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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

Let f be a function of two variables that has continuous partial derivatives and consider the points A(1,3), B(3, 3), C(1,7), and D(6, 15). The directional derivative of f at A in the direction of the vector AB is 3 and the directional derivative at A in the direction of AC is 26. Find the directional derivative of f at A in the direction of the vector AD .

To determine

To find: The directional derivative of the function f at the point A in the direction of the vector AD .

Explanation

Given:

The two variable function f and has continuous partial derivatives.

The points are A(1,3),B(3,3),C(1,7)andD(6,15) .

The directional derivative of the function f at the point A in the direction of the vector AB is 3.

The directional derivative of the function f at the point A in the direction of the vector AC is 26.

Result used:

“The directional derivative of the function f(x,y,z) at f(x0,y0,z0) in the direction of unit vector u=a,b,c is Duf(x,y,z)=f(x,y,z)u , where f(x,y,z)=fx,fy,fz=fxi+fyj+fzk .”

Calculation:

Compute the unit vector in the direction AB is as follows,

AB=OBOA=(31)i+(33)j=2i

The unit vector u ,

u=AB|AB|=2i22=i

Thus, the unit vector in the direction AB is i .

Compute the unit vector in the direction AC is as follows,

AC=OCOA=(11)i+(73)j=4j

The unit vector u ,

u=AC|AC|=4j42=j

Thus, the unit vector in the direction AC is j

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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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