Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 14: Partial Derivatives - Practice Exercises - Page 864: 8

Answer

Domain: all real numbers Range: $0\lt f(x,y,z)\leq1$ Level surfaces: $f(x,y,z)=c$ (constant in $(0,1]$)

Work Step by Step

Function: $f(x,y,z)=\frac{1}{x^{2}+y^{2}+z^{2}+1}$ Domain: x, y, and z can be any real numbers Range: $0\lt f(x,y,z)\leq1$ Level surfaces: $f(x,y,z)=c$ (constant in $(0,1]$) $x^{2}+y^{2}+z^{2}=\frac{1}{c}-1$ represents a series of spheres. For example, when $c=\frac{1}{2}$, $x^{2}+y^{2}+z^{2}=1$ is a sphere shown in the figure.
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