MATH 23: Calculus III – Unit 2 Exam Fall Semester 2006
Instructions. Attempt all questions. Answers must be justified in order
to gain full credit. Calculators are not permitted.
1. Let f(x, y, z)=xeysin z.
(a) Find a vector at the point (0,0,π/2) pointing in the direction in which f
(i) (4 points) Increases fastest (ii) (2 points) Decreases fastest.
(b) (4 points) Is there a direction at the point (0,0,π/2) in which the function does not change
initially? If so, find a vector pointing in that direction.
2. (5 points) Find the directional derivative of f(x, y, z )=xy +z2at (1,1,1) in the direction
of
i+2
j+3
k.
3. (4 points) Use the chain rule to find ∂z/∂u and ∂z/∂v where
z=cos(x2+y2)with x=ucos vand y=usin v
4. (4 points) Find the quadratic Taylor polynomial Q(x, y)about (0,0) for the
function f(x, y)=ln(1+x2−y).
5. Let
f(x, y)=⎧
⎨
⎩
xy
√x2+y2,(x, y)=(0,0)
0,(x, y)=(0,0)
(a) (5 points) Is fdifferentialable at all points (x, y )=(0,0)? Explain.
(b) (5 points) Is fdifferentiable at (0,0)? Explain.
6. A closed rectangular box has volume 32 cm3.
x
y
z
(a) (3 points) Find an expression for the surface area S(x, y)of the box.
(b) (2 points) What is the domain of S?
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