Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 14 - Partial Derivatives - Review - Concept Check - Page 1021: 13


a) See the explanation below. b) See the explanation below.

Work Step by Step

a) The directional derivatives of $f$ at $(x_0,y_0)$ in the direction of a unit vector $\over{u}$can be expressed as: $D_uf(x_0,y_0)=\lim\limits_{n \to 0}\dfrac{f(x_0+na,y_0+nb)-f(x_0,y_0)}{l}$ The interpretation of the directional derivative can be defined as the rate of change of $f$ at $(x_0,y_0)$ in the direction of a unit vector $\over{u}$. b) The expression for $D_uf(x_0,y_0)$ in terms of the first partial derivatives of $f$ that is, $f_x,f_y$ is: $D_uf(x_0,y_0)=f_x(x_0,y_0)a+f_y(x_0,y_0)b$
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