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Chapter 1 Test Bank

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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 + 3cos v 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

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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

4. 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 ( ) 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

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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

6. 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 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

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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 = π .

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10. Use the rectangles in the following graph to approximate the area of the region bounded by y = cos x, y = 0, x = 2 π − , and x = 2 π . 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.

2 5 – 5 lim – 20 + 75 x x x xx 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 12. Complete the table and use the result to estimate the limit.

–7 + 12 5 lim + 7 x x x → − 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 13. Complete the table and use the result to estimate the limit.

5 1 1 – + 3 8 lim – 5 x x xx 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

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14. Complete the table and use the result to estimate the limit. 2 2 0 sin lim x x xx –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 1 lim –4 x x x → − 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 5

x→ −x

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17. Let 3 , ( ) 0 1 1 x x f x x − ≠  =  =  .

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

1

lim ( )

xf x

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

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

(

2

)

2

lim 3

xx +

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19. Let 2 4, 1 ( ) 1, 1 x x f x x  + ≠ =  =  .

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

1

lim ( )

xf x

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

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

3

1 lim

3

xx

A) 0 B) Does not exist C) 3 D) –3 E) –6 21. Let f x( ) –3 = x– 5 and g x( )=x2. Find the limits:

(a) –4 lim ( ) xf x (b) xlim ( )→–2g x (c) lim ( ( ))x→4g f x A) 12 , 4 , –23 D) 12 , –2 , 25 B) –4 , –2 , 16 E) 7 , 4 , 289 C) –8 , 2 , –23

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22. Let f x( )=x2 + 1 and g x( ) 3= x. Find the limits: (a) li 5 m ( ) xf x (b) lix→m ( )2g x (c) lim ( ( ))x→–3g f x A) 2 , 3 , –26 B) 5 , 2 , 27 C) 26 , 6 , 30 D) 5 , 2 , 1 E) 6 , 2 , –8 23. Let f x( ) 5 + = x2 and g x( )= x + 3. Find the limits:

(a) li

2

m ( )

xf x (b) lix→m ( )5g x (c) lim ( ( ))x→5g f x

A) 9, 8, 33 B) 2, 5, 5 C) 6,5, 30 D) 4, 3, 8 E) 6, 5, 33 24. Let f x( ) 5 + 3 – 2= x2 x and g x( )=3 x – 4. Find the limits:

(a) li

2

m ( )

xf x (b) lix→m ( )2g x (c) lim ( ( ))x→5g f x

A) 24, 2, – 134 3 3 D) 24, – 2, 134 3 3

B) 28, – 2, – 134 3 3 E) None of the above

C) 28, 2, 134 3 3

25. Find the limit:

3 4 lim sin x x π → A) 1/ 2 2 2 − B) 1/ 2 2 2 C) –1/ 2 2

4 D) Does not exist E)

–1/ 2

2 4 −

26. Find the limit:

5 lim cos 6 x x π →       A) 3–1/ 2 4 B) 1/ 2 3 2 C) 1/ 2 3 – 2 D) 0 E) –1/ 2 3 – 4 27. Find the limit:

5 lim tan 6 x x π →      

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28. Suppose that lim ( ) –7 and l

x cf x = x cim ( ) 4→ g x = . Find the following limit: ( )

lim ( )g x x c→ f x 

A) –11 B) –3 C) 0 D) –28 E) 2401 29. Suppose that lim ( ) –8 and

x cf x = lim ( ) 7x cg x = . Find the following limit:

[

]

lim ( ) ( )

x cf x +g x

A) –56 B) –15 C) 0 D) –1 E) 7 30. Suppose that lim ( ) –9 and li

x cf x = x c→m ( ) 14g x = . Find the following limit:

[

]

lim ( ) ( )

x cf xg x

A) 0 B) 5 C) –23 D) –126 E) –9 31. Suppose that lim ( ) –7 and li

x cf x = x c→m ( ) 13g x = . Find the following limit:

[

]

lim –9 g( )

x cx

A) –117 B) 63 C) –9 D) 13 E) –7 32. Suppose that lim ( ) 14 and li

x cf x = x c→m ( ) 15g x = . Find the following limit:

[

]

lim ( ) ( )

x cf x g x

A) 14 B) 29 C) –1 D) 210 E) –15 33. Suppose that lim ( ) 6 and

x cf x = lim ( ) 9x cg x = . Find the following limit:

( ) lim ( ) x c f x g 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 + 512 lim + 8 x x x

A) 192 , x2 – 8 + 64x D) Limit does not exist.

B) 64 , x2 + 8 + 64x E) –192 , x2 – 8 + 64x

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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 + 154 lim – 14 x x x x

A) Does not exist. D) –157 , –12 + 11x

B) 179 , 12x + 11 E) 157 , 12x – 11 C) –179 , –12 – 11x

36. Find the limit (if it exists):

–8 lim x→ 2 – – 8 64 x x − A) 32 B) – 1 16 C) 1 16 D) –8 E) 1 32 37. Find the limit (if it exists):

(

) (

2

)

(

2

)

0 – – 6 – – 6 lim x x x x x x x x ∆ → + ∆ + ∆ − ∆ A) 1 31 2 – 6 3x 2x x B) 32 6 x x x C) 0 D) 2 – 1x E) x2 – – 6x

38. Determine the limit (if it exists):

(

)

8 0 sin 1 cos lim –2 x x x x → −

A) 0 B) 1 C) Does not exist. D) –2 E) 8 39. Determine the limit (if it exists):

(

)

0 –2 1 cos lim x x x → −

A) 0 B) –4 C) –8 D) Does not exist E) 4 40. Determine the limit (if it exists):

5 4 0 sin lim x x x

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

(i)

4

lim ( )

x→ + f x (ii) xlim ( )→4− f x (iii) lim ( )x→4 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 x=–3.

(i)

–3

lim ( )

x→ + f x (ii) xlim ( )→–3− f x (iii) xlim ( )→–3 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

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

(i)

–4

lim ( )

x→ + f x (ii) xlim ( )→–4− f x (iii) xlim ( )→–4 f x

A) 2 , 0 , Does not exist , Not continuous D) –4 , 0 , Does not exist , Not continuous B) 0 , 2 , Does not exist , Not continuous E) 2 , –2 , Does not exist , Not continuous C) 0 , 2 , 0 , Continuous

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

(

)

+ 8

lim –4 – 7

x→   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 – 11 f x = x x A) Continuous everywhere D) 7 – 6 x= . Not removable. B) x=–11. Removable E) both B and C

C) 7

– 6

x= . Removable. 46.

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

x f 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

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47.

Find the x-values (if any) at which the function ( ) 2 – 5 – 6 + 5

x f x

x x

= is not continuous. Which of the discontinuitites are removable?

A) No points of discontinuity.

B) x=5 (Not removable), x=1 (Removable) C) x=5 (Removable), x=1 (Not removable) D) No points of continuity.

E) x=5 (Not removable), x=1 (Not removable) 48. Find constants a and b such that the function

9, –5 ( ) , –5 1 –9, 1 x f x ax b x x ≤   = + < < 

is continuous on the entire real line.

A) a = 3 , b = 0 D) a = –3 , b = 6

B) a = 3 , b = –6 E) a = –3 , b = –6 C) a = 3 , b = 6

49. Find the constant a such that the function sin 4 , ( ) + 15 , 0 x x f x x a x x  ⋅ <  =    0

is continuous on the entire real line. A) –4 B) 4 C) –15 D) 15 E) 1 50. Determine whether ( ) 2 2 81 x f x x =

approaches ∞ or −∞ as x approaches –9 from the left and from the right by completing the tables below.

x –9.5 –9.1 –9.01 –9.001 f(x) x –8.999 –8.99 –8.9 –8.5 f(x) A) –9 lim ( ) x→ − f x = ∞ D) xlim ( )→–9+ f x = ∞ B) –9 lim ( ) – x→ − f x = ∞ E) Both A and C C) –9 lim ( ) – x→ + f x = ∞ F) Both B and D

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51.

Find the vertical asymptotes (if any) of the function ( ) 2 2 25 – 15 + 50 x f 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 ( ) 3 2 2+ 8 + 12 + – 26 + 24 x x f 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 30 k x= + π (k=0, 1, 2, ...)± ± D) No vertical asymptotes B) 2 1 15 k x= + π (k=0, 1, 2, ...)± ± E) 2 30 k x= π (k=0, 1, 2, ...)± ± C) 2 15 k x= π (k=0, 1, 2, ± ± ...) 54. Find the limit:

6 – 8 lim – + 6 x x x + → A) –∞ B) C) 0 D) –1 E) 1 ∞ 55. Find the limit:

(

)

(

)

2 2 12 – 12 lim 144 – 12 x x x x x → + A) –24 B) – 1 24 C) 24 D) 1 24 E) 12 56. Find the limit:

4 0 1 lim – x→ − x x       A) 0 B) –∞ C) 1 D) –1 E) ∞

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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

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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

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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

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