Is a function differentiable?

[floralcloud]

I am to determine if the function j(k)= tan|k| is differentiable... I tried to determine it graphically, but still wasn't sure, so I decided to try to find it algebraically, so I set it up like this

Comments

[supamikeymon]

First, if you really want to use the definition of the limit, I would consult some trig identities, and see if anything fits. And then pray to euler and hope for an algebraic manipulation. I'm trying to say that it looks hairy, but it's a good thought, and you should see what you can do with your idea.

I would look at tan|k| and see if there are any asymptotes (areas where there's no continuity) then just determine the left and right limits on each end of the asymptote to be infinity and negative infinity - which don't match and cause you to be not continuous, which also makes it not differentiable. That is, if my idea is possible and works... That's a pretty 'fun' function...

I haven't graphed it - but tan|k| is just tan(x) if x>0 and then mirrored on the other side of the y-axis. You may have some weird stuff (a sharp change of slope?) occur at x=0 because that's where the mirror occurs, but I actually think it may be continuous there, but I would check that first.

What do you think?

[floralcloud]

I already tried to graph it, but I couldn't tell if it was differentiable or not- weird, right? everytime I zoomed in, it just looked...fuzzy?

The trig identities do look hairy and I suspect it's simpler than I think...so I'll stick with evaluating the limit as h approaches zero..thanks for the help, though! LOL praying to euler xD

[supamikeymon]

It's really funny to me, cause that's my cat's name, too. :D :D :D

[supamikeymon]

Oh I found this neat graphing thingy online!

http://www.walterzorn.com/grapher/grapher_e.htm

Enter tan(abs(x)) the box as you clear out the other data and hit 'plot graph'

You can see a "spike" at x=0. So it's clearly not differentiable because the slope of the graph coming from both sides is not equal.

See, I wasn't sure if tan(x) had a flattening out at x=0, which would actually keep it differentiable. Luckily, this is not the case.

[supamikeymon]

Also, j(k) ?

function j of k.

that amuses me, lol :D

[floralcloud]

Thank you, it's clear to me now!