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\begin{center}
{\Large FINAL

Math 111 - 2}
\end{center}
\vspace{.1in}
You have 2 hours to complete the test.
Show all relevant work and make your work neat (or else no credit).
Use the back of these pages if necessary.

\vspace{.1in}

Name \rule{5in}{.2pt}
\vspace{.1in}

Code-Name \rule{5in}{.2pt}
\vspace{.1in}

\begin{enumerate}

\item  (8 points) Find the limits:
\begin{enumerate}
\item  
\[
\lim_{x \rightarrow \infty} \frac{17x^{17}-2x^{12}-1}{6x^{17}-x^2+7}
\]
\vspace{1.1in}
\item  \[\lim_{t \rightarrow 12^+} \frac{(t-12)^2}{t+12} \]
\vspace{1.1in}
\end{enumerate}
%\newpage

\item  (12 points)  Given the function $f$ below, 
what do $a$ and $b$ need to be so that $f$ is continuous?
\[
f(x)=\left\{ \begin{array}{ll}
-2x+5  & \mbox{ if } x \leq 5 \\
ax+b & \mbox{ if } 5 < x < 10 \\
x-12  & \mbox{ if } x \geq 10 \\
\end{array}
\right.
\]
%\vspace{3in}
\newpage

\item  (10 points)  Give me two different examples of 
functions that are continuous, but not differentiable.
Explain why they are not differentiable.
And, I want your examples to have different reasons for not being
differentiable.
(Draw the graph of the functions)
\vspace{3in}

\item  (12 points)  
\begin{enumerate}
\item  \label{limit} Find the limit:
\[
\lim_{h \rightarrow 0} \frac{(x+h)^2-x^2}{h}
\]
\vspace{2.5in}

\item  You should have known the limit in number~\ref{limit} 
without computing it.  Why?
\end{enumerate}
%\vspace{2in}
\newpage

\item  (20 points)  The following is a graph of $f'$.  
Answer the following questions about $f$.
\begin{enumerate}
\item  Where is $f$ increasing?
\vspace{.2in}
\item  Where is $f$ decreasing?
\vspace{.2in}
\item  Where are the local maximums of $f$?
\vspace{.2in}
\item  Where are the local minimums of $f$?
\vspace{.2in}
\item  Where is $f$ concave up?
\vspace{.2in}
\item  Where is $f$ concave down?
\vspace{.2in}
\item  Where are the inflection points of $f$?
\vspace{.2in}
\item  Draw a possible graph of $f$.
\vspace{.2in}
\end{enumerate}

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\newpage


\item  (20 points)  Graph the following function labeling 
all of the following (if any):
local extremum ($x$ and $y$ values),
inflection points ($x$ and $y$ values)
and asymptotes.
\[
g(x)=\frac{1}{3}x^3-3x^2-7x+10
\]

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\item  (15 points)  
You are standing on top of a 13 foot ladder which is leaning against the wall.
The bottom of the ladder starts moving away from the wall 
at a rate of 2 feet per second.
How fast are you moving down when you are 5 feet above the ground?
%\vspace{4.4in}
\newpage

\item  (15 points)  
You are in your VW bus, traveling at 80 feet per second 
(around 55 MPH).
Suddenly you see a fellow hippie up ahead 256 feet.
(I don't know how you know its 256 feet, but you do.)
You want to stop and pick them up.
Slamming on your brakes causes you to decelerate at 
8 feet per second per second. (Its a VW bus remember?)

Do you stop before or after the hippie?  
\vspace{.2in}

If before, how many feet before?
\vspace{.2in}

If after, how fast are you going when you pass the hippie?
%\vspace{4.2in}
\newpage

\item  (10 points)  What is the fundamental theorem of calculus?
Explain all symbols you use.
\vspace{2in}
%\newpage

\item  (15 points)  
\begin{enumerate}
\item \label{int} Find the integral
\[
\int_{0}^{2\pi} \sin x \, dx
\]
\vspace{2.2in}

\item  How can you interpret your answer in number~\ref{int} graphically?
\end{enumerate}
%\vspace{2in}
\newpage

\item (15 points)  Compute a Riemann Sum of $f(x)=2x^2+1$ on the interval
$[1,5]$, subdividing the interval into 4 equal subintervals.
(You may choose the $\overline{x_i}$ any way you want. 
Left hand or right hand for example)
\vspace{2.9in}

\item  (12 points)  Evaluate:
\[
\frac{d}{dx}
\int_{4x^3}^{\pi} \tan y \, dy 
\]
\vspace{2.8in}
%\newpage

\item  (12 points)  Find the average value of $6x^5-1$ on the interval $[-1,2]$
%\vspace{3.7in}
\newpage

\item (15 points)  Find the indefinite integral (anti-derivative):
\[
\int \cos(3\sin (7x)) \cos (7x) \, dx
\]
\vspace{3.8in}

\item  (9 points)
\begin{enumerate}
\item  Which parts of this course did you enjoy the most and why?
\vspace{2in}
\item  Which parts of this course did you dislike the most and why?
(What would you change?)
\end{enumerate}








\end{enumerate}



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