(a) Explanation Theorem used: “Suppose f is a differentiable function of two variables. The rate of change is defined as D u f ( x , y ) = ∇ f ( x , y ) ⋅ u where u is unit vector.” Calculation: The given temperature function is, T ( x , y , z ) = 200 e − x 2 − 3 y 2 − 9 z 2 . The direction vector P Q → form the point P ( 2 , − 1 , 2 ) towards Q ( 3 , − 3 , 3 ) is, P Q → = 〈 3 − 2 , − 3 − ( − 1 ) , 3 − 2 〉 = 〈 1 , − 2 , 1 〉 Then the unit vector u is computed as follows, u = 1 1 2 + ( − 2 ) 2 + 1 2 〈 1 , − 2 , 1 〉 = 1 1 + 4 + 1 〈 1 , − 2 , 1 〉 = 1 6 〈 1 , − 2 , 1 〉 Thus, the unit vector u is 1 6 〈 1 , − 2 , 1 〉 . The gradient of T ( x , y , z ) is as follows. ∇ T ( x , y , z ) = 〈 T x , T y , T z 〉 = 〈 ∂ ∂ x ( 200 e − x 2 − 3 y 2 − 9 z 2 ) , ∂ ∂ y ( 200 e − x 2 − 3 y 2 − 9 z 2 ) , ∂ ∂ z ( 200 e − x 2 − 3 y 2 − 9 z 2 ) 〉 = 〈 [ 200 ( e − x 2 − 3 y 2 − 9 z 2 ( − 2 x ) ) ] , [ 200 ( e − x 2 − 3 y 2 − 9 z 2 ( − 6 y ) ) ] , [ 200 ( e − x 2 − 3 y 2 − 9 z 2 ( − 18 z ) ) ] 〉 = 〈 [ − 400 ( e − x 2 − 3 y 2 − 9 z 2 ( x ) ) ] , [ − 400 ( (b) To determine To find: The direction in which the temperature increases fastest at P ( 2 , − 1 , 2 ) . (c) To determine To find: The maximum rate of change of T ( x , y , z ) = 200 e − x 2 − 3 y 2 − 9 z 2 at the point P ( 2 , − 1 , 2 ) .

Bundle: Multivariable Calculus, 8th + WebAssign Printed Access Card for Stewart's Calculus, 8th Edition, Single-Term

8th Edition

ISBN: 9781305779198

Author: James Stewart

Publisher: Cengage Learning

See similar textbooks

Question

Chapter 14.6, Problem 32E

(a)

To determine

To find: The rate of change of the temperature T(x,y,z)=200e−x2−3y2−9z2 at the point P(2,−1,2) in the direction of (3,−3,3) .

(b)

To determine

To find: The direction in which the temperature increases fastest at P(2,−1,2) .

(c)

To determine

To find: The maximum rate of change of T(x,y,z)=200e−x2−3y2−9z2 at the point P(2,−1,2) .

Expert Solution & Answer

Chapter 14 Solutions

Bundle: Multivariable Calculus, 8th + WebAssign Printed Access Card for Stewart's Calculus, 8th Edition, Single-Term

Ch. 14.1 - In Example 2 we considered the function W = f(T,...Ch. 14.1 - The temperature-humidity index I (or humidex, for...Ch. 14.1 - Prob. 3E Ch. 14.1 - Verify for the Cobb-Douglas production function...Ch. 14.1 - Prob. 5E Ch. 14.1 - The wind-chill index W discussed in Example 2 has...Ch. 14.1 - The wave heights h in the open sea depend on the...Ch. 14.1 - A company makes three sizes of cardboard boxes:...Ch. 14.1 - Let g(x, y) = cos(x + 2y). (a) Evaluate g(2, 1)....Ch. 14.1 - Let F(x,y)=1+4y2. (a) Evaluate F(3,1). (b) Find...

Knowledge Booster

The rate of change of the temperature T ( x , y , z ) = 200 e − x 2 − 3 y 2 − 9 z 2 at the point P ( 2 , − 1 , 2 ) in the direction of ( 3 , − 3 , 3 ) .

To find: The rate of change of the temperature T(x,y,z)=200e−x2−3y2−9z2 at the point P(2,−1,2) in the direction of (3,−3,3) .

To find: The direction in which the temperature increases fastest at P(2,−1,2) .

To find: The maximum rate of change of T(x,y,z)=200e−x2−3y2−9z2 at the point P(2,−1,2) .

Chapter 14 Solutions