Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 15 - Differentiation in Several Variables - 15.5 The Gradient and Directional Derivatives - Exercises - Page 802: 44



Work Step by Step

Given $$x^{2}+3 y^{2}+4 z^{2}=20, \quad P=(2,2,1)$$ Consider $f(x,y,z)=x^{2}+3 y^{2}+4 z^{2}-20$, since \begin{align*} \nabla f&=\left\langle\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right\rangle\\ &=\left\langle 2x,6y,8z\right\rangle\\ \nabla f_{P}&=\langle 4,12,8\rangle \end{align*} Then the equation of the tangent plane at $P$ is given by \begin{align*} \nabla f_{P} \cdot\langle x-x_1, y-y_1, z-z_1\rangle&= 0\\ \langle 4,12,8\rangle \cdot\langle x-2, y-2, z-1\rangle&=0\\ 4(x-2)+12(y-2)+8(z-1)&=0\\ 4x+12y+8z&=40 \end{align*}
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