Quiz 1
Math 263
Spring 2022
Instructions: Show work! You must convince me that you understand how to do the
problem and how the answer was obtained to get credit. In particular, answers (even
correct ones) unsupported by understandable work will receive no credit.
1. (10 points) If z = f (x,y), where x = r cosθ and y = r sinθ, find
(a) (3 points)
∂z
∂r
(b) (3 points)
∂z
∂θ
(c) (4 points)
∂
2
z
∂r∂θ
2. (10 points) Let f (x,y, z) = e
x
2+2y
2+3z
2
.
(a) (2 points) Let ⃗v =
⟨
3
13
,
4
13
,
12
13⟩
. Find D⃗v
f (1,1,1).
(b) (3 points) In which direction is f increasing fastest at the point (1,1,1)?
(c) (2 points) What is the rate of increase of f in the direction of fastest increase?
(d) (3 points) Find the equation of the plane tangent to the ellipsoid
x
2 + 2y
2 + 3z
2 = 6
at the point (1,1,1).
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3. (10 points) Suppose f : R 7→ R is continuous and f (0) = 0. Consider the limit
lim
(x,y)→(0,0)
yf (x)
y
2 + f (x)
2
.
Does this exist for all such f , not exist for any such f , or exist for some such f ?
Explain.
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4. (10 points) Find the points on the surface
x
2
y
2
z = 1
closest to the origin.
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