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Lecture

# Calc Hw 11.pdf

5 Pages
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Department
Mathematics
Course
M 408C
Professor
Gary, Berg
Semester
Fall

Description
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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