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\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


\urladdr{http://www.lehigh.edu/\textasciitilde{}dlj0}

\maketitle

\subsection{The Fundamental Theorem of Calculus}

\begin{thm}
\textbf{\textup{{[}Fundamental Theorem of Calculus, Part 1{]}.}} Let \( f \)
be a continuous function on an open interval containing \( [a,b] \). Set\[
G(x):=\int _{a}^{x}f(t)dt.\]
 Then, \( G(x) \) is differentiable on \( [a,b] \) and its derivative is \( f \),
that is, \( G'(x)=f(x). \)
\end{thm}
\textbf{\emph{Proof:}}

Define \( G \) as in the statement. Then, compute the derivative:\begin{eqnarray*}
G'(x) & := & \lim _{h\rightarrow 0}\frac{G(x+h)-G(x)}{h}\\
 & = & \lim _{h\rightarrow 0}\frac{\int _{1}^{x+h}f(t)dt-\int _{1}^{x}f(t)dt}{h}\\
 & = & \lim _{h\rightarrow 0}\frac{\int _{1}^{x+h}f(t)dt+\int _{x}^{1}f(t)dt}{h}\\
 & = & \lim _{h\rightarrow 0}\frac{\int _{x}^{1}f(t)dt+\int _{1}^{x+h}f(t)dt}{h}\\
 & = & \lim _{h\rightarrow 0}\frac{\int _{x}^{x+h}f(t)dt}{h}\\
 & = & \lim _{h\rightarrow 0}\frac{f(c)((x+h)-x)}{h},\, \textrm{by the MVT},\, \textrm{integral form}.\\
 & = & \lim _{h\rightarrow 0}f(c)\\
 &  & f(x),\, \textrm{since}\, c\, \textrm{is between }x\, \textrm{and }x+h.
\end{eqnarray*}

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