Making a piecewise defined function differentiable

[brh2113]

I have to find the values of a and b in terms of c so that this function is differentiable. Attached is the problem and my work, but I think that there's an error somewhere in my attempt. Any advice?

 

[Dick]

Yes, check your derivative of 1/|x| at x=c. It's not zero. You made an algebraic mistake.

 

[brh2113]

I see I forgot to distribute a negative sign on the left side's derivative, but that's trivial, because as h-->0, (-h) and (h) both approach 0.

Is there something else I'm missing? I've re-done the rest of the algebra, and I'm still getting 0.

EDIT: I see what went wrong. I moved the h up to the top of the fraction, instead of keeping it on the bottom.

 

[brh2113]

I think I've solved it (see attached). My only concern is that I've ignored the absolute value signs. Is this a problem? Or should I go back and work it through with two cases, one when X>0 or equal to 0 and one when X<0?

That seems to me the better way, but I'm wondering if it's necessary?

 

[Dick]

You posted an attachment, so I can't see it yet, but yes, you should probably do two cases. That's kind of what absolute values are all about.

 

[HallsofIvy]

Since f(x)= 1/|x| only for |x|> C for some positive number C, the derivative of 1/|x| at x=0 doesn't matter (fortunately)! What is crucial is the value and derivative of 1/|x| at x= C and x= -C.