- #1

Buri

- 273

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## Homework Statement

f: R^2 -> R f(x,y) = |x| + |y|

**(a) Find all directional derivates at (0,0) in the direction of u not equal to zero if they exist. And evaluate when they do.**

**(b) Do the partial derivatives exist at (0,0)?**

**(c) Is it differentiable at (0,0)?**

**(d)Is it continuous at (0,0)?**

**!Solutions!**

**(a)**No they do not exist. I have:

f'(0;u) = lim [t->0] 1/t[f((0,0) + t(h,k)) - f(0,0)] = lim [t->0] 1/t(|th| + |tk|)

Which then equals |h| + |k| and -|h| - |k| for t > 0 and t < 0 respectively. So the "left and right" limits aren't equal. Therefore, the limit cannot exist.

**(b)**No, because the directional derivatives don't.

**(c)**No it is not differentiable. Since all directional deritaves at 0 don't exist it implies it is not differentiable at 0.

**(d)**Yes it is continuous at 0.

Note that

lim [ (x,y) -> (0,0) ] |x| = 0 and lim [ (x,y) -> (0,0) ] |y| = 0

These are both proved by setting delta = epsilon. And therefore the summation of the two exists, so we have:

lim [ (x,y) -> (0,0) ] f(x,y) = lim [ (x,y) -> (0,0) ] |x| + |y| = lim [ (x,y) -> (0,0) ] |x| + lim [ (x,y) -> (0,0) ] |y| = 0 + 0 = 0.

**Can someone verify whether I'm right or wrong on any of the parts? I'd really appreciate it!**