Sciences
11th Edition
Hoffmann Bradley Sobecki and Price 0073532371
9780073532370
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Chapter 6
hard Section: 6.1 3. Use integration by parts to evaluate the integral (13)
Page 105
1. Evaluate /3 x xedx . A) /3/3 1 3 xx xeeC −+ C) /3/339xx xeeC −+ B) /3/311 39 xx xeeC −+ D) /3/339xx xeeC ++ Ans: C Difficulty: hard Section:
2. Evaluate /9 x xedx . A) /9/9981xx xeeC −+ C) /9/9 981 xx xee C ++ B) /9/9981xx xeeC ++ D) /9/9 981 xx xee C −+ Ans: A Difficulty:
A) (14) x xeC−+ C) (13) x xeC−+ B) (14) x xeC++ D) (14) x xeC−−+ Ans: A Difficulty:
Section:
6.1
x xedx
.
moderate
6.1
8.
9.
Chapter 6 Page 106
Evaluate 2 (23) x xedx A) 2 31 24 x exC −+ B) 2x xeC + C) 2 3 2 x xeC + D) 2 3 x eC−+ Ans: A Difficulty: hard Section: 6.1
Evaluate (3)ln(2) xxdx . A) 2(3) ln(2) 2 x x + C C) 2 6(6)ln(2) 2 x xxxx −+− + C B) 2 (6)ln(2) 3 42 xxxx x −+ + C D) 2 (6)ln(2) 2 x xxx+− + C Ans: B Difficulty: hard Section: 6.1
True or False: 22 ln(2)2ln(2)24 xx xxdxxxC +=−+−++ A) True B) False Ans: A Difficulty: hard Section: 6.1
Evaluate 9(2) xxdx A) 10 1 55(2)(10)xxC −++ C) 10 1 55(2)(51)xxC −++ B) 10 1 110(2)(52)xxC −++ D) 10 1 110(2)(101)xxC −++
C Difficulty:
Section: 6.1
4.
5.
6.
7.
Ans:
hard
Evaluate
A) 2 2829 1 16144(3)(3) x xxC −−−+ C) 2 28(3) 16 x xC −+ B) 2 2829 1 1672(3)(3) x xxC −−−+ D) 283(3) 24 xx C + Ans: A Difficulty: hard Section: 6.1
327 (3) xxdx
Evaluate
8(7) xxdx +
. Ans: ( ) ( ) 91011 77 990 xxxC +−++ Difficulty: hard Section: 6.1
10. Evaluate 747 (6) xxdx
Ans: ( )( )8 44 1 436 144 xxC +−+
Difficulty: hard Section: 6.1
11. Evaluate 8(9) xxdx +
Ans: 910(9)(9) 990 xxx C ++ −+
Difficulty: hard Section: 6.1
12. Evaluate 11611 (2) xxdx
Ans: 6 612613 1 72936(2)(2) x xxC −−−+
Difficulty: hard Section: 6.1
13. Use integration by parts to evaluate the integral 20 x dx x + .
A) 1/2 2 3(20)(40)xxC −++
B) 1/2 2 3(20)(40)xxC +−+
Ans: B Difficulty: moderate Section: 6.1
3(20)(40) 2 xxC +−+
2 3(20)(40)xxC +−+
14. True or False: The area of the finite region bounded by the curves x yxe = , y = 0, and x = 4 equals 4 15e .
A) True B) False
Ans: A Difficulty: hard Section: 6.1
15. Evaluate ln xxdx .
A) 22 ln 48 xxx + C B) x + C C) 2 ln xxC + D) 2 ln 4 xx C +
Ans: A Difficulty: hard Section: 6.1
16. Evaluate 2 ln xxdx
Ans: 33 ln 39 xxxC −+
Difficulty: hard Section: 6.1
Chapter 6 Page 107
C) 1/2
D) 3/2
17. Evaluate 2 ln3x dx x .
Ans: 1ln3x C xx −−+
Difficulty: hard Section: 6.1
18. Evaluate 7 ln2x dx x .
Ans: 66 ln21 636 x C xx −−+
Difficulty: hard Section: 6.1
19. Evaluate ( )2 ln xxdx .
Ans: ( )2 2 22 ln ln 422 xx xxx C −++
Difficulty: hard Section: 6.1
20. Evaluate ( )2 4 ln xxdx
Ans: ( ) 555 2 22 lnln 525125 xxxxxC −++
Difficulty: hard Section: 6.1
21. Evaluate 2 3 x xedx .
Ans: ( ) 2 2 1 1 2 x xeC −+
Difficulty: hard Section: 6.1
22. Evaluate 4 7 x xedx .
Ans: 4 4 (1) 4
x e xC −+
Difficulty: hard Section: 6.1
23. Evaluate 25 x dx x + .
Ans: ( ) 5ln25 24 x xC−++
Difficulty: hard Section: 6.1
Chapter 6 Page 108
24. Evaluate 73 x dx x + .
Ans: 3 749ln|73| x xC−++
Difficulty: hard Section: 6.1
25. Use the formula 1 (ln)(ln)(ln) nnn xdxxxnxdx =−
Ans: ( ) ( ) 32 ln3ln6ln6 xxxxxxxC −+−+
Difficulty: moderate Section: 6.1
26. True or False: ( ) ( ) 22 lnln2ln2 xdxxxxxxC =−++
A) True B) False
Ans: A Difficulty: hard Section: 6.1
to evaluate 3(ln)xdx
.
27. A manufacturer has found that marginal cost is 0.02 (21) q qe + dollars per unit when q units have been produced. The total cost of producing the first 10 units is $100. What is the total cost of producing the first 30 units?
A) $1,124.68 B) $545.39 C) $300.00 D) $635.39
Ans: D Difficulty: hard Section: 6.1
28. After t hours on the job, a factory worker can produce 0.5 70 t te units per hour. How many units does the worker produce during the first 2 hours? Round to two decimal places.
A) 206.01 units B) 73.99 units C) 486.01 units D) 125.49 units
Ans: B Difficulty: moderate Section: 6.1
29. True or False: If, after t hours on the job, a factory worker can produce 0.02 10 t te units per hour, then the worker produces 130 units during the first 5 hours.
A) True B) False
Ans: B Difficulty: hard Section: 6.1
30. From time t = 0 to t = 3 an object's speed is given by the function 3 () t stte = . Compute the distance travelled by the object during this time interval. Round your answer to two decimals.
A) 7,213.11 B) 7,198.98 C) 7,304.18 D) 7,202.85
Ans: D Difficulty: moderate Section: 6.1
31. An object moving in a straight line has velocity () t vtte = meters per second. Is it true that in the first 4 seconds the object will have travelled 2 412 e + meters?
A) True B) False
Ans: A Difficulty: hard Section: 6.1
Chapter 6 Page 109
32. True or False: After t seconds, an object is moving at the speed of 1 t te meters per second. If the object begins at 0 when t = 0, then the distance the object travels as a function of time is expressed 1 ()(1) t stet=−+ .
A) True B) False
Ans: A Difficulty: hard Section: 6.1
33. After t weeks, a charity is raising money at the rate of 5,000 t ln(t + 1) dollars per week. How much money is raised during the first 10 weeks? Round to the nearest ten dollars.
A) $493,000 B) $493,480 C) $493,520 D) $493,550
Ans: B Difficulty: hard Section: 6.1
34. It is projected that t years from now the population of a city will be changing at the rate of 20.01t te thousand people per year. If the current population is 1 million, what will the population be 4 years from now?
A) 1,028,000 B) 1,000,022 C) 1,028,800 D) 1,021,984
Ans: D Difficulty: hard Section: 6.1
35. Given an initial population, 0 19,000 P = , a renewal rate, R = 200, and a survival function, 0.3 () t Stte = , with time t measured in years, determine the population at the end of 20 years. Round to two decimal places.
A) 1,358.95 B) 138.00 C) 9,814.35 D) 3,125.59
Ans: D Difficulty: moderate Section: 6.1
36. Find the average value of the function /6 () x fxxe = over the interval from 0 to 5. Round to two decimal places.
A) 11.65 B) 22.98 C) 1.46 D) 7.32
Ans: C Difficulty: moderate Section: 6.1
37. Evaluate
A) 0 B) 1 C) 1 3 D) Diverges
Ans: D Difficulty: hard Section: 6.2
38. Evaluate
1 dx
A) Diverges B) 0 C) 6 5 D) 5 6
Ans: A Difficulty: hard Section: 6.2
Chapter 6 Page 110
3 1 dx x
6 1
x
Chapter 6 Page 111 39. Evaluate 2 5 1 dx xx . 2 111 Hint: 1 xxxx =− Ans: 5 ln 4 Difficulty: hard Section: 6.2 40. Evaluate 2 3 (1) dx xx . Ans: Diverges Difficulty: hard Section: 6.2 41. Evaluate 5 1 x edx . A) 5 1 5e B) 1 5e C) 5 e D) 5 5e Ans: A Difficulty: moderate Section: 6.2 42. Given that 2 0 π 2 x edx = , evaluate 2 9 0 x edx . Ans: π 6 Difficulty: hard Section: 6.2 43. Evaluate 3 1 x edx . A) 3 1 3e B) 1 3e C) e3 D) 3 3e Ans: A Difficulty: hard Section: 6.2 44. True or False: 3 0 11 (1)8 dx x = + A) True B) False Ans: B Difficulty: hard Section: 6.2 45. Evaluate the improper integral: 2 32 2 (9) x dx x + A) 51 B) 1 34 C) 1 53 D) 1 51 Ans: D Difficulty: moderate Section: 6.2
47.
True or
A) True B) False Ans: B Difficulty: hard Section: 6.2
.
A) 0 B) 1 2ln2 C) 1 2ln2 D) Diverges Ans: D Difficulty: hard Section: 6.2
49. Evaluate the improper integral: 10
1 (3)ln(3) dx xx
Round your answer to two decimal places, if necessary.
A) 0.67 B) 1.95 C) 2.30 D) Undefined Ans: D Difficulty: moderate Section: 6.2
50. Evaluate the improper integral: 4 3/5 0 x xedx
Round to two decimal places, if necessary.
A) 1.88 B) 2.5 C) 1.25 D) Undefined Ans: C Difficulty: moderate Section: 6.2
A) True B) False Ans: A Difficulty: hard Section: 6.2
A) True B) False Ans: A Difficulty: hard Section: 6.2
Chapter 6 Page 112
46.
False: 2 0 2 0 1 x dx x = +
2 1 2(ln2)
1 2ln2 D) 2 1 2(ln2) Ans:
Evaluate 3 2 1 (ln) dx xx
. A) 1 2ln2 B)
C)
B Difficulty: hard Section: 6.2
48. Evaluate 2 1 ln dx xx
3 2 0 1 3 x x
e =
51. True or False:
dx
3 2 0 1 3 x x
e =
52. True or False:
dx
53. A certain nuclear power plant produces radioactive waste at the rate of 500 pounds per year. The waste decays exponentially at the rate of 1.5% per year. How many pounds of radioactive waste from the plant will be present in the long run? Round to two decimal places, if necessary.
Ans: 33,333.33 lbs
Difficulty: hard Section: 6.2
54. Demographic studies conducted in a certain city indicate that the fraction of the residents that will remain in the city for at least t years is /50 () t fte = . The current population of the city is 500,000, and it is estimated that new residents will be arriving at the rate of 1,000 people per year. If this estimate is correct, what will be the population in the long run?
A) 50,000 B) 40,500 C) 40,800 D) 550,000
Ans: A Difficulty: hard Section: 6.2
55. A hospital patient receives 4 units of a certain drug per hour intravenously. The drug is eliminated exponentially, so that the fraction that remains in the patients body for t hours is /12 () t fte = . If treatment is continued indefinitely, approximately how many units of the drug will be in the patient's body in the long run?
A) 44 units B) 48 units C) 52 units D) 56 units
Ans: B Difficulty: hard Section: 6.2
56. A hospital patient receives intravenously 5 units of a certain drug per hour. The drug is eliminated exponentially, so that the fraction that remains in the patients body for t hours is /14 () t fte = . If the treatment is continued indefinitely, approximately how many units of the drug will be in the patient's body in the long run? Round to two decimal places, if necessary.
A) 19 units B) 65 units C) 70 units D) 0.36 units
Ans: C Difficulty: moderate Section: 6.2
57. The long run capitalized cost of an asset that initially cost 0C dollars is given by 0 0 () rt
CCAtedt =+ where A(t) is the annual cost of maintenance and r is the annual rate of interest, compounded continuously. Find the long run capitalized cost, in dollars, in the case where 0 $5,000,000 C = , A(t) = 5, 000(1 + 3t), and r = 0.08.
Ans: 7,406,250
Difficulty: hard Section: 6.2
58. Approximate the integral 2 1/ 1 x edx using (a) the trapezoidal rule and (b) Simpson's rule, both with 6 subintervals. Round your answer to five decimal places.
Ans: (a) 2.02536
(b) 2.02019
Difficulty: hard Section: 6.3
Chapter 6 Page 113
59. Approximate the integral
using (a) the trapezoidal rule and (b) Simpson's rule, both with 4 subintervals. Round your answer to five decimal places.
Ans: (a) 1.49068
(b) 1.46371
edx
Difficulty: hard Section: 6.3
60. Determine how many subintervals are required to guarantee accuracy to within 0.00001 for the approximation of the integral
ln(1)xdx +
using (a) the trapezoidal rule and (b) Simpson's rule.
Ans: (a) 46
(b) 7
Difficulty: hard Section: 6.3
61. Assume a 9-year franchise is expected to generate profit at the rate of 2 3 13,000 t + dollars per year. If, over the next 9 years, the prevailing annual interest rate remains fixed at 6%, compounded continuously, what is the present value of the franchise? Use Simpson's rule with n = 6 to approximate the integral. Round your answer to two decimal places.
A) $473.20 B) $283.92 C) $946.41 D) $315.47
Ans: D Difficulty: hard Section: 6.3
62. Shortly after leaving on a road trip, two math majors realize that the car's odometer is broken. To estimate the distance they travel between 8 PM and 9 PM, they record speedometer readings every 10 minutes:
Time8:008:108:208:308:408:509:00
Speed(mph)58704065307960
Using Simpson's rule and only the information in the table, get the best possible estimate of the distance they traveled between 8 PM and 9 PM. Round your answer to one decimal place.
Ans: 61.9 miles
Difficulty: easy Section: 6.3
Chapter 6 Page 114
2 1 0 x
2 2 1