Name:
APPM 2350 Exam #2 Summer 2008
Be sure to include your name and a grading table on the front of your blue book. You must
work all of the problems on this exam. Show ALL of your work and BOX IN YOUR
FINAL ANSWERS. A correct answer with no relevant work may receive no credit, a
wrong answer with no work will receive no credit, and an incorrect answer accompanied
by some correct work may receive partial credit. Text books, class notes, crib sheets, cell
phones, calculators, or electronic devices of any kind are NOT permitted. Please clearly
indicate the start of each new problem. Good luck!
1. (20 points) Consider f(x, y) = x2−y2
xy
(a) Where is f(x, y) continuous?
(b) What is lim
(x,y)→(0,0) f(x, y)?
(c) Find all first partial derivatives of f(x, y). Simplifying your answers will be help-
ful.
(d) If x(θ) = sin(2θ) and y(θ) = cos(2θ), find df
dθ .
2. (20 points) Let f(x, y) = sin(x−y).
(a) Determine the linearization of f(x, y) at the origin.
(b) Bound the error in this approximation if −1
5≤x≤1
5and −1
5≤y≤1
5.
(c) What is the quadratic approximation to f(x, y) at the origin?
3. (20 points) A large heated storage room is shaped like a rectangular box and has a
volume of 8000 cubic meters. Because warm air rises, the heat loss per unit area
through the ceiling is five times as great as the heat loss through the floor. If the
heat loss through the four walls is three times as great as the heat loss through the
floor, determine the room dimensions that will minimize heat loss and thus minimize
the heating costs. In order to receive full credit for this question, you must
demonstrate the solution minimizes the heat loss by evaluating the objective function
at another point on the constraint-surface.
