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\begin{document}
\centerline{Mathematics 136, section 2 -- Calculus 2}
\centerline{Information on Final Examination}
\centerline{December 5, 2016}
\vskip 10pt
\noindent
{\it General Information}
\vskip 10pt
\begin{itemize}
\item  The final examination for this class will be
given during the scheduled period 11:30am to 2:00pm on Tuesday, December 13.
\item  Like the midterms, this one will be given
{\it in our regular classroom}, Swords 321.
\item  The final will be similar in format to the 
midterm exams but perhaps two times as long.  I expect that
if you are well prepared and you work steadily, then you should 
be able to finish the exam in about 1.75 hours.  However, you will have 
the full 2.5 hour period to work on the exam if you need that 
much time.
\item  As on the midterms, you may use a calculator, but
no graphing features.
\item  No cell-phones, computers, or other electronic
devices beyond a basic calculator may be used during the exam.  Please do not bring them
with you; they will be subject to confiscation for the period
of the exam if you use them. 
\item I'm happy to try to find a time for a pre-final
review session.  We can discuss this in 
class on Wednesday, December 7 or Friday, December 9.
\item I will also be available during regular office hours
during the week of December 5 and the following time on Monday, December 12
for help as you prepare:
\item Monday, December 12, 8:00am - 12:00noon
\item Tuesday, December 13, 8:00am - 11:00am
\end{itemize}
\vskip 10pt
\noindent
{\it What Will Be Covered}
\vskip 10pt
\begin{itemize}
\item This will be a {\it comprehensive} final -- 
it will cover all the topics we have studied this semester,
with about 30\% each devoted to the material on each of the midterm exams, 
plus about 10\% on the material from Problem Set 9 (sequences and series).  
\item See the review sheets for the three midterms
for a detailed breakdown of the topics we studied earlier.
Those review sheets are now reposted on the course homepage
if you need another copy of any of them.
\end{itemize}
\vskip 10pt
\noindent
{\it Philosophical Comments and Suggestions on How to Prepare}
\vskip 10pt
\begin{itemize}
\item  The reason we give final exams  
in almost all mathematics classes is to encourage students
to ``put whole courses together'' in their minds. 
Also, preparing for the final should help to 
make the ideas ``stick'' so you will have the material at your 
disposal to use in later courses.  {\it This is especially 
important if you are preparing to continue to MATH 241 or other
advanced mathematics courses -- everything
is based on material from this semester's course and 
it will be difficult or impossible to do well in that course unless
you have the material from this one under good control.}
\item It may not be necessary to say this, but
here goes anyway:  {\it You should take this exam seriously} -- 
it is worth $25\%$ of your course average and it can 
pull your course grade up {\it or down} depending on how you do.  
\item  Get started reviewing early
and do some work on this {\it every day} between now 
and the date of the final.  Don't try to ``cram'' at the end.  
There's too much stuff that you need to know to approach 
preparing that way!
\item Review videos and your class notes in addition to the
text, especially for topics where you lost points on the midterms.  
There are a lot of worked-out examples and discussions of all of 
the topics we have covered there.
\item 
Look over the midterm exams with the solutions. Go over your corrected problem sets.
If there were questions where you lost a lot 
of points, be sure you understand why what you did was not
correct, and how to solve those questions. 
\item  Be sure you actually do 
enough practice problems so that you have the 
facility to solve exam-type questions in a limited
amount of time.   {\it Even if you 
have saved solutions for practice problems from the midterms},
it is going to be much more beneficial to do practice problems
starting ``from scratch'' rather than just reading old
solutions.  Remember, the goal of the course is to get
you to be able to develop solutions to these problems
yourselves, not just to understand solutions that someone
else (that includes you, one or more months ago!) has written down.  
Another analogy -- as most of you know from your study
of languages, it's much easier to understand 
another language passively than it is to actually 
use a language actively yourself (for instance, to form your
own complete, grammatically correct sentences).  
The goal of this course is to make you reasonably
proficient ``calculus speakers'' and there's no substitute
for active practice on those skills.
\item 
The following is (a slightly edited version of) the final I gave
in Calculus 2 in spring 2014.  It's a good guide for what our exam will look like, but
of course, different topics, different types of questions, etc. might also appear.
\emph{For instance, you should be prepared to solve separable or first order linear
differential equations as well.}
\end{itemize}

\noindent
I.  
\begin{enumerate}
\item[(A)]  Let 
$\ds
f(x) = \begin{cases} 1 & \text{ if } 0 \le x \le 2\\
                    x-1 & \text{ if } 2 < x \le 4.\\
                   11-2x & \text{ if } 4 < x \le 6
\end{cases}$  
whose graph is shown here:
\begin{center}
\includegraphics[width=2.5truein,height=2.5truein]{132FIS.png}
\end{center}
Let $F(x) = \int_0^x f(t)\, dt$.  Complete the following table of values for $F(x)$:
\vskip 1.0truein
\[
\begin{array}{|c||r|r|r|r|r|r|r|}
\hline
x          & 0 & 1 & 2 & 3 & 4 & 5 & 6 \strut \\
\hline
F(x)       & \quad 0 & \qquad  & \qquad & \qquad & \qquad & \qquad & \qquad \strut \\
\hline
\end{array}
\]
\item[(B)] (10) Compute the derivative of the function $\ds g(x) = \int_0^{2x} \frac{\cos(t)}{t^2}\, dt$.
\end{enumerate}
\vskip 10pt
\noindent
II.  Compute the following integrals. 
\begin{enumerate}
\item[(A)]  $\ds \int \frac{x^4 - 3\pi^3 + \sqrt{x}}{x^{2/3}}\, dx$
\item[(B)]  $\ds \int x^3 e^{x^2}\, dx$  (use integration by parts)
\item[(C)]  $\ds \int \frac{\csc^2(5x)\, dx}{\cot(5x) + 7}$
\item[(D)]  $\ds \int_1^e x^5 \ln(x)\, dx$.  
\item[(E)]  $\ds \int \frac{1}{\sqrt{16 + x^2}}\, dx$
\item[(F)]  $\ds \int \frac{1}{(x^2+1)(x+1)}\, dx$
\end{enumerate}
\vskip 10pt
\noindent
III.  
\begin{enumerate}
\item[(A)]  The integral
$$\int_1^3 \sqrt{1 + 9x^4}\ dx$$ 
would compute the length of the curve
$y = x^3$ from $x = 1$ to $x = 3$.
Use a midpoint Riemann sum with $n = 4$ to approximate its value.
\item[(B)] Given:  The graph of the function in the arclength integral
is concave up on the interval $[1,3]$.  Check the appropriate box:
\item[]The midpoint approximation is a overestimate $\Box$/underestimate $\Box$.  
\end{enumerate}
\vskip 10pt
\noindent
IV.  A region $R$ in the plane is bounded by the graphs $y = 9 - x^2$, $y = 2x$, $x = 0$ and $x = 1$.
\begin{enumerate}
\item[(A)]  Compute the area of the region $R$.
\item[(B)]  Compute the volume of the solid obtained by rotating $R$ about the $x$-axis.
\item[(C)]  Set up integral(s) to compute the volume of the solid
obtained by rotating $R$ about the $y$-axis.  \emph{You do not need to compute the value.}
\end{enumerate}
\vskip 10pt
\noindent
V.  The daily solar radiation $x$ per square meter (in hundreds of calories) in Florida
in October has a probability density function $f(x) = c (x - 2)(6 - x)$ if 
$2 \le x \le 6$, and zero otherwise.  Find the value of $c$, 
and then compute the probability that the daily solar
radiation is $> 4$ hundred calories per square meter.
\vskip 10pt
\noindent
VI.  An avian flu epidemic has broken out in Birdsburgh, a large city 
with total population 10 million.  Let $N$ be the number, in millions,
of people who have been infected, as a function of time $t$ in weeks.  The Birdsburgh
Public Health Department determines that the rate of change of $N$ is 
proportional to the product of the number of infected people ($N$) and 
the number of people not yet infected.   
\begin{enumerate}
\item[(A)]  Write the statement above as a differential equation, calling
the constant of proportionality $k$.
\item[(B)]  The function $N(t) = \frac{10}{1 + 9999 e^{-t}}$ should be a solution of your 
differential equation from part A.   What is the value of $k$? 
\item[(C)]   If the epidemic proceeds according the function given in part B,
how many weeks will pass before the number of infected people reaches 1 million?
\end{enumerate}
\vskip 10pt
\noindent
VII.  
\begin{enumerate}
\item[(A)]  Does the infinite series $\ds \sum_{n=0}^\infty \frac{1}{4^n}$ converge?
If so, what is the sum?
\item[(B)]  Use the Integral Test to determine whether the infinite series
$\ds \sum_{n=3}^\infty \frac{1}{n (\ln(n))^3}$ converges.
\item[(C)]  Use the Ratio Test, then consider the endpoints separately, to determine the interval of convergence of the power series $\ds \sum_{n=0}^\infty \frac{x^n}{n^2}$.  
\end{enumerate}
\end{document}


  
