In section 2.2 on "What is a manifold?" under
equation (2.8) he says ф(x)=|x3| is a C2 function because
it is infinitely differentiable everywhere except at x=0 where it is
differentiable twice but not three times.
I had to think about that one and had a quick look at this video on the Khan academy to refresh my memory. I then got to use my own graph paper to test the assertion for myself. What fun! Now I have put it in Excel. Here is the result.
I had to think about that one and had a quick look at this video on the Khan academy to refresh my memory. I then got to use my own graph paper to test the assertion for myself. What fun! Now I have put it in Excel. Here is the result.
ф(x)=|x3| is in blue. It's gradient is obviously
negative for x<0 and positive for x>0.
So
For x<0, ф'=-3x2 and for x>=0 ф'=3x2.
The gradient of ф' is always positive.
So ф''=|6x|. We can now see the problem at x=0 in green.
It is hardly necessary to plot ф''' which is -6 for x<0,
6 for x>0, but undefined at x=0.
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