## The Fundamental Theorem of Calculus, Part I (Theoretical Part) |
HELP |

Make a function `f(t)` | Make its corresponding area function `A(x)` | Differentiate `A(x)` |

The Fundamental Theorem of Calculus goes like this:

[`d/dx int_a^x f(t) dt = f(x)`]

We'll try to unravel it with this exploration. The gist of the FTC is that differentiation
"undoes" integration; in a sense, they are reverse processes of each other. There are three steps.
- We start with a function `f`, and suppose it depends on the variable `t`. (It doesn't really matter what the variable is, but we'll want to use `x` later.)
- Fix a value `a`, then consider the function `A(x) = int_a^x f(t) dt`. We imagine `f` and `a` unchanging, so this is a function of `x`; as `x` changes, the value of `A` changes.
- Finally, we differentiate `A(x)`, and surprise! we get `f(x)`.

- If you haven't done so already, get familiar with the area function applet that comes before this one.
- Verify that ...
- selecting a function `f(t)` and a starting point `a` in the leftmost panel defines a function `A(x)` in the middle panel,
- the function `A(x)` in the middle really is the area function for `f(t)`,
- as `x` changes, the red point in the middle panel shows the value of the area function evaluated at `x`.

- Describe how changing `a` affects the function `A(x)`. Explain why this happens.
- In the middle panel, check the box for "Show Tangent" and see what happens as you change `x`. How does changing `a` affect the slope?
- Now we'll graph the derivative of `A(x)` by plotting the slope of the tangent lines. Check "Trace Slope" in the rightmost panel and slowly move `x` around. Confirm that indeed, the value of the slopes are being graphed and that the graph of `d/dx A(x)` emerges.
- Describe how the graph of `d/dx A(x)` in the rightmost panel compares to the graph of `f(t)` in the leftmost panel.
- Confirm that the Fundamental Theorem of Calculus holds for several examples.

Do you think it is intuitive that the problem of finding the area between a function
and the `x`-axis is related to the problem of finding the derivative of a function?