   Chapter 14, Problem 45RE

Chapter
Section
Textbook Problem

Find the directional derivative of f at the given point in the indicated direction.45. f(x, y) = x2e−y , (−2,0), in the direction toward the point (2. −3)

To determine

To find: The directional derivative of the function f(x,y)=x2ey at the point (2,0) in the direction of (2,3) .

Explanation

Given:

The functionis f(x,y)=x2ey .

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:

Let the points be P=(2,0) and Q=(2,3) .

The directional derivative is defined as, Duf(x,y,z)=f(x,y,z)u (1)

The value of f(x,y) is computed as follows.

f(x,y)=fx,fy=x(x2ey),y(x2ey)=2x(ey),(x2)(ey)=2xey,(x2ey)

Thus, the value of f(x,y)=2xey,(x2ey)

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