calculus

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) ∂2z

∂r∂θ

2. (10 points) Let f (x,y,z) = ex 2+2y2+3z2.

(a) (2 points) Let v⃗ = ⟨ 3 13

, 4 13

, 12 13

⟩ . Find Dv⃗f (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

x2 + 2y2 + 3z2 = 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)

y2 + 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

x2y2z = 1

closest to the origin.

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