Skip to Content
OpenStax Logo
Precalculus

Chapter 12

PrecalculusChapter 12
  1. Preface
  2. 1 Functions
    1. Introduction to Functions
    2. 1.1 Functions and Function Notation
    3. 1.2 Domain and Range
    4. 1.3 Rates of Change and Behavior of Graphs
    5. 1.4 Composition of Functions
    6. 1.5 Transformation of Functions
    7. 1.6 Absolute Value Functions
    8. 1.7 Inverse Functions
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Linear Functions
    1. Introduction to Linear Functions
    2. 2.1 Linear Functions
    3. 2.2 Graphs of Linear Functions
    4. 2.3 Modeling with Linear Functions
    5. 2.4 Fitting Linear Models to Data
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    7. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 3.1 Complex Numbers
    3. 3.2 Quadratic Functions
    4. 3.3 Power Functions and Polynomial Functions
    5. 3.4 Graphs of Polynomial Functions
    6. 3.5 Dividing Polynomials
    7. 3.6 Zeros of Polynomial Functions
    8. 3.7 Rational Functions
    9. 3.8 Inverses and Radical Functions
    10. 3.9 Modeling Using Variation
    11. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 4.1 Exponential Functions
    3. 4.2 Graphs of Exponential Functions
    4. 4.3 Logarithmic Functions
    5. 4.4 Graphs of Logarithmic Functions
    6. 4.5 Logarithmic Properties
    7. 4.6 Exponential and Logarithmic Equations
    8. 4.7 Exponential and Logarithmic Models
    9. 4.8 Fitting Exponential Models to Data
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Trigonometric Functions
    1. Introduction to Trigonometric Functions
    2. 5.1 Angles
    3. 5.2 Unit Circle: Sine and Cosine Functions
    4. 5.3 The Other Trigonometric Functions
    5. 5.4 Right Triangle Trigonometry
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    7. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Periodic Functions
    1. Introduction to Periodic Functions
    2. 6.1 Graphs of the Sine and Cosine Functions
    3. 6.2 Graphs of the Other Trigonometric Functions
    4. 6.3 Inverse Trigonometric Functions
    5. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    6. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 7.1 Solving Trigonometric Equations with Identities
    3. 7.2 Sum and Difference Identities
    4. 7.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 7.4 Sum-to-Product and Product-to-Sum Formulas
    6. 7.5 Solving Trigonometric Equations
    7. 7.6 Modeling with Trigonometric Equations
    8. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 8.1 Non-right Triangles: Law of Sines
    3. 8.2 Non-right Triangles: Law of Cosines
    4. 8.3 Polar Coordinates
    5. 8.4 Polar Coordinates: Graphs
    6. 8.5 Polar Form of Complex Numbers
    7. 8.6 Parametric Equations
    8. 8.7 Parametric Equations: Graphs
    9. 8.8 Vectors
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 9.1 Systems of Linear Equations: Two Variables
    3. 9.2 Systems of Linear Equations: Three Variables
    4. 9.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 9.4 Partial Fractions
    6. 9.5 Matrices and Matrix Operations
    7. 9.6 Solving Systems with Gaussian Elimination
    8. 9.7 Solving Systems with Inverses
    9. 9.8 Solving Systems with Cramer's Rule
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 10.1 The Ellipse
    3. 10.2 The Hyperbola
    4. 10.3 The Parabola
    5. 10.4 Rotation of Axes
    6. 10.5 Conic Sections in Polar Coordinates
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Sequences, Probability and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 11.1 Sequences and Their Notations
    3. 11.2 Arithmetic Sequences
    4. 11.3 Geometric Sequences
    5. 11.4 Series and Their Notations
    6. 11.5 Counting Principles
    7. 11.6 Binomial Theorem
    8. 11.7 Probability
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Introduction to Calculus
    1. Introduction to Calculus
    2. 12.1 Finding Limits: Numerical and Graphical Approaches
    3. 12.2 Finding Limits: Properties of Limits
    4. 12.3 Continuity
    5. 12.4 Derivatives
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    7. Exercises
      1. Review Exercises
      2. Practice Test
  14. A | Basic Functions and Identities
  15. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  16. Index

Try It

12.1 Finding Limits: Numerical and Graphical Approaches

1.

a=5, a=5, f( x )=2 x 2 4, f( x )=2 x 2 4, and L=46. L=46.

2.

a. 0; b. 2; c. does not exist; d. 2; 2; e. 0; f. does not exist; g. 4; h. 4; i. 4

3.

lim x0 ( 20sin(x) 4x )=5 lim x0 ( 20sin(x) 4x )=5

Table showing that f(x) approaches 5 from either side as x approaches 0 from either side.
4.

does not exist

12.2 Finding Limits: Properties of Limits

1.

26

2.

59

3.

10

4.

64 64

5.

3 3

6.

1 50 1 50

7.

1 8 1 8

8.

2 3 2 3

9.

−1 −1

12.3 Continuity

1.
  1. removable discontinuity at x=6; x=6;
  2. jump discontinuity at x=4 x=4
2.

yes

3.

No, the function is not continuous at x=3. x=3. There exists a removable discontinuity at x=3. x=3.

4.

x=6 x=6

12.4 Derivatives

1.

3

2.

f (a)=6a+7 f (a)=6a+7

3.

f (a)= 15 ( 5a+4 ) 2 f (a)= 15 ( 5a+4 ) 2

4.

3 2 3 2

5.

0

6.

2 , 2 , 0, 0, 3 3

7.
  1. After zero seconds, she has traveled 0 feet.
  2. After 10 seconds, she has traveled 150 feet east.
  3. After 10 seconds, she is moving eastward at a rate of 15 ft/sec.
  4. After 20 seconds, she is moving westward at a rate of 10 ft/sec.
  5. After 40 seconds, she is 100 feet westward of her starting point.
8.

The graph of f f is continuous on ( ,1 )( 1,3 )( 3, ). ( ,1 )( 1,3 )( 3, ). The graph of f f is discontinuous at x=1 x=1 and x=3. x=3. The graph of f f is differentiable on ( ,1 )( 1,3 )( 3, ). ( ,1 )( 1,3 )( 3, ). The graph of f f is not differentiable at x=1 x=1 and x=3. x=3.

9.

y=19x16 y=19x16

10.

–68 ft/sec, it is dropping back to Earth at a rate of 68 ft/s.

12.1 Section Exercises

1.

The value of the function, the output, at x=a x=a is f( a ). f( a ). When the lim xa f( x ) lim xa f( x ) is taken, the values of x x get infinitely close to a a but never equal a. a. As the values of x x approach a a from the left and right, the limit is the value that the function is approaching.

3.

–4

5.

–4

7.

2

9.

does not exist

11.

4

13.

does not exist

15.

Answers will vary.

16.

Answers will vary.

17.

Answers will vary.

18.

Answers will vary.

19.

Answers will vary.

20.

Answers will vary.

21.

Answers will vary.

23.

7.38906

25.

54.59815

27.

e 6 403.428794, e 6 403.428794, e 7 1096.633158, e 7 1096.633158, e n e n

29.

lim x2 f(x)=1 lim x2 f(x)=1

31.

lim x3 ( x 2 x6 x 2 9 )= 5 6 0.83 lim x3 ( x 2 x6 x 2 9 )= 5 6 0.83

33.

lim x1 ( x 2 1 x 2 3x+2 )=2.00 lim x1 ( x 2 1 x 2 3x+2 )=2.00

35.

lim x1 ( 1010 x 2 x 2 3x+2 )=20.00 lim x1 ( 1010 x 2 x 2 3x+2 )=20.00

37.

lim x 1 2 ( x 4 x 2 +4x+1 ) lim x 1 2 ( x 4 x 2 +4x+1 ) does not exist. Function values decrease without bound as x x approaches –0.5 from either left or right.

39.

lim x0 7tanx 3x = 7 3 lim x0 7tanx 3x = 7 3

Table shows as the function approaches 0, the value is 7 over 3 but the function is undefined at 0.
40.
Table shows as the function approaches 4, the value does not exist since approaching the limit value from the left is negative infinity and approaching the limit value from the right is positive infinity.
43.

lim x0 e e 1 x 2 =1.0 lim x0 e e 1 x 2 =1.0

45.

lim x 1 | x+1 | x+1 = (x+1) (x+1) =1 lim x 1 | x+1 | x+1 = (x+1) (x+1) =1 and lim x 1 + | x+1 | x+1 = (x+1) (x+1) =1; lim x 1 + | x+1 | x+1 = (x+1) (x+1) =1; since the right-hand limit does not equal the left-hand limit, lim x1 | x+1 | x+1 lim x1 | x+1 | x+1 does not exist.

47.

lim x1 1 ( x+1 ) 2 lim x1 1 ( x+1 ) 2 does not exist. The function increases without bound as x x approaches 1 1 from either side.

49.

lim x0 5 1 e 2 x lim x0 5 1 e 2 x does not exist. Function values approach 5 from the left and approach 0 from the right.

51.

Through examination of the postulates and an understanding of relativistic physics, as vc, vc, m. m. Take this one step further to the solution,

lim v c m= lim v c m o 1( v 2 / c 2 ) = lim v c m= lim v c m o 1( v 2 / c 2 ) =

12.2 Section Exercises

1.

If f f is a polynomial function, the limit of a polynomial function as x x approaches a a will always be f( a ). f( a ).

3.

It could mean either (1) the values of the function increase or decrease without bound as x x approaches c, c, or (2) the left and right-hand limits are not equal.

5.

10 3 10 3

7.

6

9.

1 2 1 2

11.

6

13.

does not exist

15.

12 12

17.

5 10 5 10

19.

108 108

21.

1

23.

6

25.

1

27.

1

29.

does not exist

31.

6+ 5 6+ 5

33.

3 5 3 5

35.

0

37.

3 3

39.

does not exist; right-hand limit is not the same as the left-hand limit.

41.

2

43.

Limit does not exist; limit approaches infinity.

45.

4x+2h 4x+2h

47.

2x+h+4 2x+h+4

49.

cos(x+h)cos(x) h cos(x+h)cos(x) h

51.

1 x(x+h) 1 x(x+h)

53.

1 x+h + x 1 x+h + x

55.

f( x )= x 2 +5x+6 x+3 f( x )= x 2 +5x+6 x+3

57.

does not exist

59.

52

12.3 Section Exercises

1.

Informally, if a function is continuous at x=c , x=c , then there is no break in the graph of the function at f( c ), f( c ), and f( c ) f( c ) is defined.

3.

discontinuous at a=3 a=3 ; f(3) f(3) does not exist

5.

removable discontinuity at a=4 a=4 ; f(4) f(4) is not defined

7.

Discontinuous at a=3 a=3 ; lim x3 f(x)=3 , lim x3 f(x)=3 , but f(3)=6 , f(3)=6 , which is not equal to the limit.

9.

lim x2 f(x) lim x2 f(x) does not exist.

11.

lim x 1 f(x)=4; lim x 1 + f(x)=1 lim x 1 f(x)=4; lim x 1 + f(x)=1 . Therefore, lim x1 f(x) lim x1 f(x) does not exist.

13.

lim x 1 f(x)=5 lim x 1 + f(x)=1 lim x 1 f(x)=5 lim x 1 + f(x)=1 . Thus lim x1 f(x) lim x1 f(x) does not exist.

15.

lim x 3 f(x)=6 lim x 3 f(x)=6 , lim x 3 + f(x)= 1 3 lim x 3 + f(x)= 1 3

Therefore, lim x3 f(x) lim x3 f(x) does not exist.

17.

f( 2 ) f( 2 ) is not defined.

19.

f( 3 ) f( 3 ) is not defined.

21.

f( 0 ) f( 0 ) is not defined.

23.

Continuous on (,) (,)

25.

Continuous on (,) (,)

27.

Discontinuous at x=0 x=0 and x=2 x=2

29.

Discontinuous at x=0 x=0

31.

Continuous on (0,) (0,)

33.

Continuous on [4,) [4,)

35.

Continuous on (,) (,) .

37.

1, but not 2 or 3

39.

1 and 2, but not 3

41.

f( 0 ) f( 0 ) is undefined.

43.

(,0)(0,) (,0)(0,)

45.

At x=1, x=1, the limit does not exist. At x=1, x=1, f( 1 ) f( 1 ) does not exist.

At x=2, x=2, there appears to be a vertical asymptote, and the limit does not exist.

47.

x 3 +6 x 2 7x ( x+7 )( x1 ) x 3 +6 x 2 7x ( x+7 )( x1 )

49.

The function is discontinuous at x=1 x=1 because the limit as x x approaches 1 is 5 and f( 1 )=2. f( 1 )=2.

12.4 Section Exercises

1.

The slope of a linear function stays the same. The derivative of a general function varies according to x. x. Both the slope of a line and the derivative at a point measure the rate of change of the function.

3.

Average velocity is 55 miles per hour. The instantaneous velocity at 2:30 p.m. is 62 miles per hour. The instantaneous velocity measures the velocity of the car at an instant of time whereas the average velocity gives the velocity of the car over an interval.

5.

The average rate of change of the amount of water in the tank is 45 gallons per minute. If f( x ) f( x ) is the function giving the amount of water in the tank at any time t ,t, then the average rate of change of f( x ) f( x ) between t=a t=a and t=b t=b is f(a)+45(ba). f(a)+45(ba).

7.

f (x)=2 f (x)=2

9.

f (x)=4x+1 f (x)=4x+1

11.

f (x)= 1 (x2) 2 f (x)= 1 (x2) 2

13.

16 ( 3+2x ) 2 16 ( 3+2x ) 2

15.

f (x)=9 x 2 2x+2 f (x)=9 x 2 2x+2

17.

f (x)=0 f (x)=0

19.

1 3 1 3

21.

undefined

23.

f (x)=6x7 f (x)=6x7

25.

f (x)=9 x 2 +4x+1 f (x)=9 x 2 +4x+1

27.

y=12x15 y=12x15

29.

k=10 k=10 or k=2 k=2

31.

Discontinuous at x=2 x=2 and x=0. x=0. Not differentiable at –2, 0, 2.

33.

Discontinuous at x=5. x=5. Not differentiable at -4, –2, 0, 1, 3, 4, 5.

35.

f( 0 )=2 f( 0 )=2

37.

f( 2 )=6 f( 2 )=6

39.

f ( 1 )=9 f ( 1 )=9

41.

f ( 1 )=3 f ( 1 )=3

43.

f ( 3 )=9 f ( 3 )=9

45.

Answers vary. The slope of the tangent line near x=1 x=1 is 2.

47.

At 12:30 p.m., the rate of change of the number of gallons in the tank is –20 gallons per minute. That is, the tank is losing 20 gallons per minute.

49.

At 200 minutes after noon, the volume of gallons in the tank is changing at the rate of 30 gallons per minute.

51.

The height of the projectile after 2 seconds is 96 feet.

53.

The height of the projectile at t=3 t=3 seconds is 96 feet.

55.

The height of the projectile is zero at t=0 t=0 and again at t=5. t=5. In other words, the projectile starts on the ground and falls to earth again after 5 seconds.

57.

36π 36π

59.

$50.00 per unit, which is the instantaneous rate of change of revenue when exactly 10 units are sold.

61.

$21 per unit

63.

$36

65.

f'(x)=10a1 f'(x)=10a1

67.

4 ( 3x ) 2 4 ( 3x ) 2

Review Exercises

1.

2

3.

does not exist

5.

Discontinuous at x=1( lim xa f(x) does not exist),x=3(jump discontinuity), and x=7( lim xa f(x) does not exist). Discontinuous at x=1( lim xa f(x) does not exist),x=3(jump discontinuity), and x=7( lim xa f(x) does not exist).

7.

lim x2 f(x)=0 lim x2 f(x)=0

9.

Does not exist

11.

-35-35

13.

11

15.

66

17.

500500

19.

-67-67

21.

At x=4, x=4, the function has a vertical asymptote.

23.

At x=3, x=3, the function has a vertical asymptote.

25.

Removable discontinuity at a=9a=9

27.

Removable discontinuity at x=5x=5

29.

Removable discontinuity at x=5x=5, discontinuity at x=1x=1

31.

Removable discontinuity at x=-2x=-2, discontinuity at x=5x=5

33.

33

35.

1 (x+ 1)(x+h+1) 1 (x+ 1)(x+h+1)

37.

e2x+2h-e2x h e2x+2h-e2x h

39.

10x-3 10x-3

41.

The function would not be differentiable at however, 0 is not in its domain. So it is differentiable everywhere in its domain.

Practice Test

1.

3

3.

0

5.

−1 −1

7.

lim x 2 f(x)= 5 2 a lim x 2 f(x)= 5 2 a and lim x 2 + f(x)=9 lim x 2 + f(x)=9 Thus, the limit of the function as x x approaches 2 does not exist.

10.

-150 -150

12.

1 1

14.

Removable discontinuity at x=3 x=3

16.

f'(x)=- 3 2a32 f'(x)=- 3 2a32

18.

Discontinuous at −2, 0, not differentiable at −2, 0, 2

20.

Not differentiable at x=0 x=0 (no limit)

22.

The height of the projectile at t=2 t=2 Seconds

24.

The average velocity from t=1 t=1 t=2 t=2

26.

13 13

28.

0 0

29.

2

30.

x=1 x=1

32.

y=14x18 y=14x18

34.

The graph is not differentiable at x=1 x=1 (cusp).

36.

f ' (x)=8x f ' (x)=8x

38.

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

40.

f ' (x)=3 x 2 f ' (x)=3 x 2

42.

f'(x)= 1 2 x1 f'(x)= 1 2 x1

Citation/Attribution

Want to cite, share, or modify this book? This book is Creative Commons Attribution License 4.0 and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/precalculus/pages/1-introduction-to-functions
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/precalculus/pages/1-introduction-to-functions
Citation information

© Sep 25, 2020 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License 4.0 license. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.