APPM 2350: Section exam 2, version A
June 29, 2012.
ON THE FRONT OF YOUR BLUEBOOK write: (1) your name, (2) your instructor’s name, and
(3) a grading table. Text books, class notes, and calculators are NOT permitted.
Problem 1: (25 points) Short Answers:
(a) True of False: If f(x, y) is a surface function and fx(a, b) = 0 and fy(a, b) = 0, then f(x, y )
has either a local minimum or local maximum at (a, b). Explain.
(b) Level curves of the function z=f(x, y) = x
1+x2+y2are circles. Find the center and radius of
the circle defined by the level curve for z=1
4.
(c) Find the second-order Taylor polynomial for f(x, y) = exsin yat the origin.
Problem 2: (25 points) After an oil spill occurs at the origin, the density of oil in the ocean water
is described by the function
Density = f(x, y) = 1
1 + x2+y2.(1)
Freddie the fish swims from (0,2) to (2,0) via the path ~r(t) = ht, 2−ti.
(a) Using the chain rule, find the rate of change of the density of oil along the path.
(b) At what point (in (x, y) coordinates) along the path is the density maximum and what is the
maximum?
(c) Integrate your answer from part (a) to obtain f(t) along the path.
Hint: you can check your answers to parts (a) and (c) by plugging x(t)and y(t)into Eq. (1), but
your final answer for (a) must be obtained by the using the chain rule and your final answers for
(b) and (c) must use your answer for part (a).
Problem 3: (25 points) Consider the function f(x, y) = x2y−yon the domain D={(x, y)
x2+
y2≤4}.
(a) Locate and classify all critical points on the interior of D.
(b) Use Lagrange multipliers to locate the extreme values of f(x, y) on the boundary of D.
(c) What are the absolute maximum and absolute minimum values of f(x, y) on D, and where
are they located?
Problem 4: (25 points) A bug walks on a surface Sthat is the graph of f(x, y) = x4+x3y−4x.
At some time t∗(and only at this time), the~
iand ~
jcomponents of the bug’s position and velocity
are ~r(t∗) = 2
~
i+~
jand ~v(t∗) = 3
~
i−4~
j.
(a) Compute a normal vector to the surface Sat the bug’s location.
(b) What is the slope of the surface along the bug’s path at time t∗?
(c) In what direction should the bug head so the elevation decreases most rapidly?
