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{\Large TEST 3

Math 111 - 2,4}
\end{center}
\vspace{.1in}
You have 1 hour to complete the test.  
You may not use a calculator.  
Show all relevant work and make your work neat (or else no credit).
Use the back of these pages if necessary.

\vspace{.2in}

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

\begin{enumerate}

\vspace{.1in}

\item  (12 points)  True or false, circle the correct answer:

\begin{tabular}{lll}
T & F & We can apply the mean value theorem to the function $|x|$ on the interval $[-1,4]$.\\
\\
T & F & If $f''(2)=-\sqrt{2}$ then there is a local maximum at 2. \\
\\
T & F & Local maximums are always global maximums. \\
\\
T & F & Global maximums are always local maximums. \\
\\
T & F & If $f'(x)=h'(x)$ then $f(x)=h(x)$.
\end{tabular}
\vspace{.1in}

\item  (15 points) Find the limits:
\begin{enumerate}
\item  \[\lim_{x \rightarrow \infty} \frac{4x^3-2x\sqrt{x}-1}{6x^3-\sqrt{x}}\]
\vspace{1.1in}
\item  \[\lim_{t \rightarrow 3^+} \frac{3+t}{3-t} \]
\vspace{1.1in}
\item  \[\lim_{a \rightarrow \infty} \frac{\sin x^2 \cos x^2 }{x^2}\]
\end{enumerate}
%\vspace{1.1in}
\newpage

\item  (15 points)  
Can the Mean Value Theorem be applied to the function $f(x)=\sqrt{x}$
on the interval [4,36]?  If so, find the point $c$ in the theorem.
\vspace{3in}

\item  (20 points)  
You draw a rectangle with two corners on the $x$-axis and the other two 
on the parabola $y=15-3x^2$, with $y \geq 0$.  
What are the dimensions of the rectangle of this type with maximum area?
(Hint: draw the picture)
\newpage

\item  (20 points)  The following is a graph of $f'$.  
Answer the following questions about $f$ and draw a possible graph of $f$.
\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}
\end{enumerate}

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\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.
\[
g(x)=\frac{x}{x^2+1}
\]

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\end{enumerate}



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