Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 14 - Section 14.6 - Directional Derivatives and the Gradient Vector - 14.6 Exercise - Page 957: 26


$\dfrac{3}{5},\lt \dfrac{2}{5},\dfrac{1}{5},\dfrac{2}{5} \gt$

Work Step by Step

Formula to calculate the maximum rate of change of $f$: $D_uf=|\nabla f(x,y)|$ $\nabla f(x,y)=\lt \dfrac{qr}{1+(pqr)^2},\dfrac{pr}{1+(pqr)^2},\dfrac{pq}{1+(pqr)^2} \gt$ $\nabla f(1,2,1)=\lt \dfrac{2}{5},\dfrac{1}{5},\dfrac{2}{5} \gt$ $|\nabla f(1,2,1)|=\sqrt{(\dfrac{2}{5})^2+(\dfrac{1}{5})^2+(\dfrac{2}{5})^2}=\sqrt {\dfrac{9}{25}}=\dfrac{3}{5}$ Hence, the required answers are: $\dfrac{3}{5},\lt \dfrac{2}{5},\dfrac{1}{5},\dfrac{2}{5} \gt$
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