**Question 8**Given function $ w = f(x, y, z) = x^2 - 2y^2 + 3z^2 $(a) Find $\text{grad } f(2, -1, 1)$(b) Find the equation of the tangent plane at the point $ (2, -1, 1) $ to the level surface $ f(x, y, z) = 5 $.(c) Find the unit vector in the direction of the largest increase of $ f $ at the point $ (2, -1, 1) $.(d) Find the largest rate of change of $ f $ at the point $ (2, -1, 1) $.

Answered: 8. Given function w = f(x, y, z) = x -…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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The equation of grad f for the function w=f(x.y,z) is given by,

$\nabla w = \nabla f = [\begin{matrix} \frac{\partial f}{\partial x} \\ \frac{\partial f}{\partial y} \\ \frac{\partial f}{\partial z} \end{matrix}]$ which can also be written in the form $<\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}>$

To find the equation of tangent plane at a point $(x_{0}, y_{0}, z_{0})$ to the level surface f(x,y,z)= $w_{0}$ is ,

$f_{x} (x_{0}, y_{0}, z_{0}) . (x - x_{0}) + f_{y} (x_{0}, y_{0}, z_{0}) . (y - y_{0}) + f_{z} (x_{0}, y_{0}, z_{0}) . (z - z_{0}) = 0$

We have the direction of largest increase of f is in the direction of grad f. Also the unit vector in the direction of grad f at point $(x_{0}, y_{0}, z_{0})$ is,

$\frac{\nabla f (x_{0}, y_{0}, z_{0})}{∥ \nabla f ∥} w h e r e ∥ \nabla f ∥ = \sqrt{{f_{x_{0}}}^{2} + {f_{y_{0}}}^{2} + {f_{z_{0}}}^{2}}$