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\begin{document}
\Huge
\begin{center} Test 1A, MTH 1410
\end{center}

\normalsize

\textbf{Name:} \underline{\hspace{3in}}

\textbf{ID Number:} \underline{\hspace{3in}}

\begin{center}

I pledge that I have neither given nor received any unauthorized
assistance on this exam.

\vspace{.4in}

$\overline{\hspace{2.4in} (signature) \hspace{2.4in}}$

\vspace{1in}

\Large \textbf{DIRECTIONS} \normalsize

\end{center}

\begin{enumerate}
\item Show all of your work. A correct answer with insufficient
work will lose points, as will an answer with incorrect notation.

\item Clearly indicate your answer by putting a box around it.

\item Calculators are allowed on this exam.

\item Give all answers in exact form, not decimal form (that is,
put $\pi$ instead of 3.1415, $\sqrt{2}$ instead of 1.414, etc) unless otherwise stated.

\item Make sure you sign the pledge and write your ID on both pages.

\item Number of questions = 8.  Total Points = 100.

\end{enumerate}

\newpage

\normalsize
\hspace{.4in}
\textbf{ID Number:} \underline{\hspace{3in}}\\
\begin{enumerate}

\item (16 points) Use the \underline{limit definition} of the derivative to
calculate $f'(x)$ if $$f(x)=\frac{3}{x^2}.$$
Make sure you use correct notation.

\newpage


\item (9 points) For this question, pay close attention to which one is $f$ and which one is $f'$.  The following is the graph of $f'(x)$.

\vspace{2in}

On what interval(s) is $f(x)$:

\hspace{.5in} (a) Increasing

\vspace{.5in}

\hspace{.5in} (b) Decreasing

\vspace{.5in}

\hspace{.5in} (c) Concave down

\vspace{.5in}


\item (6 points) Is the statement below true or false? If true, briefly explain
why. If false, give a counterexample or explain your reasoning.

(a) $\lq \lq$ If $\d{\lim_{x \to 3} f(x)=f(3)}$, then $f'(3)$ exists. "

\newpage


\item (16 points) Use the \underline{derivative laws} (i.e. \underline{not} the definition) to calculate the equation
of the tangent line to $$\d{f(x)=\frac{x^3+2\sqrt{x^3}-7}{x}}$$ at $x=1$.

\newpage


\item (15 points) Find all horizontal and vertical asymptotes of 

$$\frac{3x^2-3x-60}{x^2-16}.$$
Explain your answer, and make sure to use correct notation.
\newpage


\item (16 points) Find the value of $c$ which will make the
following function continuous at $x=9$.  Make sure to use correct notation and explain your reasoning:

\begin{equation*}
f(x) =
\begin{cases}
\d{\left[\frac{\sqrt{x}-3}{x-9}\right]} & \text{if $x> 9$}\\
2c & \text{if $x\le 9$}
\end{cases}
\end{equation*}

\vspace{3.5in}

\newpage


\item (10 points) For the given graph, find the value for the expressions below.  If the
expression does not exist, give a (brief) explanation of why not.

\vspace{2.2in}

(i) $\d{\lim_{x\to (-1)^-}f(x)}$

\vspace{1.2in}

(ii) $\d{\lim_{x\to 2^+}f(x)}$

\vspace{1.2in}

(iii) $\d{\lim_{x\to (-\infty)}f(x)}$

\vspace{1.2in}

(iv) $\d{\lim_{x\to 3}f(x)}$

\vspace{1.2in}

(v) $f'(1)$

\newpage


\item (12 points) Sketch the graph of a \underline{continuous} function $f(x)$ that satisfies the following conditions.  Label all local maximums, local minimums, and inflection points on your graph:

    \b $f(-2)=-3$  and $\d{\lim_{x\to 2}f(x)=4}$

    \b $f'(-2)=0$ and $f'(2)=0$

    \b $f'(x)<0$ on the intervals $(-\infty, -2)$ and $(2, \infty)$;  $f'(x)>0$ on the interval $(-2,2)$.

    \b $f''(x)<0$ on the intervals $(-\infty, -3)$ and $(0, \infty)$;  $f''(x)>0$ on the interval $(-3, 0)$

    \b $\d{\lim_{x\to -\infty}f(x)=1}$

\vspace{6in}

{\textbf{Extra Credit}}(2 points) Use the derivative rules (\underline{not} the limit definition) to find $f'(x)$ if $f(x)=e^7$

\end{enumerate}
\end{document}
