$35.00

Ch06 Techniques of Integration

1. Identify u and dv for finding the integral using integration by parts.

A)

B)

C)

D)

E)

Ans: A

2. Identify u and dv for finding the integral using integration by parts.

dx

A) dx; dx

B)

C)

D)

E)

Ans: B

3. Use integration by parts to evaluate

A)

B)

C)

D)

E)

Ans: A

4. Find the indefinite integral.

A)

B)

C)

D)

E)

Ans: E

5. Use integration by parts to evaluate . Note that evaluation may require integration by parts more than once.

A)

B)

C)

D)

E)

Ans: A

6. Use integration by parts to find the integral below.

A)

B)

C)

D)

E)

Ans: D

7. Find the indefinite integral.

A)

B)

C)

D)

E)

Ans: A

8. Use integration by parts to evaluate .

A)

B)

C)

D)

E)

Ans: E

9. Find the indefinite integral.

A)

B)

C)

D)

E)

Ans: B

10. Find the indefinite integral.

A)

B)

C)

D)

E)

Ans: E

11. Find the indefinite integral.

A)

B)

C)

D)

E)

Ans: C

12. Find the indefinite integral.

A)

B)

C)

D)

E) none of the above

Ans: B

13. Evaluate the definite integral . Round your answer to three decimal places.

A) 828.111

B) 207.823

C) 118.290

D) 207.811

E) 79.245

Ans: C

14. Find the definite integral.

A)

B)

C)

D)

E) none of the above

Ans: B

15. Use integration by parts to find the integral below.

A)

B)

C)

D)

E)

Ans: D

16. A model for the ability M of a child to memorize, measured on a scale from 0 to 10, is where t is the child’s age in years. Find the average value predicted by the model for a child’s ability to memorize between first and second birthdays. Round your answer to three decimal places.

A) 3.318

B) 2.218

C) 4.118

D) 1.318

E) 2.018

Ans: E

17. Present Value of a Continuous Stream of Income. An electronics company generates a continuous stream of income of million dollars per year, where t is the number of years that the company has been in operation. Find the present value of this stream of income over the first 9 years at a continuous interest rate of 12%. Round answer to one decimal place.

A) $143.7 million

B) $81.6 million

C) $182.7 million

D) $343.2 million

E) $85.8 million

Ans: B

18. Use a table of integrals to find the indefinite integral .

A)

B)

C)

D)

E)

Ans: A

19. Use a table of integrals with forms involving to find the integral.

A)

B)

C)

D)

E)

Ans: E

20. Find the integral below using an integral table.

A)

B)

C)

D)

E)

Ans: E

21. Use a table of integrals with forms involving a + bu to find

A)

B)

C)

D)

E)

Ans: C

22. Use a table of integrals to find the indefinite integral .

A)

B)

C)

D)

E)

Ans: B

23. Use a table of integrals with forms involving to find

A)

B)

C)

D)

E)

Ans: A

24. Use a table of integrals to find the indefinite integral .

A)

B)

C)

D)

E)

Ans: C

25. Use a table of integrals to find the indefinite integral .

A)

B)

C)

D)

E)

Ans: A

26. Use the Trapezoidal Rule and Simpson’s Rule to approximate the value of the definite integral for the indicated value of n. Compare these results with the exact value of the definite integral. Round your answers to four decimal places.

A) a. Exact:

b. Trapezoidal Rule:

c. Simpson’s Rule:

B) a. Exact:

b. Trapezoidal Rule:

c. Simpson’s Rule:

C) a. Exact:

b. Trapezoidal Rule:

c. Simpson’s Rule:

D) a. Exact:

b. Trapezoidal Rule:

c. Simpson’s Rule:

E) a. Exact:

b. Trapezoidal Rule:

c. Simpson’s Rule:

Ans: E

27. Approximate the value of the definite integral using (a) the Trapezoidal Rule and (b) Simpson’s Rule for the indicated value of n. Round your answers to three significant digits.

A) a. Trapezoidal Rule:

b. Simpson’s Rule:

B) a. Trapezoidal Rule:

b. Simpson’s Rule:

C) a. Trapezoidal Rule:

b. Simpson’s Rule:

D) a. Trapezoidal Rule:

b. Simpson’s Rule:

E) a. Trapezoidal Rule:

b. Simpson’s Rule:

Ans: B

28. The rate of change in the number of subscribers to a newly introduced magazine is modeled by where is the time in years. Use Simpson’s Rule with to estimate the total increase in the number of subscribers during the first 6 years.

A) 1870 subscribers

B) 1780 subscribers

C) 1800 subscribers

D) 1878 subscribers

E) 1987 subscribers

Ans: D

29. A body assimilates a 12-hour cold tablet at a rate modeled by where is measured in milligrams per hour and is the time in hours. Use Simpson’s Rule with to estimate the total amount of the drug absorbed into the body during the 12 hours.

A) 58.915 mg

B) 68.915 mg

C) 38.915 mg

D) 48.915 mg

E) 78.915 mg

Ans: A

30. Evaluate the definite integral . Round your answer to three decimal places.

A) 20.210

B) 16.873

C) 32.580

D) 26.395

E) 4.504

Ans: E

31. Evaluate the definite integral dx. Round your answer to three decimal places.

A) 6.144

B) 7.207

C) 4.541

D) 5.644

E) 4.881

Ans: D

32. The probability of recall in an experiment is modeled by where x is the percent of recall. What is the probability of recalling between 60% and 90%? Round your answer to three decimal places.

A) 0.432

B) 0.270

C) 0.710

D) 0.219

E) 0.936

Ans: A

33. Use the table of integrals to find the average value of the growth function over the interval , where N the size of a population and t is the time in days. Round your answer to three decimal places.

A) 248.346

B) 1057.983

C) 246.346

D) 680.477

E) 682.451

Ans: C

34. The revenue (in dollars per year) for a new product is modeled by where t the time in years. Estimate the total revenue from sales of the product over its first 3 years on the market. Round your answer to nearest dollar

A) $6579

B) $3291

C) $10,821

D) $15,830

E) $1138

Ans: B

35. Approximate the definite integral “by hand,” using the Trapezoidal Rule with trapezoids. Round answer to three decimal places.

A) 11.425

B) 11.381

C) 5.691

D) 8.569

E) 15.175

Ans: D

36. Use the Trapezoidal Rule to approximate the value of the definite integral . Round your answer to three decimal places.

A) 6.7643

B) 2.7931

C) 2.7955

D) 4.6552

E) 4.6615

Ans: C

37. Decide whether the integral is proper or improper.

A) The integral is improper.

B) The integral is proper.

Ans: B

38. Determine the amount of money required to set up a charitable endowment that pays the amount each year indefinitely for the annual interest rate compounded continuously.

A) $210,000

B) $200,000

C) $220,000

D) $240,000

E) $230,000

Ans: B

39. The capitalized cost of an asset is given by where is the original investment, is the time in years, is the annual interest rate compounded continuously, and is the annual cost of maintenance (in dollars). Find the capitalized cost of an asset (a) for 5 years, (b) for 10 years, and (c) forever.

A) a. For $253,901.30

b. For $807,922.43

c. For $4,466,666.67

B) a. For $453,901.30

b. For $807,922.43

c. For $1,466,666.67

C) a. For $453,901.30

b. For $2807,922.43

c. For $4,466,666.67

D) a. For $453,901.30

b. For $807,922.43

c. For $4,466,666.67

E) a. For $453,901.30

b. For $807,922.43

c. For $466,666.67

Ans: D

40. Approximate the integral using Simpson’s Rule: , n = 6. Round your answer to three decimal places.

A) 1.161

B) 1.284

C) 0.850

D) 0.652

E) 1.017

Ans: D

41. Use Simpson’s Rule to approximate the revenue for the marginal revenue function with n = 4. Assume that the number of units sold, x, increases from 12 to 16. Round your answer to one decimal place.

A) $1602.40

B) $678.36

C) $1346.14

D) $1439.03

E) $1230.54

Ans: D

42. Use the error formulas to find n such that the error in the approximation of the definite integral is less than 0.0001 using the Trapezoidal Rule.

A) 25

B) 26

C) 24

D) 22

E) 23

Ans: E

43. A body assimilates a 16-hour cold tablet at a rate modeled by , where t is measured in milligrams per hour and t is the time in hours. Use Simpson’s Rule with n = 16 to estimate the total amount of the drug absorbed into the body during the 16 hours.

A) 58.88

B) 34.88

C) 54.33

D) 46.88

E) 64.90

Ans: C

44. Decide whether the following integral is improper.

A) no

B) yes

Ans: B

45. Evaluate the improper integral if it converges, or state that it diverges.

A)

B)

C)

D)

E) diverges

Ans: D

46. Evaluate the improper integral if it converges, or state that it diverges.

A)

B)

C)

D)

E) diverges

Ans: E

47. Determine whether the improper integral diverges or converges. Evaluate the integral if it converges.

A) converges to 0

B) converges to 3

C) converges to

D) diverges to

E) diverges to

Ans: C

48. Suppose the mean height of American women between the ages of 30 and 39 is 68.5 inches, and the standard deviation is 2.7 inches. Use a symbolic integration utility to approximate the probability that a 30-to 39-year-old woman chosen at random is between 5 and 6 feet tall.

A) 0.1772

B) 0.9017

C) 0.5707

D) 0.8547

E) 0.4257

Ans: B

49. A business is expected to yield a continuous flow of profit at the rate of $1,000,000 per year. If money will earn interest at the nominal rate of 8% per year compounded continuously, what is the present value of the business forever?

A) $12,600,000

B) $12,510,000

C) $12,500,000

D) $1,250,000

E) $1,350,000

Ans: C

50. Find the capitalized cost C of an asset forever. The capitalized cost is given by where is the original investment, t is the time in years, r = 12% is the annual interest rate compounded continuously, n is the total time in years over which the asset is capitalized, and is the annual cost of maintenance (measured in dollars). Round your answer to two decimal places.

A) $1,525,000.00

B) $1,275,000.00

C) $1,108,333.33

D) $1,299,218.75

E) $1,247,222.22

Ans: E

Ch07 Functions of Several Variables

1. Find the coordinates of the point that is located two units behind of the yz-plane, seven units to the left of the xz-plane, and four units above of the xy-plane.

A)

B)

C)

D)

E)

Ans: A

2. Find the the distance between the two points and .

A) 1 units

B) 11 units

C) units

D) 3 units

E) 5 units

Ans: C

3. Find if the midpoint of the line segment joining the two points and is .

A)

B)

C)

D)

E)

Ans: E

4. Find the lengths of the sides of the triangle with the given vertices, and determine whether the triangle is a right triangle, an isosceles triangle, or neither.

A) ; obtuse triangle

B) ; right triangle

C) ; right triangle

D) ; acute triangle

E) ; acute triangle

Ans: B

5. Find the center and radius of the sphere.

A) Center:

Radius:

B) Center:

Radius:

C) Center:

Radius:

D) Center:

Radius:

E) Center:

Radius:

Ans: A

6. Find the standard equation of the sphere whose center is and whose radius is 4.

A)

B)

C)

D)

E)

Ans: A

7. Find the equation of the sphere that has the points and as endpoints of a diameter.

A)

B)

C)

D)

E)

Ans: D

8. Find the center and radius of the sphere whose equation is . Round your answer to two decimal places, where applicable.

A) center: ; radius: 2.89

B) center: ; radius: 2.89

C) center: ; radius: 8.35

D) center: ; radius: 2.89

E) center: ; radius: 8.35

Ans: B

9. Sketch the yz-trace of the equation:

A)

B)

C)

D)

E)

Ans: B

10. Sketch the trace of the intersection of plane z = 4 with the sphere:

.

A)

B)

C)

D)

E)

Ans: D

11. Find the intercepts of the plane given by .

A) The -intercept is .

The -intercept is .

The -intercept is .

B) The -intercept is .

The -intercept is .

The -intercept is .

C) The -intercept is .

The -intercept is .

The -intercept is .

D) The -intercept is .

The -intercept is .

The -intercept is .

E) The -intercept is .

The -intercept is .

The -intercept is .

Ans: A

12. Sketch the graph of the plane given by y 5.

A)

B)

C)

D)

E)

Ans: B

13. Find the intercepts of the plane given by .

A) The -intercept is .

The -intercept is .

B) The -intercept is .

The -intercept is .

C) The -intercept is .

The -intercept is .

D) The -intercept is .

The -intercept is .

E) The -intercept is .

The -intercept is .

Ans: E

14. The two planes and are perpendicular.

A) false

B) true

Ans: A

15. The two planes and are parallel.

A) true

B) false

Ans: B

16. Describe the trace of the surface given by the function below in the xy-plane.

A) circle

B) ellipse

C) parabola

D) hyperbola

E) line

Ans: B

17. Describe the trace of the surface given by the function below in the xz-plane.

A) circle

B) parabola

C) line

D) ellipse

E) hyperbola

Ans: E

18. Identify the quadric surface.

A) The graph is a hyperboloid of one sheet.

B) The graph is hyperboloid of two sheets.

C) The graph is an elliptic cone.

D) The graph is an elliptic paraboloid.

E) The graph is an ellipsoid.

Ans: B

19. Identify the quadric surface.

A) The graph is an elliptic cone.

B) The graph is hyperboloid of two sheets.

C) The graph is a hyperboloid of one sheet.

D) The graph is an ellipsoid.

E) The graph is an elliptic paraboloid.

Ans: D

20. Because of the forces caused by its rotation, a planet is actually an oblate ellipsoid rather than a sphere. The equatorial radius is 3963 miles and the polar radius is 3953 miles. Find an equation of the ellipsoid. Assume that the center of a planet is at the origin and the xy- trace corresponds to the equator.

A)

B)

C)

D)

E)

Ans: C

21. Use the function to find

A)

B)

C)

D)

E)

Ans: C

22. Evaluate the function at the given values of the independent variables.

A)

B)

C)

D)

E)

Ans: A

23. Use the function to find

A)

B)

C)

D)

E)

Ans: E

24. Evaluate the function at

A)

B)

C)

D)

E)

Ans: E

25. Find the domain and range of the function.

A) Domain: all point such that

Range:

B) Domain: all point such that

Range:

C) Domain: all point such that

Range:

D) Domain: all point such that

Range:

E) Domain: all point such that

Range:

Ans: B

26. A manufacturer estimates the Cobb-Douglas production function to be given by

.

Estimate the production levels when and .

A) 135,540 units

B) 122,560 units

C) 131,601 units

D) 145,330 units

E) 112,745 units

Ans: A

27. Describe the domain and range of the function.

A) domain: The disk

range: The interval (0,4)

B) domain: The disk

range: The interval [0,4]

C) domain: The disk

range: The interval [0,4)

D) domain: The disk

range: The interval [0,4]

E) domain: The disk

range: The interval [0,4)

Ans: D

28. Describe the level curves of the function. Sketch the level curves for the given c-values.

, c = 0, 2, 4, 6

A)

B)

C)

D)

E)

Ans: B

29. Sketch the level curves for the function below for the given values .

A)

B)

C)

D)

E)

Ans: E

30. Describe the level curves for the function for the c-values given by .

A)

B)

C)

D)

E)

Ans: B

31. If find and

A)

B)

C)

D)

E)

Ans: D

32. If find

A)

B)

C)

D)

E)

Ans: B

33. Evaluate and for the function at the point .

A) –25 and –50

B) –45 and –80

C) –25 and –80

D) 100 and –30

E) –25 and 70

Ans: C

34. Evaluate and for the function at the point . Round your answer to two decimal places.

A) 4.03 and 1.19

B) 3.53 and 1.19

C) 3.88 and 5.82

D) 4.03 and 0.69

E) 3.88 and 5.82

Ans: A

35. Find the first partial derivatives with respect to x, y, and z.

A)

B)

C)

D)

E)

Ans: D

36. For , find all values of x and y such that and simultaneously.

A)

B)

C) (0,0)

D) Both B and C

E) Both A and B

Ans: D

37. Find the slopes of the surface in the x- and y- directions at the point .

A) slope in x-direction: 2

slope in y-direction: –17

B) slope in x-direction: –6

slope in y-direction: –8

C) slope in x-direction: –17

slope in y-direction: 2

D) slope in x-direction: –14

slope in y-direction: –14

E) slope in x-direction: –8

slope in y-direction: –6

Ans: B

38. Find the four second partial derivatives. Observe that the second mixed partials are equal.

A)

B)

C)

D)

E)

Ans: C

39. A company manufactures two types of wood-burning stoves: a freestanding model and a fireplace-insert model. The cost function for producing x freestanding and y fireplace-insert stoves is . Find the marginal costs ( and ) when and . Round your answers to two decimal places.

A)

B)

C)

D)

E)

Ans: D

40. The value of an investment of $18,000 after t years in an account for which the interest rate 100r% is compounded continuously is given by the function dollars. Write the partial derivative

A)

B)

C)

D)

E)

Ans: A

41. The utility function is a measure of utility (or satisfaction) derived by a person from the consumption of two products x and y. Suppose the utility function is . Determine the marginal utility of product x.

A)

B)

C)

D)

E)

Ans: B

42. Test for relative extrema and saddle points.

A) saddle point at

B) saddle point at

C) saddle point at

D) relative minimum at

E) relative minimum at

Ans: B

43. Examine the function for relative extrema.

A) relative maximum at

B) saddle point at

C) relative maximum at

D) relative minimum at

E) relative maximum at

Ans: A

44. Examine the function for relative extrema.

A) relative at

B) relative at

C) relative at

D) relative at

E) no relative extrema

Ans: A

45. Examine the function for relative extrema and saddle points.

A) saddle point: ; relative minimum:

B) relative minimum: ; relative maximum:

C) saddle points: ,

D) saddle point: ; relative minimum:

E) relative minimum: ; relative maximum:

Ans: A

46. Examine the function given below for relative extrema and saddle points.

A) The function has a relative maximum at .

B) The function has a relative minimum at .

C) The function has a saddle point at .

D) The function has a relative maximum at .

E) The function has a relative minimum at .

Ans: A

47. Examine the function given below for relative extrema and saddle points.

A) The function has a saddle point at .

B) The function has a relative maximum at .

C) The function has a relative minimum at .

D) The function has a saddle point at .

E) The function has a relative maximum at .

Ans: A

48. Determine whether there is a relative maximum, a relative minimum, a saddle point, or insufficient information to determine the nature of the function at the critical point .

Given:

A) at

B) at

C) at

D) at

Ans: B

49. Find the critical points of the function , and, from the form of the function, determine whether a relative maximum or a relative minimum occurs at each point.

A) relative minimum at

B) relative maximum at

C) relative minimum at

D) relative maximum at

E) no relative extrema

Ans: C

50. Find the critical points of the function , and, from the form of the function, determine whether a relative maximum or a relative minimum occurs at each point.

A) relative minima at , , , where and are arbitrary real numbers

B) relative maxima at , , , where and are arbitrary real numbers

C) relative minimum at

D) relative maximum at

E) no relative extrema

Ans: B

51. Find three positive numbers x, y, and z whose sum is 15 and product is a maximum.

A) x = 2.5, y = 2.5, z = 10

B) x = y = z = 5

C) x = 3.75, y = 3.75, z = 7.5

D) x = 2, y = 3, z = 5

E) x = 1, y = 6, z = 8

Ans: B

52. Find three positive numbers x, y, and z whose sum is 33 and the sum of the squares is a maximum.

A) x = y = z = 11

B) x = 5.5, y = 5.5, z = 22

C) x = 8.25, y = 8.25, z = 16.5

D) x = 4.4, y = 6.6, z = 11

E) x = 2.2, y = 13.2, z = 17.6

Ans: A

53. The sum of the length (denote by z) and the girth (perimeter of a cross section) of packages carried by a delivery service cannot exceed 72 inches. Find the dimensions of the rectangular package of largest volume that may be sent.

A) x = 9, y = 9, z = 24

B) x = 6, y = 6, z = 48

C) x = 12 , y = 12 , z = 24

D) x = 4.8, y = 7.2, z = 48

E) x = 2.4, y = 14.4, z = 38.4

Ans: C

54. A company manufactures two types of sneakers: running shoes and basketball shoes. The total revenue from x1 units of running shoes and y1 units of basketball shoes is:

,

where x1 and x2 are in thousands of units. Find x1 and x2 so as to maximize the revenue.

A)

B)

C)

D)

E)

Ans: A

55. Use Lagrange multipliers to maximize the function subject to the following constraint:

Assume that x, y, and z are positive.

A)

B)

C)

D)

E) no absolute maximum

Ans: C

56. Use Lagrange multipliers to find the given extremum. In each case, assume that and are positive.

Maximize

Constraint

A)

B)

C)

D)

E)

Ans: B

57. Use Lagrange multipliers to find the given extremum. Assume that and are positive.

Minimize

Constraint

A)

B)

C)

D)

E)

Ans: A

58. Use Lagrange multipliers to find the given extremum. In each case, assume that and are positive.

Maximize

Constraints

A)

B)

C)

D)

E)

Ans: C

59. A rectangular box is resting on the -plane with one vertex at the origin. The opposite lies in the plane Find the dimensions that maximize the volume.

(Hint: Maximize subject to the constraint ).

A) 15 units 12 units 5 units

B) 11 units 9 units 5 units

C) 10 units 9 units 6 units

D) 15 units 10 units 6 units

E) 12 units 11 units 7 units

Ans: D

60. A microbiologist must prepare a culture medium in which to grow a certain type of bacteria. The percent of salt contained in this medium is given by

where and are the nutrient solutions to be mixed in the medium. For the bacteria to grow, the medium must be 13% salt. Nutrient solutions and

cost $1, $2, and $3 per liter, respectively. How much of each nutrient solution should be used to minimize the cost of the culture medium?

A)

B)

C)

D)

E)

Ans: A

61. Use Lagrange multipliers to minimize the function subject to the following constraint.

Assume that x, y, and z are positive.

A) 81

B) 243

C) 162

D) 486

E) 729

Ans: B

62. Use Lagrange multipliers to find the minimum distance from the circle to the point Round your answer to the nearest tenth.

A) 13.6

B) 107.0

C) 184.4

D) 7.8

E) 60.4

Ans: D

63. A manufacturer has an order for 800 units of fine paper that can be produced at two locations. Let and be the numbers of units produced at the two plants. Find the number of units that should be produced at each plant to minimize the cost if the cost function is given by .

A) units and units

B) units and units

C) units and units

D) units and units

E) units and units

Ans: C

64. A rancher plans to use an existing stone wall and the side of a barn as a boundary for two adjacent rectangular corrals. Fencing for the perimeter costs 25 per foot. To separate the corrals, a fence that costs 20 per foot will divide the region. The total area of the two corrals is to be square feet. Use Lagrange multipliers to find the dimensions that will minimize the cost of the fencing.

A) dimensions: feet by feet

B) dimensions: feet by feet

C) dimensions: feet by feet

D) dimensions: feet by feet

E) dimensions: feet by feet

Ans: A

65. Find the sum of the squared errors for the linear model and the quadratic model using the given points.

A) ;

B) ;

C) ;

D) ;

E) ;

Ans: B

66. Find the least squares regression line for the given points. Then plot the points and sketch the regression line.

A)

B)

C)

D)

E)

Ans: A

67. Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the given points.

A)

B)

C)

D)

E)

Ans: D

68. An agronomist used four test plots to determine the relationship between the wheat yield (in bushels per acre) and the amount of fertilizer (in pounds per acre). The results are shown in the table.

(a) Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the data.

(b) Estimate the yield for a fertilizer application of 160 pounds per acre.

A)

B)

C)

D)

E)

Ans: C

69. Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

Data that are modeled by have a positive correlation.

A) True

B) False; The data modeled by have a positive correlation.

Ans: B

70. Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

When the correlation coefficient is , the model is a good fit.

A) False

B) True

Ans: B

71. Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

A linear regression model with a positive correlation will have a slope that is greater than 0.

A) True

B) False

Ans: A

72. Find the least squares regression line for the points (1,0) , (3,3) , (8,6).

A)

B)

C)

D)

E) none of the above

Ans: A

73. Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the points

. Round your answer to three decimal places.

A)

B)

C)

D)

E)

Ans: C

74. A store manager wants to know the demand y for an energy bar as a function of price x. The daily sales for three different prices of the energy bar are shown in the table.

Price, x $ 1.00 $ 1.25 $ 1.54

Demand, y 450 335 300

(i) Use the regression capabilities of a graphing utility to find the least squares regression line for the data.

(ii) Use the model to estimate the demand when the price is $1.39.

A) (i) ; (ii) 357.97832

B) (i) ; (ii) –436.133926

C) (i) ; (ii) 525.880529

D) (i) ; (ii) –436.133926

E) none of the above

Ans: E

75. Evaluate the following integral.

A)

B)

C)

D)

E) none of the above

Ans: A

76. Evaluate the double integral .

A) 92.00

B) 102.00

C) 112.00

D) 29.50

E) 17.00

Ans: B

77. Evaluate the double integral . Round your answer to two decimal places, where applicable.

A) 48.50

B) 68.50

C) 49.50

D) 58.50

E) 24.00

Ans: D

78. Evaluate the double integral . Round your answer to two decimal places, where applicable.

A) 69.75

B) 184.25

C) 192.25

D) 191.25

E) 184.75

Ans: D

79. Evaluate the double integral .

A)

B)

C)

D)

E)

Ans: E

80. Evaluate the double integral .

A)

B)

C)

D)

E)

Ans: C

81. Sketch the region whose area is given by the following double integral.

A)

B)

C)

D)

E)

Ans: D

82. Use a double integral to find the area of the region bounded by the graphs of and .

A)

B)

C)

D)

E)

Ans: E

83. Use a double integral to find the area of the region bounded by the graphs of and .

A)

B)

C)

D)

E)

Ans: A

84. Use a symbolic integration utility to evaluate the double integral.

A) 8.1747

B) 9.1211

C) 6.2031

D) 7.88.7522

E) 9.4362

Ans: A

85. Sketch the region of integration .

A)

B)

C)

D)

E)

Ans: C

86. Set up an integral for both orders of integration, and use the more convenient order to evaluate the integral below over the region R.

A)

B)

C)

D)

E)

Ans: D

87. Use a double integral to find the volume of the indicated solid.

A)

B)

C)

D)

E) none of the above

Ans: C

88. Use a double integral to find the volume of the solid bounded by the graphs of the equations.

A)

B)

C)

D)

E)

Ans: D

89. A firm’s weekly profit (in dollars) in marketing two products is given by

where and represent the numbers of units of each product sold weekly. Estimate the average weekly profit when varies between 40 and 50 units and varies between 45 and 50 units.

A)

B)

C)

D)

E)

Ans: C

90. Use a double integral to find the volume of the solid bounded by the graphs of the equations .

A)

B)

C)

D)

E)

Ans: C

91. The population density (in people per square mile) for a coastal town can be modeled by where x and y are measured in miles. What is the population inside the rectangular area defined by the vertices and ? Round to the nearest integer.

A) 12,833 people

B) 32,500 people

C) 21,667 people

D) 11,833 people

E) 10,833 people

Ans: E

92. Find the average value of over the region R: square with vertices .

A)

B)

C)

D)

E)

Ans: A

93. A company sells two products whose demand functions are given by and . So, the total revenue is given by . Estimate the average revenue if the price varies between $45 and $70 and the price varies between 45 and 70.

A) $ 52,725

B) $ 54,875

C) $ 52,223

D) $ 53,740

E) $ 55,285

Ans: D

94. The Cobb-Douglas production function for an automobile manufacturer is where x is the number of units of labor and y is the number of units of capital. Estimate the average production level if the number of units of labor x varies between 250 and 300 and the number of units of capital y varies between 250 and 300.

A) 20.99

B) 21.10

C) 10.99

D) 31.44

E) 31.24

Ans: C

Only ** 0 ** units of this product remain