Formula in Three Dimensions The distance between the points P( x1 , y1 , z1 ) and Q( x2 , y2 , z2 ) is

Example 1

.

is the plane that contains the y- and z-axes.

Any point P in space can be located by a unique

aaaaaa

as

is the plane that contains the x- and z-axes. These three coordinate planes divide space

into eight parts called

aa a

aa aaa aaaaaa

as being vertical.

The three coordinate axes determine the three

The

and labeled the

Find the distance between the points (1, 3, −5) and (2, 1, 0).

III. The Equation of a Sphere An equation of a sphere with center C(h, k, l) and radius r is

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a

.

140

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CHAPTER 9

Example 2

Vectors in Tw o and Three Dimensions

Find the radius and center of the sphere with equation ( x  2)2  ( y  9)2  ( z  14)2  25 . aaaaaaa aaaa aa aaaa aaaaaaa a

The intersection of a sphere with a plane is called the

aaaaa

of the sphere in a plane.

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SECTION 9.4

|

141

Vectors in Three Dimensions

Name ___________________________________________________________ Date ____________

9.4

Vectors in Three Dimensions

I. Vectors in Space If a vector v is represented in space with initial point P(x1 , y1 , z1 ) and terminal point Q(x2 , y2 , z2 ), then the component form of a vector in space is given as v =

a

The magnitude of the three-dimensional vector v  a1 , a2 , a3

is

.

a

.

II. Combining Vectors in Space If u  a1 , a2 , a3 , v  b1 , b2 , b3 , and c is a scalar, then complete each of the following algebraic operations on vectors in three dimensions.

uv 

a

uv  cu 

.

a

.

a

.

The three-dimensional vectors i = 1,0,0 , j = 0,1, 0 , and k = 0,0,1 are examples of The vector v  a1 , a2 , a3 can be expressed in terms of i, j, and k by v  a1 , a2 , a3 

aaaa aaaaaa a

.

III. The Dot Product for Vectors in Space If u  a1 , a2 , a3 and v  b1 , b2 , b3 are vectors in three dimensions, then their dot product is defined by

a

.

Let u and v be vectors in space and θ be the angle between them. Then cos =

In particular, u and v are perpendicular (or orthogonal) if and only if

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.

u v u v

.

a

.

142

CHAPTER 9

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Vectors in Tw o and Three Dimensions

IV. Direction Angles of a Vector The direction angles of a nonzero vector v  a1i  a2 j  a3k are the angles α, β, and γ in the interval [0, π] that aaa aaaaaa a aaaaa aaaa aaa aaaaaaaa aaa aaa aaa aaaaaa The cosines of these angles, cos  , cos  , and cos  , are called the

aaaaaaaaa aaaaaaa aa

. If v  a1i  a2 j  a3k is a nonzero vector in space, the direction angles α, β, and

aaa aaaaaa a γ satisfy

.

cos  

a1 v

.

If v  1 , then the direction cosines of v are simply

aaa aaaaaaaaaa aa a

.

The direction angles α, β, and γ of a nonzero vector v in space satisfy the following equation

This property indicates that if we know two of the direction cosines of a vector, we can find aaaaaaaaa aaaaaa aa aa aaa aaaa

aaa aaaaa

.

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SECTION 9.5

143

The Cross Product

Name ___________________________________________________________ Date ____________

9.5

The Cross Product

I. The Cross Product If u  a1 , a2 , a3 and v  b1 , b2 , b3 are three-dimensional vectors, then the cross product of u and v is the vector

a

.

The cross product u  v of two vectors u and v, unlike the dot product, is a Note that u  v is defined only when u and v are vectors in

aaaa aaa a aaaaaa

aaaaa aaaaaaaaaa

To help us remember the definition of the cross product, we use the notation of

.

.

aaaaaaaaaaaa

.

A determinant of order three is defined in terms of second-order determinants as

a1 a2 b1 b2 c1 c2

a3 b3 c3

Each term on the right side of the third-order determinant equation involves a number ai in the first row of the determinant, and ai is multiplied by the second-order determinant obtained from the left side by

aaaaaaa

aaa aaa aaa aaaaaa aa aaaaa aa aaaaaaa

.

We can write the definition of the cross-product using determinants as

II. Properties of the Cross Product The Cross Product Theorem states that the vector u  v is

aaaaaaaaaa aaaaaaaaaaaaaa

to both

u and v. If u and v are represented by directed line segments with the same initial point, then the Cross Product Theorem says that the cross product u  v points aaaaaaa a aaa a hand rule:

aa a aaaaaaaaa aaaaaaaaaa aa aaa aaaaa . It turns out that the direction of u  v is given by the right-

aa aaa aaaaaaa aa aaaa aaaaa aaaa aaaa aa aaa aaaaaaaaa aa aaaaaa aaaaaaaa aa aaaaa

aaaa aaaa aaaaa aaaa a aa aa aaaa aaaa aaaaa aaaaaa aa aaa aaaaaaaaa aa a

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Vectors in Tw o and Three Dimensions

If θ is the angle between u and v (so 0     ), then the length of the cross product of u and v is given by

In particular, two nonzero vectors u and v are parallel if and only if

a

.

III. Area of a Parallelogram The length of the cross product u  v is the area of aa a aaa a

aaa aaaaaaaaaaaa aaaaaaaaaa

.

IV. Volume of a Parallelepiped The product u ( v  w) is called the

aaaaaa aaaaa aaaaaaa

of the vectors u, v, and

w. The scalar triple product can be written as the following determinant

The volume of the parallelepiped, a three-dimensional figure having parallel faces, determined by the vectors u, v, and w, is the magnitude of their scalar triple product

a

if the volume of the parallelepiped is 0, then the vectors u, v, and w are

. In particular, aaaaaaaa

.

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SECTION 9.6

|

145

Equations of Lines and Planes

Name ___________________________________________________________ Date ____________

9.6

Equations of Lines and Planes

I. Equations of Lines A line L in three-dimensional space is determined when we know a point P0 ( x0 , y0 , z0 ) on L and aaaaaaaaa aa a

. In three dimensions the direction of a line is described by

a aaaaaaaa aa a

. The line L is given by the position vector r, where r =

t  , and r0 is the position vector of P0 . This the vector

aaa a aaaaaa

a

aaaaaaaa aa a aaaa

for .

A line passing through the point P( x0 , y0 , z0 ) and parallel to the vector v = a, b, c is described by the parametric equations a a where t is any real number.

II. Equations of Planes Although a line in space is determined by a point and a direction, the “ direction” of a plane cannot aaaaaaaaa aa a aaaaaa aa aaa aaaaa different directions. But a vector

aa

. In fact, different vectors in a plane can have aaaaaaaaaaaa

to a plane does completely specify the

direction of the plane. Thus a plane in space is determined by

a aaaaa a aa aaa aaaaa aaa a

aaaaaa a aaaa aa aaaaaaaaaa aa aaa aaaaa

. This orthogonal vector n is

called a

aaaaaa aaaaaa

.

The plane containing the point P( x0 , y0 , z0 ) and having the normal vector n = a, b, c is described by the equation

a

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Vectors in Tw o and Three Dimensions

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Name ___________________________________________________________ Date ____________

Chapter 10

Systems of Equations and Inequalities

10.1 Systems of Linear Equations in Two Variables I. Systems of Linear Equations and Their Solutions A system of equations is

a aaa aa aaaaaaaaa aaaa aaaaaaa aaa aaaa aaaaaaa

.

A system of linear equations is

a aaaaaa aa aaaaaaaaa aa aaaaa aaaa aaaaaa aa aaaaaa

A solution of a system is

aa aaaaaaaaaa aa aaaaaa aaa aaa aaaaaaaaa aa aaaaa aaaa aaaaaaaa

aa aaa aaaaaa aaaa aaaaaa

. To solve a system means to

.

a aaa aaaaaaaaa aa aaa

.

II. Substitution Method In the substitution method, we start with

aaa aaaaaaaa aa aaa aaaaaa aaa aaaaa aaa aaa

aaaaaaaa aa aaaaa aa aaa aaaaa aaaaaaaa

.

Describe the substitution method procedure. aa aaaaa aaa aaa aaaaaaaaa aaaaaa aaa aaaaaaaaa aaa aaaaa aaa aaa aaaaaaaa aa aaaaa aa aaa aaaaa aaaaaaaaa aa aaaaaaaaaaa aaaaaaaaaa aaa aaaaaaaaaa aaa aaaaa aa aaaa a aaaa aaa aaaaa aaaaaaaa aa aaa aa aaaaaaaa aa aaa aaaaaaaaa aaaa aaaaa aaa aaaa aaaaaaaaa aa aaaaaaaaaaaaaaaa aaaaaaaaaa aaa aaaaa aaa aaaaa aa aaaa a aaaa aaaa aaa aaaaaaaaaa aaaaa aa aaaa a aa aaaaa aaa aaa aaaaaaaaa aaaaaaaaa

III. Elimination Method To solve a system using the elimination method, we try to

aaaaaaa aaa aaaaaaaaa aaaaa aaaa

aa aaaaaaaaaaa aa aa aa aaaaaaaaa aaa aa aaa aaaaaaaaa

.

Describe the elimination method procedure. aa aaaaaa aaa aaaaaaaaaaaaa aaaaaaaa aaa aa aaaa aa aaa aaaaaaaaa aa aaaaaaaaaaa aaaaaaa aa aaaa aaa aaaaaaaaaaa aa aaa aaaaaaaa aa aaa aaaaaaaa aa aaa aaaaaaaa aa aaa aaaaaaaaaaa aa aaa aaaaa aaaaaaaaa aa aaa aaa aaaaaaaaaa aaa aaa aaa aaaaaaaaaaa aaa aaaaaaaaa aaaa aaaaa aaa aaa aaaaaaaaa aaaaaaaaa aa aaaaaaaaaaaaaaaa aaaaaaaaaa aaa aaaaa aaaa aaa aaaaa aa aaaa a aaaa aaaa aaa aa aaa aaaaaaaa aaaaaaaaaa aaa aaaaa aaa aaa aaaaaaaaa aaaaaaaaa

IV. Graphical Method In the graphical method, we use Note Taking Guide for Stewart/Redlin/Wats on

a aaaaaaaa aaaaaa aa aaaaaaa aaaaaa aa aaaaaaaaa Precalculus:

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Mathematics for Calculus, 7 Edition

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CHAPTER 10

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Systems of Equations and Inequalities

Describe the graphical method procedure. aa aaaaa aaaa aaaaaaaaa aaaaaaa aaaa aaaaaaaa aa a aaaa aaaaaaaa aaa aaa aaaaaaaa aaaaaaaaaa aa aaaaaaa aaa a aa a aaaaaaaa aa aa aaaaa aaa aaaaaaaaa aa aaa aaaa aaaaaaa aa aaaa aaa aaaaaaaaaaaa aaaaaaa aaa aaaaaaaaa aaa aaa aa aaa aaaaaaaaaaaaa aa aaa aaaaaa aa aaaaaaaaaaaaa

V. The Number of Solutions of a Linear System in Two Variables The graph of a linear system in two variables is graphically, we must find

a aaa aa aaaaa

, so to solve the system

aaa aaaaaaaaaaaa aaaaaa aa aaa aaaaa

.

For a system of linear equations in two variables, exactly one of the following is t rue concerning the number of solutions the system has. 1. aaa aaaaaa aaa aaaaaaa aaa aaaaaaaaa

2. aaa aaaaaa aaa aa aaaaaaaaa

3. aaa aaaaaa aaa aaaaaaaaaa aaaa aaaaaaaaaa A system that has no solution is said to be solutions is called

aaaaaaaaaa

aaaaaaaaa

. A system with infinitely many

.

If solving a system of equations eliminates both variables and results in a statement which is false, such as 0 = 7, then the system has

aa aaaaaaaa

.

VI. Modeling with Linear Systems State the guidelines for modeling with systems of equations. aa aaaaaaaa aaa aaaaaaaaaa aaaaaaaa aaa aaaaaaaaaa aaaa aaa aaaaaaa aaaa aaa aa aaaaa aaaaa aaa aaaaaaa aaaaaaaaaa aa a aaaaaaa aaaaaaa aa aaa aaaaaaaa aaaaa aa aaa aaa aa aaa aaaaaaaa aaaaaaaaa aaaaaaaa aaa aaa aaaaaaaaa aaaaa aaaa a aaa a aa aaaa aaaaa aaaaaaaaa aa aaaaaaa aaa aaaaaaa aaaaaaaaaa aa aaaaa aa aaa aaaaaaaaaa aaaa aaa aaaaaaa aaaaaa aaa aaaaaaa aaa aaa aaaaaaaaaa aaaaaaaaa aa aaa aaaaaaa aa aaaaa aa aaa aaaaaaaaa aaa aaaaaaa aa aaaa aa aa aaa aa a aaaaaa aa aaaaaaaaaa aaaa aaa aaaaaaa aaaaa aa aaa aaaaaaa aaaa aaaa aaa aaaaaaaaaaaaa aaaaaaa aaa aaaaaaaaaaa aaa aaaaa aa aaaa aa aaa aa a aaaaaa aa aaaaaaaaa aaa a aaaaaa aaaa aaaaaaaaa aaaaa aaaaaaaaaaaaaa aa aaaaa aaa aaaaaa aaa aaaaaaaaa aaa aaaaaaaa aaaaa aaa aaaaaa aaa aaaaa aa aaaa aa aaaaa aaaa aaaaaaaaaa aaa aaaaa aaaa aaaaa aaaaaa aa a aaaaaaaa aaaa aaaaaaa aaa aaaaaaaa aaaaa aa aaa aaaaaaaa

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SECTION 10.2

|

149

Systems of Linear Equations in Sev eral Variables

Name ___________________________________________________________ Date ____________

10.2 Systems of Linear Equations in Several Variables A linear equation in n variables is

aaa aaaaaaaa annnnn

aaa aaa aa aaa aa

aaa aaaa a aaaaa a aaa a aaa aaaa aaaaaaa aaa a aaa aaa aaaaaaaaa

.

I. Solving a Linear System A linear system in the three variables x, y, and z is in triangular form if

aaa aaaaa aaaaaa aaaaaaaa

aaa aaaaa aaaaaaaaaa aaa aaaaaa aaaaaaaa aaaa aaa aaaaaaa aa aaa aaa aaaaa aaaaaaaa aaaa aaaaaa a aaa a It is easy to solve a system that is in triangular form by using

aaaaaaaaaaaaaaaaa

.

To change a system of linear equations to an equivalent system, that is, a system with aaaaaaaaa aa aaa aaaaaaaa aaaaaa

, we use the

.

aaa aaaa

aaaa aaaa aaaaaa

.

List the operations that yield an equivalent system. aa aaa a aaaaaaa aaaaaaaa aa aaa aaaaaaaa aa aaaaaaaa aa aaaaaaaa aa aaaaaaaa aa a aaaaaaa aaaaaaaaa aa aaaaaaaaaaa aaa aaaaaaaaa aa aaa aaaaaaaaaa

To solve a linear system, we use these operations to change the system to an equivalent

aaaaaaaaa

aaaaaa

aaaaaaaa

, and then use back-substitution to complete the solution. This process is called

aaaaaaaaaaa

.

II. The Number of Solutions of a Linear System The graph of a linear equation in three variables is

a aaaaa aa aaaaa aaaaaaaaaa aaaaa

A system of three equations in three variables represents of the system are

aaaaa aaaaaa aa aaaa

aaa aaaaaa aaaaa aaa aaaaa aaaaaa aaaaaaaaa

. The solutions .

For a system of linear equations, exactly one of the following is true concerning the number of solutions the system has. 1. aaa aaaaaa aaa aaaaaaa aaa aaaaaaaaa

2. aaa aaaaaa aaa aa aaaaaaaaa

3. aaa aaaaaa aaa aaaaaaaaaa aaaa aaaaaaaaaa

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Systems of Equations and Inequalities

A system with no solutions is said to be solutions is said to be with

aaaaaaaaaaa

aaaaaaaaa

a aaaaa aaaaaaaa

, and a system with infinitely many . A linear system has no solution if we end up

after applying Gaussian elimination to the system.

III. Modeling Using Linear Systems Linear systems are used to model situations that involve

aaaaaaa aaaaaaa aaaaaaaaaa

.

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SECTION 10.3

|

Matrices and Systems of Linear Equations

151

Name ___________________________________________________________ Date ____________

10.3 Matrices and Systems of Linear Equations A matrix is simply

a aaaaaaaaaaa aaaaa aa aaaaaaa

. Matrices are used to

organize information into categories that correspond to aaaaaa

aaa aaaa aaa aaaaaaa aa aaa

.

I. Matrices An m  n matrix is a rectangular array of numbers with We say that the matrix has

a

m  n . The numbers aij are the

aaaaaaaaa

the matrix. The subscript on the entry aij indicates that it is in Example 1

rows and

a

columns. aaaaaaa

aaa aaa aaa aaa aaa aaa aaaaa

of .

Give the dimension of the following matrix.  1 3 0 1 5  2 3 5 4 2     0 4 1 1 4 

II. The Augmented Matrix of a Linear System We can write a system of linear equations as a matrix, called the augmented matrix of the system, by aaaaaaa aaaa aaa aaaaaaaaa aaa aaaaaaaaa aaaa aaaaaa aa aaa aaaaaaaaa

Example 2

2 x  15 y  3z  12  Write the augmented matrix for the linear system 4 x  3 y  15 z  3 .  x  10 y  5 z  10 

III. Elementary Row O perations List the elementary row operations. aa aaa a aaaaaaaa aa aaa aaa aa aaaaaaaa aa aaaaaaaa a aaa aa a aaaaaaa aaaaaaaaa aa aaaaaaaaaaa aaa aaaaa

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Performing any of these operations on the augmented matrix of a system

aaaa aaa aaaaaa

its

solution. Give a description of each of the following notations that are used to represent elementary row operations.

Ri  kR j  Ri

aaaaaa aaa aaa aaa aa aaaaaa a aaa aaa a aa aaa aaa aaaa aaa aaa aaaaaa aaaa aa aaa a

a

kRi

aaaaaaaa aaa aaa aaa aa a

a

Ri  R j

aaaaaaaaaaa aaa aaa aaa aaa aaa

a

IV. Gaussian Elimination To solve a system of linear equations using its augmented matrix,

aaa aaaaaaaaaa aaa aaaaaaaaaa aa

aaaaaa aa a aaaaaa aa a aaaaaaa aaaa aaaaaa aaaaaaaaaaa aaaa

.

List the conditions for which a matrix is considered to be in row-echelon form. aa aaa aaaaa aaaaaaa aaaaaa aa aaaa aaa aaaaaaaa aaaa aaaa aa aaaaaa aa aa aaaa aa aaaaaa aaa aaaaaaa aaaaaa aa aaa aaaaaaa aaaaa aa aaaa aaa aa aa aaa aaaaa aa aaa aaaaaaa aaaaa aa aaa aaa aaaaaaaaaaa aaaaa aaa aa aaa aaaa aaaaaaaaaa aaaaaaaa aa aaaaa aaa aa aaa aaaaaa aa aaa aaaaaaa

A matrix is in reduced row-echelon form if it is in row-echelon form and also satisfies the condition that aaaaa aaaaaa aaaaa aaa aaaaa aaaa aaaaaaa aaaaa aa a

.

Describe a systematic way to put a matrix in row-echelon form using elementary row operations. aaaaa aa aaaaaaaaa a aa aaa aaa aaaa aaaaaaa aaaa aaaaaa aaaaa aaaaa aaaa a aa aaaaaa aaaaaaaaaaa aaaaaaaaa aa aaa aaaaa aaa aa aaa aaaa aaaaa aaa aaaaa aaaaaa a aaaaaaa a aa aaa aaaa aaaa aaa aaaa aaaaaa aaaaa aaaaa aaaa aa aa aaaa aaaaa aaaa aaaa aaaa aaaaa aaaaaaa aaaaa aa aa aaa aaaaa aa aaa aaaaaaa aaaaa aa aaa aaa aaaaa aaaaaaaaaaaa aaa aaaa aa aaaaaaaaaa aaaaaaaa aaaa aaaaaaa aaaaa aaa aaaaaa aa a aaaaaa aa aaaaaaaaaaa aaaaa

Once an augmented matrix is in row-echelon form, the corresponding linear system may be solved aaaaaaaaaaaaaaaaa

. This technique is called

aaaaa

aaaaaa aaaaaaaaaaa

.

List the steps for solving a system using Gaussian elimination. aa aaaaaaaaa aaaaaaa aaaaa aaa aaaaaaaaa aaaaaa aa aaa aaaaaaa aa aaaaaaaaaaa aaaaa aaa aaaaaaaaaa aaa aaaaaaaaaa aa aaaaaa aaa aaaaaaaaa aaaaaa aa aaaaaaaaaaa aaaaa aa aaaaaaaaaaaaaaaaaa aaaaa aaa aaa aaaaaa aa aaaaaaaaa aaaa aaaaaaaaaaa aa aaa aaaaaaaaaaa aaaa aa aaa aaaaaaaaa aaaaaa aaa aaaaa aa aaaaaaaaaaaaaaaaaa

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|

Matrices and Systems of Linear Equations

153

V. Gauss-Jordan Elimination If we put the augmented matrix of a linear system in reduced row-echelon form, then we don’t need to aaaaaaa aa aaaaa aaa aaaaaa

aaaaa

.

List the steps for putting a matrix in reduced row-echelon form. aaa aaa aaaaaaaaaa aaa aaaaaaaaaa aa aaa aaa aaaaaa aa aaaaaaaaaaa aaaaa aaaaaa aaaaa aaaaa aaaa aaaaaaa aaaaa aa aaaaaa aaaaaaaaa aa aaa aaa aaaaaaaaaa aaaa aaaaa aa aaa aaaa aaaaa aaa aaaaa aaaa aaa aaaa aaaaaaa aaaaa aaa aaaa aaa

Using the reduced row-echelon form to solve a system is called

aaaaaaaaaaaa aaaaaaaaaaa

.

VI. Inconsistent and Dependent Systems A leading variable in a linear system is one that

aaaaaaaaaaa aa a aaaaaaa aaaaa aa aaa aaaaaaaaaa

aaaa aa aaa aaaaaaaaa aaaaaa aa aaa aaaaaa

.

Suppose the augmented matrix of a system of linear equations has been transformed by Gaussian elimination into row-echelon form. Then exactly one of the following is true concerning the number of solutions the system has. aa aa aaaaaaaaa aa aaa aaaaaaaaaaa aaaa aaaaaaaa a aaa aaaa aaaaaaaaaa aaa aaaaaaaa a a aa aaaaa a aa aaa aaaaa aaaa aaa aaaaaa aaa aa aaaaaaaaa aa aaa aaaaaaaaa aa aaaa aaaaaaaa aa aaa aaaaaaaaaaa aaaa aa a aaaaaaa aaaaaaaaa aaaa aaa aaaaaa aaa aaaaaaa aaa aaaaaaaaa aaaaa aa aaaa aaaaa aaaaaaaaaaaaaaaaa aa aaaaaaaaaaaa aaaaaaaaaaaa aa aaaaaaaaaa aaaa aaaaaaaaaa aa aaa aaaaaaaaa aa aaa aaaaaaaaaaa aaaa aaa aaa aaa aaaaaaa aaaaaaaaa aaa aa aaa aaaaaa aa aaa aaaaaaaaaaaaa aaaa aa aaa aaaaaaaaaa aaaa aaaaaaaaaa aa aaaa aaaa aaa aaaaaa aa aaaaaa aaaaaaaaaa aa aaaaa aaa aaaaaa aa aaaaaaa aaa aaaaaa aa aaaaaaa aaaaaaaaaaa aaaa aaa aaaa aaaaaaaaaa aaa aaaaaaa aaaaaaaaa aa aaaaa aa aaa aaaaaaaaaa aaaaaaaaaa aaa aaaaaaaaaa aaaaaaaaa aaa aaaa aa aaa aaaa aaaaaaa aa aaaaa aaaaaaa

A system with no solution is called

aaaaaaaaaaaa

.

If a system in row-echelon form has n nonzero equations in m variables (m > n), then the complete solution will have

a

nonleading variables.

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SECTION 10.4

|

155

The Algebra of Matrices

Name ___________________________________________________________ Date ____________

10.4 The Algebra of Matrices I. Equality of Matrices Two matrices are equal if

aaaa aaaa aaa aaaa aaaaaaa aa aaa aaaa aaaaaaaaa

.

The formal definition of matrix equality states that matrices A   aij  and B  bij  are equal if and only if they have

aaa aaaa aaaaaaaaa a a a a

, and corresponding entries are equal, that is

, that is, for i = 1, 2, . . . , m and j = 1, 2, . . . , n.

II. Addition, Subtraction, and Scalar Multiplication of Matrices Let A   aij  and B  bij  be matrices of the same dimension m  n , and let c be any real number. 1. The sum A + B is the m  n matrix obtained by A+B=

a

aaaaaa aaaaaaaaaaaaa aaaaaaa aa a aaa a

.

.

2. The difference A − B is the m  n matrix obtained by a aaa a

aaaaaaaaaaa aaaaaaaaaaaaa aaaaaaa aa

. A−B=

a

.

3. The scalar product cA is the m  n matrix obtained by cA =

a

aaaaaaaa

a aaaa aaaaa aa a aa a

.

Let A, B, and C be m  n matrices and let c and d be scalars. State each of the following properties of matrix arithmetic. Commutative Property of Matrix Addition:

a

.

Associative Property of Matrix Addition:

a

.

Associative Property of Scalar Multiplication:

a

.

Distributive Properties of Scalar Multiplication:

a

.

a

.

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III. Multiplication of Matrices The product AB of two matrices A and B is defined only when aa aaa aaaaaa aa aaaa aa a side,

. This means that if we write their dimensions side by

aaa aaa aaaaa aaaaaaa aaaa aaaaa aa a

AB is a matrix of dimension

If a1

aaa aaaaaa aa aaaaaaa aa a aa aaaaa

a2

. If this is true, then the product .

 b1  b  an  is a row of A, and if  2  is a column of B, then their inner product is the number     bn  a .

The definition of matrix multiplication states that if A   aij  is an m  n matrix and B  bij  is an n  k matrix, then their product is the

, where cij is the inner product

a a a aaaaaa a

of the ith row of A and the jth column of B. We write the product as

a a aa

.

The definition of matrix product says that each entry in the matrix AB is obtained from a and a

aaaaaa

of B as follows:

aa a aaaa aaa aaaaaaaaaaaaa aaaaaaa aa .

Example 1

Suppose A is a 2  6 matrix and B is a 4  2 matrix. Which of the following products is possible: A  B , B  A , both of these, or neither of these?

Example 2

Consider the matrix product that is possible in Example 1. What is the dimension of the possible product(s)?

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of A

aaa aaaaa a aa aaa aaa aaa aaa aaa aaaaaa aa

aaa aaaaaa aa aa aaaaaaaa aa aaaaaaaaaaa aaa aaaaaaa aa aaa aaa aaa aaa aaa aaaaaa aa a aaa aaaaaa aaa aaaaaaa

aaa

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157

The Algebra of Matrices

IV. Properties of Matrix Multiplication Let A, B, and C be matrices for which the following products are defined. List the properties of matrix multiplication that are true. aaaaaaaaaaa aaaaaaaaaaaaaaa a aaaaaa aaaaaaaaaaaa aaaaaaaaaaaaa a aa a aa a aaa aaaaa a aaa a aa a aaa

Matrix multiplication

aa aaa

commutative.

V. Applications of Matrix Multiplication Give an example of an application of matrix multiplication. aaaaaaa aaaa aaaaa

A matrix in which the entries of each column add up to 1 is called

aaaaaaaaaa

VI. Computer Graphics Briefly describe how matrices are used in the digital representation of images. aaaaaaa aaaa aaaaa

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SECTION 10.5

|

Inv erses of Matrices and Matrix Equations

159

Name ___________________________________________________________ Date ____________

10.5 Inverses of Matrices and Matrix Equations I. The Inverse of a Matrix The identity matrix In is the n  n matrix for which aaaaa aaa aaaaa aaaaaaa aaa a

aaaa aaaa aaaaaaaa aaaaa aa a a aaa aaa .

Identity matrices behave like

in the sense that A I n = A and I n B = B ,

aaa aaaaaa a

whenever these products are defined. If A and B are n  n matrices, and if AB = BA = I n , then we say that B is and we write

a

aaa aaaaaaa aa a

,

.

Let A be a square n  n matrix. The definition of the inverse of a matrix states that if there exists an n  n matrix A1 with the property that AA1  A1 A  I n , then we say that A1 is the

aaaaaa

of A.

II. Finding the Inverse of a 2 × 2 Matrix The following rule provides a simple way for finding the inverse of a 2  2 matrix, when it exists.

a b  If A   , then A1   c d 

If

a

, then A has no inverse.

The quantity ad  bc that appears in the rule for calculating the inverse of a 2  2 matrix is called the aaaaaaaaaaa

of the matrix. If the determinant is 0, then

aaa aaaa aa aaaaaaa aaaaaa aa aaaaaa aaaaaa aa aa

aaa aaaaa aaaa .

III. Finding the Inverse of an n × n Matrix Describe the procedure for finding the inverse of a 3 × 3 or larger matrix. aa a aa aa a a a aaaaaaa aa aaaaa aaaaaaaaa aaa a a aa aaaaaa aaaa aaa aaa aaaaaaa aa a aa aaa aaaa aaa aa aaa aaaaaaaa aaaaaa aa aa aaa aaaaaaa aaaa aaaa aaa aaa aaaaaaaaaa aaa aaaaaaaaaa aa aaaa aaa aaaaa aaaaaa aa a aaaaaa aaa aaaa aaaa aaaa aaa aaaaaaaa aaaaaaa aaaaa aaaaa aaaa aa aaa aaaaaaaa aaa aaaaa aaaaaa aa aaaaaaa aa aaaaaaaaaaa aaaaaa aaa aaaaa aaaa aa aaaaaaaaaaa aaaaaaaaaaaaa aaaa a aa

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Graphing calculators are also able to calculate matrix inverses. List the steps required to do so for your graphing calculator. aaaaaaa aaaa aaaaa

If we encounter a row of zeros on the left when trying to find an inverse, then

aaa aaaaaa aaaaaa

aaaa aaa aaaa aa aaaaaa

aaaaaaaa

. A matrix that has no inverse is called

.

IV. Matrix Equations  a1 x  b1 y  c1 z  d1  For the system of linear equations a2 x  b2 y  c2 z  d 2 , write the corresponding matrix equation, in the form a x  b y  c z  d 3 3 3  3 AX = B , where the matrix A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.

1 If A is a square n  n matrix that has an inverse A and if X is a variable matrix and B a known matrix, both

with n rows, then the solution of the matrix equation AX = B is given by

a

.

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SECTION 10.6

Determinants and Cramer’s Rule

161

Name ___________________________________________________________ Date ____________

10.6 Determinants and Cramer’s Rule A square matrix is one that has

aaa aaaa aaaaaa aa aaaa aa aaaaaaa

.

I. Determinant of a 2 × 2 Matrix We denote the determinant of a square matrix A by the symbol If A = [a] is a 1 × 1 matrix, then det(A) =

a The determinant of the 2 × 2 matrix A   c

a

aaaaaa

or

a a

.

.

b is det(A) = | A | = d 

a

.

II. Determinant of an n × n Matrix Let A be an n × n matrix. The minor M ij of the element aij is

aaa aaaaaaaaaaa aa aaa aaaaaa aaaaaaaa aa aaaaaaaa aaa

aaa aaa aaa aaa aaaaaa aa a

.

The cofactor Aij of the element aij is

a

.

Note that the cofactor of aij is simply the minor of aij multiplied by either 1 or 1 , depending on whether

i + j is

aaaa aa aaa

.

Fill in the + and – sign pattern associated the minors of a 4 × 4 matrix.

     

             

If A is an n × n matrix, then the determinant of A is obtained by

aaaaaaaaaa aaaa aaaaaaa aa

aaa aaaaa aaa aa aaa aaaaaaaa aaa aaaa aaaaaa aaa aaaaaa

det(A) =

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. In symbols, this is

a

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This definition of determinant used the cofactors of elements in the first row only. This is called aaa aaaaaaaaaaa aa aaa aaaaa aaa

aaaaaaaaa

. In fact, we can expand the determinant by any row or column in

the same way and obtain

aaa aaaa aaaaaa aa aaaa aaaa

.

The Invertibility Criterion states that if A is a square matrix, then A has an inverse if and only if a

.

III. Row and Column Transformations If we expand a determinant about a row or column that contains many zeros, our work is reduced considerably because

aa aaaaa aaaa aa aaaaaaaa aaa aaaaaa aa aaa aaaaaaaa aaaa aaa aaaa

.

The principle of row and column transformations of a determinant states that i f A is a square matrix and if the matrix B is obtained from A by adding a multiple of one row to another or a multiple of one column to another, then

aaaaaa a aaaaaa

.

This principle often simplifies the process of finding a determinant by aaaaaaa aaaaaaaa aaa aaaaa

aaaaaaaaaaa aaaaa aaaa aa

.

IV. Cramer’s Rule ax  by  r Cramer’s Rule for Systems in Two Variables states that the linear system  has the solution cx  dy  s

x=

a

.

and

y=

a

, provided that

a

b

c

d

0.

ax  by  r For the linear system  , complete the notation for each of the following and give a description. cx  dy  s a b  D  c d 

aaaa aa aaa aaaaaaaaaaa aaaaaaa

a b  Dx    c d 

aaaa aaaa aaaaaa aa aaaaaaaaa aaa aaaaa aaaaaa aa a aa a aaa aa

a b  Dy    c d 

aaaa aaaa aaaaaa aa aaaaaaaaa aaa aaaaaa aaaaaa aa a aa a aaa aa

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|

Determinants and Cramer’s Rule

163

ax  by  r Write the solution of the system  using D, Dx , and D y . cx  dy  s

Cramer’s Rule states that if a system of n linear equations in the n variables x1 , x2 , . . . , xn is equivalent to the matrix equation DX = B , and if D  0 , then its solutions are x1 =

Dxi is

Dx1 D

, x2 

Dx2 D

, . . . , xn =

aaa aaaaaa aaaaaaaa aa aaaaaaaaa aaa aaa aaaaaa aa a aa aaa a a a aaaaaa a

V. Areas of Triangles Using Determinants If a triangle in the coordinate plane has vertices (a1 , b1 ) , (a2 , b2 ) , and (a3 , b3 ) , then its area is

Area =

a1 1  a2 2 a3

b1 1 b2 1 b3 1

where the sign is chosen to make the area positive.

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Dxn D

, where .

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SECTION 10.7

|

165

Partial Fractions

Name ___________________________________________________________ Date ____________

10.7 Partial Fractions Some applications in calculus require expressing a fraction as the sum of simpler fractions called aaaaaaaaa

aaaaaa

.

Let r be the rational function r ( x) =

P( x) where the degree of P is less than the degree of Q. After we have Q( x)

completely factored the denominator Q of r, we can express r ( x) as a sum of partial fractions of the form

a

and

aaaaaaaaaaaaa

a

. This sum is called the

aaaaaaa aaaaaaaa

of r.

I. Distinct Linear Factors Case 1: The Denominator is a Product of Distinct Linear Factors Suppose that we can factor Q( x) as Q( x) = (a1 x + b1 )(a2 x + b2 )

(an x + bn ) with no factor repeated. In this

case the partial fraction decomposition of P( x) / Q( x) takes the form

P( x) = Q( x)

In your own words, describe the process of finding a partial fraction decomposition. aaa aa aaa aaaaaaa aaaaaaaa aaaaaaaaaaaaa aaaa aaa aaaaaaa aaaaaaaaa aa aa aa a a a a aaaa aa aaaaaaaa aaaa aaaa aa aaa aaaaaaaaa aaaaaaaa aa aaa aaaaaa aaaaaaaaaaaa aaaaaaaa aaa aaaaaaaaaa aaaa aa aaa aaaaaaaaa aaa aaaaaa aaaaaaaaaaaaa aaaa aaaaa a aaa aa aaaaaa aaaaaaaaa aaaa aaaa aaaaaa aaaa a aaaaaa aaaaaaaa aaaaaaaaa aaaa aaa aaaaaaa aaaaaaaa aaaaaaaaaaaaa aaa aaaa aaa aa aaaaaaaaaaa

II. Repeated Linear Factors Case 2: The Denominator is a Product of Distinct Linear Factors, Some of Which Are Repeated k Suppose the complete factorization of Q( x) contains the linear factor ax + b repeated k times; that is, (ax + b)

is a factor of Q( x) . Then, corresponding to each such factor, the partial fraction decomposition for P( x) / Q( x)

contains

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III. Irreducible Q uadratic Factors Case 3: The Denominator Has Irreducible Q uadratic Factors, None of Which Is Repeated Suppose the complete factorization of Q( x) contains the quadratic factor ax2 + bx + c , which can’t be factored further. Then, corresponding to this, the partial fraction decomposition of P( x) / Q( x) will have a term of the

form

a

.

IV. Repeated Irreducible Q uadratic Factors Case 4: The Denominator Has a Repeated Irreducible Q uadratic Factor Suppose the complete factorization of Q( x) contains the factor (ax2 + bx + c)k , where ax2 + bx + c can’t be factored further. Then the partial fraction decomposition of P( x) / Q( x) will have the terms

The techniques described in this section apply only to rational functions P( x) / Q( x) in which

. If this isn’t the case, we must first

aa a aa aaaa aaaa aaa aaaaa aa a aaaa aaaaaaaa aa aaaaaa a aaaa a

aaa aaaaaa aaa

.

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SECTION 10.8

|

Systems of Nonlinear Equations

167

Name ___________________________________________________________ Date ____________

10.8 Systems of Nonlinear Equations I. Substitution and Elimination Methods To solve a system of nonlinear equations, we can use the

aaaaaaaaaaaa aa aaaaaaaaa aaaaaa

.

Describe the process for solving a system of nonlinear equations using the substitution method. aa aaaaa a aaaaaa aa aaaaaaaaa aaaaaaaaa aaaaa aaa aaaaaaaaaaaa aaaaaaa aaaaa aaaaa aaa aa aaa aaaaaaaaa aaa aaa aaaaaaaaa aaaa aaaaaaaaaa aaa aaaa aaaaaaaa aa aaa aaaaa aaaaaaaa aaa aaaaa aaa aaa aaaaaaaaa aaaaaaaaa aaaaaaaa aaaaaaaaaaaaaaa aa aaaa aaa aaaaaaaaaa

Describe the process for solving a system of nonlinear equations using the elimination method. aaaaaa aaaaa aaaaaaaa aa aaaaaaaaa aaa aaaaaaaa aaaa aaaaaaaa aa aa aaaaaaaaaaa aaaaaa aa aaaa aaa aaaaa aaaaaaaaaa aaaa aaaaaaaa aaaa a aaa aa a aaaa aaa aaaaaaaaa aaa aaaaaa aaaaa aaa aaaaaaaaa aaaaaaaa aaa aaa aaaaaaaaa aaaaaaaa aaa aaaa aaaaaaaaaaaaaaa aa aaaa aaa aaaaa aaaaaa

II. Graphical Method Describe the process for solving a system of nonlinear equations with the graphical method. aaaaaa aaaaa aaaa aaaaaaaaa aa aaa aaaa aaaaaaaaa aaaa aaaa aaa aaaaaaaaaaa aa aaa aaaaaaaaaaaa aaaaaaa aaaa aaa aa aaaa aaaaa a aaaaaaaa aaaaaaaaaaa aaaaa aaaaaa aa aaaaaaaaaaaa aaa aaa aaaaaaaaa aa aaa aaaaaaa

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Additional notes

y

y

x

y

y

x

y

x

y

x

y

x

y

x

x

y

x

x

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SECTION 10.9

|

Systems of Inequalities

169

Name ___________________________________________________________ Date ____________

10.9 Systems of Inequalities I. Substitution and Elimination Methods The graph of an inequality, in general, consists of a region in the plane whose boundary is

aaa aaaaa aa

aaa aaaaaaaa aaaaaaaa aa aaaaaaaaa aaa aaaaaaaaaa aaaa aaaa aa aa aaaa To determine which side of the graph gives the solution set of the inequality, we need only aaaaaa

. aaaaa aaaa

.

List the steps for graphing an inequality. aa aaaaa aaaaaaaaa aaaaa aaa aaaaaaaa aaaa aaaaaaaaaaa aa aaa aaaaaaaaaaa aaa a aaaaaa aaaaa aaa a aa a aaa a aaaaa aaaaa aaa a aa aa aa aaaaa aaa aaaaaaaaaaa aaa aaaaa aa aaa aaaaaaaaaa aaaaaaaa aa aaa aaa aaaaaa aa aaa aaaa aa aaa aaaaa aaaa aa aaaaaaa aa aaaa aa aa aaa aaaa aaaaaa aa aaaaaa aaaa aa aaa aaaaa aa aaaaaaaaa aaaaaaa aaa aaaaaa aa aaaa aaaa aaaaaaa aaa aaaaaaaaaaa aa aaa aaaaa aaaaaaaaa aaa aaaaaaaaaaa aaaa aaa aaa aaaaaa aa aaaa aaaa aa aaa aaaaa aaaaaaa aaa aaaaaaaaaaa aa aaaa aaaaa aaaaa aaaa aaaa aa aaa aaaaa aa aaaaaaaa aaaa aa aa aaaa aa aaa aaaaaa aa aaa aaaa aaaaa aaaa aaa aaaaaaa aaa aaaaaaaaaaa aaaa aaa aaaaaa aaaaa aaaa aa aaa aaaaaa

II. Systems of Inequalities The solution of a system of inequalities is the set of all points in the coordinate plane that aaaaaaaaaa aa aaa aaaaaa

. The graph of a system of inequalities is

aaaaaaaa aaa

.

aaaaaaa aaaaa aaa aaaaa aa aaa

List the steps for graphing the solution of a system of inequalities. aaaaa aaaa aaaaaaaaaaa aaaaa aaaa aaaaaaaaaa aa aaa aaaaaa aa aaa aaaa aaaaaa aaaaaaa aaa aaaaaaaa aa aaa aaaaaaa aaaaa aaa aaaaaa aaaaa aaa aaaaaa aa aaa aaa aaaaaaaaaaaa aaaaaaaaaa aaa aaa aaaaaa aa aaaa aaaaaa aaaaaaa aaaa aaaaaaaaaaa aa aaaa aaaaaa aa aaa aaaaaaaa aa aaa aaaaaaa aaaaaa aaa aaaaaaaaa aaaaa aaa aaaaaaaa aa aaa aaaaaa aaaa aaa aaaaaa aa aaaa aa a

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III. Systems of Linear Inequalities If an inequality can be put into one of the forms ax  by  c , ax  by  c , ax + by > c , or ax + by < c , then the inequality is

aaaaaa

.

Describe the process of solving a system of linear inequalities. aaaaa aaaaa aaa aaaaa aaaaa aa aaa aaaaaaaaa aaaa aaaaaaaaaa aa aaaa aaaaaaaaaaa aa aaaaaaaaa aaa aaaaaa aa aaa aaaaaa aaaaaaaaaaaaa aa aaaa aa aaaaa aaaa aaa aaaa aaaaaa aaaaaa aaa aaaaaaaaaaaaa aa aaaaaaaa aaaa aaa aaaaaaaaaaa aa aaaa aaaaaa aaa aaaaaaaa aa aaaaaaaaaaaaaa aaaaaaa aaa aaaaaaaaa aa aaa aaaaa aaaa aaaaaaaaa aa aaaa aaaaaaa aaaaaaa aaaaaaaaa aaaaaaa aaaa aaaaaa aa aaaa aa aaa aaaaaaaa aaaa

When a region in the plane can be covered by a sufficiently large circle, it is said to be A region that is not bounded is called “ fenced in”—it extends

aaaaaaaaa

aaaaaaa

.

. An unbounded region cannot be

aaaaaaaaaa aaa aa aa aaaaa aaa aaaaaaaaa

.

IV. Application: Feasible Regions Give an example of variable constraints in applied problems. aaaaaaa aaaa aaaaa

Constraints or limitations such as these can usually be expressed as

aaaaaaa aa aaaaaaaaaa

When dealing with applied inequalities, we usually refer to the solution set of a system as a aaaaaa

, because the points in the solution set represent

aaa aaa aaaaaaaaaa aaaaa aaaaaaa

. aaaaaaaa

aaaaaaaa aaa aaaaaaaaa aaaaaa

.

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Name ___________________________________________________________ Date ____________

Chapter 11

Conic Sections

11.1 Parabolas I. Geometric Definition of a Parabola A parabola is the set of points in the plane that are

aaaaaaaa aaaa a aaaaa aaaaa a aaaaaaa aaa

aaaaaa aaa a aaaaa aaaa a aaaaaaa aaa aaaaaaaaaa The vertex V of the parabola lies

.

aaaaaaa aaaaaaa aaa aaaa aaa aaa aaaaaaaaa

and the axis of symmetry is the line that runs aaa aaaaaaaaa

,

aaaaaaa aaa aaaaa aaaaaaaaaaaaa aa

.

II. Equations and Graphs of Parabolas The graph of the equation x 2 = 4 py is a parabola with its focus is aa a

aaaa aa

, and its directrix is

or downward if

aaa

a a aa

to the right if

, its focus is

aaaa aa

aaa

or to the left if

axis. Its vertex is

aaaa aa

,

. The parabola opens upward if

.

2 The graph of the equation y = 4 px is a parabola with

aaaa aa

aaaaaaa

aaaaaaaaaa

, and its directrix is aaa

axis. Its vertex is a a aa

. The parabola opens

.

Describe how to find the focus and directrix of a parabola from its equations. aa aaaa aaa aaaaa aaa aaaaaaaaaa aaa aaa aaaaa aaaaaaaa aa aaaaaaaa aaaaa aaaaaaaaa aaaa aa aaa aaaaaaa a a aaaaaaaa a a aaa aa a a aaaa aaaa aaa aaaaa aa a aaa aaaaaaaa aaa aaaaaaaaaaa aaa aaa aaaaa aaa aaa aaaaaaaa aa aaa aaaaaaaaaa

The line segment that runs through the focus of a parabola perpendicular to the axis, with endpoints on the parabola, is called the

aaaaa aaaaaa

the parabola. The focal diameter of a parabola is

, and its length is the a aa a

aaaaaaaaaaaa

of

.

III. Applications Parabolas have the important property that light from a source placed at the focus of a surface with parab olic cross section will be reflected in such a way that it This property makes parabolas very useful as

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Precalculus:

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aaaaaaa aaaaaaaa aa aaa aaaa a aaaaaaaa aaaaaaaaa aaa aaaaa aaa aaaaaaaaaa

. .

th

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171

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Conic Sections

Conversely, light approaching a parabolic reflector in rays parallel to its axis of symmetry is aa aaa aaaaa aa aaa aaaaaaaa

aaaaaaaaaaaa

.

y

y

x

y

y

x

y

x

y

x

y

x

y

x

x

y

x

x

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SECTION 11.2

173

Ellipses

Name ___________________________________________________________ Date ____________

11.2 Ellipses I. Geometric Definition of an Ellipse An ellipse is the set of points in the plane

aaa aaa aa aaaaa aaaaaaaaa aaaa aaa aaaaa aaaaaa

a aaa a aa a aaaaaaaa

. These two fixed points are the

aaa

of

the ellipse. If the foci are on the x-axis, the ellipse crosses the x-axis at These points are called the the

aaaaaaaa

aaaaa aaaa and

aaaaa aaaa

aa

aaa aa

, and it has length

minor axis. The origin is the

and

aaa aa

.

of the ellipse, and the segment that joins them is called

. Its length is

aaa a

aaa aa

. The ellipse also crosses the y-axis at

. The segment that joins these points is called

aaa

aa

the

aaaaaa

. The major axis is

aaaaaa aaaa

of the ellipse.

If the foci of the ellipse are placed on the y-axis rather than on the x-axis, then aaa aaaaaaaa aa aaa aaaaaaaaa aaaaaaaaaa

aaa aaaaa aa a aaa a

, and we get a

aaaaaaa

ellipse.

II. Equations and Graphs of Ellipse s The graph of the equation

x2 a2

+

y2 b2

Its vertices are located at aa at

= 1 is an ellipse with center at the

aaaa aa

. Its minor axis is aaaa aa

aaaaaaaa

x2 b

2

+

y2 a2

. Its minor axis is

located at

with length

aaa aaa

aa

b.

with length

.

aaaaaa

. The major axis is

, where a

aaaaaaaa

with length

and have the relationship

a

. The foci are located

a

aaaaaaaaaa

, where a

aaaaaaaaa

= 1 is an ellipse with center at the

aaa aaa

Its vertices are located at aa

. The major axis is

and have the relationship

The graph of the equation

aaaaaa

a

aa

a

b.

with length . The foci are

.

III. Eccentricity of an Ellipse For the ellipse

x2 a

2

+

y2 b

2

= 1 or

x2 b

2

+

y2 a2

= 1 (with a > b > 0), the eccentricity e is the number

2 2 where c  a  b . The eccentricity of every ellipse satisfies

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aaaaa

Mathematics for Calculus, 7th Edition

a

.

,

174

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CHAPTER 11

Conic Sections

If e is close to 1, then the ellipse is the ellipse is

aaaaaaaaa

aaaaa aa a aaaaaa

in shape. If e is close to 0, then

in shape. The eccentricity is a measure of

aaaaaaaaaaa aaa aaaaaaa aa

.

Ellipses have an interesting

aaaaaaaaaa aaaaaaaa

aaa

that leads to a number of practical

applications. If a light source is placed at one focus of a reflecting surface with elliptical cross sections, then aaa aaa aaaaa aaaa aa aaaaaaaaa aaa aaa aaaaaaa aa aaa aaaaa aaaaa

.

Give an example of a practical application of the reflection property of an ellipse and explain how it works. aaaaaaa aaa aaaaa aaaaaaaaaaaa a aaaaaaaaa aa aaaaaa aaaaaaa aa aaa aaaaaaaa aaa aaaaaaaaaa aaaaaaaaa aa aaaaaaa aaaaaaaa

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SECTION 11.3

175

Hyperbolas

Name ___________________________________________________________ Date ____________

11.3 Hyperbolas I. Geometric Definition of a Hyperbola A hyperbola is the set of points in the plane

aaa aaaaaaaaaa aa aaaaa aaaaaaaaa aaaa aaa

aaaaa aaaaaa a aaa a aa a aaaaaaaa aaaa

. These two fixed points are the

of the hyperbola.

If the foci are on the x-axis, the hyperbola has x-intercepts at These points are called the

aaaaaaaa

aaa aa

and

aaa aa

of the hyperbola.

If the hyperbola’s foci are on the x-axis, does the graph of the hyperbola intersect the y-axis? A hyperbola consists of two parts called its vertices on the separate branches is the origin is called its

aaaaaaaa

aa

. The segment joining the two

aaaaaaaaaa aaaa

aaaaaa

.

of the hyperbola, and the

.

If the foci of the hyperbola are placed on the y-axis rather than on the x-axis, this has the effect of

aaaaaaaa

aaa aaaaa aa a aaa a aa aaa aaaaaaaaaa aa aaa aaaaaaaaa aaaaaaaa

a

aaaa a aaaaaaaa aaaaaaaaaa aaaa

. This leads to a

aaaaa

.

II. Equations and Graphs of Hyperbolas The graph of the equation

x2 a

2



 1 , a > 0, b > 0, is a hyperbola with center at the aaaaaa b2 aaaa aa . The transverse axis is aaaaaaaaa

vertices are located at length

aa

are located at

. This hyperbola has asymptotes given by aaaa aa

The graph of the equation

y2 a2

are located at

aa

a

and have the relationship



x2 b2

a

. The transverse axis is

and have the relationship

The asymptotes are lines that the hyperbola

aaaaaa aaaaaaa

a

a

with

. The foci

.

aaaa aa aa aaaaaaaaa

. A convenient way to find the asymptotes for a hyperbola with horizontal

Note Taking Guide for Stewart/Redlin/Wats on Precalculus: Copyright © Cengage Learning. All rights reserved.

. Its

aaaaaaaaaa aaa aaaa aaaaaa aa a aaa a

Asymptotes are an essential aid for graphing a hyperbola because they aaa aaaaa

with

.

. This hyperbola has asymptotes given by

aaa aaa

. Its

. The foci

 1 , a > 0, b > 0, is a hyperbola with center at the

aaa aaa

vertices are located at

length

y2

Mathematics for Calculus, 7th Edition

.

176

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Conic Sections

transverse axis is to first plot the points

aaa aaa aaaa aaa aaa aaa a aaa aaa

. Then

sketch horizontal and vertical segments through these points to construct a rectangle called the of the hyperbola. The slopes of the diagonals of the central box are ±b / a ,

aaaaaaa aaa so by extending them, we obtain

aaa aaaaaaaaaa aa aaa aaaaaaaaa

.

List the steps for sketching a hyperbola. aa aaaaaa aaa aaaaaaa aaaa aaaa aa aaa aaaaaaaaa aaaaaaaa aa aaa aaaaaaa aaaa aaaaa aaaaaaaa aa aaa aaaaa aaaa aaaaaaa aaa aaaa aa aaa aaa aaaaa aa aaa aa aaaaaa aaa aaaaaaaaaaa aaaaa aaa aaa aaaaa aaaaaaaa aa aaaaaaaaa aaa aaaaaaaaa aa aaa aaaaaaa aaaa aa aaaa aaa aaaaaaaaa aaaaa aaa aaa aaa aaaaaaaaaaaa aa aaa aaa aaaaaaaaaaaaa aa aaaaaa aaa aaaaaaaaaa aaaaa aa a aaaaaaa aaa aaaaaa a aaaaaa aa aaa aaaaaaaaaa aaaaaaaaaaa aaa aaaaaaaaaaa aaaaaa aaa aaaaa aaaaaa aa aaa aaaa aaaa

Like parabolas and ellipses, hyperbolas have an interesting reflection property: light aimed at one focus of a hyperbolic mirror is

aaaaaaaaa aaaaaa aaa aaaaa aaaaa

.

Give an example of how the hyperbola’s reflection property is used in real life.

aaaaaaa aaa aaaaa aaaaaaaa aaaaaaa aa aaa aaaaaaaaaaaaaaa aaaaaaaaa aaa aaa aaaaa aaaaaaa

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SECTION 11.4

Shifted Conics

177

Name ___________________________________________________________ Date ____________

11.4 Shifted Conics I. Shifting Graphs of Equations If h and k are positive real numbers, then replacing x by x  h or x + h and replacing y by y  k or y + k has the following effect (s) on the graph of any equation in x and y. Replacement

How the graph is shifted

1. x replaced by x  h

.

aaaaa a aaaaa

2. x replaced by x + h

.

aaaa a aaaaa

3. y replaced by y  k

.

aaaaaa a aaaaa

4. y replace by y + k

.

aaaaaaaa a aaaaa

. . . .

II. Shifted Ellipses If we shift an ellipse so that its center is at the point (h, k), instead of at the origin, then its equation becomes

( x  h) 2



a2

( y  k )2 b2

1

III. Shifted Parabolas If we shift a parabola so that its center is at the point (h, k), instead of at the origin, then its equation becomes a

for a parabola with a vertical axis or

a

for

a parabola with a vertical axis.

IV. Shifted Hyperbolas If we shift a hyperbola so that its center is at the point (h, k), instead of at the origin, then its equation becomes

( x  h) 2 a2 ( x  h) 2 a2





( y  k )2 b2 ( y  k )2 b2

1 for a hyperbola with a horizontal transverse axis or

1 a

Note Taking Guide for Stewart/Redlin/Wats on Precalculus: Copyright © Cengage Learning. All rights reserved.

for a hyperbola with a vertical transverse axis.

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Conic Sections

V. The General Equation of a Shifted Conic If we expand and simplify the equations of any of the shifted conics, then we will always obtain an equation of the form

a

, where A and C are not both 0. Conversely,

if we begin with an equation of this form, then we can

aaaaaaaa aaa aaaaaa aa a aaa a

to

see which type of conic section the equation represents. In some cases the graph of the equation turns out to be just a pair of lines or a single point, or there may be no graph at all. These cases are called

aaaaaaaaaa aaaaaa

.

The graph of the equation Ax2 + Cy 2 + Dx + Ey + F = 0 , where A and C are not both 0, is a conic or aaaaaaaaaa aaaaa . In the nondegenerate cases the graph is 1. a(n)

aaaaaaaa

2. a(n)

aaaaaaa

3. a(n)

aaaaaaaaa

a

if A or C is 0, if A and C have the same sign (or a circle if

aa a

),

if A and C have opposite signs.

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SECTION 11.5

|

Rotation of Axes

179

Name ___________________________________________________________ Date ____________

11.5 Rotation of Axes I. Rotation of Axes If the x- and y-axes are rotated through an acute angle  about the origin to produce a new pair of axes, these new axes are called the

aa aaa aaaaaa

(x, y) in the old system has coordinates

. A point P that has coordinates

aaa aa

in the new system.

Suppose the x- and y-axes in a coordinate plane are rotated through the acute angle  to produce the X- and Y-axes. Then the coordinates (x, y) and (X, Y) of a point in the xy- and XY-planes are related as follows: x= y=

a a

.

X=

a

.

.

Y=

a

.

If the coordinate axes are rotated through an angle of 45°, describe how to find the XY-coordinates of the point with xy-coordinates (1, 5). aaa a a aaaa a a aa aaa a a aa aaaa aaaaaaaaaa aaaaa aaaaaa aaaa aaa aaaaaaaa aa aaaa aaaaaaaa aa aaaa aaa aaaaaaaaaaaaaaa

II. General Equation of a Conic 2 2 To eliminate the xy-term in the general conic equation Ax + Bxy + Cy + Dx + Ey + F = 0 , rotate the axes

through the acute angle  that satisfies

a

.

III. The Discriminant 2 2 The graph of the equation Ax + Bxy + Cy + Dx + Ey + F = 0 is either a conic or a degenerate conic. In the nondegenerate cases, the graph is

1. a(n)

aaaaaaaa

2. a(n)

aaaaaaa

3. a(n)

aaaaaaaaa

2 if B  4 AC  0 , 2 if B  4 AC  0 , 2 if B  4 AC  0 .

2 The quantity B  4 AC is called the

aaaaaaaaaaaa

The discriminant is unchanged by any rotation and, thus, is said to be

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of the equation. aaaaaaa aaaaa aaaaaaaa

Mathematics for Calculus, 7th Edition

.

180

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Conic Sections

Additional notes

y

y

x

y

y

x

y

x

x

y

x

x

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SECTION 11.6

|

181

Polar Equations of Conics

Name ___________________________________________________________ Date ____________

11.6 Polar Equations of Conics I. A Unified Geometric Description of Conics Let F be a fixed point (the

aaaaa

be a fixed positive number (the

),

aaaaaaaaaaa

of the distance from P to F to the distance from P to

is the constant e is a

a

, an ellipse if

aaaaaaaaa

), and let e

). The set of all points P such that the ratio

the set of all points P such that aa a

a fixed line (the

aaa

aaaaa

. That is,

is a conic. The conic is a parabola if

, or a hyperbola if

aaa

.

II. Polar Equations of Conics A polar equation of the form r =

ed ed or r = represents a conic with one focus at the origin and 1 ± e cos  1 ± e sin 

with eccentricity e. The conic is: 1. a parabola if

aaa

,

2. an ellipse if

a aaaa

,

3. a hyperbola if

aaa

.

To graph the polar equation of a conic, we first

aaaaaaaaa aaa aaaaaaaa aa aaa aaaaaaaaa aaaa aaa

aaaa aa aaa aaaaaaaa

. For a parabola, the

perpendicular to the directrix. For an ellipse, the directrix. For a hyperbola, the

aaaa aa aaaaaaaa

aaaaa aaaa

aaaaaaaaaa aaaa

To graph a polar conic, it is helpful to plot the points for which θ =

is

is perpendicular to the is perpendicular to the directrix. aa aaaa aa aaa aaaa

.

Using these points and a knowledge of the type of conic (which we obtain from the eccentricity), we can easily get a rough idea of

aaa aaaaa aaa aaaaaaaa aa aaa aaaaa

.

When we rotate conic sections, it is much more convenient to use polar equations than aaaaaaaaa

. We use the fact that the graph of r  f (   ) is the graph of

aaaaaaa aaaaaaaaaaaaaaaa aaaaa aaa aaaaaa aaaaaaa aa aaaaa a When e is close to 0, an ellipse is as

aaaaaaaaa

a aaaaaaaaa

increases beyond 1, the conic is

aaaaaa aaaaaaaa . When e = 1, the conic is

. , and it becomes more elongated a aaaaaaaa

aa aaaa aaaaaaa aaaaaaaaa

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a a aaaa

Mathematics for Calculus, 7th Edition

. As e .

182

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|

Conic Sections

Additional notes

/2 y

/2 y



0 x



3/2 /2 y

/2 y

x0



3/2 /2 y



0 x



3/2 /2 y

3/2 /2 y

x0



3/2 /2 y



0 x

3/2



x0

3/2 /2 y

x0

3/2

0 x



x0

3/2

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Chapter 12

Sequences and Series

12.1 Sequences and Summation Notation I. Sequences A sequence is

a aaaaaaaa a aaaaa aaaaaa aa aaa aaa aa aaaaaa aaaaaaa

The values f(1), f(2), f(3), . . . are called the

aaaaa

.

of the sequence.

To specify a procedure for finding all the terms of a sequence,

aa aaa aaaa a aaaaaaa aaa aaa

aaa aaaa _______aaa aaaaaaaa

.

The presence of (1) n in the sequence has the effect of aaaaaaaa aaa aaaaaaaa

aaaaaa aaaaaaaaaa aaaaa aaaaaaaaaaa

.

II. Recursively Defined Sequences A recursive sequence is a sequence in which

aaa aaa aaaa aa aaa aaaaaaaa aaa aaaaaa aa

aaaa aa aaa aa aaa aaaaa aaaaaaaaa aa

.

The Fibonacci sequence, given as

aa aa aa aa aa aa aaa aaa aa aaa a a a

was named

th

after the 13 century Italian mathematician who used it to solve a problem about the breeding of rabbits.

III. The Partial Sums of a Sequence For the sequence a1 , a2 , a3 , a4 ,

, an ,

the partial sums are

S1 = S2 = S3 = S4 =

Sn =

S1 is called the

aaaaa aaaaaaa aaa

so on. S n is called the called the

. S 2 is the

Note Taking Guide for Stewart/Redlin/Wats on

Precalculus:

Copyright © C engage Lear ning. All rights res er ved.

, and

. The sequence S1 , S 2 , S3 , . . . , S n , . . . is

aaa aaaaaaa aaa

aaaaaaaa aa aaaaaaa aaaa

aaaaaa aaaaaaa aaa

.

th

Mathematics for Calculus, 7 Edition

183

184

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|

Sequences and Series

IV. Sigma Notation Given a sequence a1 , a2 , a3 , a4 , aaaaaaaa

, we can write the sum of the first n terms using

, which derives its name from the Greek letter

a

aaaaaaaaa aa aaaaa .

n

The notation is used as follows:

a

k



k 1

The left side of this expression is read as is called the

aaaa aaa aa aa a aaa a a a aa a a aa

aaaaa aa aaaaaaaaa

, or the

the idea is to replace k in the expression after the sigma by

aaaaaaaaa aaaaaaaa aaaaaaaa aa aa aa a a a a a

. The letter k , and , and

add the resulting expressions. Let a1 , a2 , a3 , a4 , and b1 , b2 , b3 , b4 , be sequences. Then for every positive integer n and any real number c, complete each of the following properties of sums. n

1.

 (a

k

 bk ) 

k 1

n

2.

 (a

k

 bk ) 

k 1 n

3.

 ca

k



k 1

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SECTION 12.2

|

Arithmetic Sequences

185

Name ___________________________________________________________ Date ____________

12.2

Arithmetic Sequences

I. Arithmetic Sequences An arithmetic sequence is a sequence of the form The number a is the

aaaaa aaaa

aa a a aa a a aaa a a aaa a a aaa a a a

, and d is the

The nth term of an arithmetic sequence is given by The number d is called the common difference because aaaaaaaa aaaaaa aa a

a

of the sequence. .

aaa aaa aaaaaaaaaaa aaaaa aa aa aaaaaaaaaa

.

An arithmetic sequence is determined completely by aaaaaaaaaa a

aaaaaa aaaaaaaaaa

aaa aaaaa aaaa a aaa aaa aaaaaa

. Thus, if we know the first two terms of an arithmetic sequence, then

aaaa a aaaaaaa aaa aaa aaa aaaa

.

II. Partial Sums of Arithmetic Sequences For the arithmetic sequence an  a  (n  1)d the nth partial sum

Sn  a  (a  d )  (a  2d )  (a  3d )  1. Sn 

.

 [a  (n  1)d ] is given by either of the following formulas.

n [2a  (n  1)d ] 2

 a  an  2. Sn  n    2 

Give an example of a real-life situation in which a partial sum of an arithmetic sequence is used. aaaaaaa aaaa aaaaa

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aa aaa

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Sequences and Series

Additional notes

y

y

x

y

y

x

y

x

x

y

x

x

Homework Assignment Page(s) Exercises

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SECTION 12.3

Geometric Sequences

187

Name ___________________________________________________________ Date ____________

12.3

Geometric Sequences

I. Geometric Sequences A geometric sequence is generated when we start with a number a and aaaaaaa aaaaaaaa a

.

An geometric sequence is a sequence of the form is the

aaaaaaaaaa aaaaaa aa a aaaaa

aaaaa aaaa

a

, and r is the

geometric sequence is given by

aaaaaa aaaaa

a

. The number a of the sequence. The nth term of a

.

The number r is called the common ratio because

aaa aaaaa aa aaa aaa aaaaaaaaaa aaaaa aa a

aaaaaaaaa aaaaaaaa aa a

.

Give a real-life example of a geometric sequence. aaaaaaa aaaa aaaaa

II. Partial Sums of Geometric Sequences For the geometric sequence an  ar n 1 the nth partial sum Sn  a  ar  ar 2  ar 3  ar 4  is given by

a

 ar n1 (r  1)

.

III. What Is an Infinite Series? 

An expression of the form

a

k

 a1  a2  a3  a4 

is called an

aaaaaaa aaaaaa

.

k 1

If the partial sum S n gets close to a finite number S as n gets large, we say that the infinite series aaa aa aaaaaaaa

. The number S is called the

If an infinite series does not converge, we say that the series

aaaaaaaaa

aaa aa aaa aaaaaaaa aaaaaa aaaaaa aaa aa aaaaaaaaaa

. .

IV. Infinite Geometric Series An infinite geometric series is a series of the form

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a

.

188

CHAPTER 12

|

Sequences and Series



If r < 1 , then the infinite geometric series

 ar

k 1

 a  ar  ar 2  ar 3 

aaaaaaaaa

and

k 1

has the sum

a

If r  1 , then the series

. aaaaaaaa

.

Additional notes

y

y

x

y

x

x

Homework Assignment Page(s) Exercises

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|

SECTION 12.4

Mathematics of Finance

189

Name ___________________________________________________________ Date ____________

12.4

Mathematics of Finance

I. The Amount of an Annuity An annuity is

a aaa aa aaaaa aaaa aa aaaa aa aaaaaaa aaaaa aaaaaaaa

The payments are usually made at of an annuity is

.

aaa aaa aa aaa aaaaaa aaaaaaaa

. The amount

aaa aaa aa aaa aaa aaaaaaaaaa aaaaaa aaaa aaa aaaa aa aaa aaaaa aaaaaaa

aaaaa aaa aaaa aaaaaaa aa aaaaa aaaaaaaa aaaa aaa aaa aaaaaaaa In general, the regular annuity payment is called the R. We also let i denote the aa aaaaaaaa the

. aaaaaaa aaaa

aaaaaaaa aaaa aaa aaaa aaaa

and is denoted by and let n denote

aaa aaaaaa

. We always assume that the time period in which interest is compounded is equal to aaaa aaaaaaa aaaaaaaa

. The amount A f of an annuity consisting of n

regular equal payments of size R with interest rate i per time period is given by

a

.

II. The Present Value of an Annuity The present value of an annuity is the amount Ap that

aaaa aa aaaaaaaa aaa aa aaa

aaaaaaaa aaaa a aaa aaaa aaaaaa aa aaaaa a aaaaaaaaa aaaa aa aaaaaa a

.

The present value Ap of an annuity consisting of n regular equal payments of size R and interest rate i per time

period is given by

a

.

III. Installment Buying When you buy a house or a car by installment, the payments that you make are aaaaaaa aaaaa aa aaa aaaaaa aa aaa aaaa

aa aaaaaaa aaaaa

.

For an installment buying situation, if a loan Ap is to be repaid in n regular equal payments with interest rate i

per time period, then the size R of each payment is given by

a

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CHAPTER 12

|

Sequences and Series

Additional notes

y

y

x

y

x

x

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SECTION 12.5

|

191

Mathematical Induction

Name ___________________________________________________________ Date ____________

12.5

Mathematical Induction

I. Conjecture and Proof What is a conjecture? aaaaaaa aaa aaaaa aaa aaa aa aaaaaa aaaaaaaaaa aa a aaaaaa aa aaaaa aaaa aaaaa aaaaaaaaa aaaaaaa aaaaaaaaaa aaaaaa

A mathematical proof is aaaaaa aaaaa

a aaaaa aaaaaaaa aaaa aaaaaaaaaa aaa aaaaa aa a aaaaaaaaa .

II. Mathematical Induction In mathematical induction, the induction step leads us aa aaa aaaa

aaaa aaa aaaaa aa aaa aaaaaaa aa aaa aaaaa

.

The Principle of Mathematical Induction states that for each natural number n, let P(n) be a statement depending on n. Suppose that the following two conditions are satisfied. 1.

aaaa aa aaaa

2.

.

aaa aaaaa aaaaaaa aaaaaa aa aa aaaa aa aaaa aaaa aaa a aa aa aaaa

Then P(n) is true for

aaa aaaaaaa aaaaaaa a

.

.

Describe how to apply the Principle of Mathematical Induction. aa aaaaa aaaa aaaaaaaaaa aaaaa aaa aaa aaaaaa aaaa a aaaaa aaaa aaaa aa aaaaa aaaa a aaaaaa aaaa aaaa aa aaaaa aaa aaa aaaa aaaaaaaaaa aa aaaaa aaaa aaa a aa aa aaaaa

Notice that we do not prove that P(k) is

aaaa

. We only show that if P(k) is true, then

aaa a aa aa aaaa aaa the

aaaaaaaaa aaaaaaaaaa

. The assumption that P(k) is true is called .

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192

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CHAPTER 12

Sequences and Series

Complete each of the following formulas for the sums of powers. n

0.

1 

a

.

k 1 n

1.

k 

a

.

k 1 n

2.

k

2



3



a

.

k 1 n

3.

k

a

.

k 1

y

y

x

y

x

x

Homework Assignment Page(s) Exercises

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SECTION 12.6

|

The Binomial Theorem

193

Name ___________________________________________________________ Date ____________

12.6

The Binomial Theorem

An expression of the form a + b is called a

aaaaaaaa

.

I. Expanding (a + b)n List some of the simple patterns that emerge from the expansion of (a + b)n . aa aaaaa aaa a a a aaaaa aa aaa aaaaaaaaaa aaa aaaaa aaaaa a a aaa aaa aaaa aaaaa a a a aa aaa aaaaaaaaa aa a aaaaaaaa aa a aaaa aaaa aa aaaaa aaaaa aaa aaaaaaaaa aa a aaaaaaaa aa aa aa aaa aaa aa aaa aaaaaaaaa aa a aaa a aa aaaa aaaa aa aa

Write the first nine rows of Pascal’s Triangle. a a a a a

a a

a a

a a

a

a

a

a a aa aa a a a a aa aa aa a a a a aa aa aa aa a a a a aa aa aa aa aa a a a

The key property of Pascal’s Triangle is that every entry (other than a 1) is aaaaaaa aaaaaaaaaa aaaaa aa

aaa aaa aa aaa aaa

.

Describe how to use Pascal’s Triangle to expand a binomial (a + b)n . aaa aaaaa aaaa aa aaa aaaaaaaaa aa a a a aaa aaa aaaa aaaa aa a a a aaaaa aaa aaaa aaaa aaa aaaaaaaa aa a aaaaaaaaa aa a aaaa aaaa aa aaaa aaa aaaa a aaaaaaaaa aa a aaaa aaaa aa aaaaa aa aaa aaaaa aaa aaaa aa aaa aaaaaaaaaa aaa aaaaaaaaaaa aaaaaaaaaaaa aa aaa aaaaa aa aaa aaaaaaaaa aaaaaa aa aaa aaa aaa aa aaaaaaaa aaaaaaaaa

II. The Binomial Coefficients Although Pascal’s Triangle is useful in finding the binomial expansion for reasonably small values of n, it isn’t practical for finding (a + b)n for large values of n because

aaa aaaaaa aa aaa aaa aaaaaaa aaaaaaaaaa

aaaa aa aaaaaaaa aaaaaaaa aa aaaaaaaaaa aa aaaa aaa aaaaa aaa aa aaaa aaaaaa aa aaaa aaaaa aaaa aaa aaaaaaa aa aaaa Note Taking Guide for Stewart/Redlin/Watson Precalculus: Mathematics for Calculus, 7th Edition Copyright © Cengage Learning. All rights reserved.

.

194

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Sequences and Series

The product of the first n natural numbers is denoted by n! = 0! =

a

aa

and is called

a aaaaaaa

.

. a

.

n Let n and r be nonnegative integers with r  n . The binomial coefficient is denoted by   and is defined by r

a

.

n The binomial coefficient   is always a r

aaaaaaa

n number. Also,   is equal to r

a

.

The key property of the binomial coefficients is that for any nonnegative integers r and k with r  k ,

a

.

III. The Binomial Theorem

The Binomial Theorem states that (a + b)n =

The general term that contains a r in the expansion of (a + b)n is

a

a

.

.

Homework Assignment Page(s) Exercises

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Chapter 13

Limits: A Preview of Calculus

13.1 Finding Limits Numerically and Graphically I. Definition of Limit We write lim f ( x)  L and say “ the limit of f(x), as x approaches a, equals L” if

aa aaa aaaa

x a

aaa aaaaaa aa aaaa aaaaaaaaaaa aaaaa aa a aaa aaaaa aa a aa aa aaaaa aa aaaaaa a aa aa aaaaaaaaaaaa aaaaa aa aa aaa aaa aaaaa aa a

.

This says that the values of f ( x) get closer and closer to

aaa aaaaaa a aa a aaaa aaaaaa

aaa aaaaaa aa aaa aaaaaa a aaaaa aaaaaa aaaa aa aa aaa a a a

.

II. Estimating Limits Numerically and Graphically Describe how to estimate a limit numerically. aaaaaaa aaa aaaaa aa aaaaaaaa aaa aaaaa aa a aaaaaaaa aaaa aa a aaaaa a aaaaaaaaaaaa aaaaaaaaa a aaaaa aaaaaa aaaaaa aa aaa aaaaaaaa aaa aaaaaa aa a aaaa aaaaaaaa a aaaa aaa aaaa aaa aaaa aaa aaaaaa aaa aaaa aaa aaa aaaaa aa aa aaaa aa aaa aa aaa aaaaaa aa aaa aaaaaaaa aaaa aa aaaaaaaa a aaaaaaaaaa aaaaaa aa a aaaa aaaaaa aaa aaaaaa aa aa

Describe how to estimate a limit graphically. aaaaaaa aaa aaaaa aa aaaaaaaa aaa aaaaa aa a aaaaaaaa aaaa aa a aaaaa a aaaaaaaaaaaa aaaaa aaa aaaaaaaa aa a aaaaaaaa aaaaaaa aaa aaa aaaa aaa aaaaa aaaaaaaa aa aaa a aaaaaa aaaa aa aaaaaa aa aaa aaaaaa aaaa aaa a aaaaa aaaa a aaaaaa aa aa aa a aaaa aaaaaa aaa aaaaaa aa aa

III. Limits That Fail to Exist Functions do not necessarily approach a finite value at every point. In other words, it’s possible for aaa aa aaaaa

a aaaaa

.

Describe three situations in which a limit may fail to exist. a aaaaa aaa aaaa aa aaaaa aa aa aaaaaaaa a aaaaaaaa aaaa a aaaaa a aaaaaaaa aaaa aaaaaaaaaaa aa a aaaaaaaa aaaa a aaaaaaaa aaaaaaaaaa

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CHAPTER 13

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Limits: A Prev iew of Calculus

To indicate that a function with a vertical asymptote has a limit that fails to exist, we use the notatio n

lim f ( x)   , which expresses the particular way in which the limit does not exist: f ( x) can be made as large

x a

as we like by

aaaaaa a aaaaa aaaaaa aa a

.

IV. O ne -Sided Limits We write lim f ( x)  L and say

aaa aaaaaaaaaa aaaaa aa aaaa aa a aaaaaaaaaa aa aaa aaa

x a

aaaaaa aa aaaa aa a aaaaaaaaaa a aaaa aaa aaaaaa aaaaa aa a

if we can make the

values of f(x) arbitrarily close to L by taking x to be sufficiently close to a and x less than a. Similarly, if we require that x be greater than a, we get

aaa aaaaaaaa aaaaa aa aaaa aa

, and we write lim f ( x)  L .

a aaaaaaaaaa a aa aaaaa aa a

x a

By comparing the definitions of two-sided and one-sided limits, we see that the following is true: lim f ( x)  L x a

if and only if

a aaa a

limits are different, the two-sided limit

y

. Thus if the left-hand and right-hand

aaaa aaa aaaaa

.

y

x

y

x

x

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SECTION 13.2

|

Finding Limits Algebraically

197

Name ___________________________________________________________ Date ____________

13.2 Finding Limits Algebraically I. Limit Laws Suppose that c is a constant and that the following limits exist: lim f ( x) and lim g ( x) , then give each of the x a

x a

following limits. 1. Limit of a Sum:

a

.

2. Limit of a Difference:

a

3. Limit of a Constant Multiple:

4. Limit of a Product:

.

a

.

a

5. Limit of a Quotient:

.

a

.

6. Limit of a Power:

a aaaaa a aa a aaaaaaaa aaaaaaa

.

7. Limit of a Root:

a aaaaa a aa a aaaaaaaa aaaaaaa

.

State each of these five laws verbally. aa aaa aa aaa aa aaa aa aaa aa aaa aaa aa aaa aa aaa

aaaaa aaaaa aaaaa aaaaa aaaaa

aa a aa a aa a aa a aa a

aaa aa aaa aaa aa aaa aaaaaaa aaaaaaaaaa aa aaa aaaaaaaaaa aa aaa aaaaaaa aaaaaaaa aaaaa a aaaaaaaa aa aaa aaaaaaaa aaaaa aaa aaaaa aa aaa aaaaaaaaa aaaaaaa aa aaa aaaaaaa aa aaa aaaaaaa aaaaaaaa aa aaa aaaaaaaa aa aaa aaaaaa aaaaaaaaa aaaa aaa aaaaa aa aaa aaaaaaaaaaa aa aaa

aaaaa aa a aaaaa aa aaa aaaaa aa aaa aaaaaa aaaaa aa a aaaa aa aaa aaaa aa aaa aaaaaa

II. Applying the Limit Laws Complete each of the following special limits. 1. lim c  x a

n 3. lim x  x a

a

22. lim x 

. a

a aaaaa a aa a aaaaaaaa aaaaaaa

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a

x a

.

Mathematics for Calculus, 7th Edition

.

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4. lim n x 

Limits: A Prev iew of Calculus

a aaaaa a aa a aaaaaaaa aaaaaaa aaa a a a

x a

.

If f is a polynomial or a rational function and a is in the domain of f, then the limit of f may be found by direct substitution, that is, lim f ( x) 

aaaa

x a

property are called

aaaaaaaaaa aa a

. Functions with this direct substitution .

III. Finding Limits Using Algebra and the Limit Laws Example 1

16  8 y  y 2 . y 4 y4

Evaluate lim

a

IV. Using Left- and Right-Hand Limits A two-sided limit exists if and only if

aaaa aa aaa aaaaaa aaaaaa aaaaa aaa aaa aaaaa

When computing one-sided limits, use the fact that aaaaaa

Example 2

.

aaa aaaaa aaaa aaaa aaaa aaa aaaaaaaaa

.

  x 2  3x if x  3 Let f ( x)   . Determine whether lim f ( x) exists. x 3 if x  3  x aa

Example 3

  x 2  3x if x  0 Let f ( x)   . Determine whether lim f ( x) exists. x 0 if x  0  x

aaa

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SECTION 13.3

|

199

Tangent Lines and Deriv ativ es

Name ___________________________________________________________ Date ____________

13.3 Tangent Lines and Derivatives I. The Tangent Problem A tangent line is a line that

aaaa aaaaaaa a aaaaa

.

We sometimes refer to the slope of the tangent line to a curve at a point as the aaaaa aa aaa aaaaa looks

aaaaa aa aaa

. The idea is that if we zoom in far enough toward the point, the curve aaaaaa aaaa a aaaaaaaa aaaa

.

The tangent line to the curve y = f ( x) at the point P(a, f (a)) is the line through P with slope m=

a

, provided that this limit exists.

Another expression for the slope of a tangent line is m =

a

.

II. Derivatives

The derivative of a function f at a number a, denoted by f (a) , is

a

if

this limit exists.

We see from the definition of a derivative that the number f (a) is the same as aaaa aa aaa aaaaa a aa aaa aaaaa a

aaa aaaaa aaaa aaaaaaa

.

III. Instantaneous Rates of Change If y = f ( x) , the instantaneous rate of change of y with respect to x at x = a is the limit of the average rates of

change as x approaches a: instantaneous rate of change =

a

List two different ways of interpreting the derivative. aa

a

aa aaa aaaaa aa aaa aaaaaaa aaaa aa

a

aa a a aa

aa a aa aaa aaaaaaaaaaaaa aaaa aa aaaaaa aa a aaaa aaaaaaa aa a aaa a a aa

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Limits: A Prev iew of Calculus

In the special case in which x = t = time and s = f(t) = displacement (directed distance) at time t of an object traveling in a straight line, the instantaneous rate of change is called the

aaaaaaaaaaa aaaaaaaa

.

Additional notes

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y

x

y

x

x

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SECTION 13.4

201

Limits at Infinity; Limits of Sequences

Name ___________________________________________________________ Date ____________

13.4 Limits at Infinity; Limits of Sequences I. Limits at Infinity We use the notation lim f ( x)  L to indicate that the values of f(x) become

aaaaaa aaa aaaaaa aa

x 

a aa a aaaaaaa aaaaaa aaa aaaaaa

.

Let f be a function defined on some interval (a, ) . Then by the definition of limit at infinity lim f ( x)  L x 

means that the values of f(x) can be made

aaaaaaa aaaaa aa a aa aaaaaa a aaaaaaaaaaaa aaaaa

.

List various ways of reading the expression lim f ( x)  L . x 

aaaa aaaaa aa aaaaa aa a aaaaaaaaaa aaaaaaaaa aa aa aaaa aaaaa aa aaaaa aa a aaaaaaa aaaaaaaaa aa aa aaaa aaaaa aa aaaaa aa a aaaaaaaaa aaaaaaa aaaaaa aa aa

Let f be a function defined on some interval (, a) . Then by the definition of a limit at negative infinity

lim f ( x)  L means that the values of f(x) can be made

aaaaaaaaaaa aaaaa aa a aa aaaaaa a

x 

aaaaaaaaaaaa aaaaa aaaaaaaa

.

lim f ( x)  L ?

How can one read the expression

x 

aaaa aaaaa aa aaaaa aa a aaaaaaaaaa aaaaaaaa aaaaaaaaa aa aa

The line y = L is called

a aaaaaaaaaa aaaaaaaaa

of the curve y = f ( x) if either

lim f ( x)  L or lim f ( x)  L .

x 

x 

The Limit Laws studied earlier in this chapter If k is any positive integer, then lim

x 

1 x

k



Note Taking Guide for Stewart/Redlin/Wats on Precalculus: Copyright © Cengage Learning. All rights reserved.

aaaa aaaa a

for limits at infinity. and lim

x 

1 xk



Mathematics for Calculus, 7th Edition

a

.

202

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Limits: A Prev iew of Calculus

II. Limits of Sequences A sequence a1 , a2 , a3 , a4 ,

has the limit L and we write lim an  L or an  L as n   if the nth term an of n 

the sequence can be made exists, we say the sequence sequence

by taking n sufficiently large. If lim an

aaaaaaaaaaa aaaaa aa a

n 

aaaaaaaaa aaa aa aaaaaaaaaaa

aaaaaaaa aaa aa aaaaaaaaaa

. Otherwise, we say the

.

If lim f ( x)  L and f (n) = an when n is an integer, then lim an 

a

n 

x 

y

y

x

y

.

y

x

y

x

x

y

x

x

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SECTION 13.5

|

Areas

203

Name ___________________________________________________________ Date ____________

13.5 Areas I. The Area Problem Describe the area problem found in calculus. aaa aa aaa aaaaaaa aaaaaaaa aa aaaaaaaa aa aaa aaaa aaaaaaaa aaaa aaa aaaa aa aaa aaaaaa a aaaa aaaa aaaaa aaa aaaaa a a aaaa aaa aaaaa aaa aaaaaa aaaa a a a aa a a aa

Describe the approach taken in calculus to finding the area of a region S with curved sides. aa aaaaa aaaaaaaaaaa aaa aaaaaa a aa aaaaaaaaaaa aaa aaaa aa aaaa aaa aaaaa aa aaa aaaaa aa aaaaa aaaaaaaaaa aa aa aaaaaaaa aaa aaaaaa aa aaaaaaaaaaa

II. Definition of Area The area A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles:

Using sigma notation, we write this as follows:

In using this formula for area, remember that x is the is the

aaaaa

of the kth rectangle, and f ( xk ) is its

aaaaa aaaaaaaa

of an approximating rectangle, xk aaaaaa

Complete each of the following formulas.

x =

a

.

xk =

a

.

f ( xk ) =

a

.

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Limits: A Prev iew of Calculus

Additional notes

y

y

x

y

y

x

y

x

x

y

x

x

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Name ___________________________________________________________ Date ____________

Chapter 14

Probability and Statistics

14.1 Counting I. The Fundamental Counting Principle Suppose that two events occur in order. The Fundamental Counting Principle states that if the first event can occur in m ways and the second can occur in n ways (after the first has occurred), then the two events can occur in order in

a

ways.

Explain how the Fundamental Counting Principle may be extended to any number of events. aa aaa aaa a a a a aa aaa aaaaaa aaaa aaaaa aa aaaaa aaa aa aa aaa aaaaa aa aa aaaaa aa aa aa aaaaa aaa aa aaa aaaa aaa aaaaaa aaa aaaaa aa aaaaa aa a aaaaa

A set with n elements has

aa

different subsets.

II. Counting Permutations A permutation of a set of distinct objects is

aa aaaaaaaa aa aaaaa aaaaaaa

The number of permutations of n objects is

aa

.

.

In general, if a set has n elements, then the number of ways of ordering r elements from the set is denoted by P(n, r) and is called the

aaaaaa aa aaaaaaaaaaa aa a aaaaaaa aaaaa a aa a aaaa

The number of permutations of n objects taken r at a time is given by

.

a

.

III. Distinguishable Permutations In general, in considering a set of objects, some of which are the same kind, then two permutations are distinguishable if one cannot be obtained from the other by aaaaaaaa aa aaa aaaa aaaa

aaaaaaaaaaaaa aaa aaaaaaaaa aa

.

If a set of n objects consists of k different kinds of objects with n1 objects of the first kind, n2 objects of the second kind, n3 objects of the third kind, and so on, where n1  n2  n3   nk  n , then the number of

distinguishable permutations of these objects is

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.

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Mathematics for Calculus, 7 Edition

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206

CHAPTER 14

|

Probability and Statistics

IV. Counting Combinations When counting combinations, order

aa aaa

A combination of r elements of a set is aaaaaa

important. aaa aaaaaa aa a aaaaaaa aaaa aaa aaa aaaaaaaa aaaaaa aa

. If the set has n elements, then the number of combinations of r elements is denoted by

C(n, r) and is called the

aaaaaa aa aaaaaaaaaaaa aa a aaaaaaaa aaaaa a aa a aaaa

The number of combinations of n objects taken r at a time is given by The key difference between permutations and combinations is

.

a aaaaa

.

.

V. Problem Solving with Permutations and Combinations List the guidelines for solving counting problems. aa aaaaaaaaaaa aaaaaaaa aaaaaaaaaa aaaa aaaaaaaaaaa aaaaaaa aaa aaaaa aaaaa aaa aaa aaaaaaaaaaa aaaaaaaa aaaaaaaaaa aa aa aa aa

aaaa aaaaa aaaaaaa aaaa aa aaaa aa aaaa aaa aaaaaa aa aaaa aa aaaaaaa a aaaaaaa aaaa a aaaaaaaa aa aaaa aaa aaaaaaaaaa aaaaa aaa aaaaa aa aaaaa aa aaaa aaa aaaaaaa aaaaaaaa aaa aaaaa aaaaaaaa aa aaa aaaaaaaaaaaaa aaa aaaaa aaaaaaa aaaaaaa aa aaa aaaaaaaaaaaaa

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SECTION 14.2

207

Probability

Name ___________________________________________________________ Date ____________

14.2 Probability I. What is Probability? An

aaaaaaaaaa aaaaaaaa

is a process, such as tossing a coin, that gives definite results, called the of the experiment. The sample space of an experiment is

aaaaaaaa aaaaaaaa

aaa aaa aa aaa

.

We will be concerned only with experiments for which all the outcomes are If S is the sample space of an experiment, then an event E is

aaaaaaa aaaaaa

.

aaa aaaaaa aa aaa aaaaaa aaaaa

.

Let S be the sample space of an experiment in which all outcomes are equally likely, and let E be an event. Then

the probability of E, written P(E), is

a

.

The probability P(E) of an event is a number between

a aaa a

. The closer the

probability of an event is to 1, the

aaaa aaaaaa

the event is to happen; the closer the

probability of an event is to 0, the

aaaa aaaaaa

the event is to happen. If P(E) = 0, then E is

called the

aaaaaaaaaa aaaaa

.

II. Calculating Probability by Counting To find the probability of an event, we do not need to list all the elements aaa aaa aaaaa The

. We need only

aaa aaaaaa aa aaaaaaaa

aaaaaaaa aaaaaaaaa

aa aaa aaaaaa aaaaa in these sets.

learned earlier will be very useful here.

III. The Complement of an Event The complement of an event E is the set of outcomes in the sample space that are not in E. We denote the complement of E by

aa

.

Let S be the sample space of an experiment and let E be an event. Then the probability of E, the complement of E, is

a

.

IV. The Union of Events If E and F are events in a sample space S, then the probability of E or F, that is the union of these events, is a

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Probability and Statistics

Two events that have no outcome in common are said to be

aaaaaaaa aaaaaaaaa

words, the events E and F are mutually exclusive if

a

mutually exclusive, then P( E  F ) 

.

a

. In other

. So if the events E and F are

If events E and F are mutually exclusive then the probability of the union of these two mutually exclusive events is

a

.

V. Conditional Probability and the Intersection of Events Let E and F be events in a sample space S. The conditional probability of E given that F occurs is

P( E | F ) =

a

.

If E and F are events in a sample space S, then the probability of E and F, that is, the intersection of these two events, is P( E  F ) 

a

.

When the occurrence of one event does not affect the probability of the occurrence of another event, we say that the events are

aaaaaaaaaaa

a

. This means that the events E and F are independent if

and

a

.

If E and F are independent events in a sample space S, then the probability of E and F, that is, the probability of

the intersection of independent events, is

a

.

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SECTION 14.3

209

Binomial Probability

Name ___________________________________________________________ Date ____________

14.3 Binomial Probability I. Binomial Probability A binomial experiment is one in which there are

aaa

outcomes, which are called

“ success” and “ failure.” Binomial Probability An experiment has two possible outcomes called “ success” and “ failure,” with P(success) = p and P(failure) =

a

. The probability of getting exactly r successes in n independent trials of the

experiment is P(r successes in n trials) =

a

.

II. The Binomial Distribution The function that assigns to each outcome its corresponding probability is called a aaaaaaaaaaaa called a

aaaaaaaaaaa

. A bar graph of a probability distribution in which the width of each bar is 1 is aaaaaaaaaaa aaaaaaaaa

.

A probability distribution in which all outcomes have the same probability is called a(n) aaaaaaaaaaaa aaaaaaaaaaaa

. The probability distribution of a binomial experiment is called a(n)

aaaaaaa aaaaaa

.

The sum of the probabilities in a probability distribution is

a

, because the sum is the

probability of the occurrence of any outcome in the sample space.

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SECTION 14.4

|

Expected Value

211

Name ___________________________________________________________ Date ____________

14.4 Expected Value I. Expected Value A game gives payouts a1 , a2 , this game is

, an with probabilities p1 , p2 ,

, pn . The expected value (or expectation) E of

a

.

Give an example of a real-life application of expected value. aaaaaaa aaaa aaaaa

II. What is a Fair Game? A fair game is

a aaaa aaaa aaaaaaaa aaaaa aaaa

times you would expect, on average, to

aaaaa aaaa

. So if you play a fair game many .

Describe the use of fair games in casinos. aaaaaaa aaaa aaaaa

Invent a simple fair game and show that it is fair. aaaaaaa aaaa aaaaa

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SECTION 14.5

213

Descriptiv e Statistics (Numerical)

Name ___________________________________________________________ Date ____________

14.5 Descriptive Statistics (Numerical) The things that a data set describes are called

aaaaaaaaaa

that is described by the data is called a data, in which

. The property of the individuals

aaaaaaaa

. We will be studying one-variable

aaaa aaa aaaaaaaa aa aaaa aaaaaaaaaa aa aaaaaaaa

The first goal of statistics is

.

aa aaaaaaaa aaaa aaaa aa aaaa aa aaaaaaa aaaaa

summary statistic is

.A

a aaaaaa aaaaaa aaaa aaaaaaaaaa aaaa aaaaaaaa aa aaa aaaa

.

I. Measures of Central Tendency: Mean, Median, Mode One way to make sense of data is to find

a aaaaaaaa aaaaaa

the data. Any such number is called a measure is the Let x1 , x2 ,

or the

aaaaaaaa

aaaaaaa aa aaaaaaa aaaaaaaa

aaaaaaa aa aaaa

of

. One such

.

, xn be n data points. The mean (or average), denoted by x , is the sum divided by n:

Another measure of central tendency is the

aaaaaa

, which is the middle

number of an ordered list of numbers. Let x1 , x2 ,

, xn be n data points, written in increasing order. If n is odd, the median is

aaaaaa

. If n is even, the median is

aaa aaaaaa

aaa aaaaaaa aa aaa aaa aaaaaa aaaaaaa

If a data set includes a number far away from the rest of the data, that data point is called

. aa aaaaaa

.

If a data set has outliers, which is a better indicator of the central tendency of the data, the median or the mean? aaaaaa The mode of a data set is

aaa aaaaaaa aaaa aaaaaaa aaaa aaaaa aa aaa aaaa

The mode has the advantage of

aaa aaaaa aaaaaaa aa aaaaaaaaa aaaa

A data set with two modes is called

aaaaaaa

. A data set such as 3, 5, 7, 9, 11 has

. . aa

.

mode.

II. O rganizing Data: Frequency Tables and Stemplots A frequency table for a set of data is

a aaaaa aaaa aaaaaa aaaa aaaaaaaaa aaaa aaaaa aaa aaa aaaaaa

aa aaaaa aaaa aaaaa aaaaaa aa aaa aaaa

. The

aaaa

is most easily determined from

a frequency table.

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A stemplot (or stem-and-leaf plot) organizes data by aa aaa aaaa

aaaaa aaa aaaaaa aa aaa aaaaaaa

. Each number in the data is written as a stem consisting of

and a leaf consisting of

aaa aaaaaaaa aaaaa

aaaaaaaa aa a aaa

, with the stem written

aaaaaaaaa aaaaaa

. Numbers with the same stem are aaaa aaaa

.

aaaaaaa .

III. Measures of Spread: Variance and Standard Deviation Measures of spread (also called measures of dispersion) describe aaaa aaaaaa a aaaaaaa aaaaa Let x1 , x2 ,

aaa aaaaaa aa aaaaaaaaaaa aa aaa

.

, xN be N data points and let x be their mean. The standard deviation of the data is

a

. The variance is

a

, the square of the standard

deviation.

IV. The Five Number Summary: Box Plots A simple indicator of the spread of data is the location of Other indicators of spread are the

aaaaaaa

The median of the lower half of the data is called upper half of the data is called good picture of

aaa aaaaaaa aaa aaaaaaa aaaaaa . The median divides a data set aaa aaaaa aaaaaaaaa a

aaa aaaaa aaaaaaaaa a

. aa aaaa

.

. The median of the . Together these values give a

aaa aaaaaa aaa aaa aaaaaa aa aaa aaaa

.

The five-number summary for a data set are the five numbers below, written in the indicated order. aaaaaaaa aa aaaaaaa aa aaaaaaa

.

A simple indicator of spread is the range, which is aaa aaaaaaa aaaaaa

aaa aaaaaaaaaa aaaaaaa aaa aaaaaaa

. This can be compared to the spread of the middle of the data as

measured by the interquartile range (IQ R), which is aaaaaaaaa

aaa aaaaaaaaaa aaaaaaa aaa aaaaa aaa aaaaa

.

A box plot (also called a box -and-whisker plot) is a method for aaaaaaa

. The plot consists of

aaaaaaaaaa aa aaa a

aaaaaaaaaa aaaaaaaaaa a aaaaaaaaaaa a aaaaaaaaa aaaaa aaaaa aaa aaaaaaaaaa aaaaa

. The box is divided by a line segment at

The whiskers are

aaa aaaaaaaa aa aaa aaaaaa

.

aaaa aaaaaaaa aaaa aaaaaa aaaa aaaa aaa aa aaa aaa aa aaa aaaaaaaa aa aaa

aaaaaaa aaa aaaaaaa aaaaaa

.

When working with quartiles, the median of the data is also called

Note Taking Guide for Stewart/Redlin/Wats on

aaa aaaaaa aaaaaaaaa a

.

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SECTION 14.6

215

Descriptiv e Statistics (Graphical)

Name _____________________________________________________ ______ Date ____________

14.6 Descriptive Statistics (Graphical) I. Data in Categories Numerical data is

aaaa aaaaa aaaaaa aaa aaaa aaaaaaa

Categorical data is aaaaaaaa

.

aaaa aaaa aaaaa aa aaaa aaaaaaa aa aaaaaa aaaa aa aaaa aaa aaaaaa aa a aaaaaaaaaa . Categorical data can be represented by

A bar graph consists of

aaa aaaaaa aa aaa aaaaaa

aaaaaaaa aaaaa aaa aaa aaa aaaa aaaaaaa

each bar is proportional to So the y-axis has a

. . The height of

aaa aaaaaa aa aaaaaaaaaaa aaaaa aaaaaaaa aaaaaaaaa aaaaa

corresponding to the number or the proportion of the

individuals in each category. The labels on the x-axis describe A pie chart consists of

.

a aaaaaaaaa aaaaaaaaaa

.

a aaaaaa aaaaaaa aaaa aaaaaaaa aaaaaaa aaa aaaa aaaaaaaa

central angle of each sector is proportional to Each sector is labeled with

. The

aaa aaaaaaa aaaaaaaaaaa aa aaaa aaaaaaaa

aaa aaaa aa aaa aaaaaaaaaaaaa aaaaaaaa

.

.

II. Histograms and the Distribution of Data To visualize the distribution of one-variable numerical data, we first aaaaaaaaa aaaa

aaaaaaa aaa aaaa aaaa

. To do so, we divide the range of the data into

aaaaaaaa aaaaaaaaa aa aaaaa aaaaaaa

, called bins. To draw a histogram of the data we first label

the bins on the x-axis, and then

aaaaa a aaaaaaaaa aa aaaa aaa

also the area) of each rectangle is proportional to

; the height (and hence

aaa aaaaaa aa aaaa aaaaaa aa aaaa aaa

A histogram gives a visual representation of how the data are

aaaaaaaaa

histogram allows us to determine if the data are

aaaaaaaaa

has a long “ tail” on the right, we say the data are

aaaaaa aa aaa aaaaa

long tail on the left, the data are

aaaaaaaaaaaaaaa

aaaaaa aa aaa aaaa

.

in the different bins. The

about the mean. If the histogram . Similarly, if there is a

. Since the area of each bar in the

histogram is proportional to the number of data points in that category, it follows that the median of the data is located at

aaa aaaaaaa aaaa aaaaaaa aaa aaaa aa aaa aaaaaaaaa aa aaaa

.

III. The Normal Distribution Most real-world data are distributed in a special way called a

aaaaaa aaaaaaaaaaaa

The standard normal distribution (or standard normal curve) is modeled by the function

a

.

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This distribution has mean

Probability and Statistics

a

and standard deviation

a

. Normal

distributions with different means and standard deviations are modeled by transformations (shifting and stretching) of the above function. Specifically, the normal distribution with mean  and standard deviation 

is modeled by the function

a

.

All normal distributions have the same general shape, called a

aaaa aaaaa

.

For normally distributed data with mean  and standard deviation  , we have the following facts, called the Empirical Rule.  Approximately aaa of the data are between    and  +  

Approximately

aaa

of the data are between   2 and  + 2



Approximately

aaaaa

of the data are between   3 and  + 3

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SECTION 14.7

217

Introduction to Statistical Thinking

Name ___________________________________________________________ Date ____________

14.7 Introduction to Statistical Thinking In statistical thinking we make judgments about an entire population based on population consists of

aaa aaa aaaaaaaaaa aa a aaaaa

aa aaa aaaaaaaaaa

a aaaaaa

.A

. A sample is

aaa aaaaaa

.

In survey sampling, data are collect through

aaaaaaaaa aa aaaaaaaaaaaaaa

observational studies, data are collected by are collected by

aaaaaaaaaaa

aaaaaaaaaa aaaaaaaaaaa

. In

. In experimental studies, data

.

I. The Key Role of Randomness A random sample is one that is selected we expect a random sample to

aaaaaaaaa aaa aaaaaaa aaaa aaaaa aaaaaaaaaaaa aaa aaaa aaaaaaaaaa

The larger the sample size, the more closely the sample properties aaaaaaaaaa

. In statistical thinking, as the population. aaaaaaaaaaa aaaaa aa aaa

.

In general, non-random samples are generated when there is

aa aa aaa aaaaaaaaa aaaaaaa

Such samples are useless for statistical purposes because

aaa aaaaaa aa aaa aaaaaaaaaaaaaa aa aaa

aaaaaaaaaa

.

A simple random sample is one in which aa aaaaa aaaaaa aa aaaa

.

aaaaa aaaaaaaaaa aaa aaaaaaaaaa aaa aaa aaaa aaaaaaaaaaa

. To satisfy this requirement the sample method used must be

aaaa

with respect to the property being measured.

List four common types of sampling bias. aa aaaaaaaaaa aaaa aaa aaaaaaaaa aaaaaa aa aaaaa aaaa aa aaa aaaaaaaaaa aa aaaaaaaa aaaa aaa aaaaaaaa aaaaaaaa aa aaaaaaaa aaaaa aa aaaaa aaa aaaaaaa aa a aaaaaaaaaaaaa aa aaa aaaaaaaa aaa aaaaaa aaaaaaaa aa aaaaaaaa a aaaaaaaaaa aaaaaaaaa aa aaaaaaaaaaaa aaaaa aa aaaaa aaaaaaaaaaa aaaa a aaaaaa aaaaaaaaaaaaaa aaa aaaaaaaaa aa aaaaaaa aa aaaaaaa aa a aaaaaaaaaaaaaa aa aaaaaaaaaaaaaa aaaa aaa aaaaaaaaa aaaaaaaa aaaaa aa aaaaa aaaaaaaaaaa aaaaaa aaaaaaaaaa aaa aaaaaaaaaa aaa aaa aaaaaaa

Many of these biases are a result of convenience sampling, in which individuals are sampled aaaa aaa aaaaaa aa aaaaaa aaaaaaaaaa Note Taking Guide for Stewart/Redlin/Wats on Precalculus: Copyright © Cengage Learning. All rights reserved.

. Mathematics for Calculus, 7th Edition

aaaa aaaaaaa

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Probability and Statistics

II. Design of Experiments In observational studies the researcher has no control over the factors affecting the property being studied. Extraneous or unintended variables that systematically affect the property being studied are called aaaaaaaaaaa aaaaaaaaa aaa aaaaaaaaaaa aaaa aa aaaaaaa aaaaaaaaaa or

aaa aa

. Such variables are said to confound,

, the results of the study.

To eliminate or vastly reduce the effects of confounding variables, researchers often conduct experiments so that such variables can be

aaaaaaaaaa

a treatment group (in which which are called

. In an experimental study, two groups are selected—

aaaaaaaaa aaa aaaaa a aaaaaaaaa

) and a control group (in

aaaaaaaaaaa aaa aaa aaaaa aa aaaaaaaaa

). The individuals in the experiment

aaaaaaaa aa aaaaaaaaaaaa aaaaa

subjects to the treatment—that is,

. The goal is to measure the response of the

aaaaaaa aa aaa aaa aaaaaaaaa aaa aa aaaaaa

The next step is to make sure that the two groups are as similar as possible, except for

. aaa aaaaaaaa

.

If the two groups are alike except for the treatment, then any statistical difference in response between the groups can be confidently attributed to

aaa aaaaaaaaa

A common confounding factor is the

.

aaaaaa aaaaaa

, in which patients who think they are

receiving a medication report an improvement even though the “ treatment” they received was a placebo, aaaaaaaaa aa aaaaa aaaaaaaaa aaaaaaaaa aaaaaa a aaaaaa aaaaa

a

.

III. Sample Size and Margin of Error Statistical conclusions are based on probability and are always accompanied by a

aaaaaaaaa aaaaa

.

The 95% confidence level means that there is less than a 5% chance (or 0.05 probability) that the result obtained from the sample

aaaaa aa aaaaaaaa aa aaaaaa aaaaa

accompanied by

a aaaaaa aa aaaaa

. In the popular press, poll results are

.

At the 95% confidence level, the margin of error d and the sample size n are related by the formula

a

.

IV. Two-Variable Data and Correlation Two-variable data measures

aaa aaaaaaaaa aa aaaa aaaaaaaaaa

be graphed in a coordinate plane resulting in a

. Two-variable data can

aaaaaaa aaaa

. Analyzing such data

mathematically by finding the line that best fits the data is called finding the Associated with the regression line is a

aaaaaaaaaaa aaaaaaaaa

aaaaaaaaa aaaa

.

, which is a measure of how

well the data fit along the regression line, or how well the two variables are correlated. The correlation coefficient r is between

a

and

a

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SECTION 14.7

have

219

Introduction to Statistical Thinking

. The closer r is to 1 or 1 , the closer

aaaaaa aaaaaaaaaaa

aa aaa aaaaaaaaaa aaaa

|

aaa aaaa aaaaa aaa

.

In statistics, a question of interest when studying two-variable data is whether or not the correlation is statistically significant. That is, what is the probability that the correlation in the sample is due to aaaaa

aaaaaa

? If the sample consists of only three individuals, even a strong correlation coeffici ent

aaa aa aaaaaaaaaaa

aaa

. On the other hand, for a large sample, a small correlation coeffici ent may be

significant. This is because if there is no correlation at all in the population, it’s very unlikely that a large random sample would produce data that have a linear trend, whereas a small sample is more likely to produce correlated data by Correlation is

aaaaaa aaaaa aaa aaa aaaa aa

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Additional notes

y

y

x

y

y

x

y

x

x

y

x

x

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SECTION 14.8

Introduction to Inferential Statistics

221

Name ___________________________________________________________ Date ____________

14.8 Introduction to Inferential Statistics The goal of inferential statistics is to infer information about an entire population by aaaaaa aaaa aaaa aaaaaaaaaa

aaaaaaaaa a aaaaaa

.

I. Testing a Claim about a Population Proportion Intuitively In statistics a hypothesis is a statement or claim about

a aaaaaaaa aa aa aaaaaa aaaaaaaa

.

In this section we study hypotheses about the true proportion p of individuals in the population with aaaaaaaaaa aaaaaaaaaaaaaa

a

.

Suppose that individuals having a particular property are assumed to form a proportion p0 of a population. To answer the question of how this assumption compares to the true proportion p of these individuals, we begin by stating

. The null hypothesis, denoted H 0 , states the

aaa aaaaaaaa aaaaaaaaaa

“ assumed state of affairs” and is expressed as hypothesis (also called the

. The alternative

), denoted by H1 , is the proposed substitute

aaaaaaaa aaaaaaaaa

to the null hypothesis and is expressed as

aa aaaaaaaaaaa a

To test a hypothesis, we examine a either

aa aaaaaaaaa a

aaaaaa aaaaaa

aaaaaaaaa aa aaaaaaa

. from the population. The claim is

by data from that random sample. If, under the assumption that

the null hypothesis is true, the observed sample proportion is very unlikely to have occurred by chance alone, we

aaaaaa aaa aaaa aaaaaaaaaa

. Otherwise, the data does not provide us with enough

evidence to reject the null hypothesis, so we

aaaa aa aaaaaa aaa aaaa aaaaaaaaa

.

II. Testing a C laim about a Population Proportion Using Probability: The P-value The

aaaaaaa

associated with the observed sample is the probability of obtaining a

random sample with a proportion at least as extreme as the proportion in our random sample, given that H 0 is true. A “ very small” P-value tells us that it is “ very unlikely” that the sample we got was obtained by chance alone, so we should

aaaaaa aaa aaaa aaaaaaaaaa

the null hypothesis is called the

aaaaaaaaaaaa aaaaa

. The P-value at which we decide to reject of the test, and is denoted by  .

List the steps for testing a hypothesis. aa aaaaaaaaa aaa aaaaaaaaaaa aaaaa aaaaa a aaaa aaaaaaaaaa aaa aa aaaaaaaaaaa aaaaaaaaaaa aa aaaaaa a aaaaaa aaaaaaa aaaaaa a aaaaaa aaaa aa aaaaaaaa aaaaaaaaaaa aaa aaaaaaa aaaaa aa aaaaaaaaa aaa aaaaaaaa aaa a aaaaaaaaaa aa aaaa aaa aaaaaaa aaaaaaaaaa aaaa aaa aaaaaa aaaaaaaaa

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aa aaaa a aaaaaaaaaaa a aaaaaaa aaaaaaa aaaa aaa aaaaaaaaaaaa aaaaa aa aaa aaaa aaaaaaaaa aaaa aa aaaaaa aaaaaa aaa aaaa aaaaaaaaaaa aaaaaaaaaa aa aaaa aa aaaaaa aaa aaaa aaaaaaaaaaa

III. Inference about Two Proportions Most statistical studies involve a comparison of two groups, usually called aaaaaaa aaaaa

aaa aaaaaaaaa aaaaa aaa aaa

. For example, when testing the effectiveness of an investigational

medication, two groups of patients are selected. The individuals in the treatment group are aaaaaaaaaa

; the individuals in the control group are

aaaaa aaa

aaa aaaaa aaa aaaaaaaaa

The proportions of patients that recover in each group are compared. The goal of the study is

. aa

aaaaaaaaa aaaaaaa aaa aaaaaaaaaa aa aaa aaaaaaaaaa aaaa aaaaaaaa aa aaaaaaaaaaaaa aaaaaaaaaaa aaaa aaa aaaa aaa aaaaaa aa aaaa aaaaaaa

.

Let p1 and p2 be the true proportions of patients that recover in each group. The null hypothesis is t hat the medication or treatment has treatment does have an effect:

aa aaaaaaa a a

between the proportions in the two samples is due to (less than the significance level of the test), we

. The alternative hypothesis is that the . The P-value is the probability that the difference aaaaaa aaaaa aaaaaa aaa aaaa aaaaaaaaaa

. If the P-value is small .

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