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 801: 15



Work Step by Step

Given $$ f(x, y)=x-x y, \quad \mathbf{r}(t)=\left\langle t^{2}, t^{2}-4 t\right\rangle, \quad t=4 $$ since $\mathbf{r}(4)=\langle 16,0\rangle$ and $$ \begin{aligned} \nabla f &=\left\langle\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right\rangle \\ &=\langle1-y,-x\rangle \\ \mathbf{r}^{\prime}(t) &=\left\langle 2t, 2t-4\right\rangle \end{aligned} $$ Then \begin{aligned} \left.\frac{d}{d t} f(\mathbf{r}(t))\right|_{t=4}&=\nabla f_{\mathbf{r}(4)} \cdot \mathbf{r}^{\prime}(4)\\ &=\langle1,-16\rangle \cdot\langle 8,4\rangle \\ &= 8-64=-56 \end{aligned}
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