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: 6

Answer

Domain: $x: (-\infty,\infty); y: (-\infty,\infty); z: (-\infty,\infty); $ Range: $f(x,y,z): (-\infty,\infty); $ Level surfaces: $f(x,y,z)=c$ (constant)

Work Step by Step

Function: $f(x,y,z)=x^{2}+4y^{2}+9z^{2}$ Domain: x, y , and z can be any real numbers; Range: $f(x,y,z)\geq0$ Level surfaces: $f(x,y,z)=c$ (constant) $x^{2}+4y^{2}+9z^{2}=c$ represents a series of ellipsoids. For example, when $c=9$, $x^{2}+4y^{2}+9z^{2}=9$ is an ellipsoid as shown in the figure.
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