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Math 112 - Winter 2006
Exam 2
February 23, 2006
Name:
Section:
Student ID Number:
TA’s Name:
Total 50
- Check that your exam contains all the problems listed above.
- You must show your work on all problems. The correct answer with no supporting work may result in no credit.
- You may use a calculator for arithmetic purposes only (such as plugging into the quadratic formula or plugging into a function). ALL other work must be written and demonstrated on your exam. No credit will be given for guess and check or calculator methods, even if they give the correct answer.
- If you need more room, use the backs of the pages and indicate to the grader that you have done so.
- There are multiple versions of the exam. Any student found engaging in academic miscon- duct will receive a score of 0 on this exam.
GOOD LUCK!
- (10 points) Compute the following derivatives and partial derivatives as indicated. The cor- rect answer with no supporting work receives no points. You do not have to simplify your final answer.
(a) (5 points) Let g(x) =
5 ln(x) + 1 x^2
. Find g′(x).
ANSWER: g′(x) = (b) (5 points) Let f (x, y) = x^2 e^2 x^ + y^3 x. Find fx(x, y) and fy(x, y).
ANSWER: fx(x, y) = fy(x, y) =
- (10 points) Consider the function
f (x) = 140 ln(x) + x^2 − 47 x + 120.
(a) (5 points) Find all critical numbers of f (x).
ANSWER: list of critical numbers: x = (b) (5 points) Use the second derivative to determine whether each of the critical numbers of f (x) give a local minimum, local maximum, or neither. Clearly indicate your answers. (If you classify a critical number without using the second derivative and showing your work, you will receive no credit.)
- (10 points) Suppose p = G(r, s) = 2s^2 − 6 rs + 12r.
(a) (2 points) Find the partial derivatives Gr(r, s) and Gs(r, s).
ANSWER: Gr(r, s) = Gs(r, s) = (b) (4 points) Find all candidates for local maximum and local minimum of G(r, s).
ANSWER: list of candidates: (r, s) = (c) (4 points) If you fix r to be 3, then p = G(3, s) becomes a function of only one variable, the variable s. Find all critical numbers for the function G(3, s).
ANSWER: s =
