###### Definition 4.1 Antiderivative of a Function

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

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

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\newcommand{\indefiniteintegral}[2]{\int#1\,d#2}
\newcommand{\lt}{<}
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\)

This subsection demonstrates a few more features.

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.

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*}

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.

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}

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}

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 in the HTML output there is no opportunity to provide an answer.

Consult Definition 4.1 and the The Fundamental Theorem of Calculus to assist you with the following problem.

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