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 \(f\text{.}\)
Section4Antidifferentiation
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.
The Fundamental Theorem of Calculus in one of the high points of a course in single-variable course.
Theorem4.2The Fundamental Theorem of Calculus
If \(f(x)\) is continuous, and the derivative of \(F(x)\) is \(f(x)\text{,}\) then
\begin{equation*} \definiteintegral{a}{b}{f(x)}{x}=F(b)-F(a) \end{equation*}Proof
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.
Corollary4.3
Suppose \(f(x)\) is a continuous function. Then
\begin{equation} \frac{d}{dx}\definiteintegral{a}{x}{f(t)}{t}=f(x)\label{equation-alternate-FTC}\tag{4.1} \end{equation}Proof
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.
1Graphical Antiderivatives
Consult Definition 4.1 and the The Fundamental Theorem of Calculus to assist you with the following problem.
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 ptxcsstesting 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.