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MATH 112
Final Exam March 13, 2004
Name
Student ID #
Section
Total 100
- Check that you have a complete exam. Your exam consists of one cover sheet, followed by seven problems on eight pages.
- There are multiple versions of the exam. It will be apparent if you copy someone else’s work. Students found engaging in academic misconduct will receive a 0 on this exam.
- You are allowed to use a calculator, a ruler, and one sheet of handwritten notes.
- When rounding is necessary, you may round your final answer to 2 digits after the decimal.
- We can only give you credit for computations that appear on your exam. Show all your work.
- If you use a trial and error method when an algebraic method is available, you will not receive full credit.
- Write your answers in the specified locations.
- If you need more room, use the backs of the pages and indicate to the reader that you have done so. If you still need more paper, please ask for some.
- Raise your hand if you have a question.
- You have 3 hours to complete the exam.
GOOD LUCK!!
- Freida is taking a learning test in which the time she takes to memorize items from a given list is recorded. Let M (t) be the number of items she can memorize in t minutes. The graph of her instantaneous learning rate is shown below. (NOTE: This is not the graph of M (t).)
items per minute
time (in minutes)
M ′(t)
(a) (2 points) Find all times at which the graph of M (t) has a horizontal tangent.
ANSWER: t = minutes (b) (2 points) Find all times at which the graph of M ′(t) has a horizontal tangent.
ANSWER: t = minutes (c) (4 points) Estimate the value of M ′′(50) and use it to determine whether M (t) is concave up or concave down at t = 50.
ANSWER: M ′′(50) = ; M (t) is concave at t = 50
(d) (3 points) Approximate the value of
10
M ′(t) dt.
ANSWER:
20
M ′(t) dt =
(e) (3 points) Approximately how many items can Freida memorize in the first 20 minutes?
ANSWER: items
- You sell Items. You determine that the marginal cost is M C(q) = 3q^2 − 24 q +48 (q in items, M C in dollars) and that the marginal revenue is M R(q) = 120 + 40q − 1. 2 q^2 (q in items, M R in dollars).
(a) (3 points) What quantity will yield the largest profit? (Your answer need not be a whole number of items.)
ANSWER: q = items (b) (2 points) Find the formula for total revenue, T R(q). (You may assume that T R(0) = 0.)
ANSWER: T R(q) = (c) (4 points) The total cost of producing 3 Items is $235. Find the formula for T C(q).
ANSWER: T C(q) = (d) (3 points) What is the maximum possible profit? (As always, show all work.)
ANSWER: dollars
- A car moves on a long, straight road in such a way that its position (i.e. distance) is given by a function f (t), where t is in seconds and f (t) is in feet. We don’t have a formula for f (t), but we know that for all a and h, f (a + h) − f (a) = 2h^2 + 4ah + 5h.
(a) (3 points) What is the car’s net change in position (how far did the car go) between the times t = 2 and t = 3?
ANSWER: feet (b) (4 points) What is the car’s average speed over the interval from t = 1 to t = 4?
ANSWER: feet per second (c) (4 points) If we know that at time t = 0 the car’s position is f (0) = 1, what is f (4)?
ANSWER: feet (d) (3 points) Compute f ′(1).
ANSWER: f ′(t) = (e) (3 points) Find a formula for f ′(t).
ANSWER: f ′(t) =
(d) You know the value of a function h(x) at three different values of x.
x h(x) 1 1. 1.3 1. 1.5 1.
Estimate the value of h′(1).
(e) Evaluate the anti-derivative
t^4 /^5 − 9 +
t
dt.
(f) Find the area of the following region.
y = −x^2 + 10x + 12
- Below are the graphs for total cost and total revenue from running a business that produces and sells toy cars. The quantity is written as q and is in units of hundreds of cars. You are not given formulas for the T C and T R functions.
hundreds of dollars
quantity (in hundreds of cars)
TR
TC
(a) (3 points) Estimate the cost of the 301st car.
ANSWER: $
(b) (3 points) Are there values of q at which marginal revenue is positive? ANSWER: (circle one) yes no If yes, give the largest interval on which marginal revenue is positive. ANSWER: from q = to q = (c) (3 points) Are there intervals over which marginal revenue is increasing? ANSWER: (circle one) yes no If yes, give the largest interval over which marginal revenue is increasing. ANSWER: from q = to q = (d) (2 points) Estimate the quantity that maximizes profit.
ANSWER: q = (e) (2 points) Give the largest interval over which M R > M C.
ANSWER: from q = to q =
