Calculus 8th Edition

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

Chapter 14 - Partial Derivatives - Review - Exercises - Page 1024: 47


$\dfrac{\sqrt{145}}{2},\lt 4,\dfrac{9}{2} \gt$

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: $f(x,y)=x^2y+\sqrt y$ $\nabla f(x,y)=\lt 2xy, x^2+\dfrac{1}{2\sqrt y} \gt$ From the given data, we have $f(x,y)=f(2,1)$ $\nabla f(2,1)=\lt (2)(2)(1),2^2+\dfrac{1}{2\sqrt 1} \gt=\lt 4,\dfrac{9}{2} \gt$ $|\nabla f(2,1)|=\sqrt{4^2+(\dfrac{9}{2})^2}=\dfrac{\sqrt{145}}{2}$ Therefore, the maximum rate of change of $f(x,y)$ and the direction is: $\dfrac{\sqrt{145}}{2},\lt 4,\dfrac{9}{2} \gt$
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