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{\Large\textbf{MATH 182}} $\;$
{\Large\textbf{TEST I}}
         \hfill \textbf{February 12, 1998}
         \hfill \textbf{Name}\underline{\hspace{2in}} \\
\vspace{0in}

{\Large\textbf{SHOW ALL WORK}} \hfill \textbf{Instructor/Section}
\underline{\hspace{2in}}\\

\begin{center}
\setlength{\tabcolsep}{15pt}
\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|}
        \hline
        Problem & 1  & 2  & 3  & 4  & 5  & 6  & 7    & Total  \\
        \hline
        Points  & 14 & 42 & 8 &  8 &  8  & 8  & 12   & 100     \\ 
        \hline
\end{tabular}
\end{center}

\begin{enumerate}
     \item Determine whether the following integrals are convergent
     or divergent.  For those integrals that are convergent, evaluate
     them.

     \begin{tabular}{rlrl}
               &  &  & \\
                a) & ${\displaystyle \int_{1}^{\infty}\frac{1}{x^{3/2}}\;dx}$ &

                b) & ${\displaystyle \int_{0}^{1}\frac{1}{x^{2/3}}\;dx}$ \\
                   & \hspace{3.5in}  &  &  \\[22ex]

     \end{tabular}

     \item Evaluate the following integrals.

     \begin{tabular}{rlrl}

                a) & ${\displaystyle \int_{0}^{1}xe^{-2x}\;dx}$ &

                b) & ${\displaystyle \int e^{2x}\cos x\;dx}$ \\
                   & \hspace{3.25in}  &  &  \\[30ex]

                c) & ${\displaystyle \int_{\pi/4}^{\pi/2}\cot x\;dx}$ &

                d) & ${\displaystyle \int \arcsin x \;dx}$ \\ [32ex]

                e) & ${\displaystyle \int (2x+1)\cos(x^{2}+x)\;dx}$ &

                f) & ${\displaystyle \int_{0}^{1} g(t)\;dt}$, where
                     $g(t)$ is an even function, \\
                     & & & \\
                     & & &
                     ${\displaystyle \int_{-2}^{2} g(t)\;dt = 8}$,
                     and ${\displaystyle \int_{1}^{2} g(t)\;dt = 1}$. \\

     \end{tabular}


\newpage
$\;$
\vspace{-.5in}

     \item Express $\displaystyle {\lim _{n\rightarrow \infty} \;
     \sum_{i=1}^{n}\; \left( \frac{i}{n}\right) ^{10}\cdot
     \frac{1}{n}}\;\;$
     as an integral on the interval $[0,1]$.

     \vspace{1in}

     \item Approximate the integral ${\displaystyle \int_{0}^{2}
     x^{2}\;dx}\;$ by using a Riemann sum taking the sample points to
     be right-hand endpoints and $n=4$.

     \vspace{1in}

     \item The left, right, trapezoidal, and midpoint rule
     approximations were used to estimate
     ${\displaystyle \int_{0}^{2} f(x)\;dx},$ where $f(x)$ is the
     function whose graph is shown.  The estimates were 6.1, 6.2,
     6.3, and 6.4, and the same number of subintervals were used in
     each case.  Fill in the blanks below.\\

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     \begin{enumerate}
        \item  The midpoint approximation is
        \underline{\hspace{.5in}}. \\

        \item  The trapezoidal approximation is
        \underline{\hspace{.5in}}. \\

        \item  The left endpoint approximation is
        \underline{\hspace{.5in}}. \\

        \item  The right endpoint approximation is
        \underline{\hspace{.5in}}.
     \end{enumerate}




\item Suppose ${\displaystyle g(x)=\int_{-x^{2}}^{0} \cos
     t\;dt}.\;$ Find $g'(x) dx$.

     \vspace{1in}

\item Let ${\displaystyle g(x)= \int_{0}^{x} f(t)\;dt},\;$ where
     $f(t)$ is the function whose graph is shown below.  Fill in the
     blanks.

     \begin{enumerate}
        \item  Is $g(4)$ positive or negative? \underline{\hspace{1in}}
        \\

        \item  $g(0)=$ \underline{\hspace{1in}}. \\

        \item  Is the function $g$ increasing or decreasing on the
        interval $(5,6)$? \underline{\hspace{1in}} \\

        \item  $g$ has a local maximum, local minimum, or neither at
        $x=10$. \underline{\hspace{1in}} \\

        \item  The absolute maximum of $g$ occurs at $x=$
        \underline{\hspace{1in}}. \\

        \item  $g$ has a local maximum, local minimum, or neither at
        $x=7$. \underline{\hspace{1in}}\\

     \end{enumerate}

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\end{enumerate}
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