Answered: Determine whether the statement " If…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

Determine whether the statement " If f(x, y) = x + y, then −1 ≤ D_u f(x, y) ≤ 1" is true or false. If it is false, explain why or give an example that shows it is false.

Expert Solution

Step 1

If the statement "If $f (x, y) = x + y$ , then $- 1 \leq D_{u} f (x, y) \leq 1$ for any unit vector u" is true or false.

Step 2

Examine the stated function:

$f (x, y) = \sqrt{1 - x^{2} - y^{2}}$

Thus, first find the gradient of the stated function as:

$\begin{array}{rcl} \nabla f (x, y) & = & f_{x} (x, y) i + f_{y} (x, y) j \\ = & \frac{d}{d x} [x + y] i + \frac{d}{d y} [x + y] j \\ = & i + j \end{array}$

Thus, the direction of maximum increase is stated by the unit vector:

$\begin{array}{rcl} \vec{u} & = & \frac{i + j}{\sqrt{1^{2} + 1^{2}}} \\ = & \frac{i + j}{\sqrt{2}} \end{array}$

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