Question

2. Consider the function \\ $\phi(x, y, z) = xy + yz + xz.$ \\ (a) Find its rate of change in the direction $(1, 2, 3)$ at the point \\ $(1, 3, -1)$. \\ (b) At this same point, $(1, 3, -1)$, find the rate of change in the direction in which $\phi$ increases the most rapidly.

          2. Consider the function \\
$\phi(x, y, z) = xy + yz + xz.$ \\
(a) Find its rate of change in the direction $(1, 2, 3)$ at the point \\
$(1, 3, -1)$. \\
(b) At this same point, $(1, 3, -1)$, find the rate of change in the direction in which $\phi$ increases the most rapidly.

2. Consider the function

ϕ(x, y, z) = xy + yz + xz.

(a) Find its rate of change in the direction (1, 2, 3) at the point

(1, 3, -1).

(b) At this same point, (1, 3, -1), find the rate of change in the direction in which ϕ increases the most rapidly.

Added by Randy N.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Faizanullah Kazmi

02/27/2022

Step 1

The gradient of a function is given by the vector of its partial derivatives. The partial derivatives of \phi(x, y, z) are: \frac{\partial \phi}{\partial x} = y + z \frac{\partial \phi}{\partial y} = x + z \frac{\partial \phi}{\partial z} = x + y Evaluating Show more…

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Consider the function phi (x,y,z)=xy+yz+xz. (a) Find its rate of change in the direction (1,2,3) at the point (1,3,-1). (b) At this same point, (1,3,-1), find the rate of change in the direction in which phi increases the most rapidly.