I am having some difficulty with the following problems:

1. For the function

use definition

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:

Simplifying I get:

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

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

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.