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\centerline{Mathematics 136, section 2 -- Advanced Placement Calculus}
\centerline{Exam 2 -- November 4, 2009}
\vskip 10pt
\noindent
{\it Directions}
\vskip 10pt
Do questions I - V and \emph{either} part A \emph{or} 
part B of question V on the back.  If you submit work for both parts of V,
I will count it for Extra Credit.  There are 100 regular points
and 15 possible Extra Credit points.
Do all work in the blue exam booklet. 

\begin{enumerate}
\item[I.]  Consider the function $f(x) = x^3 + 6x^2 - 15x$.  
\begin{enumerate}
\item[A)] (10) Find all the critical numbers of $f$ and classify
each as a local maximum, local minimum, or neither.
\item[B)] (5)  What are the absolute (global) maximum and minimum
values of $f(x)$ on the interval $[0,3]$? 
\end{enumerate}
\vskip 10pt
\item[II.]  (10) Evaluate the following limit:
\[
\lim_{\theta\to 0} \frac{\cos(5 \theta) - 1}{\theta^2}.
\]
\vskip 10pt
\item[III.]  The velocity of a accelerating car is measured at each 
of the following times, yielding a table of values:  
\[
\begin{array}{lccccccc}
       t = {\rm time (sec)}& \vline &1&3&5&7&9&11\\ \hline
       v(t) = {\rm velocity (ft/sec)}& \vline &1&3.5&5.1&7.8&9.0&11.2
\end{array}
\]
\begin{enumerate}
\item[A)] (5) Give an estimate for the total distance traveled by the car between $t = 1$ and $t = 11$ using a {\it left-hand} Riemann
sum for the velocity function and $\Delta t = 2$.  
\item[B)] (5)  Is your result from A) less than or greater than
the actual total distance traveled, assuming $v$ is always increasing
between $t = 1$ and $t = 11$?  Explain.
\end{enumerate}
\vskip 10pt
\item[IV.] Compute each of the following integrals.  Show all work for A and B.
\begin{enumerate}
\item[A)] (5)  $\displaystyle{\int_1^4 x^2 - 3\sqrt{x} + 4\ dx}$
\item[B)] (10)  $\displaystyle{\int \sec^2(2x) {\root 3 \of {\tan(2x) + 4}}\ dx}$ 
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\item[] Integrate using the indicated method and the table (say which entry you
are using if you use one).
\item[C)]  (10) Use parts:  $\displaystyle{\int x^2 e^{4x}\ dx}$  (full credit
only for showing all work; you can check your result by the table)
\item[D)]  (12.5) Use a trigonometric substitution:
$\displaystyle{\int \frac{\sqrt{25 - x^2}}{x}\ dx.}$
\item[E)]  (12.5) Use partial fractions:
\[
\int \frac{2x + 19}{(x + 1)(x^2 + 16)} \ dx.
\]
\end{enumerate}
\vskip 10pt
\item[V.]
\item[A)]  (15)  A fish swimming against a current of $u$ ft/sec (constant)
for a distance $L$ (constant) will expend energy 
$$E(v) = \frac{Lv^3}{v-u}$$
if it swims at $v$ ft/sec ($u < v$).  What $v$ minimizes the total energy expended?
\item[B)]  (15)  Using the definition (\emph{not} the Evaluation Theorem) 
compute 
\[
\int_2^3 x^2 + 4x\ dx.
\]
The following formulas may be useful:
\[
\sum_{i=1}^n i = \frac{n(n+1)}{2} \qquad \sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}.
\]
\end{enumerate}
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