If you start walking north from the point (0, −1000) will your elevation increase or decrease? At what rate? The elevation of a valley above sea level (in feet) is modeled by the functionf(x, y) = -2000 + 0.004(x – 1000)² + 0.001(y + 1000)2

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Description: If you start walking north from the point (0, −1000) will your elevation increase or decrease? At what rate? The elevation of a valley above sea level (in feet) is modeled by the functionf(x, y) = -2000 + 0.004(x – 1000)² + 0.001(y + 1000)2

Answered: The elevation of a valley above sea…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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If you start walking north from the point (0, −1000) will your elevation increase or decrease? At what rate?

The elevation of a valley above sea level (in feet) is modeled by the function
f(x, y) = -2000 + 0.004(x – 1000)² + 0.001(y + 1000)2

Expert Solution

Step 1

Given the function,

$f (x, y) = - 2000 + 0.004 {(x - 1000)}^{2} + 0.001 {(y + 1000)}^{2}$

in the direction $\vec{u} = (0, - 1000)$

The gradient of the function $f (x, y)$ is,

$\begin{array}{rcl} \nabla f (x, y) & = & <f_{x}, f_{y}> \\ = & <0.008 (x - 1000), 0.002 (y + 1000)> \end{array}$

Converting $u$ into a unit vector,

$\begin{array}{rcl} \vec{v} & = & \frac{\vec{u}}{||\vec{u}||} \\ = & \frac{(0, - 1000)}{1000} \end{array}$

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