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\begin{document}
\centerline{Mathematics 134 -- Calculus 2 with Fundamentals}
\centerline{Information on Final Examination}
\centerline{May 3, 2018}
\vskip 10pt
\noindent
{\it General Information}
\vskip 10pt
\begin{itemize}
\item  The final examination for this class will be
given during the scheduled period 8:00am to 10:30am on Thursday, May 10.
\item  Like the midterms, this one will be given
{\it in our regular classroom}, Swords 302.
\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.  Tables for the trigonometric
reduction formulas, normal curve areas, and time value of money formulas will be provided. 
\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 Tori will run a final exam review session on Wednesday, May 9, 8 to 10pm.
\item I will also be available for office hours the last day of classes and 
during the reading period as follows:
\begin{itemize}
\item Monday, May 7: 10am to 12 noon and 2pm to 3pm (but not after that because
of Honors presentations in the department)
\item Tuesday, May 8: 8am to 10am (sorry, but that is the {\it only} time I can be
available due to multiple other things going on that day in the department and otherwise)
\item Wednesday, May 9: 10am to 12 noon as usual (I won't be available in the afternoon
due to a long meeting with Public Affairs about redesigning our department web site.) 
\end{itemize}
\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 somewhere around 25\% each devoted to the material on each of the midterm 
exams (but of course topics from earlier in the semester might appear in parts
of questions about things we did later). 
\item See the review sheets for the four 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}
\vfill\eject
\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 $20\%$ 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.  
The fact that you reviewed just recently to prepare for Exam 4
should actually {\it help you here}(!)
\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, but a few different topics, different types of questions, etc. might also appear.
\emph{For instance, you should be prepared to solve separable differential equations as well and
there will be a question about the ``time value of money'' concepts from the last weeks of our
course like VII B.}
\item Your knowledge of the trig substitution and partial fractions methods will be tested with questions 
like the ones from the midterms -- i.e. I will ask you about each step of the method
separately rather than asking you to work out integrals ``start to finish.'' The integrals 
in question II below are not set up that way, so this part of your exam will look somewhat
different.  
\end{itemize}
\vskip 10pt
\noindent
{\it Sample Final Exam}
\vskip 10pt
\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 $F(x) = \int_{-1}^x f(t)\, dt$ (there's a part of the graph with $x < 0$
that you're not seeing).  Complete the following table of values for $F(x)$:
\[
\begin{array}{|c||r|r|r|r|r|r|r|}
\hline
x          & 0 & 1 & 2 & 3 & 4 & 5 & 6 \strut \\
\hline
F(x)       & \quad 3 & \qquad  & \qquad & \qquad & \qquad & \qquad & \qquad \strut \\
\hline
\end{array}
\]
\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)] Use a right-hand Riemann sum with $n = 4$ (that is, the $R_4$ approximation) 
to approximate the value of the integral
$$\int_1^3 \sqrt{1 + 9x^4}\ dx.$$ 
\item[(B)] Given:  The graph of the function in the integral
is increasing on the interval $[1,3]$.  Check the appropriate box:
\item[]The $R_4$ 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 units of 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 $\displaystyle{N(t) = \frac{10}{1 + 9999 e^{-t}}}$ 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)]  Find the sum of the finite geometric series with alternating signs:
$$3 - \frac{3}{2} + \frac{3}{2^2} - \frac{3}{2^3} + \cdots + \frac{3}{2^6}$$
What happens if you continue adding terms of the same form indefinitely?  That is, 
what is the sum 
$$\sum_{k = 0}^\infty \frac{(-1)^k 3}{2^k}?$$
\item[(B)]  What would be the monthly payment on a $\$500,000$ 30-year fixed-rate
mortgage at $3.5\%$ annual interest.
\end{enumerate}
\end{document}


  
