Given the divided difference coefficients from the Hermite Spreadsheet we construct the interpolating fourth degree polynomial:

p[x_]:=1+x( 1+x (-1+x(-1+ 2x)))

To the functions:

f[x_]:=(x^5)-(x^(3 ))-(x^2)+x+1

We can see that the two functions agree at 0 and 2 and the contact at 0 is of higher order, not sure you can “see” that it is 3rd order but you can defintiely see it is at least 2nd.

Plot[{f[x],p[x]},{x,-1,2.2}]

Plot[Abs[f[x]-p[x]],{x,-1,2.2}]

And here is a plot of the error:

Expand[p[x]]

Here is a plot for Section 4.6 problem 2 of the function and the polynomial:

Plot[{Sin[x],x- ((x^2)/π)},{x,-π/4,5π/4}]

And here is the error:

Plot[{Sin[x]-x+ ((x^2)/π)},{x,-π/4,5π/4}]

p[0]

f[0]

p'[0]

f'[0]

p''[0]

f''[0]