Final v1 04

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```					Spring 03
1. The table below shows the number of NCAA Division I men's basketball games. If
the average rate of change continues at the same rate as between 2000 and 2003, project
the number of games in the year 2007.

Year                 2000      2001      2002        2003
Number of games         8773      8967      9024        9103

A) 10341 games
B) 9873 games
C) 9543 games
D) 11083 games
E) 9419 games

2. Let H ( x)  ( g ( x)) 4 . Given g (5)  3 and g ' (5)  4 , find H '(5).

A) -432
B) -1728
C) -108
D) -48
E) 256

3. An animal population grows exponentially. At time t = 0 there are 500 animals.
Two years later, there are 800. Find the population of animals at time t = 3. Round your
answer to the nearest whole number.

A) 2048 animals
B) 1100 animals
C) 950 animals
D) 1012 animals
E) 2150 animals
4. After drinking a cup of coffee, the amount Q of caffeine in the body is given by a
function Q = f(t) where Q is in milligrams and t is the time in hours after drinking the
coffee. When t = 4 there are 50 milligrams, and when t = 4 Q is changing at a rate of
-2 milligrams per hour. Estimate the amount of caffeine in the body 4 hours and 15
minutes after drinking the coffee.

A) 80.0 milligrams
B) 47.0 milligrams
C) 49.8 milligrams
D) 49.5 milligrams
E) 20.0 milligrams

5. Find the area between y  3x 2  2 x  3 and y  8 x  6 .

A) 30
B) 30.5
C) 31
D) 31.5
E) 32

6. A rumor spreads among a group of 600 people. The number of people, N(t), who
have heard the rumor by time t in hours since the rumor started to spread can be
600
approximated by a function of the form N (t )                   . Approximately when is the
1  599e 0.5t
rumor spreading fastest, i.e., for which value of t does the derivative, N ’(t), have the
maximum value? Round your answer to the nearest whole number

A) t = 13
B) t = 14
C) t = 15
D) t = 16
E) t = 17
The problems 7 and 8 use the following graph of the derivative f ' ( x) of a function
f(x).

y

y=f'(x)

0       1       2      3          4       5        6     7       8   9    10   x

7. Select a correct statement.

A) f(x) has a local maximum at       x = 1 only.
B) f(x) has a local maximum at       x = 1 and x = 7 only.
C) f(x) has a local maximum at       x = 5 only.
D) f(x) has a local maximum at       x = 9 only.
E) f(x) has a local maximum at       x = 3 only.

8. Select a correct statement

A) f(x) has an inflection point at   x = 1, x = 5, and x = 9 only.
B) f(x) has an inflection point at   x = 1 only.
C) f(x) has an inflection point at   x = 2 and x = 5 only.
D) f(x) has an inflection point at   x = 1 and x = 7 only.
E) f(x) has an inflection point at   x = 1, x = 3, and x = 7 only.
9. The total cost of producing q units of a product is given by C (q)  5000  0.02 q 2 ; the
product sells for \$160 per unit. What production level maximizes profit?

A) 4000 units
B) 5160 units
C) 912 units
D) 8000 units
E) 4840 units

10. You run a small furniture business. You sign a deal with a customer to deliver up to
500 chairs, the exact number to be determined by the customer later. The price will be
\$110 per chair up to 300 chairs, and above 300, the price will be reduced by \$.2 per chair
(on the whole order) for every additional chair over 300 ordered. What is the largest
revenue your company can make under this deal?

A) \$36,000
B) \$36,750
C) \$36,125
D) \$425
E) \$36,220

11. The marginal cost of drilling an oil well depends on the depth at which you are
drilling: drilling becomes more expensive, per meter, as you dig deeper into the earth.
The fixed cost is \$200,000, and if x is the depth in meters, the marginal costs are
C ' ( x)  400  8 x dollars/meter. Find the total cost of drilling a 500 meter well.

A) \$1,200,000
B) \$1,300,000
C) \$1,400,000
D) \$2,000,000
E) \$1,500,000
8
12. The graph of a function, f(x), is given below. Find   
0
f ( x)dx .

A) -1
B) -2
C) 5
D) 10
E) -4

y
3

y=f(x)
2

1

0
x
0                2                  4                          6        8

-1

-2

-3

13. The half-life of radioactive strontium-90 is 29 years and the initial amount is 10
grams. How long will it be until only 20% remains? Round your answer to the nearest
100th.

A) 49.76 years
B) 67.34 years
C) 146.71 years
D) 69.83 years
E) 93.65 years
14. A bicyclist is pedaling along a straight road for one hour with a velocity, v, shown in
the graph below. She starts out 50 kilometers from the lake and positive velocities take
her towards the lake. When is she closest to the lake?

A) after 1 hour
B) after 2 hours
C) after 3 hours
D) after 4 hours
E) after 5 hours

20        v

Toward lake

10

Away          0
0            1              2             3              4              5   t
from lake

-10

15. Oil is leaking out of a ruptured tanker at the rate of r (t )  2t  3 thousand liters per
minute. Using a Riemann sum with four intervals and evaluating at left endpoints,
estimate the amount of oil leaked during the first 12 minutes.

A) 48 thousand liters
B) 180 thousand liters
C) 132 thousand liters
D) 144 thousand liters
E) 192 thousand liters
16. A hot metal bar is allowed to cool. At time, t, minutes its temperature, H, in degrees
Celsius is given by H  20  980e 0.3t . Find the average temperature of the bar over the
first 10 minutes. Round your answer to the nearest whole number.

A) 330 degrees
B) 3304 degrees
C) 369 degrees
D) 688 degrees
E) 229 degrees

17. A new machine costs \$60,000 and earns profit at the continuous rate of \$20,000 per
year. How long will it take for the present value of the profit to equal the cost of the
machine? Use an interest rate of 4% per year, compounded continuously. Round your
answer to the nearest 100th.

A) 3.08 years
B) 3.92 years
C) 3.20 years
D) 2.94 years
E) 3.16 years

18. At what constant, continuous rate must money be deposited into an account if the
account is to contain \$30,000 in 6 years? The account earns 5% interest, compounded
continuously. Round your answer to the nearest dollar.

A) \$3,918
B) \$3,704
C) \$22,225
D) \$4,287
E) \$23,506
q  50
19. For a product, the supply curve is p           and the demand curve is
3
100  q
p           , where the price is in units of dollars/item. Using the equilibrium price and
2
quantity, find the consumer surplus.

A) \$400
B) \$200
C) \$50
D) \$40
E) \$250

For problems 20-25 write your answers on the front page. Only the answers written
on the front page will be graded.
20. Let f ( x)  (2 x 3  1)( x 2  K ) , where K is a constant. Given f ' (1)  5 , find K.
21. Let f ( x)  e x x 4  2 . Find f ' ( x) .

22. Find the x-value of the global maximum and the x-value of the global minimum of
f ( x)  x 3  3x 2 on the interval  0.9  x  3.1 .

23. Let b  1 be a constant. Use the Fundamental Theorem of Calculus to evaluate
b 2    4
1 ( x  x 3 )dx .
24. Find f (x) if f ' ( x)  9e 3 x  1 and f (0)  2 .

2e x
25. Find the indefinite integral       ex  3
dx .

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