The Fundamental Theorem of Calculus, Part I (Theoretical Part)
|Make a function `f(t)`
||Make its corresponding area function `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.
For Further Thought
We officially compute an integral `int_a^x f(t) dt` by using Riemann sums; that
is how the integral is defined. However, the FTC tells us that the integral `int_a^x f(t) dt`
is an antiderivative of `f(x)`. Thus, we can use our already-developed theory of
derivatives to compute integrals. Computing antiderivatives is much easier than computing
Riemann sums, right?
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?