Name(s):
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Math 148 Lab Assignment 9: §10.5–10.6
Directions: You may work in groups of 2–3 to complete this assignment. Answer
each question completely. Show all work to receive full credit, and circle your
final answer.
1. Let z = xex/y , where x = cos t and y = e2t . Find
dz
dt
when t = 0.
2. Find the directional derivative of f (x, y) = x2 y 3 −4y at the point (2, −1) in the direction
of the vector ~v = h2, 5i.
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3. Find the gradient of each function.
(a) f (x, y) = x3 − 4x2 y + y 2
(b) f (x, y) =
p
x2 + 2y
(c) f (x, y, z) = xeyz + xyez
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4. Suppose that over a certain region of space the electrical potential V is given by
V (x, y, z) = 5x2 − 3xy + xyz.
(a) Find the rate of change of the potential at P = (3, 4, 5) in the direction of the
vector ~v = h1, 1, −1i.
(b) In which direction does V change most rapidly at P ?
(c) What is the maximum rate of change at P ?
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5. Find the local extrema and saddle points of f (x, y) = x4 + y 4 − 4xy + 1.
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