Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 14 - Section 14.6 - Directional Derivatives and the Gradient Vector - 14.6 Exercise - Page 959: 55



Work Step by Step

Formula to calculate tangent plane equation is: $(x_2-x_1)f_x(x_1,y_1,z_1)+(y_2-y_1)f_y(x_1,y_1,z_1)+(z_2-z_1)f_x(x_1,y_1,z_1)=0$ At point$(1,1,-1)$ $(x-x_0)(2x)+(y-y_0)(-2y)+(z-z_0)(-2z)=0$ There must exist a constant number $p$ when the planes are parallel. Thus, $(\dfrac{p}{2})^2-(-\dfrac{p}{2})^2-(-\dfrac{p}{2})^2=-\dfrac{p^2}{4}$ which has no solution. Hence, There are no points on the hyperboloid.
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