Identifying Non-Differentiable Points Without Graphing

[lpbug]

Homework Statement

Differentiability-

Okay, so I understand that a function is not differentiable if there are either:
A. A cusp
B.A jump
C. f(x) DNE
D. Vertical tangent
E. Pretty much if there isn't a limit there is no derivative which means its not differentiable.

How would one find the points not differentiable if one didn't graph the function?

Thanks!

Homework Equations

The Attempt at a Solution

n/a

 

[armolinasf]

First take a look at the graph of abs(x) you will notice that there is a corner and this because the absolute value function is really a piecewise function where abs(x) = x if x>0 and -x if x<0. This tells you that the function is not smooth: there the slope change abruptly when crossing the y axis.

 

[lpbug]

Yeah I noticed that there is a corner, but is there a more in depth reason, maybe because the slope at 0 doesn't exist? Something of that sort...?

 

[pergradus]

Yeah I noticed that there is a corner, but is there a more in depth reason, maybe because the slope at 0 doesn't exist? Something of that sort...?

Try drawing a tangent line to the graph of |x| at x = 0. You'll see why it's not differentiable.

 

[Char. Limit]

Yeah I noticed that there is a corner, but is there a more in depth reason, maybe because the slope at 0 doesn't exist? Something of that sort...?

If you can, graph the derivative. You'll notice a noticeable jump discontinuity at x=0.

 

[Calrid]

Always graph the function if in doubt.

Any break in the line may or may not be non differentiable but I'm pretty sure you will be unable to tell if $\frac{d}{dx} x^2+y^2$ is differentiable by looking at the equation. Well unless by inspection but you get the point.

The only time you would not graph something is if you already knew it was non convergent or where you know that certain rules of differentiation rule out the likelihood of linearity.