Using an alternate way to write a Taylor series of $f(t+\tau)$ to derive numerical integration and differentiation formulas

Question

In chapter 9 section 3 of An Introduction to Mathematical Modeling, the author Edward A. Bender points out that: $$f(t+\tau)=\sum_{n=0}^\infty \frac{(\tau D)^n}{n!}$$ so that $$f(t+\tau)=e^{\tau D}f(t)$$ Where $D=\frac{d}{dt}$. This is easy enough to see once $f(t+\tau)$ has been expanded in its Taylor Series. However, Bender then makes the comment: "I can't resist the side remark that (21) can also be used to derive numerical integration and differentiation formulas. For example, see L.P Ford (1955, ch.8)". I think he meant L.R Ford but I can't get a hand on that book so can anyone explain what type of numerical integration and differentiation formulas can be derived from this and how they can be derived? Bender seems fairly excited about the prospect so I'm really interested in how this works.

Answer 1

The Euler Method and the Three Term Taylor Method spring to mind immediately but to be honest I don't see how much more useful this formula really is once you know

$$f(x+h)\approx \sum_k\frac{f^{(k)}(x)}{k!}h^k,$$

which is exactly the same as your formula with $\tau=h$, $n=k$ and well $\displaystyle D=\frac{d}{dx}$.