   Chapter 14.6, Problem 32E

Chapter
Section
Textbook Problem

The temperature at a point (x, y, z) is given byT(x, y, z) = 200 e − x 2 − 3 y 2 − 9 z 2 where T is measured in °C and x, y, z in meters.(a) Find the rate of change of temperature at the point P(2, − 1, 2) in the direction toward the point (3, −3, 3).(b) In which direction does the temperature increase fastest at P?(c) Find the maximum rate of increase at P.

(a)

To determine

To find: The rate of change of the temperature T(x,y,z)=200ex23y29z2 at the point P(2,1,2) in the direction of (3,3,3) .

Explanation

Theorem used:

“Suppose f is a differentiable function of two variables. The rate of change is defined as Duf(x,y)=f(x,y)u where u is unit vector.”

Calculation:

The given temperature function is, T(x,y,z)=200ex23y29z2 .

The direction vector PQ form the point P(2,1,2) towards Q(3,3,3) is,

PQ=32,3(1),32=1,2,1

Then the unit vector u is computed as follows,

u=112+(2)2+121,2,1=11+4+11,2,1=161,2,1

Thus, the unit vector u is 161,2,1 .

The gradient of T(x,y,z) is as follows.

T(x,y,z)=Tx,Ty,Tz=x(200ex23y29z2),y(200ex23y29z2),z(200ex23y29z2)=[200(ex23y29z2(2x))],[200(ex23y29z2(6y))],[200(ex23y29z2(18z))]=[400(ex23y29z2(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)=200ex23y29z2 at the point P(2,1,2) .

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