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: 16



Work Step by Step

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