Absolute value and tangents

[Paul A]

My apologies if this topic has been a subject of a previous thread. I
searched previous posts and didn't notice anything that matched it.
This problem came up after a discussion about discontinuity in class.
I graphed y=abs(x) and then constructed a tangent on the function. I
then measured the slope of the tangent and grabbed the point defining
the point of contact for the tangent and moved it toward x=0. The
slope remained 1. I thought this may be a problem with display digits
so I found the coordinates of the point and changed the x-coord to 0
and still had a slope of 1.
Interestingly, I defined a piecewise function corresponding to the abs
(x) and when I found the slope at x=0, no slope is displayed which I
guess is better than a slope of 1. I decided to check how the 84 and
using the Draw Tangent option got y=0x + 0.
Am I doing something wrong?

Paul A

[Nelson Sousa]

Odd...

I tried and got different answers on TI-Nspire and TI-Nspire CAS. All versions were 1.6.

So, TI-Nspire handheld, TI-Nspire computer software and TI-Nspire Teacher Edition computer software, all drew a horizontal line at x=0; TI-Nspire CAS and TI-Nspire CAS computer software jumped from y=-x to y=x. I have no idea what's the reason for this difference, on the geometry app Nspire and Npspire CAS should be exactly the same.

But defining f2(x)=nDeriv(f1(x0),x0=x) we get the same result in both versions: -1 for x<0, +1 for x>0 and 0 at x=0. 

In reality, being a numerical algorithm, there's no reason to expect any particular result for this slope, as the tangent to the curve at x=0 is undetermined.  However, it would be good to have both the numerical derivative and the graphical derivative to detect points where the derivative doesn't exist due to the two limits being different.

Nelson

[JLosse]

Paul,

If you do what you described, only place the point of tangency to the
left of x=0 and then slide it to the right toward zero, you will see
that the slope changes from -1 to +1 at x=0, again without becoming
undefined at x = 0. If you control the motion of the point with a
slider you can be sure when the point is at the origin.

This suggests to me that on the nspire the slope of the tangent is
calculated as (f(x+h)-f(x))/h, where h is always a small positive
value.

Using a slider to control the point of tangency, I noticed that, if x
= -1*10^-14, the slope of the tangent was given as -1, but at x =
-1*10^-15, the slope was given as +1. I increased the displayed
digits on the slope to 9 (the max) but never saw a slope of
0.999xxxxx....., which would have confirmed my suspicion that a small
positive h is always used.

The TI-84 used a symmetric difference quotient (f(x+h)-f(x-h))/(2h)
for numerical derivatives and tangent, which accounts for it's giving
an incorrect derivative of zero for f(x) = abs(x) at the origin.

JLosse

[Nelson Sousa]

but numeric Nspire says the slope is zero when x=0. 

That is (surprisingly) a CAS only matter?!

Nelson