Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 15 - Differentiation in Several Variables - 15.5 The Gradient and Directional Derivatives - Exercises - Page 802: 37

Answer

$f$ is increasing.

Work Step by Step

We have at $P$: $\nabla {f_P} = \left( {2, - 4,4} \right)$ and ${\bf{v}} = \left( {2,1,3} \right)$ Using Theorem 2, the rate of change of $f$ is $\frac{d}{{dt}}f\left( {{\bf{r}}\left( t \right)} \right) = \nabla T\left( {{\bf{r}}\left( t \right)} \right)\cdot{\bf{r}}'\left( t \right)$ $\frac{d}{{dt}}f\left( {{\bf{r}}\left( t \right)} \right) = \left( {2, - 4,4} \right)\cdot\left( {2,1,3} \right) = 12$ Since the rate of change is positive, $f$ is increasing.
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