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Math 112 Final Exam March 14, 2015
Name
Student ID # Section
HONOR STATEMENT
“I affirm that my work upholds the highest standards of honesty and academic integrity at the University of Washington, and that I have neither given nor received any unauthorized assistance on this exam.”
SIGNATURE:
Total 100
- Your exam should consist of this cover sheet, followed by 8 problems on 8 pages. Check that you have a complete exam.
- Turn your cell phone OFF and put it away for the duration of the exam.
- You may not listen to headphones or earbuds during the exam.
- Unless otherwise indicated, you must use the methods of this course and show all of your work. The correct answer with little or no supporting work may result in no credit. If you use a guess-and-check method when an algebraic method is available, you may not receive full credit.
- Follow the instructions on individual questions on how to round your answers.
- You have signed an honor statement. Do not cheat.
GOOD LUCK!
- (12 points) Below is the graph of the derivative of f (x). Answer the following questions about f (x).
(a) In which interval(s) is the function f decreasing? Give your answer(s) in the form a < x < b.
(b) In which interval(s) is the graph of the function f concave down? Give your answer(s) in the form a < x < b.
(c) List all values of x for which f (x) is a relative maximum.
(d) List all x-values where the graph of f (x) has an inflection point.
(e) Estimate the average rate of change of f from x = 3 to x = 8.
(f) List the following three quantities from the smallest to the largest: f (19), f (23), f (27.5). Give your answer in the form f (a) < f (b) < f (c).
- (14 points) Differentiate the following functions. You do not have to simplify your answers.
(a) f (x) = e−^0.^2 x(4x − 1)^5
(b) g(t) = ln
t + 5
√t+
(c) Compute both fx and fy for f (x, y) =
2 x − 3 y 5 x + 6y
- (11 points) A bakery makes Apple and Cherry pies. The profit from producing and selling x Apple pies and y Cherry pies is given by the joint profit function
P (x, y) = − 2. 5 x^2 + 3xy − y^2 − 4 x + 7y − 13 ,
where P (x, y) is in dollars.
(a) This function is maximized at its critical point. What is the maximum profit
(b) The bakery has an order for 23 pies for Pi-Day. They have to make 10 Apple and 12 Cherry pies, but they are free to pick the type of the 23rd. Should they make an Apple pie or a Cherry pie? Use partial derivatives to explain your choice.
- (14 points) The following are graphs showing the statistics for a 900 day interval for the port of Constantinapolis starting at the beginning of 1451. The number of ships coming in per day is given by f (t) = 0. 185 t^3 − 3 t^2 + 16. 75 t + 38. The number of ships leaving the port is given by g(t) = − 0. 7 t^2 + 11. 3 t + 31. Initially there were 35 ships at the port. Be careful with units.
(a) Use the graph to estimate the number of ships which have come into the port in the first 100 days. Show on the graph what you computed. Give units with your answer.
(b) Compute the average number of ships leaving the port per day during the first 200 days using the equations given. Round your answer to the nearest ship.
(c) Say in words what the quantity
1
(f (t) − g(t)) dt represents. Do not compute it.
(d) Use the graph to estimate when the number of ships at port reaches its maximum value. Use integration to find the total number of ships at port that day. Round your answer to the nearest ship.
- (17 points) You produce and sell Chorks. The total revenue, in hundreds of dollars, for selling x-hundred Chorks is given by the function T R(x). You are not given the formula for T R(x), but you know that, for any value of h,
T R(x + h) − T R(x) = 4. 95 h − 0. 1 xh − 0. 05 h^2.
(a) Compute the Marginal Revenue M R(x) and the Total Revenue T R(x).
(b) Your Marginal Cost is given by M C(x) = 3 +
x + 1
and your fixed costs are $1000. What is your Total Cost function T C(x) in hundreds of dollars where x is in hundreds of Chorks? Be careful with the units.
(c) Find all positive values of x at which the graphs of T R and T C have parallel tangent lines. Do these value(s) of x give a local minimum or local maximum for the Profit function? Explain.
(d) Find the maximum profit. Round your answer to the nearest dollar.
