Math 212 Multivariable Calculus - Midterm I
October 4th, 2002
Instructions: This is a closed book, closed notes exam. Use of calculators is not permitted. Show
all your work for a full credit.
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Problem Max Points Your Score
Problem Max Points Your Score
1
10
4
15
2
20
5
20
3
15
6
20
Total
100
1
(1) Compute the volume of the parallelopiped with sides 2i + j − k, 5i − 3k, and i − 2j + k.
(2) Find the equation of the curve obtained by intersecting θ = 3π/4 and x − 2y + z = 2.
(3) Let f (u, v, w) = (eu−w , cos(v + u) + sin(u + v + w)) and g(x, y) = (ex , cos(y − x), e−y ).
Calculate f ◦ g and D(f ◦ g)(0, 0).
(4) Let f (x, y) = xy.
(a) In which direction from (1, −1) does f increase the fastest?
(b) Compute the directional derivative of f along v = (0, 1) at (1, −1).
(5) Consider the surface given by z = f (x, y) = x7 y + xy 7 + xy.
(a) Show that f is differentiable everywhere.
(b) Find the critical points of f and classify them.
(c) Find an equation of the tangent plane at each critical point.
(d) Compare the surface with z = xy and explain why they have the same type of
critical points.
(6) Let S be the closed and bounded surface obtained by intersecting x2 + y 2 ≤ 1 and
x + y + z = 1. Find the absolute maximum and the absolute minimum of f (x, y, z) =
x2 + 2y 2 + zy − y + 1 on S.
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