Suppose that $ f(x, y) = x^3y^2 $. The directional derivative of $ f(x, y) $ in the direction $\langle 3, -1 \rangle$ and at the point $(x, y) = (-3, 2)$ is $\underline{\hspace{1.5cm}}$.Question Help: Video[Submit Question Button]

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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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Suppose that $ f(x, y) = x^3y^2 $. The directional derivative of $ f(x, y) $ in the direction $\langle 3, -1 \rangle$ and at the point $(x, y) = (-3, 2)$ is $\underline{\hspace{1.5cm}}$.

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Step 1

We need to find the directional vector of f(x, y) at the point (-3,2) in the direction<3,-1>

$f (x, y) = x^{3} y^{2}$

$\frac{\partial f}{\partial x} = 3 x^{2} y^{2}, \frac{\partial f}{\partial y} = 2 y x^{3} \nabla f = \frac{\partial f}{\partial x} \hat{i} + \frac{\partial f}{\partial y} \hat{j} \nabla f = 3 x^{2} y^{2} \hat{i} + 2 y x^{3} \hat{j} A t t h e p o i n t (- 3, 2) \nabla f (- 3, 2) = 3 {(- 3)}^{2} {(2)}^{2} \hat{i} + 2 (2) {(- 3)}^{3} \hat{j} \nabla f (- 3, 2) = 108 \hat{i} - 108 \hat{j}$