PART II TEST AND TEST ANSWER KEYS Test Bank

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Test Bank . ... Chapter 14. Integration. .... Use the vertical line test to determine which of the following graphs represent y as a function of x. (a). (b). (c). (d).
PART II

TEST AND TEST ANSWER KEYS

Test Bank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Chapter 1P Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15

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Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14

Preparation for Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Limits and Their Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Applications of Differentiation . . . . . . . . . . . . . . . . . . . . . . . 66 Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Logarithmic, Exponential, and Other Transcendental Functions. . . . . . . . . . . . . . . . . . . . . . . 91 Applications of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Integration Techniques, L’Hôpital’s Rule, and Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Conics, Parametric Equations, and Polar Coordinates . . . . . . 135 Vectors and the Geometry of Space . . . . . . . . . . . . . . . . . . . . 151 Vector-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Functions of Several Variables . . . . . . . . . . . . . . . . . . . . . . . . 174 Multiple Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Vector Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

Answer Key to Chapter Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

Final Exams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Answer Key to Final Exams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

Gateway Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Answer Key to Gateway Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

25

26

Chapter P

Preparation for Calculus

Test Form A

Name

__________________________________________

Date

Chapter P

Class

__________________________________________

Section _______________________

1. Find all intercepts of the graph of y  (a) 2, 0

x2 . x3

(b) 2, 0, 3, 0



(d) 2, 0, 0, 

2 3



____________________________

(c)

0, 3, 3, 0 2

(e) None of these

2. Determine if the graph of y 

x is symmetrical with respect to the x-axis, the y-axis, or the origin. x2  4

(a) About the x-axis

(b) About the y-axis

(d) All of these

(e) None of these

(c) About the origin

3. Find all points of intersection of the graphs of x2  2x  y  6 and x  y  4. (a) 0, 6, 0, 4

(b) 10, 14, 13, 17

(d) 5, 1, 2, 6

(e) None of these

(c) 5, 9, 2, 2

4. Which of the following is a sketch of the graph of the function y  x3  1? (a)

(b)

y

y

2

4 3 x

−2

1

2

2

−1

−2

(c)

−1

(d)

y

1

2

1

2

y

2

2

x −3

−1

x −2

1 −1

−1

−2

−2

(e) None of these 5. Find an equation for the line passing through the point 4, 1 and perpendicular to the line 2x  3y  3. (a) y  23 x  1

(b) 3x  2y  2  0

(d) 3x  2y  10

(e) None of these

(c) 2x  3y  10

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x

−2

Chapter P

6. Find the domain of f(x) 

1 . 3  2x

(a)

 , 32

(b)

 32, 

(d)

 , 32  32, 

(e) None of these

(c)

32, 

7. Find f x  x for f x  x3  1. (a) x3  1  x

(b) x3  3x2x  3xx2  x3  1

(c) x3  x3  1

(d) 3x 6  1

(e) None of these 1

8. If f x  (a) (d)

x

and gx  1  x2, find f g(x).

1  x2 x 1 x

(b)

 1  x2

1 1  x2

(c) 1 

1 x

(e) None of these

9. If the point  3, 2  lies on the graph of the equation 2x  ky  11, find the value of k. 1

5 (a)  2

(b) 34

17 (c)  2

(d) 10

(e) None of these

10. Which of the following equations expresses y as a function of x? (a) 3y  2x  9  17 (d) Neither a nor b

(b) 2x2y  x  4y (e)

3y2



x2

(c) Both a and b

5

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11. Given f x  x2  3x  4, find f x  2  f 2. (a) x2  3x  4

(b) x2  x

(d) x2  3x  4

(e) None of these

(c) x2  x  8

12. Determine which function is neither even nor odd. (a) f x  tan x

(b) f x  3x5  5x3  1

(d) f x  x2  1

(e) Both a and b

(c) f x 

3 x2

13. Find the point that lies on the line determined by the points (1, 2) and (3, 1). (a) (0, 0)

(b) (5, 1)

(d) (5, 5)

(e) (2, 0)

(c) (4, 6)

14. Determine the slope of the line given by the equation 9x  5y  11. (a)

5 9

5 (b)  9 9

(d)  5

(e) 9

(c)

9 5

Test Bank

27

28

Chapter P

Preparation for Calculus

15. Describe the transformation needed to sketch the graph of y 

1 1 using the graph of f x  . x2 x

(a) Shift f x two units to the right. (b) Shift f x two units to the left. (c) Shift f x two units upward. (d) Shift f x two units downward. (e) Reflect f x about the x-axis. 16. Use the vertical line test to determine which of the following graphs represent y as a function of x. (a)

(b)

y

y

3

3

2

2

1

1 x

−3

−1

1

2

x −3

3

−2

−1

1

−1

−1

−2

−2

−3

−3

(c)

(d)

y

2

3

y 4

5

3 2 3

1 x

2 −4 −3 1

1

−2

x −3

−1

−3

3 −1

−4



1 17. Let f x  x 2x  1, (a)

x < 0 . Find f 3. x ≥ 0

1 3

(d) Undefined

(b) 1 (e)

(c) 7

22 3

18. The dollar value of a product in 1998 is $1430. The value of the product is expected to increase $83 per year for the next 5 years. Write a linear equation that gives the dollar value V of the product in terms of the year t. (Let t  8 represent 1998.) (a) V  1430  83(t  8)

(b) V  83  1430t

(d) V  83  1430(t  8)

(e) V  1430  83(t  8)

(c) V  1430  83t

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(e) None of these

Chapter P

Test Bank

19. During the first and second quarters of the year, a business had sales of $150,000 and $185,000, respectively. If the growth of sales follows a linear pattern, what will sales be during the fourth quarter? (a) $220,000

(b) $235,000

(d) $255,000

(e) None of these

(c) $335,000

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20. In order for a company to realize a profit in the manufacture and sale of a certain item, the revenue, R, for selling x items must be greater than the cost, C, of producing x items. If R  79.99x and C  61x  1050, for what values of x will this product return a profit? (a) x ≥ 55

(b) x ≥ 8

(d) x ≥ 56

(e) None of these

(c) x ≥ 18

29

30

Chapter P

Preparation for Calculus

Test Form B

Name

__________________________________________

Date

Chapter P

Class

__________________________________________

Section _______________________

1. Find all intercepts of the graph of y 



(a) 1, 0, 0, 

1 3





(d) 3, 0, 0, 

1 3

x1 . x3

(b) 1, 0



____________________________

(c) 3, 0, 1, 0

(e) None of these

2. Determine if the graph of y 

x2

x2 is symmetrical with respect to the x-axis, the y-axis, or the origin. 4

(a) About the x-axis

(b) About the y-axis

(d) All of these

(e) None of these

(c) About the origin

3. Find all points of intersection of the graphs of x2  3x  y  3 and x  y  2. (a) 5, 3, 1, 1

(b) 0, 3, 0, 2

(d) 5, 7, 1, 1

(e) None of these

(c) 5, 3, 1, 1

4. Which of the following is a sketch of the graph of the function y  x  13? (b)

y

y

2

2

1

1 x

−2

x −2

2

−1

−2

(c)

2

−2

(d)

y

y

2

2

1

1 x

−2

−1

1

2

x −2

−1

2

−2

(e) None of these

© Houghton Mifflin Company. All rights reserved.

(a)

Chapter P

Test Bank

5. Find an equation for the line passing through the point 4, 1 and parallel to the line 2x  3y  3. (a) 2x  3y  11 (d) y 

2 3x

(b) 2x  3y  5

1

(c) 3x  2y  5

(e) None of these

6. Find the domain of f x 

1 3  2x

.

(a)

 ,  23

(b)

 32, 

(d)

 ,  23   23, 

(e) None of these

(c)

 23, 

7. Find f x  x for f x  x2  2x  3. (a) x2  x  3  x

(b) x2  2xx  x2  2x  2x  3

(c) x2  2x  3  x

(d) 5

(e) None of these 8. If f x  1  x2 and gx  (a) (d)

1 x

1  x2 x 1 x

, find f g(x). (b)

 1  x2

1 1  x2

(c) 1 

1 x

(e) None of these

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9. If the point 1, 1 lies on the graph of the equation kx2  xy  y2  5, find the value of k. (a) 7

(b) 3

(c) 5

(d) 3

(e) None of these

10. In which of the following equations is y a function of x? (a) 2x  3y  1  0

(b) x2  3y2  7

(d) Both a and b

(e) Both a and c



(c) 2x2y  7



11. Given f x  x  3  5, find f 1  f 5. (a) 0

(b) 4

(d) 14

(e) None of these

(c) 14

12. Determine the even function. x3 1

(a) f x  sin x

(b) f x 

(d) f x  x3  1

(e) None of these

x2

(c) f x  3x 4  5x2  1

31

32

Chapter P

Preparation for Calculus

13. Find the point that lies on the line determined by the points 1, 3 and 2, 4. (a) 3, 2

(b) 1, 1

(d) 4, 2

(e) 4, 2

(c) 10, 0

14. Determine the slope of the line given by the equation 7x  4y  6  0. (a)

7 4

(d)  74

(b)

4 7

(e)

3 2

(c) 7

15. Describe the transformation needed to sketch the graph of y 

1 1  2 using the graph of f(x)  . x x

(a) Shift f(x) two units to the right.

(b) Shift f(x) two units to the left.

(c) Shift f(x) two units upward.

(d) Shift f(x) two units downward.

(e) Reflect f(x) about the x-axis. 16. Use the vertical line test to determine which of the following graphs does not represent y as a function of x. (a)

(b)

y

y

4

2

3 2

1

1 x

x −4

−2

1

−1

−1

1

2

−2 −3 −4

(c)

(d)

y

y

3

2

2 1 1 −2

−1

1

2

π 2

−π 2

3

−1

−1

−2 −2

−3

(e) Both a and d 17. Let f x 



x2  5, 3x  1,

x < 2 . Find f 1. x ≥ 2

(a) 4

(b) 2

(d) 2

(e) 0

(c) 4

x

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x −3

Chapter P

Test Bank

18. The dollar value of a product in 1998 is $78. The value of the product is expected to decrease $5.75 per year for the next 5 years. Write a linear equation that gives the dollar value V of the product in terms of the year t. (Let t  8 represent 1998.) (a) V  78  5.75t

(b) V  78  5.75t

(d) V  78  5.75t  8

(e) V  5.75  78t  8

(c) V  78  5.75t  8

19. A business had annual retail sales of $124,000 in 1993 and $211,000 in 1996. Assuming that the annual increase in sales follows a linear pattern, predict the retail for 2001. (a) $356,000

(b) $435,000

(d) $298,000

(e) $327,000

(c) $646,000

© Houghton Mifflin Company. All rights reserved.

20. In order for a company to realize a profit in the manufacture and sale of a certain item, the revenue, R, for selling x items must be greater than the cost, C, of producing x items. If R  69.99x and C  59x  850, for what values of x will this product return a profit? (a) x ≥ 78

(b) x ≥ 15

(d) x ≥ 13

(e) None of these

(c) x ≥ 85

33

34

Chapter P

Preparation for Calculus

Test Form C

Name

__________________________________________

Date

Chapter P

Class

__________________________________________

Section _______________________

____________________________

A graphing calculator/utility is recommended for this test. 1. Find the x-intercepts for the graph of y  3x5  4x 4  2x3. (a) 0, 0, 2, 0,  13 , 0

(b) 0, 0

(d)

(e) There are no x-intercepts.

 2, 13 

(c) 0, 0, (2, 0, 1, 0

2. Use a graphing utility to graph the equation y  x 4  3x3  20. (a)

(b)

y

y 30

6

25

4 2 x −8 −6 −4

2

6

15

8 10 5 x −1

(c)

(d)

y

1

2

1

2

3

4

y

30

25

25

20 15

15 5

10

x

5 −2 x −1

1

3

(e) None of these 3. Identify the type(s) of symmetry: x 4y2  2x2y  1  0. (a) About the x-axis

(b) About the y-axis

(d) Both a and b

(e) None of these

(c) About the origin

4. Find all points of intersection of the graphs of x2  3x  y  3 and x  y  2. (a) 2, 0 and 1, 3

(b) 5, 3 and 1, 3

(d) 5, 37 and 1, 5

(e) None of these

(c) 1, 1 and 5, 7

5. A business had annual retail sales of $110,000 in 1993 and $224,000 in 1996. Assuming that the annual increase in sales followed a linear pattern, what was the retail sales in 1995? (a) $182,000

(b) $195,000

(d) $186,000

(e) None of these

(c) $188,000

© Houghton Mifflin Company. All rights reserved.

−3

4

−5

Chapter P

Test Bank

6. Find an equation for the horizontal line that passes through the point 3, 2. (a) x  2

(b) y  2

(c) x  3

(d) y  3

(e) None of these

7. Write an equation satisfied by the graph obtained by shifting y  2x  5 three units to the left. (a) y  2x  8

(b) y  2x  1

(d) y  2x  2

(e) None of these

8. Let f x 

1 x

and g x)  2x  3. Find the domain of  f  gx.

(a) x ≥ 

3 2

(b) x > 0

(d) x > 

3 2

(e) None of these

9. Find the zero(s) of the function f x  (a) 3 and 4 (d)

(c) y  2x  11

(c) x
5

43

44

Chapter 1

Limits and Their Properties

Test Form B

Name

__________________________________________

Date

Chapter 1

Class

__________________________________________

Section _______________________

____________________________

1. Find the limit: lim 2x2  1. x→3

(c) 17

(a) 37

(b) 19

(d) ± 2

(e) None of these









2. Given lim 2x  1  3. Find  such that 2x  1  3 < 0.01 whenever 0 < x  1 < . x→1

(a) 3

(b) 0.05

(d) 1

(e) None of these

3. Use the graph at the right to find lim f x for f x  x→1

(a) 0 (c)

(c) 0.005

1 . x1

y

(b) 1



(d) The limit does not exist.

1 x

(e) None of these

−1

1

2

3

4

x2  2x  3 . x→1 x2  1

4. Find the limit: lim

(b) 1

(d) The limit does not exist.

(e) None of these

(c)



5. Find the limit: lim 9  x2. x→3

(a) 0

(b) 6

(d) The limit does not exist.

(e) None of these

(c) 32

6. If lim f x   12 and lim gx  23 , find lim  f xgx . x→c

(a)

x→c

1 6

x→c

(b)  13

(d) The limit does not exist.

(c) 1

(e) None of these

x2  5x  6 . x→1 x1

7. Find the limit: lim (a) 0 (d)



(b) 7 (e) None of these

(c)  

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(a) 0

Chapter 1

8. Find the limit: lim

x→2

x2 . x2





(a) 0

(b) 1

(d) The limit does not exist.

(e) None of these

9. Find the limit: lim

x  9  3

x

x→0

.

(a) 0 (d)

(b) 1

1 3

(c)



(e) None of these

10. Find the limit: lim csc x→5

x . 4 (b) 1

(a) 1 (d) 

(c) 2

1

(c)  2

(e) None of these

2

1  cos2 x . x→0 x

11. Find the limit: lim

(a) 1

(b) 0

(d) The limit does not exist.

(e) None of these

(c)



12. Find the limit: lim 2x  3. x→2

(a) 1, 1

© Houghton Mifflin Company. All rights reserved.

(d)

(b) 1

1 2

(c) 1

(e) None of these

1 13. Find the limit: lim . x→0 x (a) 

(b) 0

(d) The limit does not exist.

(e) None of these

14. Find the limit: lim

x→1

5 . x  12

(a) 0

(b)  

(d) 

(e) None of these



15. Find the limit: lim 2  x→1

(c)  

(c)

5 4



5 . x  12

(a)  

(b) 

(d) 2

(e) None of these

(c) 3

Test Bank

45

Chapter 1

Limits and Their Properties

16. At which values of x is f x 

x4 discontinuous? x2  x  2

(a) 4

(b) 1, 2, 4

(d) 1, 2, 4, 2

(e) None of these

17. Let f x 

(c) 1, 2

1 and gx  x  1. Find all values of x for which f g(x) is discontinuous. x



(a) 0

(b) 1

(d) 1, 1

(e) None of these

(c) 0, 1

18. Determine the value of c so that f x is continuous on the entire real line when f x  (a) 4

(b) 4

(d) 1

(e) None of these

19. Find all vertical asymptotes of f x  (a) x  2 (d) x 

1 2

x2x3,c,

x ≤ 1 . x > 1

(c) 0

2x  1 . x3

1 (b) x  , x  3 2

(c) x  3

(e) None of these

20. Find all vertical asymptotes of f x 

x2 . x2  4

(a) x  2, x  2

(b) x  2

(d) x  2

(e) None of these

(c) x  0

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46

Chapter 1

Test Bank

Test Form C

Name

__________________________________________

Date

Chapter 1

Class

__________________________________________

Section _______________________

____________________________

A graphing calculator/utility is recommended for this test. 1. Use a graphing utility to graph the function: f x  x2  4x and then estimate lim f x (if it exists). x→2

(a) 0

(b) 12

(c) 4

(d) 12

(e) None of these

2. Use the graph to find lim f x (if it exists).

y

x→1

(a) 1

2

(b) 2

1

y = f ( x) x

(c) The limit does not exist.

−2

1

(d) 1

−1

(e) 3

−2

3. Find the limit: lim

x→1

(a) 3

(b) 0

(d) The limit does not exist.

(e) None of these

x→ 2

(b)

22 

2 

2 3

 2

(c) 3

(e) None of these

6. Find the limit: lim

x→9

(d) 0

(c) 

x2 and then estimate lim f x. x→0 x3

(b) 2

(d) 0

(a) 12

1 2

(e) None of these

5. Use a graphing utility to graph the function f x  (a) 

(c) 

sin x . x

(a) 0 (d)

2

x2  x  2 . x3

4. Find the limit: lim © Houghton Mifflin Company. All rights reserved.

47

x2  6x  27 . x9 (b) The limit does not exist. (e) None of these

(c) 3

48

Chapter 1

Limits and Their Properties

1 1  x3 3 7. Find the limit: lim . x→0 x (a) 

1 9

(b) 0

(d) The limit does not exist.

(c)

1 9

(c)



(e) None of these

1  cos2 x . x→0 x

8. Find the limit: lim (a) 1

(b) 0

(d) The limit does not exist.

(e) None of these

9. Match the graph with the correct function. (a) f x  (c) f x 

x3 x1 x2

x1  2x  3

y 6

(b) f x  x  3

5

(d) f x 

x2  2x  3 x1

2 1 x

(e) None of these

−2



1 , x3 10. Find the x-values (if any) for which f is not continuous: f x  1 , 1 2 (a) 5 (b) 2

(c) 3

(1, 4)

4

−1

1

2

3

x ≤ 5 . x > 5

(d) 3, 5

(e) None of these 11. Use the graph to find the x-values (if any) at which f is not continuous. (c) 2, 3

(b) 3, 3

8 6

(d) 1, 3

4 2

(e) None of these

x −4

2 −2 −4

12. Use a graphing utility to graph f x  x3  2x  5. Then use this graph to find the interval for which the Intermediate Value Theorem guarantees the existence of at least one number c in that interval for which f c  0. (a) 1, 1

(b) 1, 2

(d) 3, 4

(e) None of these

(c) 2, 3

4

6

8

© Houghton Mifflin Company. All rights reserved.

(a) 3

y

Chapter 1

Test Bank

13. Use the graph to find the interval(s) for which the function f is continuous. (a)  , 2 and 2, 

(b)  , 

(c)  , 1 and 1, 

(d) 1, 2

y

6

3

(e) None of these

x −3

3 −3

14. Find the limit:

lim

x→ 12

2x  1.

(a) 0

(b) 2

(d) The limit does not exist.

(e) None of these

15. Find the limit: lim x→6

(c) 1

3x  18 . 6x

(a) 1

(b) 1

(d) 3

(e) None of these

(c) 3

16. Which of the following statements is not true of f x  x2  25? (a) f is continuous at x  10.

(b) f is continuous on the interval  , 5.

(c) f is continuous on the interval 5, .

(d) f is continuous on the interval 5, 5.

(e) f is not continuous at x  0.

17. Find the limit: lim x→1

(a)

2 . x1



(b)  

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(d) The limit does not exist.

18. Find the limit: lim x→3

(a)

1 3

(d) 1

49

(c) 0

(e) None of these

x2  3x  2 . x2  5x  6 (b) 

(c)  

(e) None of these

19. Find the vertical asymptote(s): f x 

x2 . x2  3x  10

(a) x  2, x  5

(b) y  1

(d) x  5

(e) x  5

(c) y  0

20. f x decreases without bound as x approaches what value from the right? f x  (a) 5

(b) 3

(d) 3

(e) None of these

4

x  35  x

(c) 5

6

50

Chapter 1

Limits and Their Properties

Test Form D

Name

__________________________________________

Date

Chapter 1

Class

__________________________________________

Section _______________________

____________________________

1. Calculate lim 2x2  6x  1. x→2









2. If lim 3x  2  7, find  such that 3x  2  7 < 0.003 whenever 0 < x  3 < . x→3

3. Find the limit: lim f x if f x  x→1

x2,  4, 2

x1 . x1

x2  3x  2 . x→1 x2  1

4. Find the limit: lim

5. Find the limit: lim 4x2  9. x→2

1 2 6. If lim f x   2 and lim gx  3 , find lim  f x  gx. x→c

x→c

x→c

x2 . x→2 x3  8

7. Find the limit: lim

x  x  x

8. Find the limit: lim

x

x→0

x→1

x1 . x1





10. Find the limit: lim cot x→5

x . 6

x . x→0 sin 3x

11. Find the limit: lim

12. Find the limit: lim 2x  1. x→2

13. Find the limit: lim  x→1

14. Find the limit: lim

x→3

1 . x1

3 . x2  6x  9

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9. Find the limit: lim

.

Chapter 1



15. Find the limit: lim 3x  2  x→0



1 . x2

16. Find the value(s) of x for which f x  removable or nonremovable.

17. Let f x 

x2 is discontinuous and label these discontinuities as x2  4

5 and gx  x4. x1

a. Find f g(x). b. Find all values of x for which f g(x) is discontinuous.



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x2, 18. Determine the value of c so that f x is continuous on the entire real line if f x  c , x 19. Find all vertical asymptotes of f x if f x 

x2  3x  1 . x7

20. Find all vertical asymptotes of f x if f x 

2x  2 . x  1x2  x  1

x ≤ 3 x > 3.

Test Bank

51

52

Chapter 1

Limits and Their Properties

Test Form E

Name

__________________________________________

Date

Chapter 1

Class

__________________________________________

Section _______________________

____________________________

A graphing calculator/utility is recommended for this test. 1. Use the graph to find lim f x (if it exists).

y

x→0

4 3

y = f ( x) 1 x −2

−1

1

2

2. Determine whether the statement is true or false. If it is false, give an example to show that it is false. If lim f x  9, then f 3  9. x→3

3. Calculate the limit: lim 2x2  6x  1. x→2

4. Find the limit: lim

x2  x  2 . x3

5. Find the limit: lim

x . cos x

x→1

x→ 

6. Use a graphing utility to graph the function f x  x3  x  5 and then estimate lim f x. x→1

x2  4 . x2

a. Use a graphing utility to graph the function. b. Use the graph to estimate lim f x. x→2

c. Find the limit by analytical methods. 8. Let f x 

x3  2x2 . x2

a. Find lim f x (if it exists). x→2

b. Identify another function that agrees with f x at all but one point. c. Sketch the graph of f x.

9. Find the limit: lim

x→0

10. Find the limit: lim

x→0

x  x  2  x  2

x

.

x  x2  2x  x  x2  2x . x

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7. Let f x 

Chapter 1

Test Bank

11. Find the x-values (if any) for which f is not continuous. f x 

3x2x 2,3x  6, 2

x < 1 x ≥ 1

12. Use the graph to find the x-values (if any) for which f is not continuous.

y 3

y = f ( x)

2 1

x −3

−2

−1

1

2

3

−2 −3

f x  tan

13. Find the interval(s) for which the function is continuous.

x 2

y 9 6 3 x 2π −3 −6 −9

14. Use the Intermediate Value Theorem to show that the function f x  x 4  2x2  3x has a zero in the interval 2, 1. 15. Use a graphing utility to graph f x 

© Houghton Mifflin Company. All rights reserved.

the function is not continuous.

16. Let f x 

x2x  1,3, 2

x2 . Then use the graph to determine x-values at which x2  4

x ≤ 0 . Find each limit (if it exists). x > 0

a. lim f x

b. lim f x

x→0

x→0

17. Use a graphing utility to find the limit: lim

x→0

18. Find all vertical asymptotes of f x 

19. Find the limit: lim x→3

20. Find the limit: lim

x→1

1 . x3

2 . 1  x2

c. lim f x x→0

sin 3x . Then verify your answer analytically. x

x2  x  2 . x2  x  6

53

54

Chapter 2

Differentiation

Test Form A

Name

__________________________________________

Date

Chapter 2

Class

__________________________________________

Section _______________________

____________________________

1. If f x  2x2  4, which of the following will calculate the derivative of f x? (a)

2x  x2  4  2x2  4 x 2x2  4  x  2x2  4 x→0 x

(b) lim

2x  x2  4  2x2  4 x→0 x

(c) lim

2x2  4  x  2x2  4 x (e) None of these (d)

2. Differentiate: y  (a) 1 (d)

1  cos x . 1  cos x

2 sin x 1  cos x2

(b) 2 csc x

(c) 2 csc x

(e) None of these

3. Find dydx for y  x3x  1. 3x2 2x  1

(b)

(d)

7x3  x2 2x  1

(e) None of these

(c) 3x2x  1

4. Find fx for f x  2x2  57. (a) 74x6

(b) 4x7

(d) 72x2  56

(e) None of these

5. Find

d 2y x3 for y  . dx2 x1

(a) 0 (d)

(c) 28x2x2  56

8 x  13

(b)

8 x  13

(e) None of these

(c)

4 x  13

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x27x  6 2x  1

(a)

Chapter 2

Test Bank

6. The position equation for the movement of a particle is given by s  t2  13 when s is measured in feet and t is measured in seconds. Find the acceleration at two seconds. (a) 342 unitssec2

(b) 18 unitssec2

(d) 90 unitssec2

(e) None of these

7. Find

(c) 288 unitssec2

dy if y2  3xy  x2  7. dx

(a)

2x  y 3x  2y

(b)

3y  2x 2y  3x

(d)

2x y

(e) None of these

(c)

2x 3  2y

8. Find y if y  sinx  y. cosx  y 1  cosx  y

(a) 0

(b)

(d) 1

(e) None of these

(c) cosx  y

9. Differentiate: y  sec2 x  tan2 x. (a) 0

(b) tan x  sec4 x

(d) 4 sec2 x tan x

(e) None of these

(c) sec2 x sec2 x  tan2 x

t 10. Find the derivative: s t  csc . 2 (a) csc

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(d)

t t cot 2 2

1 2t cot 2 2

1 t (b)  cot2 2 2

(c)

t t 1 csc cot 2 2 2

(e) None of these

11. Find an equation for the tangent line to the graph of f x  2x2  2x  3 at the point where x  1. (a) y  2x  2

(b) y  4x2  6x  5

(d) y  4x2  6x  2

(e) None of these

(c) y  2x  1

12. Find all points on the graph of f x  x3  3x2  2 at which there is a horizontal tangent line. (a) 0, 2, 2, 2

(b) 0, 2

(d) 2, 2

(e) None of these

(c) 1, 0, 0, 2

55

56

Chapter 2

Differentiation

13. Find the instantaneous rate of change of w with respect to z if w  (a)

7 6z

(d) 

(b) 14 3z3

14 z 3

7 . 3z2 (c) 

14 3z

(e) None of these

14. Suppose the position equation for a moving object is given by s t  3t2  2t  5 where s is measured in meters and t is measured in seconds. Find the velocity of the object when t  2. (a) 13 msec

(b) 14 msec

(d) 6 msec

(e) None of these

(c) 10 msec

15. A point moves along the curve y  2x2  1 in such a way that the y value is decreasing at the rate of 2 units per second. At what rate is x changing when x  32? (b) decreasing 13 unitsec

(d) increasing 72 unitsec

(e) None of these

(c) decreasing 72 unitsec

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(a) increasing 13 unitsec

Chapter 2

57

Test Form B

Name

__________________________________________

Date

Chapter 2

Class

__________________________________________

Section _______________________

1. If f x  x2  x, which of the following will calculate the derivative of f x?

x2  x  x  x2  x x→0 x

(a) lim

 x  x2  x  x  x2  x x→0 x

(b) lim (c)

 x  x2  x  x  x2  x x

x2  x  x  x2  x x (e) None of these (d)

2. Differentiate: y 

3x . x2  1

(a)

3 1  x2

(b)

3 2x

(d)

31  x2 1  x22

(e) None of these

(c)

3x2  3 1  x23

3. Find dydx for y  x 3x  1. (a)

© Houghton Mifflin Company. All rights reserved.

Test Bank

(d)

9x  1 2x 3 2x

(b)

9 x  1 2

(c) 3x

(e) None of these

4. Find fx for f x  sin3 4x. (a) 4 cos3 4x

(b) 3 sin2 4x cos 4x

(d) 12 sin2 4x cos 4x

(e) None of these

5. Find

(c) cos3 4x

d 2y x2 for y  . 2 dx x3

(a)

10 x  33

(b) 0

(d)

2 x  33

(e) None of these

(c)

10 x  33

____________________________

58

Chapter 2

6. Find (a)

Differentiation

dy if x2  y2  2xy. dx x 1y

(d) 

x y

(b)

yx yx

(c) 1

(e) None of these

7. Find y if x  tanx  y. (a) sin2 x  y (d)

1  sec2 x sec2 y

(b) sec2 x  y

(c) tan2 x  y

(e) None of these

8. Differentiate: y  csc2   cot2 . (a) cot   csc4 

(b) 0

(c) 4 csc2  cot 

(d) csc2 csc2   cot2 

(e) None of these

9. Find the derivative: s t  sec t. (a) tan2t (d) sect tant

sect tant 2t (e) None of these

(b)

(c) sec

1 1 tan 2t 2t

10. A particle moves along the curve given by y  t3  1. Find the acceleration when t  2 seconds. 2 3

(a) 3 unitssec2

(b)

unitssec2

(d)  19 unitssec2

(e) None of these

1 (c)  108 unitssec2

(a) y  4x  2

(b) 2x  y  1  0

(d) 2x  y  5

(e) None of these

(c) y  4x2  2x  1

12. Find point(s) on the graph of the function f x  x3  2 where the slope is 3. (a) 1, 3, 1, 3

(b) 1, 1, 1, 3

(d) 1, 3

(e) None of these

13. Find the instantaneous rate of change of w with respect to z for w  (a)

3 2

(b) 2

(d)

1 z2

(e) None of these

(c)

3 2, 0  

1 z  . z 2 (c)

z2  2 2z2

© Houghton Mifflin Company. All rights reserved.

11. Find an equation for the tangent line to the graph of f x  2x2  2x  3 at the point where x  1.

Chapter 2

Test Bank

14. Suppose the position equation for a moving object is given by s t  3t2  2t  5 where s is measured in meters and t is measured in seconds. Find the velocity of the object when t  2. (a) 13 msec

(b) 6 msec

(d) 14 msec

(e) None of these

(c) 10 msec

© Houghton Mifflin Company. All rights reserved.

15. A point moves along the curve y  2x2  1 in such a way that the y value is decreasing at the rate of 2 units per second. At what rate is x changing when x   32 ? (a) decreasing 72 unitsec

(b) increasing 72 unitsec

(d) decreasing 13 unitsec

(e) None of these

(c) increasing 13 unitsec

59

60

Chapter 2

Differentiation

Test Form C

Name

__________________________________________

Date

Chapter 2

Class

__________________________________________

Section _______________________

____________________________

A graphing calculator/utility is recommended for this test. 1. Determine whether the slope at the indicated point is positive, negative, or zero.

y 3

(a) Zero

2

(b) No slope

1 x

(c) Positive

−3

−2

1

2

3

−1

(d) Negative

−2 −3

(e) None of these 2. Find the slope of the graph of f x  x2  2x at the point a, f (a). (c) f a

(a) 0

(b) 2a  2

(d) a2  2a

(e) None of these

3. Use the graph to determine all x-values at which the function is not differentiable.

y

(a) a, b, c, d, e, f, and g (b) b and d d

(c) a and f

a

b

c

e f g

x

(d) a, f, and g

3x2  8 and its derivative, f, on the same coordinate axes. x2  4 Then use the graph to describe the behavior of f at that value of x where fx  0.

4. Use a graphing utility to graph f x  (a) f x  0

(b) f increases without bound.

(d) f has no tangent line.

(e) None of these

5. Find the value of the derivative of the function f t 

(c) f has a horizontal tangent line.

t3  2 at the point 2, 3. t

(a) 

9 2

(b) 

7 2

(d) 

11 16

(e) None of these

(c) 12

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(e) None of these

Chapter 2

Test Bank

6. The graph at the right represents the graph of the derivative of which of the following functions? (a) f x  2x2  1

y

(b) f x  2x  3

3 2

(c) f x  3x2  2x  1 (d) f x 

x3



1 x

x2

−3

−2

1 −1

(e) None of these

7. Find

−1

−2

dy : y  4 sin x  5 cos x  x. dx

(a) 4 cos x  5 sin x  1

(b) 4 cos x  5 sin x  1

(d) 4 cos x  5 sin x

(e) None of these

(c) 4 cos x  5 sin x  1

2 8. The position function for a particular object is s   35 2 t  58t  91. Which statement is true?

(a) The initial velocity is 35.

(b) The velocity is a constant.

(c) The velocity at time t  1 is 23.

(d) The initial position is  35 2.

(e) None of these

9. Find

dy for y  csc   cot . d

(a) 0

(b) cot2   csc  cot 

(d) csc  cot   csc2 

(e) None of these

(c) sec  tan   sec2 

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10. Find an equation of the tangent line to the graph of f   tan  at the point (a) 4x  4y    4

(b) 42x  4y    4

(d) y  x

(e) None of these

4 , 1.

(c) 4x  2y    2

1 f x 11. Let f 3  0, f3  6, g3  1 and g3  . Find h3 if hx  . 3 gx (a) 18

(b) 6

(d) 2

(e) None of these

12. Find the derivative: f x 

(c) 6

1 .  x3

3 3 

(a)

1 33  x343

(b)

(d)

x2 3  x343

(e) None of these

x2 3  x343

(c)

x2 3  x323

2

3

61

62

Chapter 2

Differentiation

13. Find the derivative: f   sin 2. cos 2 sin 2 (d) cos  (a)

(b) sec 2

(c)

cos 2 2sin 2

(e) None of these

14. Determine the slope of the graph of the relation 2x2  3xy  y3  1 at the point 2, 3. (a)  17 21 (d)

4 3

(b)

5 7

(c)  13

(e) None of these

15. A machine is rolling a metal cylinder under pressure. The radius of the cylinder is decreasing at a constant rate of 0.05 inches per second and the volume V is 128 cubic inches. At what rate is the length h changing when the radius r is 1.8 inches? Hint: V  r2h (b) 39.51 in.sec

(d) 43.90 in.sec

(e) None of these

(c) 2.195 in.sec

© Houghton Mifflin Company. All rights reserved.

(a) 2.195 in.sec

Chapter 2

Test Bank

Test Form D

Name

__________________________________________

Date

Chapter 2

Class

__________________________________________

Section _______________________

____________________________

1 1. Use the definition of a derivative to calculate the derivative of f x  . x 2. Differentiate: y 

3. Find

2x . 1  3x2

dy for y  2x  1 x3. dx

4. Find fx for f x  cot3x.

5. Calculate

d 2y 1x for y  . dx2 2x

6. The position equation for the movement of a particle is given by s  t3  12 where s is measured in feet and t is measured in seconds. Find the acceleration of this particle at 1 second. 7. Find y if y 

8. Find

x . xy

dy if x  cos y. dx

9. Find the derivative: f   sec  2. 10. Differentiate: y  sin2 x  cos2 x. © Houghton Mifflin Company. All rights reserved.

63

11. Find an equation for the tangent line to the graph of f x  x  1 at the point where x  3. 12. Find the values of x for all points on the graph of f x  x3  2x2  5x  16 at which the slope of the tangent line is 4. 1 13. Find the instantaneous rate of change of R with respect to x if R  2x2  . x 14. An object is thrown (straight down) from the top of a 220-foot building with an initial velocity of 26 feet per second. a. Write the position equation for the movement described. b. What is the velocity at 1 second? 15. As a balloon in the shape of a sphere is being blown up, the volume is increasing at the rate of 4 cubic inches per second. At what rate is the radius increasing when the radius is 1 inch?

64

Chapter 2

Differentiation

Test Form E

Name

__________________________________________

Date

Chapter 2

Class

__________________________________________

Section _______________________

____________________________

A graphing calculator/utility is recommended for this test.

1. At each point indicated on the graph, determine whether the value of the derivative is positive, negative, zero, or the function has no derivative.

y

a

b c d

e

x

4 2. Let f x  . x a. Use the definition of the derivative to calculate the derivative of f. b. Find the slope of the tangent line to the graph of f at the point 2, 2. c. Write an equation of the tangent line in part b. d. Use a graphing utility to graph f and the tangent line on the same axes. Then sketch the graphs. 3. Find the point(s) on the graph of y 

1 where the graph is parallel to the line 4x  9y  3. x

4. A coin is dropped from a height of 750 feet. The height, s, (measured in feet), at time, t (measured in seconds), is given by s  16t2  750. b. Find the instantaneous velocity when t  3. c. How long does it take for the coin to hit the ground? d. Find the velocity of the coin when it hits the ground.

 3 r . How fast is the 3 changing volume with respect to changes in r when the radius is equal to 2 feet?

5. The volume of a right circular cone of radius r and height r is given by V 

6. Find the derivative: f x  5 sec x tan x. 7. Evaluate the derivative for the function f t 

t  2 at the point , . cos t 3 3





© Houghton Mifflin Company. All rights reserved.

a. Find the average velocity on the interval 1, 3.

Chapter 2

Test Bank

8. Let f x  x5  5x. a. Calculate f x. b. Use a graphing utility to graph f and f on the same axes. Sketch the graphs. c. Use the graph to determine those point(s) where f has a horizontal tangent line. d. Give the value of f at each of the points found in part c. 9. The graphs of a function f and its derivative f are given on the same coordinate axes. Label the graphs as f or f and state the reasons for your choice.

y

3 2 1 x −3

2

3

10. Find f  x: f x  2x2  5. 11. Find the point(s) (if any) of horizontal tangent lines: x2  xy  y2  6.

© Houghton Mifflin Company. All rights reserved.

12. A balloon rises at the rate of 8 feet per second from a point on the ground 60 feet from an observer. Find the rate of change of the angle of elevation when the balloon is 25 feet above the ground.

65