chapter 1 test bank

Chapter 1: Limits and Their Properties

1. Decide whether the following problem can be solved using precalculus, or whether calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, use a graphical or numerical approach to estimate the solution. Find the distance traveled in 11 seconds by an object moving with a velocity of

. ( ) 6 + 3cosv t t=A) precalculus , 63.0000 D) precalculus , 66.0133 B) calculus , 66.0133 E) precalculus , 67.3633 C) calculus , 63.0000

2. Decide whether the following problem can be solved using precalculus, or whether

calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, use a graphical or numerical approach to estimate the solution. Find the distance traveled in 12 seconds by an object traveling at a constant velocity of 12 feet per second. A) precalculus , 288 D) calculus , 288 B) precalculus , 144 E) calculus , 164 C) calculus , 144

Copyright Houghton Mifflin Company. All rights reserved. 17


3. Decide whether the following problem can be solved using precalculus, or whether calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, use a graphical or numerical approach to estimate the solution. A cyclist is riding on a path whose elevation is modeled by the function ( 2( ) 0.09 18 )f x x= x where x and ( )f x are measured in miles. Find the rate of change of elevation when x = 4.5.

A) precalculus , 0.21 D) precalculus , 0.09 B) calculus , 0.09 E) calculus , 0.81 C) precalculus , 0.21


calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, use a graphical or numerical approach to estimate the solution. A cyclist is riding on a path whose elevation is modeled by the function ( ) 0.16f x x= where x and ( )f x are measured in miles. Find the rate of change of elevation when x = 4.0.

A) calculus , 1.28 D) precalculus , 0.16 B) precalculus , 1.28 E) precalculus , 0.41 C) calculus , 0.16

18 Copyright Houghton Mifflin Company. All rights reserved.


5. Decide whether the following problem can be solved using precalculus, or whether calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, use a graphical or numerical approach to estimate the solution. Find the area of the shaded region.

A) precalculus , 22 D) calculus , 15 B) calculus , 22 E) precalculus , 18 C) precalculus , 15


calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, use a graphical or numerical approach to estimate the solution. Find the area of the shaded region bounded by the triangle with vertices (0,0), (3,4), (7,0).

A) precalculus , 14.0 D) calculus , 28.0 B) precalculus , 28.0 E) precalculus , 42.0 C) calculus , 42.0



7. Consider the function ( )f x = x and the point P(100,10) on the graph of f. Consider the secant lines passing through P(100,10) and Q(x,f(x)) for x values of 97, 99, and 101. Find the slope of each secant line to four decimal places. (Think about how you could use your results to estimate the slope of the tangent line of f at P(100,10), and how to improve your approximation of the slope.) A) 0.0504 , 0.0501 , 0.0499 D) 0.0252 , 0.0251 , 0.0249 B) 0.0252 , 0.0251 , 0.0249 E) 0.0504 , 0.0501 , 0.0499 C) 0.0504 , 0.0501 , 0.0499

8. Use the rectangles in each graph to approximate the area of the region bounded by the

following. y = 9/x, y = 0, x = 1, and x = 9.

A) 30.1714 , 24.4607 D) 60.3429 , 48.9214 B) 30.1714 , 48.9214 E) 45.2571 , 36.6911 C) 60.3429 , 24.4607

9. Use the rectangles in the following graph to approximate the area of the region bounded

by y = sin x , y = 0, x = 0, and x = .

A) 1.4221 B) 3.7922 C) 0.9481 D) 1.8961 E) 1.2704



10. Use the rectangles in the following graph to approximate the area of the region bounded

by y = cos x , y = 0, x = 2 , and x =

2 .

A) 3.9082 B) 2.6055 C) 0.9770 D) 1.4656 E) 1.9541

11. Complete the table and use the result to estimate the limit.

25

5lim 20 + 75x

xx x

x 4.9 4.99 4.999 5.001 5.01 5.1 f(x)

A) 0.100000 B) 0.400000 C) 0.275000 D) 0.525000 E) 0.475000


7

+ 12 5lim + 7x

xx

x 7.1 7.01 7.001 6.999 6.99 6.9 f(x)

A) 0.223607 B) 0.348607 C) 0.223607 D) 0.390273 E) 0.473607


5

1 1 + 3 8lim

5xx

x

x 4.9 4.99 4.999 5.001 5.01 5.1 f(x)

A) 0.114375 B) 0.094375 C) 0.145625 D) 0.015625 E) 0.125625



14. Complete the table and use the result to estimate the limit. 2

20

sinlimx

xx

x 0.1 0.01 0.001 0.001 0.01 0.1 f(x)

A) 1 B) 0.5 C) 1 D) 0.5 E) 0

15. Complete the table and use the result to estimate the limit. ( )

0

cos 4 1lim

4xxx

x 0.1 0.01 0.001 0.001 0.01 0.1 f(x)

A) 1 B) 0 C) 1 D) 2 E) 2

16. Determine the following limit. (Hint: Use the graph of the function.)

( )2

lim 5x

x

A) 7 B) 5 C) 2 D) 3 E) Does not exist



17. Let 3 ,

( )0 1

1x xf x

x = = .

Determine the following limit. (Hint: Use the graph of the function.)

1lim ( )x

f x

A) 2 B) 0 C) 3 D) 4 E) Does not exist


( )22

lim 3x

x +

A) Does not exist B) 2 C) 3 D) 7 E) 0



19. Let 2 4, 1

( )1, 1

x xf x

x + = =

.

Determine the following limit. (Hint: Use the graph of the function.)

1lim ( )x

f x

A) 5 B) 1 C) 4 D) 16 E) Does not exist.


3

1lim3x x

A) 0 B) Does not exist C) 3 D) 3 E) 6

21. Let and ( ) 3 5f x x= 2( )g x x= . Find the limits:

(a)

4lim ( )x

f x (b) (c) 2lim ( )x g x 4lim ( ( ))x g f xA) 12 , 4 , 23 D) 12 , 2 , 25 B) 4 , 2 , 16 E) 7 , 4 , 289 C) 8 , 2 , 23



22. Let and . Find the limits: 2( ) + 1f x x= ( ) 3g x x= (a) li

5m ( )

xf x (b) li (c) 2m ( )x g x 3lim ( ( ))x g f x

A) 2 , 3 , 26 B) 5 , 2 , 27 C) 26 , 6 , 30 D) 5 , 2 , 1 E) 6 , 2 , 8

23. Let 2( ) 5 + f x x= and ( ) + 3g x x= . Find the limits: (a) li

2m ( )


A) 9, 8, 33 B) 2, 5, 5 C) 6,5, 30 D) 4, 3, 8 E) 6, 5, 33

24. Let and 2( ) 5 + 3 2f x x x= 3( ) 4g x x= . Find the limits: (a) li

2m ( )


A) 3 324, 2, 134 D) 3 324, 2, 134 B) 3 328, 2, 134 E) None of the above C) 3 328, 2, 134

25. Find the limit:

34

lim sinx

x

A)

1/ 222

B) 1/ 222

C) 1/ 224

D) Does not exist E) 1/ 224

26. Find the limit:

5lim cos

6xx

A)

1/ 234

B) 1/ 232

C) 1/ 232

D) 0 E) 1/ 234

27. Find the limit:

5lim tan6xx

A) 1/ 23 B) 1/ 23 C) D) Does not exist E) 1/ 26 1/ 26



28. Suppose that and llim ( ) 7x c

f x = im ( ) 4x c g x = . Find the following limit: ( )lim ( )g x

x cf x

A) 11 B) 3 C) 0 D) 28 E) 2401

29. Suppose that and lim ( ) 8x c

f x = lim ( ) 7x c g x = . Find the following limit: [ ]lim ( ) ( )

x cf x g x +

A) 56 B) 15 C) 0 D) 1 E) 7

30. Suppose that and lilim ( ) 9x c

f x = m ( ) 14x c g x = . Find the following limit: [ ]lim ( ) ( )

x cf x g x

A) 0 B) 5 C) 23 D) 126 E) 9


f x = m ( ) 13x c g x = . Find the following limit: [ ]lim 9g( )

x cx

A) 117 B) 63 C) 9 D) 13 E) 7


f x = m ( ) 15x c g x = . Find the following limit: [ ]lim ( ) ( )

x cf x g x

A) 14 B) 29 C) 1 D) 210 E) 15

33. Suppose that and lim ( ) 6x c

f x = lim ( ) 9x c g x = . Find the following limit: ( )lim( )x c

f xg x

A) 3

2 B) 54 C) 2

3 D) Does not exist. E) 54

34. Find the following limit (if it exists). Write a simpler function that agrees with the given

function at all but one point.

3

8

+ 512lim + 8x

xx

A) 192 , 2 8 + 64x x D) Limit does not exist. B) 64 , 2 + 8 + 64x x E) 192 , 2 8 + 64x xC) 64 , 2 8 64x x



35. Find the following limit (if it exists). Write a simpler function that agrees with the given function at all but one point.

2

14

12 179 + 154lim 14x

x xx

A) Does not exist. D) 157 , 12 + 11xB) 179 , 12 + 11x E) 157 , 12 11xC) 179 , 12 11x

36. Find the limit (if it exists):

8limx 2

864

xx

A) 32 B) 1

16 C) 1

16 D) 8 E) 1

32

37. Find the limit (if it exists):

( ) ( ) ( )2 2

0

6 6limx

x x x x x xx

+ +

A) 3 21 1 6

3 2x x x B) 3 2 6x x x C) 0 D) 2 1x E) 2 6x x

38. Determine the limit (if it exists):

( )

80

sin 1 coslim

2xx x

x

A) 0 B) 1 C) Does not exist. D) 2 E) 8


( )0

2 1 coslimx

xx

A) 0 B) 4 C) 8 D) Does not exist E) 4


5

40

sinlimx

xx

A) B) 1 C) 0 D) Does not exist. E) 2



41. Use the graph as shown to determine the following limits, and discuss the continuity of the function at . 4x = (i)

4lim ( )x

f x+ (ii) 4lim ( )x f x (iii) 4lim ( )x f x

A) 4 , 4, 4, Not continuous D) 3 , 3 , 3 , Continuous B) 3 , 3 , 3 , Not continuous E) 2 , 2 , 2 , Not continuous C) 2 , 2 , 2 , Continuous

42. Use the graph as shown to determine the following limits, and discuss the continuity of

the function at . 3x = (i)

3lim ( )

xf x+ (ii) 3lim ( )x f x (iii) 3lim ( )x f x

A) 1 , 1 , 1 , Not continuous D) 2 , 2 , 2 , Continuous B) 1 , 1 , 1 , Continuous E) 3 , 3 , 3 , Continuous C) 2 , 2 , 2 , Not continuous



43. Use the graph to determine the following limits, and discuss the continuity of the function at . 4x = (i)

4lim ( )

xf x+ (ii) 4lim ( )x f x (iii) 4lim ( )x f x

A) 2 , 0 , Does not exist , Not continuous D) 4 , 0 , Does not exist , Not continuousB) 0 , 2 , Does not exist , Not continuous E) 2 , 2 , Does not exist , Not continuousC) 0 , 2 , 0 , Continuous

44. Find the limit (if it exists). Note that ( )f x x= represents the greatest integer function.

( )+8

lim 4 7x

x

A) 39 B) 35 C) Does not exist D) 39 E) 35

45. Find the x-values (if any) at which the function is not continuous.

Which of the discontinuitites are removable?

2( ) 3 7 11f x x x=A) Continuous everywhere D) 7

6x = . Not removable.

B) 11x = . Removable E) both B and C C) 7

6x = . Removable.

46.

Find the x-values (if any) at which the function 2( ) + 64xf x

x= is not continuous.

Which of the discontinuitites are removable? A) 8 and -8. Not removable. D) Discontinuous everywhere B) Continuous everywhere E) None of the above C) 8 and -8. Removable.



47. Find the x-values (if any) at which the function 2

5( ) 6 + 5xf x

x x= is not continuous.

Which of the discontinuitites are removable? A) No points of discontinuity. B) 5x = (Not removable), 1x = (Removable) C) 5x = (Removable), 1x = (Not removable) D) No points of continuity. E) 5x = (Not removable), 1x = (Not removable)

48. Find constants a and b such that the function

9, 5

( ) , 5 19, 1

xf x ax b x

x

= + <


51. Find the vertical asymptotes (if any) of the function

2

2

25( ) 15 + 50xf x

x x= .

A) x = 10 B) x = 10 C) x = 5 D) x = 5 E) x = 50

52. Find the vertical asymptotes (if any) of the function

2

3 2

+ 8 + 12( )+ 26 + 24x xf x

x x x= .

A) x = 4 B) x = 1 C) x = 4 D) x = 1 E) A and B F) C and D

53. Find the vertical asymptotes (if any) of the function ( ) tan(15 )f x x= . A) 2 1

30kx += ( 0, 1, 2, ...)k = D) No vertical asymptotes

B) 2 115kx += ( 0, 1, 2, ...)k = E) 2

30kx = ( 0, 1, 2, ...)k =

C) 215

kx = ( 0, 1, 2, ...)k =

54. Find the limit:

6

8lim + 6xxx+

A) B) C) 0 D) 1 E) 1

55. Find the limit:

( )( )2

212

12lim144 12xx x

x x + A) 24 B) 1

24 C) 24 D) 1

24 E) 12

56. Find the limit:

4

0

1lim x

xx

A) 0 B) C) 1 D) 1 E)



Answer Key

1. C Section: 1.1

2. B Section: 1.1

3. E Section: 1.1

4. D Section: 1.1

5. D Section: 1.1

6. A Section: 1.1

7. C Section: 1.1

8. A Section: 1.1

9. D Section: 1.1

10. E Section: 1.1

11. A Section: 1.2

12. C Section: 1.2

13. D Section: 1.2

14. A Section: 1.2

15. B Section: 1.2

16. D Section: 1.2

17. A Section: 1.2

18. D Section: 1.2

19. A Section: 1.2

20. B Section: 1.2

21. E Section: 1.3

22. C Section: 1.3



23. A Section: 1.3

24. D Section: 1.3

25. B Section: 1.3

26. C Section: 1.3

27. A Section: 1.3

28. E Section: 1.3

29. D Section: 1.3

30. C Section: 1.3

31. A Section: 1.3

32. D Section: 1.3

33. C Section: 1.3

34. A Section: 1.3

35. E Section: 1.3

36. C Section: 1.3

37. D Section: 1.3

38. C Section: 1.3

39. A Section: 1.3

40. C Section: 1.3

41. E Section: 1.4

42. A Section: 1.4

43. B Section: 1.4

44. D Section: 1.4

45. A Section: 1.4




46. B Section: 1.4

47. C Section: 1.4

48. E Section: 1.4

49. B Section: 1.4

50. E Section: 1.5

51. B Section: 1.5

52. E Section: 1.5

53. A Section: 1.5

54. B Section: 1.5

55. D Section: 1.5

56. E Section: 1.5

chapter 1 test bank

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