Rates from numbers

You will write three functions that work with a rule $f(t)$ given as a Python function: the average rate over an interval, the rate at one moment, and the moment of fastest change. The checker tests several functions.

The idea. The average rate over $[a, b]$ is $\dfrac{f(b) - f(a)}{b - a}$. The rate at one moment is the limit of the average as the interval shrinks. A computer cannot take a limit, but a very small interval is close enough. A balanced interval, from $x - h$ to $x + h$, is the best choice.

Your tasks.

1. Finish average_rate(f, a, b).
2. Finish rate_at(f, x). Use the interval from x - h to x + h, with h = 1e-6.
3. Finish fastest(f, a, b). Return the point of the grid a, a + step, ..., b where rate_at is largest.