   Chapter 14.6, Problem 35E

Chapter
Section
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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