\documentclass [12pt] {article}

\usepackage{amsmath}

\topmargin 0pt
\advance \topmargin by -\headheight
\advance \topmargin by -\headsep
\textheight 8.9in
\oddsidemargin 0pt
\evensidemargin \oddsidemargin
\marginparwidth 0.5in
\textwidth 6.5in

\begin{document}
\begin{center}
\begin{tabular*}{6in}{c@{\extracolsep{\fill}}c@{\extracolsep{\fill}}c}
EXAM 1 &\huge TMath 125  &\normalsize PRACTICE\\
\end{tabular*}
\end{center}
\vskip .3 in
Note: This is a practice exam and is intended only for study purposes. The actual exam will contain different questions and may have a different layout.

\begin{enumerate}
\item TRUE/FALSE: Circle T in each of the following cases if the 
statement is \emph{always} true.  Otherwise, circle F.  Let $a$, $b$, and $c$ be constants.  Assume $f$ and $g$ are continuous.

\begin{tabular}{lll}
~\\
T &F & $\int_a^b cf(x)dx=c\int_a^bf(x)\,dx$\\
~\\
T &F & $\int f(x)g(x)dx=\int f(x)\,dx\int g(x)dx$\\
~\\
T &F & All continuous functions have derivatives.\\
~\\
T &F & All continuous functions have antiderivatives.\\
~\\

T &F & $\int_{-1}^1 \frac{1}{x^2} \,dx =\frac{-1}{x}\vert^1_{-1}=\frac{-1}{1}-\frac{-1}{-1}=-2$\\
~\\
\end{tabular}
\vskip .2 in
\large
Show your work for the following problems.  The correct answer with 
no supporting work will receive NO credit (this includes multiple
choice questions).

\normalsize
\item Oil leaked from a tank at a rate of $r(t)$ liters per hour.  The rate decreased as time passed and values of the rate at two hour time intervals are shown in the table.  Find lower and upper estimates for the total amount of oil that leaked out. 

\begin{center}
\begin{tabular*}{3.7in}{|l|llllll|}
\hline
$t$ (hours) & 0 & 2&4&6&8&10\\
\hline
$f(t)$ (Liters/hour) &8.7&7.6&6.8&6.2&5.7&5.3\\
\hline
\end{tabular*}
\end{center}
\vfill
\item Let $a$ be a constant (like 2.5 or something).  Find the equation of the line that is tangent to the graph of $y=\ln x$ at $x=e^a$ for some constant $a$.
\vfill

\newpage
 $f(x)=
                                \begin{cases}
                                        -\sqrt {4-x^2};  &\text {if $-2\leq x\leq 2$}\\
                                        x-2;  &\text {if $2<x$} \\
                                \end{cases} $  
 \item Refer to the above definition of $f(x)$ to answer the following questions. 

 \begin{enumerate}
 \item Carefully graph $f(x)$.
 \vskip 3in
 \item  Use your above graph to find $\int_{-2}^4 f(x) \,dx$.
 \vskip 1in
 \item Give a rough sketch the graph of $\int_{-2}^x f(t)\,dt$.
 \end{enumerate}
 



\newpage
\item  Find 

$\frac{d}{dx} \int_0^{\tan x}\sqrt{1+r^3}\,dr$ \hspace{2in} $\left(\int_x^0 e^{\arctan(t)}\,dt\right)'$
\vskip 2in

\item Evaluate the following. 

$\displaystyle{\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{x^2\sin x}{1+x^6}\,dx}$ \hspace{2in} $\displaystyle{\int x^3 \sqrt{x^2+1}\,dx}$
\vfill
 $\displaystyle{\int \tan(x) \,dx}$\hspace{2in} $\displaystyle{\int_4^7  \sqrt{2t+1}\,dt}$
\vfill


\newpage

\item Consider the line $y+1=x$ and the parabola $\frac{1}{2}y^2-3=x$. 
\begin{enumerate}
\item Find the area of the region bounded by the above two curves.
\vfill
\item Find the volume of the solid that results by rotating the above region about the line $x=5$
\vfill
\end{enumerate}
\vfill

\newpage
\item Kobayashi has won the hot dog-eating world championship six times.  Recently he challenged a giant bear to a 3 minute hot dog-eating contest.  Kobayashi found that the rate he can eat hot dogs goes down as time goes by and can be modeled by $k(t)=\frac{12}{(t+1)^2}+24$, were $t$ is measured in minutes.  The bear isn't quite as used to the system and seems to start with a slower rate that gets larger and is well modeled by $b(t)=8t^3+20$.  Find out how many hot dogs Kobayashi and the bear eat and determine who won the context.
\vfill

%example 5 from \S6.4
\item A tank has the shape of an inverted circular cone with height 10m and base 4m.  It is filled with water to a height of 8m.  Find the work required to empty the tank by pumpting all of the water to the top of the tank.  (The density of water is 1000kg/$\text{m}^3$.)
\vfill


\end{enumerate}

\end{document}
