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\title{Fundamental Theorem of Calculus}

\begin{document}

\maketitle
\begin{enumerate}
\item[Problems:]

$$ $$

%%1%%
\item [1.] State "Fundamental Theorem of Calculus".

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%%2%%
\item [2.] (1) Suppose that f is a continuous function and g is a
\item [] \hspace{1.2em} differentiable function.
\item [] \hspace{1.2em} Let $$\it H(x)=\int_a^{g(x)}f(t)dt.$$
\item [] \hspace{1.2em} Find $H'(x)$.
\item [] (2) Suppose that f is a continuous function and $g_1$, $g_2$ are
\item [] \hspace{1.2em} differentiable functions.
\item [] \hspace{1.2em} Let $$K(x)=\int_{g_1(x)}^{g_2(x)}f(t)dt.$$
\item [] \hspace{1.2em} Find $K'(x)$.

\

\

%%3%%
\item [3.] $f(x)= \left\{ \begin{array}{ll}
           x + 1   \hspace{2.2em} $if\enspace$ x < 1 \\
           3       \hspace{4.0em} $if\enspace$ x = 1  \\
           x^2 \hspace{3.5em} $if\enspace$ 1 < x.
        \end{array} \right.$
\item [] Let $$F(x)=\int_{-1}^{x}f(t)dt.$$
\item [] Find $F(x)$ and $F'(x)$.

\

%%4%%
\item [4.] $f(x)= \left\{ \begin{array}{ll}
           x + 1   \hspace{2.2em} $if\enspace$ x < 1 \\
           3       \hspace{4.0em} $if\enspace$ x = 1  \\
           x^2 + 1 \hspace{1.8em} $if\enspace$ 1 < x.
        \end{array} \right.$
\item [] Let $$F(x)=\int_{-1}^{x}f(t)dt.$$
\item [] Find $F(x)$ and $F'(x)$.





\end{enumerate}

\end{document}
