APPM 2350 EXAM 2 FALL 2007
INSTRUCTIONS: Computers, calculators, books, and crib sheets are not permitted. Write your (1)
name, (2) instructor’s name, and (3) recitation number on the front of your bluebook. Work all problems.
Show your work clearly. Note that a correct answer with incorrect or no supporting work may receive no
credit, while an incorrect answer with relevant work may receive partial credit.
1. (20 points) Using an appropriate technique from Calculus III, determine the extreme values of the
function f(x, y) = x2+y2
4, assuming that xy = 1. For each point you find, be sure to state whether
the function has a local maxima or minima at that location, and the value of f(x, y).
2. (20 points) The surface elevation, z, of a small mountain area that you have just purchased is given
by the function z=−y2−x4+ 4x2+ 54 for −3≤x≤3 and −3≤y≤3.
(a) Where on this property would you build a hut? (Hint: perhaps a level spot that is not a summit.
Clearly explain your reasoning, and show your calculations, to support your answer.)
(b) A stream emerges from a spot on the surface located at P0(0,1,53) and flows downhill following
the line of steepest descent. At the point P0, in which direction does the stream flow? (Give
your answer as a vector with i,j, and kcomponents.)
(c) By approximately how much has the water in the stream descended after it has travelled a short
distance s= 0.2 from the source P0?
3. (20 points) A hummingbird is flying along the path r(t) during the evening when the temperature
distribution is given by the function T(x, y, z). At some time t∗(and only at this particular time),
you know that the hummingbird’s position, velocity and acceleration are r(t∗) = 2 i+ 4 j+ 3 k,
v(t∗)=2i+ 2 j+5 k, and a(t∗)=4i+3 j+ 4 krespectively. Furthermore, you know that ∇T|(2,4,3) =
2i+ 1 j+ 2 k, and T(2,4,3) = 42.
(a) As the hummingbird flies past the location r(t∗) at what rate is the temperature Tchanging
with respect to distance?
(b) As it flies past location r(t∗), at what rate is the temperature Tchanging with respect to time?
(c) If the hummingbird continues on its original path r(t) for a short interval of time ∆t= 0.1, by
approximately how much does the temperature change?
(d) However, suppose at time t∗something suddenly scares the hummingbird and it starts to fly
in a direction that happens to be the direction of the greatest rate of decrease of T. Assuming
the hummingbird maintains its same speed, by approximately how much does the temperature
change if it flies for ∆t= 0.1?
4. (20 points) Consider the function f(x, y) = x+y+ cos(x+y).
(a) Calculate the second order Taylor approximation to f(x, y) near the origin.
(b) Now, calculate the first order approximation to f(x, y)near the point Plocated at π
4,π
4,
(c) Estimate the error associated with your linearization in part (b) assuming that you only use it
for values of xand ysuch that
x−π
4
≤0.2 and
y−π
4
≤0.2.
OVER
