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MATH 112
Final Exam June 4, 2005
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
- Check that your exam contains seven problems on nine pages.
- Turn your cell phone OFF and put it away for the duration of the exam.
- Unless otherwise indicated, you must show your work. The correct answer with 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.
- Put your name on your sheet of notes and turn it in with the exam.
GOOD LUCK!
- (12 points) Compute the indicated derivative. Do not simplify.
(a) y = ln(3x^2 + 4) 8 x^6
dy dx
(b) f (x, y) = xy^2 e−^10 y
fx(x, y) =
fy(x, y) =
- (12 points) Below is a blow-up of the graph of f (x) near x = a, along with the tangent line to f (x) at x = a and a secant line from x = a to x = a + h. Each of the following expressions belongs in one of the blanks below. Put each item in the correct blank.
I. f (a) II. f ′(a) III. f (a + h) IV. h V. f (a + h) − f (a)
VI.
f (a + h) − f (a) h
this distance
this distance
coordinates of this point (a, )
slope of this line
slope of this line
(a + h, ) coordinates of this point
6 6
- (10 points) Pete’s Pink Paints Shop offers two shades: Blush and Bashful. Each shade is made by mixing white paint with red paint. Blush is 75% white and 25% red; Bashful is 50% white and 50% red. Each week, Pete receives 800 gallons of white paint and 600 gallons of red paint from his supplier. He earns $6 profit on each gallon of Blush and $4.75 profit on each gallon of Bashful. Let x be the number of gallons of Blush and y be the number of gallons of Bashful Pete produces and sells each week.
(a) Give a formula for W (x, y), the number of gallons of white paint Pete needs in order to make x gallons of Blush and y gallons of Bashful.
W (x, y) = (b) Give a formula for R(x, y), the number of gallons of white paint Pete needs in order to make x gallons of Blush and y gallons of Bashful.
R(x, y) = (c) How many gallons of each shade should Pete produce in order to maximize profit? Use the method of linear programming and show all your work.
ANSWER: Blush: x = gallons Bashfull: y = gallons
Again, the rate of flow for Vat A is
a(t) = 3t^2 − 36 t + 96
and the amount in Vat B is B(t) = −t^3 + 55. 5 t^2 + 97.
(c) At t = 0, Vat A contains 25 gallons more than Vat B. Write out a formula for the amount in Vat A.
ANSWER: A(t) = (d) At what time is the level in Vat B rising most rapidly? (That is, at what time is the rate of flow into Vat B largest?)
ANSWER: t = hours (e) What is the highest level the water reaches in Vat B during the interval from t = 0 to t = 50?
ANSWER: gallons
- (16 points)
To the right are rough sketches of two functions:
f (x) = − 5 x^2 + 82x + 100 and
g(x) = −x^3 + 14. 25 x^2 + 15.
f (x)
g(x)
(a) Find a value of x at which the slope of the tangent line to f (x) is 66.
ANSWER: x =
(b) Define a new function h(x) by h(x) = g(x) x
. Is h(x) concave up or down at x = 2?
ANSWER: (circle one) concave up concave down (c) Let D(x) = f (x) − g(x). Find the smallest value of x at which the graph of D(x) has a horizontal tangent.
ANSWER: x = (d) Use the Second Derivative Test to determine whether D(x) has a local maximum or a local minimum at the value you found in part (c).
ANSWER: (circle one) local max local min
(c) Complete the following table. interval 0–1 1–2 2–3 3–4 4–5 5–6 6– area between M R and M C 6 10.5 15 12 on that interval
(d) Determine the change in profit that results from increasing quantity from q = 3 thousand Lugbos to q = 6 thousand Lugbos. Is this an increase or a decrease in profit?
ANSWER: Profit (circle one) increases decreases
by thousand dollars (e) Fixed cost is $16,500. That is, F C = 16.5 thousand dollars. Name the smallest quantity at which profit is equal to 0.
ANSWER: q = thousand Lugbos (f) What is the largest possible profit? (Again, F C = 16.5.)
ANSWER: thousand dollars
