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Subsection4.1A Few More Features

This subsection demonstrates a few more features.

The derivative and antiderivative of a function can be understood through study of their graphical relationships.

Definition4.1Antiderivative of a Function

Suppose that \(f(x)\) and \(F(x)\) are two functions such that

\begin{equation*} F^\prime(x) = f(x). \end{equation*}

Then we say \(F\) is an antiderivative of \(f\text{.}\)

The Fundamental Theorem of Calculus in one of the high points of a course in single-variable course.

Left to the reader.

We state an equivalent version of the FTC, which is less-suited for computation, but which perhaps is a more interesting theoretical statement.

We simply take the indicated derivative, applying Theorem 4.2 at (4.2).

\begin{align} \frac{d}{dx}\definiteintegral{a}{x}{f(t)}{t}&=\frac{d}{dx}\left(F(x)-F(a)\right)\label{equation-use-FTC}\tag{4.2}\\ &=\frac{d}{dx}F(x)-\frac{d}{dx}F(a)\notag\\ &=f(x)-0 = f(x)\tag{4.3} \end{align}

Subsection4.2WeBWorK Exercises

The first problem in this list is coming from the WeBWorK Open Problem Library. One implication of this is that we might want to provide some commentary that connects the problem to the text. The other two ask for essay answers, which would be graded by an instructor, so there is no opportunity to provide answer.

2Every Continuous Function has an Antiderivative

WeBWorK problems can allow for open-ended essay responses that are intended to be assessed later by the instructor. For anonymous access, no text field is provided. But if this problem were used within WeBWorK as part of a homework set, users could submit an answer.

WeBWorK Exercise
3Inverse Processes
WeBWorK Exercise