## Thomas' Calculus 13th Edition

Published by Pearson

# Chapter 14: Partial Derivatives - Practice Exercises - Page 864: 1

#### Answer

Domain: x and y can be any real numbers in the x-y plane. Range: $f(x,y) \geq 0$ Level curves: $9x^{2}+y^{2} = c$ (ellipses).

#### Work Step by Step

Function: $f(x,y)=9x^{2}+y^{2}$ Domain: x and y can be any real numbers in the x-y plane. Range: $f(x,y) \geq 0$ Level curves: $f(x,y) = c$ (constant), i.e. $9x^{2}+y^{2} = c$ which are a series of ellipses. E.G. for $c=36, 9x^{2}+y^{2} = 36$, can be plotted as an ellipse as shown in the figure. $x^{2}/4+y^{2}/36 = 1$

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