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\begin{document}
\centerline{MATH 133 -- Calculus with Fundamentals 1}
\centerline{Sample Questions for Exam 2}
\centerline{October 6, 2017}

\begin{enumerate}

\item{} An object moves along a straight line path with position given by 
$x(t) = 4t^2 + t - 7$, ($t$ in seconds, $x$ in feet).   
\begin{enumerate}
\item{}  What is the average velocity of the object over the interval $[0,5]$ of $t$-values?
\item{}  Fill in the following table with average velocities computed over the indicated intervals.  Using
this information, estimate the \emph{instantaneous velocity} at $t = 2$.  
$$\begin{array}{ccccccccc}
\hline
{\rm interval} &\vline& [2,2.5] &\vline& [2,2.05] &\vline& [2,2.005] &\vline& [2,2.0005]\\ \hline
{\rm ave. vel.}&\vline&         &\vline&       &\vline&        &\vline&      \\ \hline 
\end{array}$$
\item{}  Construct a similar table for intervals \emph{ending} at $t = 2$ and 
repeat the calculations in the previous part.  If you estimate the instantaneous
velocity at $t = 2$ using this new information, does your result agree with what you 
did before (it should!)
\end{enumerate}

\item{} 
\begin{enumerate} 
\item  What is the slope of the secant line to the graph $y = x^3 + 1$ through the points
with $x = 1$ and $x = 2$?
\item  What is the slope of the secant line to the graph $y = x^3 + 1$ through the points
with $x = 1$ and $x = 1 + h$ for a general $h$?
\item  The slope of the tangent line to $y = x^3 + 1$ at $x = 1$ would be
obtained from what limit?
\item  Estimate the limit in the previous part numerically (as in the first question).
\item  Evaluate the limit exactly using our algebraic techniques.
\end{enumerate}

\item{} The graph of a function~$f$ is shown
  below with several points marked. Find all the marked points at
  which the following are true, and give explanations for your answers.
 \begin{figure}
 \begin{center}
 \includegraphics[width=3.0in]{Ex2RevPlot.png} 
 \caption{$y = f(x)$ for Question 3.}
\end{center}
 \end{figure}

\begin{enumerate}

  \item $f$ has an infinite discontinuity

  \item $f$ has a jump discontinuity 

  \item $f$ has a removable discontinuity

  \item $f$ is continuous

\end{enumerate}


\item{} Compute the indicated limits. Show all work for full credit.

  \begin{enumerate}

  \item
     $\ds\lim_{x\to1}\frac{3x^2-5x-2}{x^2-4x+4}$
\vskip 10pt
  \item 
     $\ds\lim_{x\to 2}\frac{3x^2-5x-2}{x^2-4x+4}$
\vskip 10pt
  \item 
     $\ds\lim_{x\to\infty}\frac{3x^2-5x-2}{x^2-4x+4}$
\vskip 10pt
   \item 
     $\ds\lim_{x\to 2}\frac{\displaystyle{\frac{1}{x^2} - \frac{1}{4}}}{x - 2}$
\vskip 10pt
   \item
     $\ds\lim_{t\to 0} \frac{\sin(6t)}{\sin(7t)}$
\vskip 10pt
   \item
     $\ds\lim_{h\to 0} \frac{\sqrt{h + 9} - \sqrt{9}}{h}$

  \end{enumerate}

\item{}  Suppose you know each of the following conditions.
What can you say about $\lim_{x\to c} f(x)$ for the indicated $c$?  Why?
  
  \begin{enumerate}

    \item{}

         $x^2 + x \le f(x) \le x^3 + 3$ for all real $x$, at $c = 0$. 

    \item{} 
         
         $-x^2 + 2x \le f(x) \le x^4 - 4x^3 + 6x^2 - 4x + 2$ for all real $x$, at $c = 1$

    \item{} 

         $f(x) = x \sin\left(\frac{1}{x}\right)$ for all real $x \ne 0$, at $c = 0$
\end{enumerate}

\end{enumerate}
\end{document}
