E1:
Try to understand what the higher derivatives mean, as applied to the Taylor approximation of a function.
E2: Do that for the sine function. Why does the 4th approximation not look like the base function itself? (the 4th derivative alone is identical to it).
E3: Go through the approximations of the power function: What type of polynom is each one? (Write them down, using the factors in the number fields). Check that 7th to 9th are identical (tiny differences recognizable are calculation errors). Compare the 7th order polynoms around different model points and argue their identity. Look at the one for x0 = 0.
E4: Experiment with all base functions. Consider why and when a linear approximation, which is easy to handle analytically, is good enough. Take a look at the sequence factors.