Test Bank For Applied Calculus for Business, Economics, and the Social and Life Sciences 11th Editio

Final Test

Test Bank For For Applied Calculus for Business, Economics, and the Social and Life Sciences 11th Edition By Laurence Hoffmann, Gerald Bradley, David Sobecki, Michael Price With Midter & Final MC Answers Are at the end of Each Chapter

Chapter 1 1. Compute the indicated value of the given function.  –2t  8 if t  1 ; f (–3) f (t )   2  t  9 if t  1 A) 18 B) 0 C) 2 D) 14 2. Determine the domain of the given function. x2  1 f ( x)  x3 A) all real numbers x except x = 1, x = –1, and x = –3 B) all real numbers x C) all real numbers x except x = –3 D) all real numbers x except x = 1 and x = 4 3. Find the difference quotient,

9 x f  x  h  f  x

f  x  h  f  x h

.

f  x 

A) B)

h f  x  h  f  x h

9 x  x  h

C)

9 xh

D)

 

f  x  h  f  x h f  x  h  f  x h

=9 =1

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Final Test

4. Find the indicated composite function.

1 f (x + 6) where f ( x)  x 1 A) f (x + 6) = x  6  x 1 B) f (x + 6) = x6

C)

f (x + 6) = 1 

D)

f (x + 6) =

3 x

1 6 x

5. Plot the given point in a rectangular coordinate system.

(1, –5) A) C) y

y x

x (1, -5)

(1, -5)

(Each gridline represents one unit.)

(Each gridline represents one unit.)

B)

D) (1, -5) y

y (1, -5)

x

(Each gridline represents one unit.)

x

(Each gridline represents one unit.)

6. Find the distance between the given points.

(–2, 6) and (3, 7) A) D = 6 B) D = 26 C) D  26 D) D  6

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Final Test

7. Sketch the graph of the given function. f ( x)  x 2  2

(Tick marks are spaced one unit apart.)

8. Find the points of intersection (if any) of the given pair of curves. y  x 2 and y = 3x – 2.

A) There are no points of intersection. B) (2, 4) C) (1, 1), (2, 4) D) (0,0) 9. Find the slope (if possible) of the line that passes through the given pair of points.

(28, 7) and (8, 3).

1 A) B) The slope is undefined. C) 5 D) 0 5 10. Find the slope and y-intercept (if they exist) of the line 8y = 7x.

7 and y-intercept is 0.

A) Slope is 7 and y-intercept is 8.

C) Slope is 8 8 and y-intercept is 0.

B) Slope is D) Slope is 7 and y-intercept is 0.

7 11. Write an equation for the line through (5, 4) and parallel to the x-axis.

A) y = 4 B) y = –4 C) x = –5 D) x = 5 12. The cost of renting a backhoe at one distributor is $325, plus $35 per day. Write a linear function C(x) that describes the cost of renting the backhoe for x days, then use your function to find how much it would cost to rent it for 12 days.

A) C ( x)  35 x  313; $733 C) C ( x)  12  325  35 x  ; $8,940 B)

C ( x)  325 x  35; $3,935

D)

C ( x)  325  35 x; $745

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Final Test

13. Each unit of a certain commodity costs p = 47x + 12 cents when x units of the commodity are produced. If all units are sold at this price, express the revenue derived from the sales as a function of x.

A) 46x + 12 cents C) 48x + 12 cents B) 47 x 2  12 cents D) x(47x + 12) cents 14. A closed cylindrical can has a surface area of 120 square inches. Express the volume of the can as a function of its radius, r.

A) V(r) = 60r cubic inches C) V (r )  π(60  r 3 ) cubic inches B) V (r )  120 πr 2 cubic inches D) V (r )  πr (60  r 2 ) cubic inches 15. A closed box with a square base is to have a volume of 10 cubic meters. The material for the top and bottom of the box costs $5 per square meter, and the material for the sides costs $1 per square meter. Express the construction cost of the box as a function of the length of its base, x.

40 2 dollars A) C ( x)  10 x 2  C) C ( x)  2 x 2   10 dollars x x 10 dollars B) C ( x)  x 2  40 x  2 dollars D) C ( x)  2 x 2  x 16. Two car rental agencies are competing. One agency rents cars for 50 dollars per day and 35 cents a mile; the other agency rents cars for 35 dollars per day and 45 cents a mile.

For a 10 day trip, how many miles must you travel to have the total cost be the same with each agency? Round to the nearest whole mile , if necessary.

A) 43 miles B) 1,500 miles C) 33 miles D) 150 miles 17. Find the indicated limit if it exists. lim  x  4  x 3

A) 3

B) Does not exist

C) –1

D) 7

18. Find the indicated limit if it exists. x 2  3x  2 lim x 1 x2  1 3 A) Does not exist B) 6 C) D) 3 2 19. Find lim f  x  and lim f  x  . If the limiting value is infinite indicate whether it is + x 

x 

or –. f (x) = (5 – 9x)(x – 2) A) –– B) –0 C) ++ D) +0

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Final Test

20. Find lim f  x  and lim f  x  . If the limiting value is infinite indicate whether it is + x 

x 

or –. f  x 

4  2 x3 6 x3 – 6 x + 1

1 1 A) – B) –+ C)  ,  3 3

1 1 D)  , 3 3

21. An efficiency consultant determines that when new workers are hired to wait tables at an upscale restaurant, the average number of tables they can wait on in a 6 hour shift is given by

5.3x 2  50 x  14 0.1x  0.8 where x is the number of shifts they've worked since being hired. What happens to an average waiter's productivity in the long run (as x   )?

A) It approaches 53 tables per shift.

C) It approaches 23 tables per shift.

B) It increases without bound.

D) It approaches 14 tables per shift.

N  x 

22. Find the indicated one-sided limit. If the limiting value is infinite, indicate whether it is + or –. x 2 lim x4 x4 1 1 A)  B) C) 2 D) –2 4 4 23. Find the indicated one-sided limit. If the limiting value is infinite, indicate whether it is + or –.  x  5 if x  2 lim f ( x) where f ( x)   2 x 2  x if x  2 A) There is none B) 7 C) 0 D) 4 24. List all values of x for which f (x) is not continuous.

2x  1 . f ( x)  2 x x 1 A)  B) –1 C) None D) 0 and –1 2 25. List all the values of x for which the given function is not continuous.  x3 if x  5 f ( x)   125 if x  5 A) x = 5 B) x = ±5 C) The function is continuous for all values of x.

D) x = 0

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Final Test

Answer Key 1. D 2. C 3. A 4. B 5. A 6. C 7. D 8. C 9. A 10. C 11. A 12. D 13. D 14. D 15. A 16. B 17. D 18. A 19. A 20. C 21. C 22. B 23. B 24. D 25. C

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Final Test

Chapter 2 1. The equation of the line tangent to the graph of f ( x)  x 2  2 x at x = 7 is A) y = 16x – 7 B) y = 16x – 49 C) y = 16x – 343 D) y = 16x – 686 2. The equation of the line tangent to the graph of f ( x)  3 x at x = 1 is 1 1 3 1 1 3 3 A) y  x  B) y  x  1 C) y  x  D) y  x  2 2 2 2 2 2 2 3. For f ( x)  

3

, find the average rate of change of f (x) with respect to x as x changes x from 144 to 145. Then use calculus to find the instantaneous rate of change at x = 144.

Round your answer to six decimal places, if necessary.

A) Average rate of change: 0.000864; Instantaneous rate of change: 0.000868 B) Average rate of change: –0.000864; Instantaneous rate of change: 0.125 C) Average rate of change: 0.000864; Instantaneous rate of change: –0.125 D) Average rate of change: –0.000864; Instantaneous rate of change: 0.000868

4. Differentiate: f ( x)  x  A) 7

B)

1 2 x



7 2 x

3

7 x

C) 0 D)

1 2 x



7 2 x3

5. Find the equation of the tangent line to the graph of f ( x)  x 2  3 at the point (3, 12).

A) y = 12 B) y = 6x – 6 C) x = 3 D) Not defined 6. Find the rate of change of the given function f (x) with respect for x for the prescribed value x = –2. f (x) = x3 + 3x + 9 A) 6 B) –3 C) 15 D) 24 7. An appliance store manager estimates that for x television ads run per day, R( x)  0.01x3  x 2  3x  200 refrigerators will be sold per month. Find R (4) and interpret what it tells us about sales.

A) R (4)  4.52; sales will be increasing at about 5 refrigerators per month per ad when they're running 4 ads.

B) R (4)  203.36; the cost of refrigerators will be rising by $203.36 if they're selling 4 per day.

C) R (4)  4.52; they'll sell about 5 refrigerators if they run 4 ads per day.

D) R (4)  203.36; they'll sell about 203 refrigerators if they run 4 ads per day.

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Final Test

8. An object moves along a line in such a way that its position at time t is s(t )  t 3  9t 2  15t  2 . Find the velocity and acceleration of the object at time t. When is the object stationary?

A) v(t )  3t 2  18t  15 ; a(t) = 6t – 18; t = 3 B) v(t )  3t 2  6t  15 ; a(t) = 6t – 6; t = 1 C) v(t )  3t 2  18t  15 ; a(t) = 6t – 18; t = 1 D) v(t )  3t 2  18t  15 ; a(t) = 6t – 18; t = 1 and 5 9. Differentiate: f ( x)  ( x 2  1)( x  6) A) x 2  1

B) 3x 2  12 x  1

C) 2x + 1

10. What is the rate of change of f (t )  A)

9 4

19 16

B) 4 C)

11. Differentiate: f ( x)  A) 2x

B) –x C)

7t  5 with respect to t when t = 2? t2

17 4

D)

x2 x6 3 x 2  12 x

 x  6

D) 12x + 1

2

x 2  12 x

D)

 x  6

2

12. Find f (4) ( x ) if f ( x)  x5  8x 4  7 x3  7 x 2  7 x  4 .

A) f (4) ( x)  120 x  192 C) f (4) ( x)  x2  8x  7 B) f (4) ( x)  x2  8x D) f (4) ( x)  60 x2  192 x  42 13. Find f ( x) if f ( x) 

1

15



A) B)

3x



3  7 x2

72 5 16 x 3 x x 15 72 f ( x)   3  5 8 x 3x x

f ( x)  

3

C) D)

72 5 72 x 3x x 3 3 f ( x)   3  3 8 x 3x x f ( x)  

5

3



14. An equation for the tangent line to the curve y  (7 x 2  x  1)5 at the point where x = 0 is A) y = 5x + 1 B) y = 5x – 1 C) y = 10x – 1 D) y = 10x + 1

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Final Test

15. Find all points on the graph of the function f ( x)  horizontal.

A) (0, 0) and (–4, –8)

B) There are none.

C) (0, 0) D) (2, 1)

16. True or False: If f ( x)  1  3 x 2 , then f "( x)  A) False

x2 where the tangent line is x2

3 .

(1  3 x 2 )3/2

B) True

17. When a certain commodity is sold for p dollars per unit, consumers will buy 31,500 units per month. It is estimated that t months from now, the price of the D( p)  p commodity will be p(t )  t 2/3  5.15 dollars per unit. The approximate rate at which the monthly demand will be changing with respect to time in 27 months is A) –32 units per month C) –132 units per month B) –35 units per month D) 35 units per month 18. True or False: The function f ( x) 

x  5 will decrease by approximately 0.6 as x 2x  1

decreases from 3 to 2.7.

A) False B) True 19. The largest percentage error you can allow in the measurement of the radius of a sphere if you want the error in the calculation of its surface area using the formula S  4πr 2 to be no greater than 6 percent is about: A) 2% B) 6% C) 3% D) 1% 20. You measure the side of a cube to be 14 centimeters long and conclude that the volume of the cube is 143  2, 744 cubic centimeters. If your measurement of the side is accurate to within 2%, approximately how accurate is your calculation of this volume?

Round to two decimal places, if necessary.

A) Maximum error in volume is about ±164.64 cm3 B) Maximum error in volume is about ±0.84 cm3 C) Maximum error in volume is about ±2,304.96 cm3 D) Maximum error in volume is about ±11.76 cm3 21. An efficiency study of the morning shift at a certain factory indicates that an average worker arriving on the job at 7:00 A.M. will have assembled f ( x)   x3  7 x 2  3x transistor radios x hours later. Approximately how many radios will the worker assemble between 9:00 and 9:30 A.M.?

A) Approximately 7 radios C) Approximately 14 radios B) Approximately 13 radios D) Approximately 390 radios

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Final Test

dy , where xy 3  3 x 2  7 y . dx 6 x  y3 6x 2 3 y  6 x A) B) C) y3 3xy 2  7

22. Find

D) y3  6 x  7

23. Find the equation of the tangent line to the given curve at the specified point. x5 y 5  5xy  6 x  y  8 ; (0, 8) 1 1 A) y   x  8 B) y = –46x + 8 C) y = 46x + 8 D) y  x8 46 46 d2y for 4 x 5  11 y  100 . dx 2 80 C) 60 x 2  100 D)  x3 11

24. Use implicit differentiation to find A) 60 x 2  11

B) 80x3

25. Suppose the output at a certain factory is Q  4 x3  5x3 y 4 + 3 y3 units, where x is the number of hours of skilled labor used and y is the number of hours of unskilled labor.

The current labor force consists of 20 hours of skilled labor and 10 hours of unskilled labor. Use calculus to estimate the change in unskilled labor y that should be made to offset a 1-hour increase in skilled labor x so that output will be maintained at its current level. Round you answer to two decimal places, if necessary.

A) –2.67 hours B) 2.67 hours C) –1 hours D) –0.38 hours

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Final Test

Answer Key 1. B 2. D 3. A 4. D 5. B 6. C 7. A 8. D 9. B 10. C 11. D 12. A 13. B 14. B 15. A 16. A 17. B 18. A 19. C 20. A 21. A 22. A 23. B 24. D 25. D

Chapter 3 1. Find the intervals of increase and decrease for f ( x)  6 x3  9 x 2  108x  3 .

A) Increasing on x < –3 and x > 2, decreasing on –3 < x < 2 B) Increasing on –3 < x < 2, decreasing on x < –3 and x > 2 C) Increasing on x < –2, decreasing on x > 3 D) Increasing on x  –2 and x  3 , decreasing on – 2  x  3 2. Determine the critical points of the given function and classify each critical point as a relative maximum, a relative minimum, or neither. f ( x)  3 x 4  8 x 3  6 x 2  3 A) (0, 2) neither; (1, 3) relative minimum B) (0, 2) relative minimum; (1, 3) relative maximum C) (0, 3) relative minimum; (1, 4) neither D) (0, 2) relative minimum; (1, 4) neither

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Final Test

3. Determine the critical numbers of the given function and classify each critical point as a relative maximum, a relative minimum, or neither. Round numbers to two decimal places, if necessary.

1 f ( x)  2 x  8x  7 A) x = 1, 4, 7; (1, 1) relative maximum; (4, –0.11) relative minimum; (7, 1) relative maximum B) x = 1, 7; (1, 1) relative maximum; (7, 1) relative maximum C) x = 1, 7; (1, 1) relative maximum; (7, 1) relative minimum D) x = 1, 4, 7; (4, –0.11) relative maximum 4. The revenue derived from the production of x units of a particular commodity is 80 x  x 2 R( x)  2 million dollars. What level of production results in maximum x  80 revenue? What is the maximum revenue? Round numbers to two decimal places, if necessary.

A) Maximum at x = 8 and maximum revenue is R(8) = 4 (million dollars) B) Maximum at x = 8 and maximum revenue is R(8) = 32 (million dollars) C) Maximum at x = 9 and maximum revenue is R(9) = 2.67 (million dollars) D) Maximum at x = 8 and maximum revenue is R(8) = 2.67 (thousand dollars) 5. Find constants a, b, and c so that the graph of the function f ( x)  ax  bx  c has a relative maximum at (7, 34) and crosses the y-axis at (0, 6).

4 4 A) a  , b = 8, c = 6 C) a  , b = –8, c = 6 7 7 4 4 B) a   , b = –8, c = –6 D) a   , b = 8, c = 6 7 7 2

6. Determine where the graph of f ( x)  x3  6 x 2  8x  4 is concave down.

A) For x > 2 B) For x < –2 C) For x > –2 D) For x < 2 7. Determine where the graph of f ( x)  x 4  54 x 2  6 is concave up.

A) Everywhere B) For x < –3 and x > 3 C) Nowhere D) For –3 < x < 3 8. Locate all inflection points of f ( x) 

x . x 1 2

A)

 0, 0  , 1,  ,  1,  

C)

 0, 0  ,  1,  

B)

(0, 0)

D)

  3 3   3,   , (0, 0),  3,  4  4   

1  2 

1 2



1 2

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Final Test

9. Use the second derivative test to find the relative maxima and minima of the function f ( x)  2 x3  6 x 2  18x  1 .

A) Relative maximum at (–3, –8); relative minimum at (1, 55) B) Relative maximum at (–3, 55); relative minimum at (1, –9) C) Relative maximum at (–3, –9); relative minimum at (–1, 23) D) Relative maximum at (1, 55); relative minimum at (–3, –9) 10. A 5-year projection of population trends suggests that t years from now, the population of a certain community will be P(t )  t 3  15t 2  72t  55 thousand. a. At what time during the 5-year period will the population be growing most rapidly? b. At what time during the 5-year period will the population be growing least rapidly? c. At what time is the rate of population growth changing most rapidly?

A) a. t = 2 years b. t = 0 years c. t = 2 years B) a. t = 2 years b. t = 1 year c. t = 3 years C) a. t = 5 years b. t = 0 years c. t = 0 years D) a. t = 0 years b. t = 0 years c. t = 2 years 11. Name the vertical and horizontal asymptotes of the given graph. y

5

x -2

A) B) C) D)

3

Vertical asymptotes: x = –2, x = 3, x = 0; horizontal asymptotes: y = 5, y = 0 Vertical asymptotes: x = –2, x = 3, x = 0; horizontal asymptote: y = 5 Vertical asymptotes: x = –2, x = 3; horizontal asymptote: y = 5 Vertical asymptotes: x = –2, x = 3; horizontal asymptotes: y = 5, y = 0

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Final Test

12. Find all vertical and horizontal asymptotes of the graph of the given function. x 2 – 3x f  x  2 x + 5x + 6 A) Vertical asymptotes: x = –3, x = –2; horizontal asymptote: y = 1 B) Vertical asymptotes: x = –3, x = –2; horizontal asymptote: y = 3 C) Vertical asymptotes: x = –3, x = –2, x = 3; horizontal asymptote: y = 0 D) Vertical asymptotes: x = –3, x = –2, x = 3; horizontal asymptote: y = 1 13. Sketch the graph of the given function. x+5 f  x  x+2 A)

C)

y

y x

(-5, 0)

x (0, 1.5)

(0, -2.5)

B)

D) y

(-5, 0)

y

(0, 2.5) x

x

(0, -3.5)

14. Find the absolute minimum of the function f ( x)  x3  3x 2 on the interval 1  x  5.

A) 5 B) –4 C) –1 D) 0 15. Find the absolute maximum of the function f ( x)  x7  x6 on the interval –1  x  1.

A) –2 B) 0 C) 1 D) –1 16. The cost of producing x units of a certain commodity is C ( x)  3 x  5 x  5 dollars. If the price is p(x) = (41 – x) dollars per unit, determine the level of production that maximizes profit.

A) x = 3 B) x = 1 C) x = 5 D) x = 2 2

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Final Test

17. The demand function for a certain commodity is D  p   the demand inelastic?

A) p > –5 B) p > 0

C) p < –5

30 . For what values of p is p5

D) p < 0

18. Find two non-negative numbers x and y whose sum is 14 and are such that the xy2 is as large as possible.

14 28 A) 7 and 7 B) and C) 1 and 13 D) 14 and 28 3 3 19. The owner of a novelty store can obtain joy buzzers from the manufacturer for 50 cents apiece. He estimates he can sell 60 buzzers when he charges $1.00 apiece for them and that he will be able to sell 10 more buzzers for every 8 cent decrease in price. What price should he charge in order to maximize profit?

A) $0.79 B) $0.89 C) $1.09 D) $0.99 20. A cable is to be run from a power plant on one side of a river 900 meters wide to a factory on the other side, 6,000 meters downstream. The cost of running cable under the water is $5 per meter, while over land is $4 per meter. What is the most economical route over which to run the cable?

A) The cable reaches the opposite bank 6,000 meters downstream.

B) The cable reaches the opposite bank 3,000 meters downstream.

C) The cable reaches the opposite bank 1,200 meters downstream.

D) The cable reaches the opposite bank directly across stream.

21. An open box is to be made from a square piece of cardboard, 36 inches by 36 inches, by removing a small square from each corner and folding up the flaps to form the sides.

What are the dimensions of the box of greatest volume that can be constructed in this way?

A) 24 inches long by 24 inches wide by 6 inches deep B) 6 inches long by 6 inches wide by 24 inches deep C) 12 inches long by 12 inches wide by 6 inches deep D) 18 inches long by 18 inches wide by 6 inches deep

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Final Test

Answer Key 1. A 2. C 3. D 4. A 5. D 6. D 7. B 8. D 9. B 10. C 11. C 12. A 13. B 14. B 15. B 16. C 17. B 18. B 19. D 20. C 21. A

Chapter 4 1. Evaluate the given expression.

5/4

 256     625  1, 024 4 A) B) 3,125 5

C)

1, 024 625

D)

256 625

2. Find all real numbers x that satisfy the given equation.

3x23x = 13,824 A) 15 B) 12 C) 9 D) 3 3. Find all real numbers x that satisfy the given equation. x 1

1 4 – 3 x2   3  81  4 4 4 A) 0, – B) 0, C) – 3 3 3

D) 0

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Final Test

4. If $1,500 is invested at 9 percent compounded continuously, what is the balance after 13 years?

A) $4,598.71 B) $465.55 C) $1,635.00 D) $4,832.99 5. It is projected that t years from now, the population of a certain country will be P(t )  70e0.01t million. What will be the population in 20 years? Round your answer to two decimal places, if necessary.

A) 140.96 million B) 86.72 million C) 85.5 million D) 84.28 million 6. The population density x miles from the center of a certain city is D( x)  9e0.06 x thousand people per square mile. What is the population density 7 miles from the center of this city? Round your answer to two decimal places.

A) 5.91 thousand people per square mile B) 6.57 thousand people per square mile C) 13.70 thousand people per square mile D) 5.26 thousand people per square mile 7. Determine the monthly car payment for a new car costing $18,228, if there is a down payment of $6,000 and the car is financed over a 6-year period at an annual rate of 9% compounded monthly.

A) $328.57 B) $220.42 C) $231.50 D) $198.37 8. Solve for x. Round to three decimal places, if necessary. log 4 x  5 A) 1.495 B) 1.32 C) 1,024 D) 625 9. Solve for x: log5 ( x  1)  3 .

A) 125 B) 244 C) 126 D) 124 10. Solve for x: a 7 x 1  b  ln b   1 A)   ln a 

7

B) x 

1 ln b  C) x  1  1   7  ln a 

1 ln b  1   7  ln a 

11. Solve the given equation for x.

–6  –7  5e –7 x e5 ln 5 A)  B) C) –7e5 7 7

D) 

1 ln x 4  16 5 16 A) x = e B) x  e C) x  e4

D)

ln b 7 ln a

ln 5 7

12. Solve for x: 4 ln x 

D) x  e5

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Final Test

13. A radioactive substance decays exponentially. If 700 grams were present initially and 300 grams are present 100 years later, how many grams will be present after 400 years?

Round your answer to two decimal places, if necessary.

A) 22.37 grams B) 21.12 grams C) 0.00 grams D) 23.62 grams 14. An efficiency expert hired by a manufacturing firm has compiled the following data relating workers' output to their experience: Experience (months) Output (units per hour)

0 300

2 510

The expert believes that the output Q is related to experience t by a function of the form Q(t )  600  Ae kt . Find the function of this form that fits the data. Round numbers to four decimal places, if necessary.

A) Q(t )  600  510e0.6020t C) Q(t )  600  300e –0.6020t B) Q(t )  600  510e –0.6020t D) Q(t )  600  300e0.6020t 15. Differentiate the given function. f ( x )  e 6 x A) x  6 xe6 x

B) x  e6 x

C) x  6e6 x 1

16. Differentiate the given function. f ( x)  35  3e0.06 x A) 3xe0.06 x B) 0.18e0.06 x C) 0.18e0.06 x

D) x  6e6 x

D) 3xe0.06 x

17. Differentiate the given function. f ( x )  ln x 3 1 x 3 A) B) C) 3x D) 3x 3 x 6 7 18. Find the derivative of ln  ln x   .   42 42 7 A) B) C) ln(ln x) x ln x x ln x

D)

42 x  ln x

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Final Test

19. Use logarithmic differentiation to find f ( x) .

6x  5 8  6x 1 6 6  f ( x)     f ( x) 8  6x  5 8  6x  f ( x)  8

A) B)

 6x  5  f ( x)  f ( x )    8  6x 

C)

6   6 f ( x)     f ( x)  6x  5 8  6x 

D)

 6x  5  f ( x)     8  6x 

–7/8

–7/8

20. The consumer demand for a certain commodity is D( p)  8,000.00e0.57 p units per month when the market price is p dollars per unit. Express consumers' total monthly expenditure for the commodity as a function of p and determine the market price that will result in the greatest consumer expenditure.

A) $8,000.00 B) $14.04 C) $26.32 D) $1.75 21. Let f (t ) 

19 . Which value of t corresponds to a possible inflection point for 3  4et /110

f (t)?

A) There is no inflection point.

B)

3 ln(110) 4

3 C) ln   4

4 D) 110 ln   3

22. Consider the function f ( x)  e  ( x 6) /5 . For what value of x does this function attain its maximum value, and what is the maximum function value? Round maximum function value to two decimal places, if necessary.

A) x = 6, f (6) = 1 C) There is no maximum.

3 6 3 6 B) x  , f    0.00 D) x  , f    0.01 5 5 5 5 2

23. Let f ( x)  6 x5  90 ln x , for x > 0. Find the minimum value of f for x > 0.

A) 0 B) 3(35  15 ln 3) C) 18(1 – ln 3) D) 6(35  15 ln 3)

1 of the residents of a small town. The number of 14 residents who had heard about the accident t hours later is given by a function of the B 1 form , where B is the population of the town. If of the residents had heard  kt 4 1  Ce 1 about the accident after 1 hours, how long did it take for of the residents to hear the 2 news? Round your answer to two decimal places.

A) 0.87 hours B) 0.44 hour C) 1.75 hours D) 3.50 hours

24. A traffic accident was witnessed by

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Final Test

25. Suppose your family owns a rare book whose value t years from now will be V (t )  9e 0.8t dollars. If the prevailing interest rate remains constant at 6% per year compounded continuously, when will it be most advantageous for your family to sell the book and invest the proceeds? Round your answer to two decimal places.

A) 125.00 years B) 48.61 years C) 194.44 years D) 55.56 years

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Final Test

Answer Key 1. A 2. D 3. B 4. D 5. C 6. A 7. B 8. C 9. C 10. B 11. B 12. D 13. D 14. C 15. D 16. C 17. D 18. C 19. A 20. D 21. D 22. A 23. C 24. C 25. D

Chapter 5





1. Evaluate  5 x  9 x  7 dx .

A) B)

6

5x7 9 x 2  C 7 2 5x7 9 x2   7x  C 7 2

C)

30 x5  9  C

D) 5x7  9x2  7x  C

2. A study indicates that x months from now the population of a certain city will be increasing at the rate of (3  3 x) x 1/ 2 people per month. By how much will the population increase over the next 9 months?

A) 54 people B) 62 people C) 72 people D) 82 people

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Final Test

3. An object is moving so that its velocity after t minutes is v(t )  1  8t  9t 2 meters per minute. How far does the object travel from the end of minute 2 to the end of minute 3?

A) 37 meters B) 78 meters C) 157 meters D) 77 meters x  4. Evaluate    9  9 

2/9

dx .

11/9

11/9

A)

x    9 9 

B)

9 x    9 11  9 

C)

81  x    9 11  9 

D)

x2  9x  C 18

C) 25 x 5 e x  C

D) e x  C

C 11/9

C

C

5. Evaluate  5 x 4 e x dx .

5

A) e x  C 6

B) x5 e x  C 5

5

5

6. Evaluate  x 6 x 7  4dx .

A) x  4  C 7

B) C ( x 7  4)3/ 2

2( x 7  4)3/2 C C) 21

D) ( x7  4)3/2  C

7 x 4  56 x3  56 dx x5  10 x 4  40 x  4 7 7 ln x5  10 x 4  40 x  4  C ln x 4  8 x3  8  C C) 5 5 5 7 ln x5  10 x 4  40 x  4  C ln x5  10 x 4  32 x  C D) 7 5

7. Evaluate the indicated integral:  A) B)

8. Solve the given initial value problem for y = f (x). dy x +5 where y = 6 when x = 1  2 dx x + 10 x + 3 1 1 1 A) y  ln x 2 + 10 x + 3  ln 14 + 6 C) y  ln x 2 + 10 x + 3 + 6 2 2 2 1 1 1 B) y  ln x 2 + 10 x + 3  ln 14 D) y  ln x 2 + 10 x + 3 2 2 2

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Final Test

9. The slope f (x) at each point (x, y) on a curve y = f (x) is given, along with a particular point (a, b) on the curve. Use this information to find f (x).

2 f   x   xe36 x ; (–6, 1) A) B)

2 1 y  –1  e36 x 2 2 1 y   e36 x 2

C) D)

3 1 36 x2  e 2 2 2 1 y  1  e36 x 2 y

10. The velocity v(t) (t)= x(t) at time t of an object moving along the x axis is given, along with the initial position x(0) of the object.

1/2 x   t   3  3t  1 ; x(0) = 4 Find: (a) The position x(t) at time t. (b) The position of the object at time t = 1. (c) The time when the object is at x = 2.

Round answers for parts (b) and (c) to one decimal place.

2 3/2 A) (a) x  t     3t  1 3 (b) x(1) = –5.3 (c) t = –0.3 2 14 3/2 B) (a) x  t     3t  1 + 3 3 (b) x(1) = 0.7 (c) t = –7.7 2 3/2 C) (a) x  t     3t  1 3 (b) x(1) = 0.1 (c) t = –12.3 2 14 3/2 D) (a) x  t     3t  1 + 3 3 (b) x(1) = –0.7 (c) t = 0.5

2t  7 ft3/min. If the tank is empty when t = 0, how much water does it contain 8 minutes later? Express the answer to two decimal places.

A) 49.45 ft3 B) 38.37 ft3 C) 30.59 ft3 D) 0.72 ft3

11. Water flows into a tank at the rate of

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Final Test

12. In a certain section of the country, the price of chicken is currently $3 per kilogram. It is estimated that x weeks from now, the price will be increasing at a rate of 3 x  1 cents per kilogram. How much will chicken cost 4 weeks from now?

A) $4.20 per kilogram C) $2.21 per kilogram B) $3.20 per kilogram D) $0.21 per kilogram 1

13. Evaluate  x11 dx . 1

A) 

1 22

B) 0 C)

1 11

D)

1 22

14. Find the area of the region under the line y = 4x + 7 over the interval 1  x  8 .

A) 189 B) 173 C) 193 D) 175 15. Suppose the marginal cost is C ( x)  e0.4 x , where x is measured in units of 300 items and the cost is measured in units of $10,000. Find the cost corresponding to the production interval [300, 1,500]. Round to the nearest dollar.

A) $13,375 B) $5,350 C) $9,029 D) $2,140 16. It is estimated that t days from now a farmer's crop will be increasing at the rate of 0.3t 2  0.6t  1 bushels per day. By how much will the value of the crop increase during the next 10 days if the market price remains fixed at $6 per bushel?

A) $870.00 B) $138.00 C) $840.00 D) $780.00 17. Compute the area of the finite region bounded by the graphs of the functions y  5 x 2 , and y  5x 2  20 x  80 .

A) 360 B) 320 C) 280 D) 240 18. Find the average value of the function f ( x) 

x over the interval [1, 3]. Round 4x  1

your answer to two decimal places.

A) 0.28 B) 1.22 C) 0.44 D) 0.22 19. Find the Gini index for the given Lorentz curve. Round your answer to four decimal places, if necessary.

L( x)  0.6 x 2  0.3 x A) 0.3600 B) 1.7000 C) 0.3000 D) 0.2571 20. The chief economist for a meat-packing plant estimates that if L worker-hours of labor are employed each week, the number of pounds of meat made ready for sale will be given by Q ( L)  650 L3/4 . Find the average weekly output as labor varies from 480 to 790 hours. Round to the nearest pound.

A) 82,069 pounds B) 73,954 pounds C) 34,248 pounds D) 86,461 pounds

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Final Test

300 dollars per unit, find the total 0.4q  8 amount of money consumers are willing to spend to get 15 units of the commodity.

A) $419.71 B) $0.56 C) $167.88 D) $209.86

21. Given a consumer's demand function, D(q) 

22. Money is transferred continuously into an account at the constant rate of $1,100 per year. The account earns interest at the annual rate of 8% compounded continuously.

How much will be in the account at the end of 3 years?

A) $3,300.00 B) $1,746.97 C) $151,567.68 D) $3,729.68 23. Given an initial population, P0  140, 000 , a renewal rate, R = 355, and a survival function, S (t )  e 0.08t , with time t measured in years, determine the population at the end of 19 years.

A) 34,087 B) 5,408 C) 11,009,981 D) 136,533 24. Find the volume of the solid of revolution formed by rotating the region R about the xaxis.

R is the region under the curve y  49  x 2 from x = –7 to x = 7.

1,372 196 A) 343π B) π C) 49π D) π 3 3 25. A certain nuclear power plant produces nuclear waste in the form of strontium-90 at the constant rate of 200 pounds per year. The waste decays exponentially with a half-life of 28 years. How much of the radioactive waste from the nuclear plant will be present after 130 years? [Hint: Think of this as a survival and renewal problem.] Round to two decimal places.

A) 4,033.07 pounds C) 8,402.48 pounds B) 7,755.70 pounds D) 3,877.85 pounds

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Final Test

Answer Key 1. B 2. C 3. B 4. C 5. D 6. C 7. A 8. A 9. C 10. D 11. C 12. B 13. B 14. D 15. A 16. C 17. A 18. D 19. C 20. A 21. A 22. D 23. A 24. B 25. B

Chapter 6 x /9 1. Evaluate  xe dx .

A) B)

xe x /9 e x /9  C 9 81 9 xe x /9  81e x /9  C

C) D)

xe x /9 e x /9  C 9 81 9 xe x /9  81e x /9  C

2. Use integration by parts to evaluate the integral  (15  x)e dx . x

A) B)

(16  x)e x  C (16  x)e x  C

C) D)

(16  x)e x  C (15  x)e x  C

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Final Test

3. Evaluate  ( x  3) ln(2 x) dx .

A) B)

x2  x( x  6) ln(2 x) + C 2 ( x  3) 2 ln(2 x) + C 2

6x 

C) D)

x2  x( x  6) ln(2 x) + C 2 x 2 x( x  6) ln(2 x) 3x   +C 4 2

4. Evaluate  x( x  2) dx .

9

A) B)

1 ( x  2)10 (5 x  2)  C 110 1 ( x  2)10 (5 x  1)  C 55

C) D)

5. Use integration by parts to evaluate the integral  A) B)

2 ( x  16)3/2 ( x  32)  C 3 2 ( x  16)1/2 ( x  32)  C 3

C) D)

1 ( x  2)10 (10 x  1)  C 110 1 ( x  2)10 ( x  10)  C 55 x x  16

dx .

2 ( x  16)1/2 ( x  32)  C 3 3 ( x  16)1/2 ( x  32)  C 2

6. Evaluate  x ln x dx .

A)

x 2 ln x C 4

B) x 2 ln x  C

C)

x 2 ln x x 2  +C 4 8

D) x + C

0.02 q dollars per unit when q 7. A manufacturer has found that marginal cost is (2q  1)e 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) $545.39 B) $635.39 C) $1,124.68 D) $300.00 0.5t units per hour. How 8. After t hours on the job, a factory worker can produce 150te many units does the worker produce during the first 4 hours? Round to two decimal places.

A) 843.60 units B) 397.00 units C) 356.40 units D) 243.60 units

9. From time t = 0 to t = 4 an object's speed is given by the function s (t )  te . Compute the distance travelled by the object during this time interval. Round your answer to two decimals.

A) 8,330,724.81 B) 8,330,728.68 C) 8,330,738.94 D) 8,330,830.01 4t

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Final Test

10. Given an initial population, P0  11, 000 , a renewal rate, R = 100, and a survival function, S (t )  te 0.3t , with time t measured in years, determine the population at the end of 13 years. Round to two decimal places.

A) 171.99 B) 12,231.87 C) 3,895.50 D) 1,693.69 11. Find the average value of the function f ( x)  xe Round to two decimal places.

A) 0.87 B) 3.46 C) 10.86 D) 7.79 12. Evaluate 



A) 0

1 7

1

x

7 6

B)

 x /3

over the interval from 0 to 4.

dx

C)

6 7

D) Diverges



13. Evaluate  e6 x dx .

1

A) e6

B) 6e6

1 6e

C)

D)

1 6e6 

14. Evaluate the improper integral: 3 A)

1 104

B) 

15. Evaluate 2 A) 

1 102

C)

1 68

x2 dx ( x3  7) 2

D) 102

1 dx . x(ln x)3

1 2(ln 2) 2

B) 

1 2 ln 2

C)

1 2 ln 2

D)

1 2(ln 2) 2



1 dx 7 ( x  5) ln( x  5) Round your answer to two decimal places, if necessary.

A) 0.69 B) 1.95 C) Undefined D) –0.37

16. Evaluate the improper integral: 



17. Evaluate the improper integral:  x3 e  x /7 dx 4

0

Round to two decimal places, if necessary.

A) 1.75 B) Undefined C) 3.5 D) 2.63

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Final Test

18. 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 f (t )  et /50 . 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) 40,800 B) 50,000 C) 550,000 D) 40,500 19. A hospital patient receives intravenously 15 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 f (t )  et /6 . 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) 90 units B) 2.50 units C) 75 units D) 21 units 2 20. Assume a 6-year franchise is expected to generate profit at the rate of 3 t  19, 000 dollars per year. If, over the next 6 years, the prevailing annual interest rate remains fixed at 7%, 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) $1,123.47 B) $561.73 C) $374.49 D) $337.04

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Final Test

Answer Key 1. B 2. C 3. D 4. B 5. C 6. C 7. B 8. C 9. B 10. C 11. A 12. D 13. D 14. B 15. D 16. C 17. A 18. B 19. A 20. C

Chapter 7 1. Compute f (ln 2, ln 5) if f ( x, y)  e2 x  y .

A) 10 B) 9 C) 20 D) None of the above 2. Compute f (6, 0) if f ( x, y )  A) Undefined

B) 6

x . ey

C) 0 D)

1 6

3. Given the function of three variables f (x, y, z) = xy + xz + yz, evaluate f (3, 7, 7).

A) 86 B) 100 C) 91 D) 94 4. Compute f y for f ( x, y)  7 xy .

6

A) 7 y 6

B) 42 xy 5  7 y 6

C) 7x D) 42xy 5

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Final Test

5. Compute all first-order partial derivatives of the given function. f ( x, y)  (4 x  2 y)4 5 3 5 3 A) f x  16(4 x  2 y ) , f y  8(4 x  2 y ) C) f x  16(2 y ) , f y  8(4 x) B)

f x  16( x  2 y)3 , f y  8(4 x  2)3

D)

f x  16(4 x  2 y )3 , f y  8(4 x  2 y )3

6. Compute f x for f ( x, y)  e8 xy .

A) 8xye8 xy

B) 8 ye8 xy

C) 8 xe8 xy

7. Compute f yy for f ( x, y)  e

8 xy

A) 64e8 xy

B) 64 x 2 e8 xy

D) 8e8 xy

.

C) 64 x 2 y 2 e8 xy

D) 64 y 2 e8 xy

8. The only grocery store in a small rural community carries two brands of frozen apple juice, a local brand that it obtains at the cost of 18 cents per can and a well-known national brand that it obtains at the cost of 40 cents per can. The grocer estimates that if the local brand is sold for x cents per can and the national brand for y cents per can, approximately 70 – 5x + 4y cans of the local brand and 80 + 6x – 7y cans of the national brand will be sold each day. How should the grocer price each brand to maximize the profit from the sale of the juice?

A) local brand (x) at 39 cents, national brand (y) at 104 cents B) local brand (x) at 46 cents, national brand (y) at 52 cents C) local brand (x) at 44 cents, national brand (y) at 52 cents D) local brand (x) at 88 cents, national brand (y) at 50 cents 9. Given the following points in the plane, find the slope of the least squares line: (1, 2), (2, 1), (3, 3), and (5, 6) round your answer to two decimal places, if necessary.

A) 0.97 B) 1.03 C) 1.14 D) 1.09 10. Given the following points in the plane, find the y-intercept of the least squares line to two decimal places: (3, 3), (5, 5), (7, 8), and (9, 8) A) –0.90 B) –1.50 C) 0.60 D) 0.72 11. The following data shows the age and income for a small number of people.

Age (years) Incomes ($)

24 30,000

32 34,000

39 53,000

54 72,000

60 81,000

Find the best fit straight line of this data, rounding coefficients and constants to the nearest whole number. Let x represent age and y represent income.

A) y = 3,830x – 8,036 C) y = 1,484x – 8,041 B) y = 3,106x – 8,048 D) None of the above

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Final Test

12. A military radar is measuring the distance to a jet fighter. The radar has received the following measurements: time t (minutes) distance (miles)

1 280

2 289

3 294

4 299

5 308

6 310

Using a least squares fit to the data, extrapolate to the nearest tenth of a minute when the jet will be 345 miles away?

A) 10.4 minutes B) 13.7 minutes C) 12.9 minutes D) 11.5 minutes 13. Use Lagrange multipliers to find the maximum value of f (x, y) = 9xy subject to the constraint 9x + 3y = 27.  3 9  243 A) f (0, 0) = 0 C) f  ,    4 4  16  3 9  243 B) f (3, 9) = 243 D) f  ,   4 2 2 14. Find the maximum value of the function f (x, y, z) = 7x + 8y + 6z on the sphere x 2  y 2  z 2  596 .

A) 298 B) 894 C) 1,043.0 D) 149 15. A manufacturer is planning to sell a new product at the price of $230 per unit and estimates that if x thousand dollars is spent on development and y thousand dollars is 340 y 170 x  spent on promotion, approximately units of the product will be sold. The y  7 x  14 cost of manufacturing the product is $160 per unit. If the manufacturer has a total of $340,000 to spend on development and promotion, how should this money be allocated to generate the largest possible profit? [Hint: Profit equals (number of units)(price per unit minus cost per unit) minus total amount spent on development and promotion.]

A) $173,500 on development, $167,000 on promotion B) $167,000 on development, $173,000 on promotion C) $166,500 on development, $173,500 on promotion D) $173,000 on development, $167,000 on promotion 7

4

16. Evaluate the following double integral:  –4  4x 4 y 7 dy dx 1 A) 4 B) The integral can't be evaluated. C) D) 0 4

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Final Test

17. Use inequalities to describe R in terms of its vertical and horizontal cross sections.

R is the region bounded by y = ex, y = 2, and x = 0 A) Vertical cross sections: e x  x  2 C) Vertical cross sections: 0  x  ln y ex  y  2 0  y  ln 2 Horizontal cross sections: 0  y  ln 2 Horizontal cross sections: 1  y  2 1 x  2 0  x  ln 2 x B) Vertical cross sections: e  x  ln 2 D) Vertical cross sections: 0  x  ln 2 ex  y  2 Horizontal cross sections: 1  y  2 0  x  ln y

0 y2 Horizontal cross sections: 0  y  2 1  x  ln 2

18. Evaluate the given double integral for the specified region R.  8x  2 y  dA , where R is the triangle with vertices (0, 0), (2, 0), and (0, 1) R

A) 18

B) 12 C)

68 3

D)

94 3

19. Use a double integral to find the area of R. R is the triangle with vertices (–2, 1), (2, 1), and (0, –1).

 

yx – 1

A)

x2

y 1

 

x y +1

B)

1 dy dx  4

 

yx – 1

C)

x 2

1 dx dy  4

y 1

 

x y + 1

D)

x  –2 y  x – 1

y  –1 x  y – 1

1 dy dx  8

x  –2 y  x – 1

1 dx dy  8

y  –1 x  y – 1

20. At a certain factory, the output Q is related to inputs x and y by the expression Q( x, y)  2 x3  2 x 2 y 4  9 y 5 . If 0  x  8 and 0  y  10, what is the average output of the factory? (Approximate to 2 decimal places.)

A) 111,504.00 units C) 235,590.13 units B) 235,589.33 units D) 308,341.33 units

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Final Test

Answer Key 1. C 2. B 3. C 4. D 5. D 6. B 7. B 8. C 9. C 10. C 11. C 12. D 13. D 14. A 15. C 16. D 17. D 18. D 19. B 20. B

Chapter 8 1. Find the general solution of A) y = 9x + C

dy  x3  9 . dx

B) y  4 x 4  9 x  C

C) y  3 x 2  C

D) y 

x4  9x  C 4

dy  7y . dx B) y  e7 x  C C) y = 7x + C D) y  Ce 7 x

2. Find the general solution of A) y = ln x + C

dy  e7 x . dx (7 x)e B) y  C) y  7 xe7 x  C C

3. Find the general solution of A) y 

e7 x C 7

D) y 

Ce7 x 7

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Final Test

dy  x 2 , given y = 18 when x = 1. dx 3 x 53 x3 x3 53  18 D) y  B) y  C) y    3 3 3 3 3

4. Find the particular solution of A) y 

x3  53 3

5. Find the general solution of B) y  3

A) y  C

dy 19 x  2 . dx y

57 x 2 C 2

C) y  3

(38 x) 2 C 3

D) y 

19 x 2 C 2

dy x 2 y 3  , given y = 6 when x = 0. dx 5 x3 180 6 6 B) y  4 x  6 C) y  D) y  3 5 – 24 x3

6. Find the particular solution of A) y 

2 ( x  6) 2 3

7. Find the particular solution of the given differential equation that satisfies the indicated dy condition:  y 2 6  x ; y = 1 when x = 6. dx 3 2(6  x)3/2  3 A) y  C) y  2(6  x)3/2  3 3 3 3 B) y  D) y  3/2 2(6  x)  3 2(6  x)3/2  3 8. Write a differential equation describing the given situation. Define all variables you introduce. (Do not try to solve the differential equation at this time.) An investment grows at a rate of 3% of its size. dQ  0.03Q A) Let Q denote the investment and let t denote time; dt dQ  0.03t B) Let Q denote the investment and let t denote time; dt dQ  0.3Q C) Let Q denote the investment and let t denote time; dt dQ  3Q D) Let Q denote the investment and let t denote time; dt 9. Find the general solution of the given first-order linear differential equation. dy 9 y   4x dx x x2 C 4x2 C 4x2 C 11x 2 C  9 B) y   11 C) y   9  9 D) y  A) y  11 x 11 x 4 11 x x

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Final Test

10. Find the general solution of the given first-order linear differential equation. dy y   20 x19 e x dx 20 x ex A) y  1/20  C C) y  x1/20 e x ( x  1)  C x e x ( x  1)  C e x ( x  1)  C B) y  D) y  x 20 x1/20 11. Find the particular solution of the given differential equation that satisfies the given condition. dy  y  2 x 2 ; y  1 when x  3 dx 35e x 3 A) y  2 C) y  35e x  2( x 2  2 x  2) x  2x  2 B) y  35e x 3  2( x 2  2 x  2) D) y  Ce x 3  2( x 2  2 x  2) 12. The price of a certain house is currently $260,000. Suppose it is estimated that after t months, the price p(t ) will be increasing at the rate of 0.01 p(t )  1, 000t dollars per month.

True or False: In 7 months from now, to the nearest whole dollar, the price of the house will be $303,934.

A) False B) True 13. Find an equation for the orthogonal trajectories of the given family of curves.

8x 2  y  C A) y  ln x16  C B) y  xe1/16  C C) y  ln x1/16  C D) ln y  x1/16  C 14. A dead body is discovered at 8:00 A.M. on Tuesday in a basement where the air temperature is 60F. The temperature of the body at the time of discovery is 72F and 20 minutes later, the temperature is 71F. True or False: The time of death was 6:00 A.M. on Tuesday.

A) False B) True 15. The intensity of light I (d ) at a depth d below the surface of a body of water changes at a rate proportional to I. If the intensity at a depth of d  2.4 feet is half of the surface intensity I 0 , to the nearest tenth of a foot, at what depth is the intensity 15% of I 0 ?

A) 6.6 feet B) 9.9 feet C) 13.2 feet D) 7.9 feet

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Final Test

16. A desert preserve in the southwest is known for its populations of coyotes and road runners. The growth rate for each population can be modeled by this pair of differential equations: dR  2 R  0.2 RC dt dC  1.4C  0.02 RC dt where C is the number of coyotes and R is the number of road runners. Find the equilibrium populations for this model. (Hint: These are the populations for which dR dC   0 ). dt dt A) 60 coyotes, 15 road runners C) 15 coyotes, 60 road runners B) 10 coyotes, 70 road runners D) 70 coyotes, 10 road runners 17. Use Euler's method with the step size h  0.2 to estimate the solution y (1) of the given initial value problem. Round your answer to two decimal places. x y y  ; y (0)  4 4x  y A) 1.64 B) 4.92 C) 6.56 D) 3.28 18. True or False: The first five terms of the initial value problem yn  yn21 ; y0  2 are 2, 4, 8, 16, and 32.

A) False B) True 19. Find constants A and B so that the given expression yn satisfies the specified difference equation. n2 yn  nyn1  4n  2n3 ; yn  An  B A) A  2, B  2 B) A  2, B  2 C) A  2, B  2 D) A  2, B  2 20. Solve the given first-order linear initial value problem.  yn1  yn ; y0  1 A) yn  1

B) yn  (1)n1

C) yn  (1)n

D) yn  (1)n  1

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Final Test

Answer Key 1. D 2. D 3. A 4. D 5. B 6. C 7. D 8. A 9. C 10. D 11. B 12. B 13. C 14. A 15. A 16. B 17. D 18. A 19. D 20. C

Chapter 9 1. Use summation notation to write the given series in compact form.

1 2 3 4     4 16 64 256 n n     n n n 1  1   n A)  n  1   B)  n C)  n D)     4 n 1 4 n 1 4 n 1 n 1  4  2. Find the fifth partial sum S5 of the given series.  (1) n  3n n 1 121 61 40 20 A)  B)  C) – D) – 243 243 81 81

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Final Test

3. Determine whether the given geometric series converges, and if so, find its sum. 

 5    9  n 1

n

A) Converges to 

5 14

B) Converges to

5 4

C) Diverges

D) Converges to

9 14

4. Determine whether the given geometric series converges, and if so, find its sum.  3  7n n 0 21 7 7 A) Converges to B) Diverges. C) Converges to D) Converges to 8 6 2 





k 1

k 1

k 1

5. If  ak  –4 and  bk  9 , find   9ak  2bk  .

A) 18

B) –54

C) –13

D) –18

6. Express the given decimal as a fraction.

1.441441441 49 160 174 60 A) B) C) D) 111 111 143 11 7. Suppose that nationwide, approximately 91% of all income is spent and 9% is saved.

What is the total amount of spending generated by a 55 billion dollar tax rebate if savings habits do not change?

A) $611 billion B) $105 billion C) $556 billion D) $60 billion 8. A patient is given an injection of 21 units of a certain drug every 24 hours. The drug is eliminated exponentially so that the fraction that remains in the patient's body after t days is f (t )  et /5 . If the treatment is continued indefinitely, approximately how many units of the drug will eventually be in the patient's body just prior to an injection?

A) 94.85 units B) 46.65 units C) 115.85 units D) 141.50 units 9. True or False: The given series converges.  k 9  k2  6 k 1 A) False B) True 10. True or False: The given series converges.  4  6k  k k 1 A) False B) True

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Final Test

11. True or False: The given series converges.  k3  2k k 1 A) False B) True 12. True or False: the given series converges.  ln k  9k k 1 A) False B) True 13. Determine the interval of absolute convergence for the given power series. 

2 x

 k  6 

k

k 0

A) 6  x  6

B) All real x C) 1  x  1

D) 

1 1 x 6 6

14. Determine the radius of convergence for the given power series.  kx k  k 1 k 0 4 1 A) R  4 B) R   C) R  D) R  0 4 15. Determine the radius of convergence for the given power series.  k ! xk  k k 0 6 1 A) R  B) R  6 C) R   D) R  0 6 16. Find a power series for the given function. x f ( x)  2 x n     x n 1 x n 1 x n 1 x A)    B)  C)  n 1 D)  n n 0 2 n 0 2 n 0 2 n 0  2  17. Find the Taylor series for the given function at the indicated point x  a. f ( x)  e2 x  e2 x ; a = 0 

2(22 n ) 2 n x n 0 (2n)!

A) 



2(2n ) 2 n x n 0 n !

B) 

2(22 n 1 ) 2 n 1 x n  0 (2n  1)! 

C) 



2(2n ) n x n 0 n !

D) 

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Final Test

18. Find the Taylor series about x  0 for the indefinite integral x  1  5x2 dx     52 n 2 n 5n 2 n 5n 5n 2n 2 x x x 2 n 1 x A)  B)  C)  D)  n 0 2n n 0 2n n  0 2n  1 n 0 2n  2 19. Use a Taylor polynomial of specified degree n to approximate the indicated quantity.

Round to four decimal places.

36.1; n  3 A) 6.0071 B) 6.0083 C) 6.0715 D) 6.0166 20. Use a Taylor polynomial of specified degree n together with term-by-term integration to estimate the indicated definite integral. Round to six decimal places 0.4  x 2

0 e

A) 0.422396

dx; n  6 B) 1.173483

C) 0.469393

D) 0.379652

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Final Test

Answer Key 1. A 2. C 3. A 4. D 5. B 6. A 7. A 8. A 9. A 10. B 11. B 12. B 13. A 14. A 15. D 16. C 17. A 18. B 19. B 20. D

Chapter 10 1. A die is rolled two times and the number of 2s that are rolled is noted. What is the sample space for this random experiment?

A) {0; 1; 2} B) {1; 2; 3} C) {1; 2} D) {0; 1; 2; 3} 2. Consider the random experiment of tossing a fair coin five times, and let X denote the random variable that counts the number of times heads appears. Find P( X  3).

5 7 3 13 A) B) C) D) 16 8 16 16 3. Outcomes for X Probability

0 0.15

1 0.20

2 0.25

3 0.4

The outcomes and corresponding probability assignments for a discrete random variable X are given.

True or False: The expected value is 1.4.

A) False B) True

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Final Test

4. The grand prize in a lottery is $500,000. There are 6 second prizes of $50,000 and 12 third prizes of $10,000. If 1,000,000 tickets are sold, what is a fair price to pay for a ticket to this lottery?

A) $5.15 B) $0.92 C) $9.20 D) $1.84

4e4 x for x  0 is a probability density function.

5. True or False: f ( x)   for x  0 0 A) False B) True 1  for 0  x  6 is a probability density function for a particular random 6. f ( x)   6 0 otherwise variable X. Use integration to find P(3  x  4) rounded to the nearest hundredth.

A) 0.83 B) 0.11 C) 0.06 D) 0.17

7. Suppose the length of time, x, that it takes a chimpanzee to solve a simple puzzle is measured by a random variable X that is exponentially distributed with a probability density function  2 2 x /3 if x  0  e f ( x)   3  0 if x  0 where x is in minutes. Find the probability that a randomly chosen chimpanzee will take more than 6 minutes to solve the puzzle.

A) e4  1 B) e4  1 C) e4 D) e 4 8. The life span of car stereos manufactured by a certain company is measured by a random variable X that is exponentially distributed with a probability density function  0.2e0.2 x if x  0 f ( x)   if x  0 0 where x is the life span in years of a randomly selected stereo. What is the probability that the life span of a randomly selected stereo is between 1 and 11 years? Round to the nearest hundredth.

A) 0.64 B) 0.71 C) 0.35 D) 0.78

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Final Test

9. The time interval between the arrivals of successive trains at a certain station is measured by a random variable X with a probability density function  0.2e0.2 x for x  0 f ( x)   for x  0 0 where x is the time (in minutes) between the arrivals of a randomly selected pair of successive trains. What is the probability that two successive trains selected at random will arrive within 9 minutes of one another? Round to the nearest hundredth.

A) 0.17 B) 0.83 C) 0.62 D) 0.42 10. A 2-hour movie runs continuously at a local theater. You leave for the theater without first checking the show times. Use an appropriate uniform density function to find the probability that you will arrive at the theater within 10 minutes of the start of the film (before or after). Round the the nearest hundredth.

A) 0.10 B) 0.83 C) 0.17 D) 0.14  2x if 0  x  7  is a probability density function for a continuous random 11. f ( x)   49  0 otherwise variable X. Find the variance. Round to the nearest hundredth.

A) 3.26 B) 2.18 C) 1.09 D) 2.72

12. The joint probability density function for two continuous random variables X and Y is  1  x /6  y if x  0 and y  0  e e f ( x, y )   6 0 otherwise Find the expected values E ( X ) and E (Y ).

A) E ( X )  6; E (Y )  1 C) E ( X )  1; E (Y )  6 B) E ( X )  1; E (Y )  5 D) E ( X )  7; E (Y )  1 13. Suppose the time X a customer must spend waiting in line at a certain fast food restaurant is a random variable that is exponentially distributed with a density function  1  x /3 if x  0  e f ( x)   3  0 if x  0 where x is the number of minutes that a randomly selected customer spends waiting in line. Find the expected waiting time for customers at the restaurant.

A) 6 minutes B) 2.5 minutes C) 3 minutes D) 1.5 minutes

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Final Test

14. The density function of a normal random variable X is 2 1 f ( x)  e  x /32 4 2 Find the expected value E ( X ).

A) 4 B) 0 C) 2 D) 8 15. Assuming that the random variable Z has a standard normal distribution, find P( Z  2.03). Round your answer to four decimal places.

A) 0.0166 B) 0.0268 C) 0.0336 D) 0.0212 16. Find the appropriate value of b for P(b  Z  b)  0.6827, assuming that the random variable Z has a standard normal distribution.

A) 1.2 B) 1.1 C) 0.9 D) 1 17. Suppose X has a normal distribution with   16 and   3.

True or False: P( X  19)  0.4207 A) False B) True 18. True or False: If X has a normal distribution with   2, then P( X  1.76)  P( X  2.24).

A) False B) True 19. The gestation period of humans follows an approximately normal distribution with a mean of 266 days and a standard deviation of 16 days. Find the probability that a gestation period is between 260 and 276 days. Round to four decimal places.

A) 0.1901 B) 0.3802 C) 0.2851 D) 0.0950 20. Suppose that the number of peanuts in a bite-sized chocolate candy follows a Poisson distribution with a mean of 3.8 peanuts per candy. Find the probability that a candy contains exactly 4 peanuts. Round to three decimal places.

A) 0.097 B) 0.049 C) 0.194 D) 0.146

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Final Test

Answer Key 1. A 2. C 3. A 4. B 5. B 6. D 7. C 8. B 9. B 10. C 11. D 12. A 13. C 14. B 15. D 16. D 17. A 18. B 19. B 20. C

Chapter 11 1. Specify the number of degrees in the indicated angle.

A) 150

B) 210

C) 240

D) 150

2. Convert the indicated angle from degrees to radians.

185 37  37  37  A) B) C) D) None of the above 36 18 72 3. Convert the indicated angle from radians to degrees.

23 12 A) 172.5 B) 690 C) 345 D) None of the above

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Final Test

4. Evaluate the given expression.  cos 2 A) Undefined B) –1 C) 0 D) 1 5. Evaluate the given expression.  17   sec     4  2 2 A) B) 2 C) – 2 D) – 2 2 6. Evaluate the given expression.

9 tan 4 A) 1 B) 0 C) –1 D) undefined

3 7. Find sec if sin   .

5 5 4 3 A) B) C) 4 5 4

D)

4 3

8. Find all values of  in the specified interval that satisfy the given equation. sin   cos 2 ; 0    2  5 3   5    5 A)   , B)   , , C)   , D)   , , 6 6 2 6 2 6 6 2 6 6 9. Suppose a company determines that the profit P (in hundreds of dollars) obtained from sales during week t (for 0  t  52 ) of a particular year may be modeled by the trigonometric function  (t  17)  P (t )  115  40 cos    26  What is the largest weekly profit?

A) $15,500 B) $23,000 C) $11,500 D) $7,500 10. Differentiate the given function. f (t )  sin 6t  cos 4t A) 6sin 6t  4 cos 4t B) 6 cos 6t  4sin 4t

C) D)

6 cos 6t  4sin 4t cos 6t  sin 4t

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Final Test

11. Differentiate the given function. g (t )  5sin10 t  6sin t10 A) 50sin 9 t  cos t  60sin 9 t  cos t B) 5cos10 t  6cos t10

C) D)

5cos10 t  6cos t10 50sin 9 t  cos t  60t 9 cos t10

C) D)

20sin x  6 cos x  C –20sin x  6 cos x  C

12. Find the indicated integral:   20cos x  6sin x  dx A) B)

20sin x  6 cos x  C –20sin x  6 cos x  C

13. Find the indicated integral.

5 6  x sin 3  x dx





A)







C)

B)

1 cos 3  x6  C 6

D)

x6 cos 3  x 6  C 6









1  cos 3  x6  C 6 1 sin 3  x6  C 6





14. Determine the period p, the amplitude b, the phase shift d, and the vertical shift a of the given trigonometric function f(t).   f (t )  2 + 5cos   t  1  3  A) p  3, b  –5, d  1, a  2 C) p  3, b  5, d  1, a  2 B) p  6, b  5, d  1, a  2 D) p  6, b  –5, d  1, a  2 15. Solve the given differential equation subject to the specified initial condition. dy cos t  ; y (0)  3 dt 9y A) B)

2 sin t 9 2 y (t )  3  sin t 9 y (t )  3 

C) D)

2 sin t  9 9 2 y (t )  9  sin t 9 y (t ) 

16. Find the volume of the solid generated by rotating the region R about the x-axis. R is π π the region bounded by the x-axis, y  sec x , x   , and x  .

4 4 45π 9π A) B) C) 2π D) 9π 2 2

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Final Test

17. A person 6 feet tall is watching a streetlight 20 feet high while walking toward it at a speed of 4 feet per second. At approximately what rate is the angle of elevation of the person's line of sight changing with respect to time when the person is 9 feet from the base of the light? Round to four decimal places.

A) 0.2022 radians per second C) –0.0505 radians per second B) 0.1663 radians per second D) None of the above 18. A manufacturer will supply x units of a certain commodity to the market when the price is p  S ( x) dollars per unit, where S ( x)  23  3x – 2.5sin x .

What is the producers' surplus when x = 5 units?

A) $39.09 B) $113.70 C) $30.18 D) $37.50 19. Using a graphing utility, find the value of t for which the smallest value of x(t) occurs for the indicated interval. Estimate to four decimal places, if necessary. x(t )  et (4sin 3t  3cos 2t ) for 0  t   A) t  0.9605 B) t  0 C) t  0.2632 D) None of the above 20. A projectile is fired from a cannon at ground level at an angle of 24 with an initial velocity of 54 ft/sec. How far from the cannon does the projectile land? Round your answer to the nearest tenth of a foot.

Use the following formulas. x  (v0 cos )t

y  16t 2  (v0 sin )t  s0 A) 33.9 ft B) 37.1 ft C) 67.7 ft D) 135.4 ft

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Final Test

Answer Key 1. B 2. A 3. C 4. C 5. B 6. A 7. A 8. A 9. A 10. B 11. D 12. C 13. C 14. B 15. C 16. C 17. A 18. D 19. D 20. C

Final Exam 1. Find the composite function f  g  x   .

1 f (u )  , g  x   x  3 u 3 A) f (g(x)) = 1  x 1 B) f (g(x)) = x3

C)

f (g(x)) = x  3 

D)

f (g(x)) =

1 x

1 3 x

2. Find the points of intersection (if any) of the given pair of curves. y = x + 3 and y = 2x + 4 A) (–1, 2) B) (0, 3) and (1, 4) C) (1, 6) D) (1, –4)

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Final Test

3. Find the slope and y-intercept of the line whose equation is given. x y  1 2 3 1 3 A) Slope: ; y-intercept: 1 C) Slope: ; y-intercept: 1 2 2 3 1 3 B) Slope: ; y-intercept: D) Slope:  ; y-intercept: 3 2 2 2 4. Find the relative rate of change of f (x) with respect to x for the prescribed value x = 1. f (x) = 3x3 + x2 – 8 11 4 1 A) B) – C) 11 D) – 4 11 11 5. An equation for the tangent line to the curve y  ( x 2  x  1)3 at the point where x = 1 is: A) y = 2x + 1 B) y = 9x – 1 C) y = 9x D) y = 9x – 8 6. Find the slope of the tangent line to the curve x 2  3 xy  y 2  3 at the point (1, 1).

A) –5 B) 5 C) 3 D) 1 7. Find the intervals of increase and decrease for the function f ( x)  x 2  5x  4 .

5 5 A) Decreasing for x   and increasing for x   2 2 B) Decreasing for all x C) Increasing for all x 5 5 D) Decreasing for x   and increasing for x   2 2 8. The second derivative test reveals that f (x)  x2  4 has A) nothing significant at x = 4.

C) a point of inflection at x  4 .

B) a relative maximum at x  4 .

D) a relative minimum at x = 0.

9. To raise money, a service club has been collecting used bottles that it plans to deliver to a local glass company for recycling. Since the project began 90 days ago, the club has collected 45,000 pounds of glass for which the glass company currently offers 1 cent per pound. However, because bottles are accumulating faster than they can be recycled, the company plans to reduce by 1 cent each day the price it will pay for 100 pounds of used glass. Assume that the club can continue to collect bottles at the same rate and that transportation costs make more than one trip to the glass company unfeasible. What is the most advantageous time for the club to conclude its project and deliver the bottles?

A) 15 days from now B) Today C) 5 days from now D) 55 days from now

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Final Test

10. Simplify: A)

1 9

38 37

1 3

B) 3 C)

D) 315

1 the ratio in 6 the atmosphere. Approximately how old is the fossil? The half-life of 14 C is 5,730 years. Round to the nearest whole year.

A) 17,190 years B) 14,812 years C) 34,380 years D) 14,842 years

11. An archaeologist has found a fossil in which the ratio of 14 C to 12 C is

12. Find the equation of the tangent line to f (x) = ln x + 2 at x = 1.

A) y = 1 B) x = 1 C) y = x + 1 D) y = x 13. Evaluate  e

6 x 8

dx .

A) (6 x  8)e6 x8  C

B) (6 x  8)Ce6 x8

C) e6 x 8  C

D)

e 6 x 8 C 6

14. Records indicate that t hours past midnight, the temperature at the local airport was f (t )  0.3t 2  4t  10 degrees Fahrenheit. What was the average temperature at the airport between 8:00 A.M. and noon? Round your answer to one decimal place, if necessary.

A) 3.9 degrees F B) 9.6 degrees F C) 98.0 degrees F D) 19.6 degrees F 15. An investment will generate income continuously at the constant rate of $1,100 per year for 5 years. If the prevailing annual interest rate remains fixed at 6% compounded continuously, what is the present value of the investment?

A) $6,044.13 B) $47.52 C) $4,751.67 D) $475.17 16. Evaluate  (2  3x)e A) 3e2 x  C

2 x

dx .

3 2 x xe  C 2

B)

1 3 C) e 2 x  x    C 4 2

D) xe2 x  C

17. Evaluate  x ( x  3) dx .

3

A) B)

2

7

( x 2  3)8 x3 C 24 x2 2 ( x  3)8  C 16

C) D)

x2 2 1 ( x  3)8  ( x 2  3)9  C 16 144 2 x 1 ( x 2  3)8  ( x 2  3)9  C 16 72

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Final Test

18. An object moving in a straight line has velocity v(t )  te t meters per second. Is it true that in the first 4 seconds the object will have travelled 4e2  12 meters?

A) False B) True  dx 19. Evaluate 1 3 . x 1 A) 1 B) C) Diverges 3



20. Evaluate 2 A)

1 2 ln 2

1 x ln x B) 

D) 0

dx .

1 2 ln 2

C) Diverges

D) 0

21. Compute f (4, 3) if f ( x, y)  5xy3 .

A) 108 B) 192 C) 960 D) 540 22. Compute f x for f ( x, y)  7 xy 2 .

A) 14xy1

B) 7x C) 14 xy1  7 y 2

D) 7 y 2

23. A manufacturer with exclusive rights to a sophisticated new industrial machine is planning to sell a limited number of the machines to both foreign and domestic firms.

The price the manufacturer can expect to receive for the machines will depend on the number of machines made available. (For example, if only a few of the machines are placed on the market, competitive bidding among prospective purchasers will tend to drive the price up.) It is estimated that if the manufacturer supplies x machines to the domestic market and y machines to the foreign market, the machines will sell for x y y x thousand dollars apiece at home and for 50   thousand dollars 60   5 20 10 20 apiece abroad. If the manufacturer can produce the machines at a cost of $38,000 apiece, how many should be supplied to each market to generate the largest possible profit?

A) x = 41, y = 643 B) x = 40, y = 500 C) x = 100, y = 8 D) x = 80, y = 100

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Final Test

24. The accompanying table lists the high-school GPA and college GPA for a number of students: High school GPA College GPA

2.0 3.2

2.5 2.6

3.1 3.0

3.7 3.6

3.7 3.8

4.0 3.7

Using the best fit straight line, predict the college GPA (to one decimal place) for a student whose high school GPA was 3.5.

A) 3.5 B) 3.4 C) 3.3 D) 3.6 25. Use Lagrange multipliers to find the maximum value of f (x, y, z) = 6xyz subject to 2x + 5y + 2z = 60.

A) f (10, 4, 10) = 2,400 C) f (11, 2, 10) = 1,320 B) f (0, 0, 0) = 0 D) f (13, 5, 8) = 3,120 26. Use a double integral to find the area of R. R is the region bounded by y = 3x, y = ln x, y = 0, and y = 1. y 1 x  e y x e y 1 7 A) y 0 x  y 1 dx dy  e  C) x 0 y ln x 1 dy dx  e 6 3 B)

y 1

x e y

y 0

x 0

 

1 dx dy  e  1

27. Find the general solution of xe A) y  C

D)

x e

y 3 x

x 0

y  ln x

 

1 dy dx 

3e2 2

dy  ex . dx

B) y  xe x  C

C) y  Ce x

D) y  e x  C

28. Find the particular solution of the given differential equation that satisfies the indicated dy condition:  28 x3 y 2 ; y = 6 when x = 1. dx 6 6 A) y  C) y   4 43  42 x 43  42 x 4 6 6 B) y  D) y  4 43  42 x 43  42 x3 29. Write a differential equation describing the given situation: Powdered lemonade dissolves in a pitcher at a rate that is proportional to the amount of undissolved powder remaining. Let P = the total amount of powder added, D = the amount dissolved, and t = time. dD dD kPD  kP( P  D) A) C) dt dt dD dD  k ( P  D)  kPD B) D) dt dt

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Final Test

30. Find the general solution of the given first-order linear differential equation. dy x3  x2 y  3 dx C 3 C 3 3 C A) y  2  B) y   2 C) y  C  2 D) y   2 x x x x x x 31. A wooded nature preserve supports populations of silver foxes and snowshoe hares.

The growth rate for each population can be modeled by this pair of differential equations: dF  0.24 F  0.0012 FH dt dH  0.08R  0.005FH dt where H is the number of hares and F is the number of foxes. Find equilibrium dF dH populations for this model. (Hint: These are the populations for which   0 .) dt dt A) 12 foxes, 170 hares C) 30 foxes, 220 hares B) 12 foxes, 100 hares D) 16 foxes, 200 hares 32. Use Euler's method with the step size h  0.2 to estimate the solution y (2) of the given initial value problem. Round to one decimal place.

2x y  ; y (1)  2 y A) 2.6 B) 3.1 C) 2.1 D) 3.4 33. Determine whether the given geometric series converges, and if so, find its sum. 

5   7  n 1

n

A) Diverges.

B) Converges to

5 2

C) Converges to

5 12

D) Converges to

7 2

34. A ball is dropped from a height of H feet and bounces indefinitely, repeatedly rebounding to 76% of its previous height. If it travels a total distance of 80 feet, what is H? Round your answer to one decimal place.

A) 9.6 feet B) 30.4 feet C) 19.2 feet D) 10.9 feet 35. True or False: The given series converges. 

 k 2.5 k 1

A) False

B) True

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Final Test

36. True or False: The given series converges.  3k  5k  5k  7 k k 1 A) False B) True 37. Determine the radius of convergence for the given power series.  7 k 1 x k  k k 0 1 A) R  7 B) R   C) R  D) R  0 7 38. Find the Taylor series for the given function at the indicated point x  a.

1 f ( x)  ln(5 x); a  5 n n n 1 n   (1) 5  1 1  1 A)  C)  n  x   x  n 5 5  n 1 n 1 n5  

B)

5n  1  n  x  5  n 1

(1) n1  1  n5n  x  5  n 1 

n

D)

n

39. Consider the random experiment of rolling a fair die two times, and let X denote the random variable that counts the number of times 6 appears. Find P( X  2).

1 11 25 1 A) B) C) D) 36 36 36 12 40. A factory produces MP3 players. During regular inspections, 2% of the MP3 players are found to be defective. On any particular day, what is the probability that the twentieth MP3 player inspected is the first defective player found that day? Round your answer to four decimal places.

A) 0.6812 B) 0.0134 C) 0.0136 D) 0.9864  1  x /2 if x  0  e is a probability density function for a particular random 41. f ( x)   2 0 if x  0 variable X. Use integration to find P( X  4).

A) e2  1

B) e 2

C) e2

D) 1  e 2

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Final Test

1 if 2  x  8  is a probability density function for a continuous random 42. f ( x)   6  0 otherwise variable X. True or False: The expected value is 5.

A) False B) True

43. Suppose that during business hours, the time between successive wireless calls at a network switch can be represented by an exponentially distributed random variable X with expected value E ( X )  0.025 second. Using an appropriate probability density function, find the probability that the time between the arrival of successive wireless calls at the switch is between 0.03 and 0.065 second. Round to the nearest hundredth.

A) 0.63 B) 0.46 C) 0.77 D) 0.23 44. Suppose that the number of people suffering bee stings in a rural county in May follows a Poisson distribution with a mean of 1.5 bee stings per day. Find the probability that on a randomly selected day, there will be 2 bee stings. Round to three decimal places.

A) 0.151 B) 0.251 C) 0.201 D) 0.100 45. Evaluate the given expression.  2  cos     3  3 1 1 A) – B) – C) D) 2 2 2

3 2

46. Differentiate the given function. y  5 tan 2 x  x



  5  A) 10   sec  2 x  x  2 x  B) 5sec  2x  x  2

2

C) D)

  1 5cot    2x  x   5  10   tan 2 x  x sec 2 x  x 2 x 



 



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Final Test

47. Determine the period p, the amplitude b, the phase shift d, and the vertical shift a of the given trigonometric function f(t).   f (t )  1.5 + 5.5sin   t  1  3  A) p  6, b  –5.5, d  1, a  1.5 C) p  6, b  5.5, d  1, a  1.5 B) p  3, b  1.5, d  1, a  5.5 D) p  6, b  1.5, d  1, a  5.5 48. The average temperature t hours after midnight in a certain town is given by the formula   T (t )  66  13sin  (t  3)  . 12  At what rate is the temperature changing at noon? Round to one decimal place.

A) 2.4 degrees per hour C) 9.2 degrees per hour B) –9.2 degrees per hour D) –2.4 degrees per hour 49. A manufacturer will supply x units of a certain commodity to the market when the price is p  S ( x) dollars per unit, where S ( x)  21  2 x – 2.5sin x .

What is the average price per unit as the level of production increases from x  4 to x  7 ? (Round to the nearest cent.)

A) $32.53 per unit B) $32.00 per unit C) $31.47 per unit D) $42.47 per unit 50. A study shows that consumers in one city will buy q hundred bottles of a new perfume q  when the price is p  30  8q  sin   dollars per bottle. Find the consumer's surplus  3  for this product when 22,000 bottles are demanded and produced. Round to the nearest dollar.

A) $2,311 B) $1,932 C) $1,837 D) $847

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Final Test

Answer Key 1. B 2. A 3. D 4. B 5. D 6. A 7. A 8. D 9. C 10. B 11. B 12. C 13. D 14. D 15. C 16. C 17. C 18. B 19. C 20. C 21. D 22. D 23. D 24. B 25. A

26. A 27. D 28. A 29. B 30. B 31. D 32. B 33. B 34. D 35. B 36. B 37. C 38. A 39. B 40. C 41. C 42. B 43. D 44. B 45. B 46. A 47. C 48. D 49. C 50. B

Mid Term Test 1. An efficiency study of the morning shift at a certain factory indicates that an average worker who arrives on the job at 8:00 A.M. will have assembled f ( x)   x3  5x 2  16 x transistor radios x hours later. How many radios will such a worker assemble between 10:00 and 11:00 A.M.?

A) 18 B) 20 C) 22 D) 15 2. Find the points of intersection (if any) of the given pair of curves. y = x + 8 and y = 2x + 4 A) (12, 20) B) (4, 12) C) (1, –4) D) (0, 6)

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Chapter 1

3. Find the slope (if possible) of the line that passes through the given pair of points.

(7, 0) and (17, 19) 19 10 10 19 A) B)  C) D)  10 19 19 10 4. A company makes a certain product for $4 each and sells it for $8. If the company has overhead expenses of $10,000 per year, how many of its products must be made and sold to break even?

A) 20,000 B) 40,000 C) 2,500 D) 10,000 5. Find the indicated limit if it exists. x 2 lim x4 x  4 1 1 A)  B) Does not exist C) 4 4

D) 4

6. Differentiate: f ( x)  x6  7 A) 6 x7  7 x

B) 5x 5

C) 6 x5  7

D) 6x5

7. Find the equation of the tangent line to the graph of f ( x)  A) y 

3 3 x 4 2

B) y  

3 3 x 16 2

8. What is the rate of change of f (t )  A) –19

B) 19 C)

1 19

D) 

3 3 C) y   x  4 2

3  3 at the point  4,  . x  4 3 D) y  x 16

2t  7 with respect to t when t = 13? t 6

1 19

9. An equation for the tangent line to the curve y  ( x 5  x  1)5 at the point where x = 1 is A) y = 30x B) y = 5x + 1 C) y = 30x – 1 D) y = 30x – 29 10. The output at a certain plant is Q  0.06 x 2  0.15xy  0.05 y 2 units per day, where x is the number of hours of skilled labor used and y is the number of hours of unskilled labor used. Currently 60 hours of skilled labor and 150 hours of unskilled labor are used each day. Use calculus to estimate the change in unskilled labor that should be made to offset a 1 hour increase in skilled labor so that output will remain the same. Round your answer to two decimal places, if necessary.

A) No change C) A decrease of 1.24 hours B) An increase of 1.24 hours D) It cannot be determined

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Chapter 1

11. Find the intervals of increase and decrease for f ( x) 

8x  1 . Round numbers to two 2 x  10

decimal places, if necessary.

A) Increasing on x  0.12 and on x > 5, decreasing on 0.12 < x  5 B) Increasing on 0.12 < x  5, decreasing on x  0.12 and on x > 5 C) Increasing on x < 5 and x > 5 D) Increasing on x < 5, decreasing on x > 5 12. Determine where the graph of f ( x)  x3  3x 2  9 x  1 is concave down.

A) x > –1 B) x < –1 C) x < 1 D) x > 1 13. The second derivative test reveals that f ( x)  x 4  4 x 2  1 has A) neither a relative maximum nor a relative minimum.

B) a relative maximum and two relative minima.

C) a relative maximum only.

D) a relative minimum only.

14. Find the absolute maximum of the function f ( x)  x 3 on the interval  A) 1

B) –1

C) 0

D)

1  x  1.

2

1 8

15. The owner of an appliance store expects to sell 500 toasters this year. Each toaster costs her $6 dollars to purchase, and each time she orders a shipment of toasters, it costs $30.

In addition, it costs $4 a year to store each toaster. Assuming the toasters sell out at a uniform rate and that the owner never allows herself to run out of toasters, how many toasters should be ordered in each shipment to minimize the annual cost? (Round any fractional amounts.)

A) 87 B) 22 C) 44 D) 33 16. The fraction of television sets manufactured by a certain company that are still in working condition after t years of use is approximately f (t )  e0.2t . What fraction can be expected to fail before 2 years of use? Round your answer to two decimal places, if necessary.

A) 1.49 B) 0.91 C) 0.67 D) 0.33 17. Solve the given equation for x.

6  5  9e3 x e9 ln 9 ln 9 A) B) C)  3 3 3

D) 3e9

x x 2 18. True or False: If f ( x)  ( x  7e ) , then f   x   x  7e .

A) False B) True

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Chapter 1

19. Find the equation of the tangent line to f ( x)  4 ln x3 at x = e.

12 12 12 A) y  x  24 B) y = 12x C) y  x D) y  x  24 e e e 20. True or False: The function y = ln 6x is concave downward everywhere.

A) False B) True x3  3x 2  2 x3 3x 2 dx    2 ln | x | C x 3 2 B) True

21. True or False:  A) False

7x 22. Evaluate the following integral:  3e dx

A)

3 7x e C 7

B) 3e7 x  C

C) 21e7 x  C

D)

3 8x e C 8

23. Evaluate  x x 2  2dx .

A)

( x  2)

B)

x3  2x  C 3

2

3/2

C

C) D)

3( x 2  2)3/2 C 4 ( x 2  2)3/2 C 3

24. Sketch the region R and then use calculus to find the area of R. R is the region between the curve y  x3 and the line y = 18x for x  0.

A) 20.25 B) 4.5 C) 0 D) 81 25. Money is transferred continuously into an account at the constant rate of $1,000 per year. Assume the account earns interest at the annual rate of 8% compounded continuously. Compute the future value of the income stream over a 5 year period.

A) $12,295.62 B) $6,147.81 C) $491.82 D) $24,591.23

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Chapter 1

Answer Key 1. C 2. B 3. C 4. C 5. C 6. D 7. B 8. C 9. D 10. C 11. C 12. C 13. B 14. A 15. A 16. D 17. C 18. A 19. C 20. B 21. B 22. A 23. D 24. D 25. B

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