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\huge{TMATH 126:} \Huge{Quiz 1}

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\small This is an open note, open book, and open peer group quiz. You are welcome to use whatever materials you would like for this quiz except for people outside the class.  For example, you may not ask me, the TLC student workers, or other faculty for help.

Groups of 3 or 4 are mandatory.  Only one quiz should be turned in per group. If there is only one name on the top of a quiz that is turned in, I \emph{will not} mark it.  

Show \emph{all} your work (numerically, algebraically, or geometrically) for each and simplify. No credit is given without supporting work.

\normalsize
\begin{enumerate}
\item $[3]$ TRUE/FALSE: Circle T in each of the following cases if the 
statement is \emph{always} true and provide a brief justification.  Otherwise, circle F and provide a counterexample. Let $f$ and $g$ be functions each with the domain $\mathbb{R}$.

\begin{tabular}{lll}
~\\
T &F & If $\displaystyle{\lim_{x\rightarrow 3}f(x)=5}$ and $\displaystyle{\lim_{x\rightarrow 3}g(x)=0}$, then $\displaystyle{\lim_{x\rightarrow 3}\frac{f(x)}{g(x)}}$ does not exist.\\
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T &F & $\displaystyle{\int_{-2}^1x^{-4}\,dx=\left[\frac{-1}{3x^3}\right]^1_{-2}=\frac{-1}{3}-\frac{-1}{-24}=\frac{-9}{24}}$.\\
~\\
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T &F & If $0\leq g(x)<f(x)$ for $a\leq x \leq b$, then $\displaystyle{\int_a^bg(x)\,dx\leq \int_a^bf(x)\,dx}$.\\
~\\
~\\
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\end{tabular}



\item $[5]$ Sketch the graph of an example function $f$ that satisfies the following conditions:
\begin{enumerate}
\item the domain of $f$ is $\mathbb{R}$ 
\vskip .3in
\item $f'(-4)$ is not defined
\vskip .3in
\item $\displaystyle{\lim_{x\rightarrow -3}f(x)\not= f(-3)}$
\vskip .3in
\item $\displaystyle{\int_2^4f(x)\,dx= 4}$
\end{enumerate}

\vspace{-2in}\hspace{2in}\includegraphics[height=2.5in]{blank_5}

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\item $[4]$ Determine the following, if they exist:

$\displaystyle{\lim_{x\rightarrow \infty}x\sin(\frac{3\pi}{x})}$ \hspace{3in} $\displaystyle{\lim_{x\rightarrow \infty}e^{-x}\cos (x)}$
\vfill


\item $[4]$ Let $f(x)=xe^{-3x}$.
\begin{enumerate}
\item Find the equation of the line tangent to $f$ when $x=5$.
\vfill
\item Use the linear approximation you found in part (a), to estimate $f(5.2)$.
\vskip .75in
\end{enumerate}


\item $[4]$ Find the following:

$\displaystyle{\int x\cos(2x)\,dx}$ \hspace{3in} $\displaystyle{\int \frac{y+6y^7}{y^2}\,dy}$
\vfill




\end{enumerate}
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