Calculus 8th Edition

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

Chapter 14 - Partial Derivatives - 14.6 Directional Derivatives and the Gradient Vector - 14.6 Exercises - Page 997: 22


$\sqrt{17}, \lt 4,1 \gt$

Work Step by Step

Our aim is to determine the maximum rate of change of $f(x,y)$.In order to find this, we have : $D_uf=|\nabla f(x,y)|$ Given: $f(s,t)=te^{st}$ $\nabla f(x,y)=\lt t^2 e^{st},e^{st}+ste^{st} \gt $ From the given data, we have $f(x,y)=f(0,2)$ Thus, $\nabla f(0,2)=\lt 2^2 e^{0},e^{0}+s(0) \gt=\lt 4,1 \gt$ or, $|\nabla f(0,2)|=\sqrt{4^2+ 1^2}=\sqrt{17}$ Therefore, the maximum rate of change of $f(x,y)$ and the direction is:$\sqrt{17}, \lt 4,1 \gt$
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