Fundamental Theorem Part 1 Discussion

(The following is/are the initial prompt(s) for an online discussion; students may have posted responses, and prompts for further discussion may have been added, but these things are not shown.)

The Fundamental Theorem of Calculus connects integrals, antiderivatives, and derivatives. It’s usually presented in 2 parts, the first of which provides a more theoretical connection, and the second of which provides a way to evaluate integrals in terms of antiderivatives. See section 5.3 in our textbook for both parts of the theorem. This Canvas discussion is a chance to start working with and understanding the first part.

The Functions

The first part of the Fundamental Theorem deals with functions of the form

which are probably rather strange-looking functions. To get more comfortable with them, consider the specific example

Try to work out what F is for a couple of different x values, and post your results or questions. (You could do this with Riemann sums, but the integrand, t, is one for which you can also calculate F(x) just using geometric area formulas.)

Can you work out a general formula for F(x)? What is it, and how did you come up with it?

Once there’s a general formula for F(x), is the derivative of that general formula equal to x, as the Fundamental Theorem says it should be?

Using the Fundamental Theorem

How would you use part 1 of the Fundamental Theorem to find g′(x) if g is defined by...

1. \[g(x) = \int_1^x \cot t\,dt\]
2. \[g(x) = \int_{-2}^x \frac{e^t}{t^2-1}\,dt\]
3. \[g(x) = \int_x^\pi t \sin t\,dt\]