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M 408C
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Gary, Berg
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Mathematics

M 408C

Gary, Berg

Fall

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taboada (lat2278) – HW11 – berg – (56260) 1
This print-out should have 19 questions.
∂z sye − xs
Multiple-choice questions may continue on 4. =
the next column or page – ﬁnd all choices ∂t yet
before answering.
∂z 5sye2t− xs
5. = t
001 10.0 points ∂t ye
t
Use the Chain Rule to ﬁnddw when ∂z 5sye + xs
dt 6. ∂t = y et
w = xe y/z
003 10.0 points
and Find ∂z when
∂x
x = t , y = 1 − t, z = 1 + 3t.
4y 2x
xe + yz + ze = 0.
dw x 3xy
1. = 2t − − 2 ey/z
dt z z ∂z e4y+ 2ze2x
1. = − 2x
2. dw = t − x − xy ey/z ∂x e + y
dt z z 2x
∂z e − y
dw x xy 2. ∂x = e4y + 2ze2x
3. = t + + 2 ey/z
dt z z 4y 2x
3. ∂z = − e + 2ze
4. dw = t + x + xy ey/z ∂x e2x− y
dt z z
∂z e2x+ y
dw x 3xy 4. = −
5. = 2t + + 2 ey/z ∂x e4y+ 2ze2x
dt z z
∂z e2x+ y
6. dw = 2t − x− 3xy ey/z 5. = 4y 2x
dt z z ∂x e + 2ze
4y 2x
∂z e + 2ze
002 10.0 points 6. ∂x = e2x+ y
∂z
Use the Chain Rule to ﬁnd when
∂t 004 10.0 points
z = x , The temperature at a point (x, y) in the
y plane is T(x, y) C. If a bug crawls on the
and plane so that its position in the plane after t
minutes is given by (x(t), y(t)) where
x = 5se , y = 1 + se−t .
√ 1
x = 1 + t, y = 4 + t,
2t 3
∂z 5sye + xs
1. ∂t = y e t determine how fast the temperature is rising
on the bug’s path at t = 3 when
∂z sye + xs
2. =
∂t y et Tx(2, 5) = 16, T y2, 5) = 6.
∂z sye − xs
3. = t 1. rate = 7 C/min
∂t ye taboada (lat2278) – HW11 – berg – (56260) 2
◦
2. rate = 4 C/min 007 10.0 points
◦ Determine the gradient of
3. rate = 6 C/min
f(x, y) = xy 2
4. rate = 3 C/min
at the point P(3, −2).
5. rate = 5 C/min
1. ∇f|P= 4i + 12j
005 10.0 points
2. ∇f| = 4i − 12k
The radius of the base of a right circular P
cone is increasing at a rate of 3 ins/min while
its height is increasing at a rate of 4 ins/min.. ∇f|P= 12j + 4k
At what rate is the volume of the cone
changing when the radius is 3 inches and the 4. ∇f|P= 12i + 4k
height is 5 inches?
5. ∇f|P= 12j − 4k
1. rate = 42π cu. ins/min
6. ∇f| = 4i − 12j
P
2. rate = 43π cu. ins/min
008 10.0 points
3. rate = 40π cu. ins/min
The contour map given below for a function
4. rate = 41π cu. ins/min
f shows also a path r(t) traversed counter-
clockwise as indicated.
5. rate = 44π cu. ins/min
006 10.0 points
Find the directional derivativv, f , of 10
p 2
f(x,y) = 2x + 3y 3
Q
-3
at the point (6, 2) in the direction -2
v = i + j. P -1
0
R
7
1. fv = 12
3
2. fv = Which of the following properties does the
4 derivative
d
1 f(r(t))
3. fv = dt
12 have?
4. f = 5 I positive at R,
v 12
II positive at P,
1
5. fv = III positive at Q.
4 taboada (lat2278) – HW11 – berg – (56260)

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