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**Alterah** I am having some difficulty with the following problems:

1. For the function

$\displaystyle f(x,y)=\left\{\begin{array}{cc}\frac{x|y|}{\sqrt(x ^2+y^2)},&\mbox{ if }

(x,y) != (0,0)\\0, & \mbox{ if } (x,y) = (0,0)\end{array}\right.$

use definition $\displaystyle D_vf(a) = \lim_{h \to 0}\frac{f(a + hv) - f(a)}{h}$ to determine for which unit vectors **v** = v**i** + w**j** the direction derivative [tex] D_vf(0,0) exists.

I am able to get the function down to the following:

$\displaystyle \lim_{h \to 0}(\frac{hv|hw|}{\sqrt(h^2v^2+h^2w^2)} - 0)/h$

Simplifying I get:

$\displaystyle \lim_{h \to 0}\frac{h^2v|w|}{\sqrt(v^2+w^2)}$

This is where I get lost. The answer is supposed to be v|w|. The limit I have goes to 0.

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Problem 2

The depth of a lake is given by $\displaystyle f(x,y) = 400 - 3x^2y^2 $ find the direction a swimmer should swim so that there is no change in the depth. The swimmer begins at (1,-2).

This is part 2 of the problem. Part one has me find the direction the depth increases most rapidly and I got $\displaystyle \frac{1}{\sqrt5}(-2i + j)$

To be honest I am not quite sure where to begin. Would I set the gradient equal to zero at that point and go from there?

Thanks for any and all help.