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\begin{document}


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\noindent
\parbox{2in}{\bf Math 1110-102}
\hfill {\Large \bf Homework \#1} \hfill
\parbox{2in}{\bf \hfill {\bf Due:} Monday (7/11)}

\vspace{0.25in}

\noindent {\large\bf Name:} \underline{\hspace*{3in}}
\hfill {\bf Be sure to show your work!}

\vspace{0.2in}

\noindent {\bf \large \#1.} Consider the following graph of $y=f(x)$:

\vspace{-0.25in}

\begin{center}
\qquad
\begin{tikzpicture}[
  declare function={
    f(\x)= 2*(\x)^2-\x-1;
  }
]
\begin{axis}[
  axis x line=middle, axis y line=middle,
  width=8cm, height=5cm,     % size of the image
  ymin=-1.5, ymax=5.5, ytick={1,...,5}, ylabel=$y$,
  xmin=-1.25, xmax=2.25, xtick={-1,...,2}, xlabel=$x$,
]

\addplot[blue, ultra thick, domain=-1.1:2.1, smooth]{f(x)};

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\end{tikzpicture} 
\end{center}

\begin{enumerate}[(a)]
\item Draw the secant line through the point located at $x=-1$ and $x=1$. Also, draw the secant line through the points at $x=1$ and $x=2$.

\vspace{0.05in}

\item Find the average rate of change of $f(x)$ as $x$ changes from $x=-1$ to $x=1$. Then find the equation of the corresponding secant line.

\vfill

\item Find the average rate of change of $f(x)$ as $x$ changes from $x=1$ to $x=2$. Then find the equation of the corresponding secant line.

\vfill

\end{enumerate}

\noindent {\bf \large \#2.} Consider the following table of information about the function $g(x)$ and its derivative $g'(x)$:
\begin{tabular}{|r|c|c|c|c|} \hline
$x=$ \Large\strut & $-2$ & $0$ & $3$ & $6$ \\ \hline
$g(x)=$ \Large\strut & $1$ & $4$ & $-2$ & $0$ \\ \hline
$g'(x)=$ \Large\strut & $5$ & $-1$ & $0$ & $2$ \\ \hline
\end{tabular}

\vspace{-0.25in}

\begin{enumerate}[(a)]
\item Find the equation of the line tangent to $y=g(x)$ based at $x=-2$.

\vfill


\item Find the equation of the line tangent to $y=g(x)$ based at $x=3$.

\vfill

\end{enumerate}

\newpage

\noindent {\bf \large \#3.} For each of the following functions, (i) Find the derivative using rules/short cuts (like the power rule, etc.) and then (ii) Find the derivative using the limit definition: $\displaystyle f'(x) = \lim\limits_{h \to 0} \dfrac{f(x+h)-f(x)}{h}$.

\vspace{0.15in}

\begin{enumerate}[(a)]

\item $f(x) = 2x^3-x+5$ \qquad (i) $f'(x)=$ \underline{\hspace*{3in}} 

\vspace{0.1in}

(ii) $f'(x)$ by limit definition\dots

\vfill

\item $f(x) = \dfrac{1}{x}$  \qquad (i) $f'(x)=$ \underline{\hspace*{3in}} 

\vspace{0.1in}

(ii) $f'(x)$ by limit definition\dots

\vfill

\end{enumerate}

\end{document}


