Math 115 – Calculus I Practice Problems – Final Exam Fall …torres/115/practicefinal2010.pdf · Math 115 – Calculus I Practice Problems – Final Exam Fall 2010 The Final exam

Math 115 – Calculus IPractice Problems – Final Exam

Fall 2010

The Final exam is on Wednesday, December 15, 4:30 p.m. - 7:00 p.m.

The exam will cover the following sections from the textbook.

• Chapter 2, Sections 2.1, 2.2, 2.3, 2.4, 2.5, 2.6.

• Chapter 3, Sections 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7.

• Chapter 4, Sections 4.1, 4.2, 4.3, 4.4, 4.5.

• Chapter 5, Sections 5.1, 5.2, 5.4, 5.5, 5.6.

• Chapter 6, Sections 6.1, 6.3, 6.4, and 6.5 (except for problems requiring 6.2).

These problems are intended to be used as part of a review for the final exam. Do not confuse the following,however, with a sample final. The actual proportion of questions from each section in the final exam mayvary from that in the practice problems. The problems below are not a substitute for studying all thematerial covered in class and the homework assignments.

Solve the following problems and circle the right answer.

1. Let f be the function defined by f(x) =

√x+ 1

x− 2. The domain of f is

(A) (−∞, 2) and (2,∞) (B) (−1, 2] and [2,∞) (C) [−1, 2) and (2,∞) (D) [−1,∞)

(E) None of the above.

2. limx→3

(3x2 − 4) =

(A) 77 (B) 5 (C) 23 (D) Does not exist


3. limx→4

x2 − 16

x− 4=

(A) 8 (B) 0 (C) 4 (D) Does not exist.


4. limx→5

x2 − 2x− 15

x− 5=



1

5. limx→1−

x2 − 1

2x− 2=

(A) 1 (B) 0 (C) − 1 (D) Does not exist.


6. limx→∞

2x2 − 10x+ 7

5x2 − 1=

(A)7

5(B) 7 (C)

2

5(D) Does not exist.


7. limx→−∞

6x3 + 4x2 − 17x

x4 + 3x=

(A)1

6(B) 6 (C) 0 (D) Does not exist


Use the graph of the function f in Figure 1 below to answer Problems 8,9,10

1 2 3

1

2

3

-1

-2

-3

-1-2-3 x

y

y=f(x)

FIGURE 1

8. limx→0+

f(x) =

(A) 0 (B) − 1 (C) 1 (D) Does not exist.


2

9. limx→ (−1)

f(x) =



10. The function f is continuous at

(A) x = 2.5 (B) x = −1 (C) x = 0 (D) x = 2


11. Which of the following statements is true for the function represented in the graph below in Figure2?

FIGURE 2

(A) f ′ > 0 on (0, 2), f ′′ > 0 on (0, 1) and f ′′ < 0 on (1, 2).

(B) f ′ > 0 on (0, 2), f ′′ < 0 on (0, 1) and f ′′ > 0 on (1, 2).

(C) f ′ < 0 on (0, 2), f ′′ > 0 on (0, 1) and f ′′ < 0 on (1, 2).

(D) f ′ < 0 on (0, 2), f ′′ < 0 on (0, 1) and f ′′ > 0 on (1, 2).

(E) None of the above

12. Which of the following statements is true for the function represented in the graph below in Figure3?

3

FIGURE 3

(A) f ′′ > 0 on (0, 2), f has one inflection point and no critical point in (0, 2).

(B) f ′′ > 0 on (0, 2), f has no inflection point and one critical point in (0, 2).

(C) f ′′ < 0 on (0, 2), f has one inflection point and no critical point in (0, 2).

(D) f ′′ < 0 on (0, 2), f has no inflection point and one critical point in (0, 2).


13. If x3 + 2y3 + y − 5 = 0, then by implicit differentiationdy

dx=

(A) 3x2 + 6y2 (B) − 3x2 − 6y2 (C)−6y2 − 1

3x2(D) − 3x2

6y2 + 1


14. The equation of the tangent line to the curve y =3

x− 2at (3, 3) is

(A) y = −3x+ 12 (B) y = −3x+ 8 (C) y = 3x− 10 (D) y = −3


15. The equation of the tangent line to the curve y = ex2

+ x at (0, 1) is

(A) y = 2x+ 1 (B) y = −2x+ 1 (C) y = x+ 1 (D) y = −x+ 1


4

16. The equation of the tangent line to the curve y = ln(x+ 2) at (−1, 0) is

(A) y = −x+ 1 (B) y = x+ 1 (C) y = x− 1 (D) y = −x− 1


Answer Problems 17, 18 and 19 using the information below.

(Compute the answers to three decimal places.)

A marine biologist is running experiments involving the environmental temperature for a particularspecies of fish. The appetite A of the fish seems to be affected by the temperature T of the water and

can be approximated by A =5000

T 2 + 1, 50◦ ≤ T ≤ 80◦, where A = 1 is considered normal appetite.

17. The rate of change in appetite with respect to the temperature when the temperature is 60◦ is

(A) − 0.046 (B) 1.388 (C) − 1.388 (D) 41.660 (E) 0.046

18. The value of the appetite when the temperature is 70◦ is

(A) 1.020 (B) 2.020 (C) 72.412 (D) 5001.000 (E) 0

19. The average rate of change of the appetite from 50◦ to 80◦ is

(A) 0.799 (B)0.190 (C) 0.691 (D) − 1.218 (E) − 0.041

Answer Problems 20, 21, and 22 using the following information.A nature preserve is being established. A population biologist has estimated that the population ofthe deer in the preserve is given by the model

Q(t) = 150e0.05t

where t is in years and t = 0 correspond to the time when the preserve was established.

20. The population of deer when the preserve was established was

(A) 3000 deer (B)75 deer (C) 150 deer (D) 300 deer (E) 1500 deer

21. The instantaneous rate of change of the population of deer when the preserve was established was

(A) 30 deer/year (B)7.5 deer/year (C) 75 deer/year (D) 300 deer/year

(E) 150 deer/year

5

22. The average rate of change of the population of deer during the first five years after the preserve wasestablished was (computed to two decimal places)

(A) 42.61 deer/year (B)7.50 deer/year (C) 8.52 deer/year (D) 300.00 deer/year

(E) 150.00 deer/year

Answer Problems 23 and 24 using the following information.The market value of an office building located in the commercial district of a city is given by

V (t) = 300, 000e12

√t

where V (t) is measured in dollars and t is the time in years from the present.

23. The present value of the property is

(A) $300, 000. (B) $150, 000. (C) $600, 000.

(D) $424, 264. (E) $212, 132.

24. The instantaneous rate of change of the value of the property 1 year from now will be (in wholedollars/year)

(A) 300, 000 dollars/year. (B) 150, 000 dollars/year. (C) 247, 308 dollars/year.

(D) 123, 654 dollars/year. (E) 212, 132 dollars/year.

Answer Problems 25, 26, and 27 using the following information.

The demand equation for apartments in a large city is given by p = −0.6x+1, 000, where p denotesthe monthly rent for an apartment and x denotes the number of apartments rented. The monthly costassociated with renting a total of x apartments is C = 20, 000 + 400x.

25. The revenue function R is

(A) −0.6x+1, 000 (B) −0.6x2+1, 000x (C) 400.6x+19, 000 (D) −400.6x+19, 000 (E) 20, 000x+400x2


26. The profit function P is

(A) −0.6x2+1000x (B) −400.6x−19, 000 (C) 400 (D) −0.6x2+600x−20, 000


27. The marginal profit function is

(A) −1.2x+1000 (B) −400.6 (C) 0 (D) −1.2x+600 (E) None of the above.

6

28. The differential of the function f(x) = x3 + 10x− 3 is

(A) 3x2 + 10 (B)x3 + 10x− 3

x(C) (x3 + 10x− 3)dx (D) (3x2 + 10)dx


29. The function f(x) = 2x3 − 12x2 + 18x+ 1 is increasing for

(A) all real numbers (B) x in (1, 3) (C) x in (−∞, 1) (D) x in (−∞, 1) and (3,∞)


30. How many critical points does x+3

xhave?

(A) None. (B) One. (C) Two. (D) Three.


31. Consider the function f(x) = x3. The point with coordinates (0, 0) is

(A) a relative minimum. (B) a relative maximum.

(C) an inflection point. (D) an absolute minimum.

(E)None of the above.

32. Consider f(x) = x3 − 12x2 − 8. The inflection point(s) of f is (are)

(A) (0,−8) and (8,−264) (B) (6,−224) (C) (4,−136) (D) (0,−8) and (6, 224)

(E) No inflection points.

33. If f(2) = 4, f ′(2) = 0, and f ′′(2) = 15, then (2, 4) are the coordinates of

(A) a relative minimum. (B) a relative maximum.

(C) an inflection point. (D) an absolute maximum.


34. The vertical asymptote(s) for the graph of f(x) =5x2 − 39x− 8

x2 − x− 12is (are)

(A) x = 3 and x = 4 (B) y = 5 (C) y = 3 and y = −4 (D) x = 4 and x = −3


7

35. The horizontal asymptote(s) of the f(x) =5x2 − 39x− 8

x2 − x− 12is (are)

(A) y = 3 and y = 4 (B) x = 5 (C) x = 3 and x = −4 (D) y = 5


Answer Problems 36, 37, and 38 using the following function.

Let f(x) = x3 − 6x2 + 2x− 1.

36. The function f(x) is concave upward on

(A) (−∞, 2) (B) (2,∞) (C) (−∞, 2) and (2,∞) (D) (−2, 0) and (2,∞)


37. The function f(x) is concave downward on

(A) (−∞, 2) (B) (2,∞) (C) (−∞, 2) and (2,∞) (D) (−2, 0) and (2,∞)


38. The inflection point(s) of f(x) is (are)

(A) (2,−13) (B) (0,−1) (C) (0,−1) and (2,−13)

(D) (0, 1) (E) No inflection points.

39. The number of lamps a company sells is a function of the price charged, and it can be approximatedby the formula N(x) = 200 + 50x + 36.5x2 − x3, where x is the price charged for each lamp indollars (0 ≤ x ≤ 37). The price which will result in the maximum number of lamps sold is

(A) $37. (B) $13. (C) $25. (D) $20. (E) 0.

40. A company has determined that its profit (P ) depends on the amount of money (x) spent on adver-tising. The relationship is given by the equation

P (x) =2x

x2 + 4+ 70,

where P and x are measured in thousands of dollars. The amount that the company should spendon advertising to assure maximum profit is

(A) $70, 000. (B) $10, 000. (C) $2, 000. (D) $5, 000. (E) $6, 000.

41. Let f(x) =1

xand g(x) =

√x+ 1. Then f(g(4)) is

(A)3

2(B)

1

3(C)

2

3(D)

1

2


8

42. Let f be the function defined byf(x) = ln(x− 9).

The domain of f is

(A) (0,∞) (B) (0, 9) (C) (−∞,−9) (D) (9,∞)


43. The function f(x) which satisfies the initial value problem f ′(x) = 3x− 1; f(1) = 2 is

(A) f(x) =3

2x2 − x+

3

2(B) f(x) =

1

2x2 − x+

3

2(C) f(x) =

3

2x2 − x+

1

2

(D) f(x) =1

2x2 − x+

5

2(E) f(x) =

3

2x2 − 2x+

3

2

44. The function f(x) which satisfies the initial value problem f ′(x) = ex + x; f(0) = 3 is

(A) f(x) = ex +x2

2− 1 (B) f(x) = ex +

x2

2− 3 (C) f(x) = ex +

x2

2+ 2

(D) f(x) = ex +x2

2(E) f(x) = ex + x2 + 2

45. The function f(x) which satisfies the initial value problem f ′(x) = x3 + x2; f(1) = 0 is

(A) f(x) =x4

4+x3

3+

7

2(B) f(x) =

x4

4+x3

3− 7

12(C) f(x) =

x4

4+x3

3+

1

7

(D) f(x) =x4

4+x3

3− 1

7(E) f(x) =

x4

4+x3

3

46. The area (in sq. units) of the region under the graph of the function f(x) = 4x + 5 on the interval[−1, 2] is

(A) 11 (B) 21 (C) 18 (D) 17 (E) 16

47. The area (in sq. units) of the region under the graph of the function f(x) = x2 on the interval [−1, 2]is

(A) 3 (B)8

3(C) 9 (D) 2 (E) 1

48. The area (in sq. units) of the region under the graph of the function f(x) = 1x

on the interval [2, 8] is

(A) e8 (B) e8 − e2 (C) ln(8) (D) ln(2) (E) 2 ln(2)

9

49. The indefinite integral∫

3exdx is equal to

(A) ex + C (B) 3ex + C (C) ln(3x) + C (D) 3 ln(x) + C (E) 3ex + 3x+ C

50. The indefinite integral∫ (

2 +√x+ ex

)dx is equal to

(A) 2x+3

2x3/2 + ex + C (B) x+

2

3x3/2 + ex + C

(C) 2x+2

3x3/2 − ex + C (D) x2 +

2

3x3/2 + ex + C (E) 2x+

2

3x3/2 + ex + C

51. The indefinite integral∫ (

x2/5 − 1

x3

)dx is equal to

(A)7

5x7/5 +

1

2x−2 + C (B)

5

7x7/5 + 2x−2 + C (C)

5

7x7/5 +

1

2x−2 + C

(D)7

5x3/5 +

1

2x−2 + C (E)

5

7x7/5 +

1

2x−1 + C

52. The definite integral∫ 3

−3(4x+ 3)dx is equal to

(A) 0 (B) 18 (C) 54 (D) 36 (E) 9


1

1

xdx is equal to

(A) ln(4) (B) e5 − e (C) e4 (D) ln(5) (E) − 4

5


1

(1 +

1

x2

)dx is equal to

(A)17

4(B) 0 (C)

23

4(D)

15

4(E)

1

4

55. The derivative of the function ln(x2 + 1) is equal to

(A)2x

x2 + 1(B)

x2

x2 + 1(C) 2x ln(x2 + 1) (D) x2 ln(x2 + 1) (E)

x2 + 1

2x

56. The derivative of the function ex3+x is equal to

(A) ex3+x. (B) ex3+x(3x2 + 1). (C) ln(x3 + x).

(D) (3x2 + 1) ln(x3 + x). (E)3x2 + 1

x3 + x.

10

57. If 100x = 10x+3, then x =

(A) 0 (B) 4 (C) 3 (D) − 1


58. ln(xex) =(A) ln x+ x (B) x lnx (C) ex + lnx (D) ex lnx


59. If 12− e2x = 3, then x =

(A) − 3 (B)ln 9

2(C) ln 9 (D) − 3 ln 9


60. The horizontal and vertical asymptotes of the graph2x2 + 1

x2 − x− 6are

(A) x = 2, y = 3, and y = −2 (B) y = 2, x = 3, and x = −2 (C) x = 3, y = 2, and y = −2

(D) x =1

2and x = −1

2(E) y =

1

2and y = −1

2

61. The absolute maximum and absolute minimum values of f(x) =x

x2 + 4on[0, 3] are

(A)3

13and 0 (B)

1

4and 0

(C)1

4and

3

13(D)

1

5and 0


62. The price of certain stock at time t (0 ≤ t ≤ 5) is estimated by

P (t) = 0.1t3 + 0.05t2 − 3t+ 10.

Then, the stock price will have

(A) maximum price at t = 0 and minimum price at t = 5.

(B) maximum price at t = 2 and minimum price at t = 3.

(C) maximum price at t = 0 and minimum price at t = 2.

(D) maximum price at t = 0 and minimum price at t = 3


11

63. A manufacturer has a monthly cost given by the function C(x) = 65 + 23x + 0.001x2 (in dollars)where x is the number of units manufactured. The product sells for $30 per unit. Then, the profit (orloss) realized from the sale of the 101st unit is approximately

(A) (−6.8) dollars (B) 6.8 dollars (C) 700 dollars (D) 635 dollars


64. If h(x) = g(f(x)), where f(x) = e2x, and g′(1) = −2, then h′(0) =

(A) 4 (B) − 2 (C) − 4 (D) 16


65. An oil tanker is spilling its cargo into an expanding circular oil slick of area A = πR2, where the

radius R is measured in feet. The radius of the oil slick is increasing in such a way thatdR

dt= 2

ft/sec when R = 4 feet. How fast is the area of the oil slick increasing at the time when R is 4 feet?

(A) 16π ft2/sec (B) 32π ft2/sec (C) 64π ft2/sec (D) 8(√

3− 1)π ft2/sec(E) None of the above.

66. The velocity of a car ( in ft/sec) t seconds after starting from rest is given by the function

v(t) = t3/2, (0 ≤ t ≤ 25.)

Then the position of the car at time t is given by

(A) S(t) =3

2t5/2feet. (B) S(t) =

2

3t5/2feet. (C) S(t) =

5

2t5/2feet.

(D) S(t) =2

5t5/2feet. (E) S(t) =

2

5t3/2feet.

67. The average acceleration of the car in the above problem from for 1 ≤ t ≤ 25 is

(A) − 3/4 feet/s2. (B) 2/3 feet/s2. (C) 5/2 feet/s2 (D) − 5/2 feet/s2.


68. Problems 17, 19 and 23 in page 387 of the textbook.

69. Problem 17 in page 428 of the textbook.

70. Problem 46 in page 448 of the textbook.

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Math 115 – Calculus I Practice Problems – Final Exam Fall …torres/115/practicefinal2010.pdf · Math 115 – Calculus I Practice Problems – Final Exam Fall 2010 The Final exam