We address the following questions:
We can extend this argument to create the cubic approximation, etc, when f is suitably differentiable by applying the same steps with still higher derivatives. If we do this on forever, we get the "Taylor series expansion of f at argument x0."
10.1 Write down the Taylor series expansion about x0 for a general infinitely differential function f.
10.2 Write down the approximation formula of degree 5 for a general function that is 5 times differentiable, and apply it explicitly for the sine function at x0 = 0. Give the cubic approximation to the sine at x0 = 1.
10.3 The exponential function, being its own derivative, can be factored out of its Taylor series expansion. Apply that expansion around x0, to deduce the relation between exp(x) and exp(x0).