Name:

Math 122 Test 3Directions: Please write your answers, with all appropriate work, in the spaces

provided. Answers given without work shown will be counted incorrect. You may usethe backs of the pages if you require some space for scratchwork or brainstorming.

1. Suppose that the revenue function for a company selling otter plushies isR(q) = 40

q, while the cost function is C(q) = 2q + 5. For what value of q is

the profit function for this company maximized? (8 pts)

2. A rectangular fenced-in pen is being built in the corner of a barn, as illustratedbelow. Given that the farmer has 20 feet of fencing available with which to constructthis pen, what dimensions will maximize the area? (8 pts)

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3. Suppose that the cost function for an item is C(q) = q2 q + 1.(a) What is the average cost of producing 10 items? (7 pts)

(b) What is the marginal cost at q = 10? (7 pts)

(c) Is the average cost minimized at q = 10? (no work need be shown here) (5pts)

4. Estimate

20

x dx using:

(a) a left Riemann sum with 4 subintervals. (7 pts)

(b) A right Riemann sum with 4 subintervals. (7 pts)

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5. Integrate each of the following: (5 pts each)

(a)

xn dx, n 6= 1

(b)

1

xdx

(c)

ex dx

6. Suppose that the velocity function for a vehicle is given by v(t) = 80t+1

, where v isin miles per hour and t is in hours. How far does the vehicle travel from t = 0 tot = 4 hours? Give an exact answer. (Hint: 5, part (b)) (7 pts)

7. Given n 2, which is larger:1

0

xn dx, or

10

xn+1 dx? (7 pts)

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8. Integrate each of the following: (7 pts each)

(a)

xex

2

dx

(b)

(4x + 18)2014 dx

9. What is the area bounded by the curves f(x) = x and g(x) = x9 in the firstquadrant? (7 pts)

10. (Bonus) Integrate one or both of the following: (5pts each)

(a)

1

x ln(x)dx

(b)

x

x + 1dx

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