The area function by computer

You will write three functions that show the Fundamental Theorem of Calculus on the computer. The first finds an area by adding thin strips. The second uses an antiderivative. The third finds the slope of the area function and compares it with the function itself.

The idea. The area under $f$ from $a$ to $b$ is close to the sum of the strips $f(\text{middle of strip}) \times \text{width}$. The theorem says it equals $F(b) - F(a)$ for an antiderivative $F$. It also says that the slope of the area function at $x$ is $f(x)$.

Your tasks.

1. Finish area(f, a, b). Cut $[a, b]$ into N equal strips and add f(middle) * width for each strip.
2. Finish ftc(F, a, b). Return F(b) - F(a).
3. Finish slope_of_area(f, a, x). Return the slope of area(f, a, x) at x, from two nearby points x - D and x + D.