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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

The temperature at a point (x, y, z) is given by

T(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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Chapter 14 Solutions

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P-14RECh-14 P-15RECh-14 P-16RECh-14 P-17RECh-14 P-18RECh-14 P-19RECh-14 P-20RECh-14 P-21RECh-14 P-22RECh-14 P-23RECh-14 P-24RECh-14 P-25RECh-14 P-26RECh-14 P-27RECh-14 P-28RECh-14 P-29RECh-14 P-30RECh-14 P-31RECh-14 P-32RECh-14 P-33RECh-14 P-34RECh-14 P-35RECh-14 P-36RECh-14 P-37RECh-14 P-38RECh-14 P-39RECh-14 P-40RECh-14 P-41RECh-14 P-42RECh-14 P-43RECh-14 P-44RECh-14 P-45RECh-14 P-46RECh-14 P-47RECh-14 P-48RECh-14 P-49RECh-14 P-50RECh-14 P-51RECh-14 P-52RECh-14 P-53RECh-14 P-54RECh-14 P-55RECh-14 P-56RECh-14 P-57RECh-14 P-58RECh-14 P-59RECh-14 P-60RECh-14 P-61RECh-14 P-62RECh-14 P-63RECh-14 P-64RECh-14 P-65RECh-14 P-1PCh-14 P-2PCh-14 P-3PCh-14 P-4PCh-14 P-5PCh-14 P-6PCh-14 P-7PCh-14 P-8P

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