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\title{The Fundamental Theorem of Calculus}
\author{Your Name(s) Go Here}
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\date{\today}
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There are two forms for the Fundamental Theorem of Calculus.

\begin{enumerate}
\item[\bf{FTC 1}]  If $f$ is continuous on $[a,b]$, then the function $g$ defined by 
$$g(x)=\int_a^x f(t) \; dt \qquad a \leq x \leq b$$
is continuous on $[a,b]$ and differentiable on $(a,b)$ and $g'(x)=f(x)$.
\item[\bf{FTC 2}] If $f$ is continuous on $[a,b]$, then 
$$\int_a^b f(x) \; dx =F(b)-F(a)$$
where $F$ is any antiderivative of $f$, that is, a function such that $F'=f$.  


\end{enumerate}

We illustrate with the following example 

\begin{quote}
\item Find $f'(x)$, where $$f(x)=\int_2^{x^2} \frac{\sin(t)}{t} \; dt.$$

\end{quote}

We note first that $f(x)=F(x^2)-F(2)$, where $F'(t)=\frac{\sin(t)}{t}$.  We then use the chain rule to differentiate.   

$$f'(x)=2xF'(x^2)-0$$
$$=2x \frac{sin(x^2)}{x^2}.$$


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