Let $ f(x, y) = (x^2 + y^2) e^{-(x^2 + y^2 - 5)} $. Find the rate of change of $ f $ at $ X = (1, -2) $ in the direction $\mathbf{v}$ pointing toward the origin, that is, compute $\frac{d}{dt} f(X + t\mathbf{v})|_{t=0}$ for an appropriate unit length vector $\mathbf{v}$.

Answered: : (x² + y²)e-(x*+y -5), Find the rate…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

Let $ f(x, y) = (x^2 + y^2) e^{-(x^2 + y^2 - 5)} $. Find the rate of change of $ f $ at $ X = (1, -2) $ in the direction $\mathbf{v}$ pointing toward the origin, that is, compute $\frac{d}{dt} f(X + t\mathbf{v})|_{t=0}$ for an appropriate unit length vector $\mathbf{v}$.

Expert Solution

Step 1

Given that $f (x, y) = (x^{2} + y^{2}) e^{- (x^{2} + y^{2} - 5)}$

The objective is to find the rate of change of f at $X = (1, - 2)$ in the direction v pointing towards the origin.

Now, $v = (0, 0) - (1, - 2) = (- 1, 2)$ then $\hat{v} = \frac{1}{\sqrt{1 + 4}} (- 1, 2)$

Consider, $f (x, y) = (x^{2} + y^{2}) e^{- (x^{2} + y^{2} - 5)}$

Then

$\begin{array}{rcl} \frac{\partial f}{\partial x} & = & 2 x e^{- (x^{2} + y^{2} - 5)} - 2 x^{3} e^{- (x^{2} + y^{2} - 5)} - 2 x y^{2} e^{- (x^{2} + y^{2} - 5)} \\ = & 2 x e^{- (x^{2} + y^{2} - 5)} (1 - x^{2} - y^{2}) \\ \frac{\partial f}{\partial y} & = & - 2 y x^{2} e^{- (x^{2} + y^{2} - 5)} + 2 y e^{- (x^{2} + y^{2} - 5)} - 2 y^{3} e^{- (x^{2} + y^{2} - 5)} \\ = & 2 y e^{- (x^{2} + y^{2} - 5)} (- x^{2} + 1 - y^{2}) \\ = & 2 y e^{- (x^{2} + y^{2} - 5)} (1 - x^{2} - y^{2}) \end{array}$

Step by step

Solved in 2 steps

Knowledge Booster

$Differentiation$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions