Thomas' Calculus 13th Edition

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

Chapter 14: Partial Derivatives - Section 14.1 - Functions of Several Variables - Exercises 14.1 - Page 788: 39


See image. .

Work Step by Step

$ a.\quad$ To sketch the surface $z=f(x,y)=x^{2}+y^{2}$: * note that z can not be negative (sum of two squares), so the surface is on or above the xy plane * in a plane $z=k$, the trace is a circle of radius $\sqrt{k}$ * in a plane parallel to $x=0$, the trace is a parabola $z=y^{2}+k$ * in a plane parallel to $y=0$, the trace is a parabola $z=x^{2}+k$ This is an elliptical (circular) paraboloid. $ b.\quad$ In the xy plane, we equate $f(x,y)$ with several values of c, $z=x^{2}+y^{2}=c\qquad $(so only nonnnegative c apply) These are circles with radii $\sqrt{c}.$ Take $c=0,1,2^{2},3^{2}$....
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