\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,amssymb,epsfig,float,graphicx,subfig}

\markright{MONT 104N, Information on Final Exam}

\setlength{\topmargin}{-0.5in}
\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{0in}
\setlength{\textheight}{8.5in}

\newcommand{\Blank}{$\underline{\phantom{Please write your name in this
        space}}$} 
\newcommand{\Strut}{$\vphantom{\Big|}$}

\begin{document}

\centerline{MONT 104N -- Modeling the Environment}
\centerline{Information on Final Exam}
\centerline{December 6, 2019}
\vskip 10pt
\noindent
{\it General Information and Groundrules}
\vskip 10pt
As announced in the course syllabus, the
final exam for our Montserrat seminar  will be given
at the announced time for MWF 9:00am classes:
\vskip 10pt
\centerline{\it Friday, December 20, 8:00am to 10:30am} 
\vskip 10pt
\noindent
The exam will be held in our regular class room.
\begin{itemize}
\item This will be an individual exam.  No sharing of information 
in any form will be permitted during the exam.
\item You may bring (and should use) a calculator during the exam,
but no other electronic devices.  {\it Know how to compute logarithm and 
exponential values} using your calculator in addition to basic arithmetic.  
\item If you are well-prepared and work 
steadily, I expect the exam will take 1 to 1 1/2 hours
(60 to 90 minutes) to complete.  However, you will have the
full 2 1/2 hour period (150 minutes) to work if you 
need that much time. 
\item This will be a {\it comprehensive} exam.  The 
exam questions may cover concepts
and techniques from any section of the course (Chapters 1,2,3,4,5,6,7,8
in the Modeling and Data Analysis textbook).  But I will only ask about topics we have 
actively discussed in class, the problem sets, or the projects.   
\item There will be four or five mathematical
problems (each possibly with a few separate parts).  These
questions will be similar to things you have seen on the
problem sets or the group projects.
Some sample exam questions are given later in this document.
I will post solutions for those in a few days -- my recommendation
is to try working out solutions first on your own, then use
the solutions to check your work.  
\item  The exam will also include an essay question 
(worth about 1/3 of the total points) on the set topic below.  It should
take you about 20 or 30 minutes to produce a good, detailed
answer for that question.  So it will be necessary to spend
a sufficient portion of your preparation time on deciding what you want 
to say.
\item I will be available during the reading period
at the following times for ``last minute'' questions:
\begin{itemize}
\item Monday, December 16, 10am - 12noon
\item Tuesday, December 17, 1 - 3pm
\item Wednesday, December 18, 8 - 10am
\item Thursday, December 19, 2 - 4pm
\end{itemize}
{\it I will probably not be available at other times
that week due to the many things that must be completed
in a short time (sorry!)}
\end{itemize}
\vskip 10pt
\noindent
{\it Essay Topic}
\vskip 10pt
In general terms, what is a mathematical model?  
Describe in general terms what they are, 
how they are constructed, and how they are used.  Give examples of three
different types of mathematical models we have studied this semester.  
Even if mathematical models don't capture {\it every feature} of a real world
situation, why is it still important to develop them and understand the
information we get from them?  For instance, what conclusions about use
of natural resources did we derive by looking at logistic models with various 
types of harvesting in the Chapter Project from Chapter 7?  As another example, 
how are mathematical models important in understanding our choices of which 
energy sources to use?  Why is it important to understand 
how radioactive substances decay?  What type(s) of model(s) that we discussed would 
apply to describe that process?  What are some of the issues involved with
using radioactive decay to generate electricity--that is, why is this not
a ``no-brainer'' as a solution to the problem of $CO_2$ buildup in the atmosphere
from fossil fuel burning?  
\vskip 10pt
\noindent
{\it Sample Mathematical Questions}
\vskip 10pt
\noindent
{\it Note:} These were essentially the mathematical questions from the final
exam I gave in MONT 100N in Fall 2017.  The questions this year will be different, but
comparable in topics covered, in length, in level of difficulty, etc.  
\vskip 10pt
\noindent
I.  In 1990, forests covered $4.047 \times 10^9$ hectares of the Earth's surface.
By 2000, forest area had decreased to $4.038\times 10^9$ hectares.  Assuming
that the decrease in forest area is {\it linear}, and that it will continue at 
the same rate into the future,  in this problem you will develop a linear model 
for the forest area remaining as a function of $t = $ years since 1990.
\vskip 10pt
\begin{enumerate}
\item[A.]   Determine the slope for the linear model of the forest area.  What are the units of the slope?
\item[B.]   What is the linear equation modeling the forest
area as a function of $t = $ years since 1990.
\item[C.]   According to your model, in what year will
the forest area reach $4.0 \times 10^9$ hectares?
\item[D.]   According to the United Nations Food and Agriculture
Organization, the actual forest area remaining in 2010 was $4.033 \times 10^9$
hectares.  How close is the prediction your model from part B gives for
the forest area in 2010?
\end{enumerate}
\vskip 10pt
\noindent
II.  Wind power has emerged as a fast growing source of
energy for electrical power generation in recent years.
In 2016, the generating power of wind turbines installed around
the world was about
$301$ gigawatts and it was increasing at about $33.2\%$ per
year.
\begin{enumerate}
\item[A.] The typical English unit of power is the horsepower.
$1 {\rm\ horsepower} = 7.457 \times 10^{-7}$ gigawatts.  Convert $301$ gigawatts
to the equivalent number of horsepower.
\item[B.] Construct an exponential model for $WP = $
wind power generation as a function of
$t = $ years after 2016.  \emph{Use units of $10^2$ gigawatts for WP} -- see
the entry for 2016 in the table below.
\item[C.]  Fill in the table of values for $WP$ below with
values predicted by
your model for the years $2017 - 2022$.   Round to 2 decimal places.
About how many years will it take for $WP$ to reach approximately 
double the 2016 level?
\vskip 10pt
$$
\begin{array}{c|c|c|c|c|c|c|c|}
Year & 2016 & 2017 & 2018 & 2019 & 2020 & 2021 \\
\hline
WP  & 3.01 &\qquad\qquad &\qquad\qquad & \qquad\qquad &\qquad\qquad &\qquad\qquad  \\
\hline
\end{array}
$$
\vskip 10pt
\item[D.] How many years will it take for wind power generation
to reach $20 \times 10^2$ gigawatts according to your model?
\end{enumerate}
\vskip 10pt
\noindent
III.  Suppose that a population of fast-reproducing insects in an area
has a natural net growth rate of $5\%$ per month from births and deaths, and
that there is a net migration \emph{loss} of 20 individuals per month.
\vskip 10pt
\begin{enumerate}
\item[A.]   Write a difference equation that models this situation.
\item[B.]   Using an initial value $P(0) = 500$, determine the populations
in months $1,2,3,4,5$ according to the model you stated in part A and
record the values in the following table (round any decimal values to the nearest
whole number)
\vskip 10pt
$$
\begin{array}{c|c|c|c|c|c|c|}
n & 0 & 1 & 2 & 3 & 4 & 5  \\
\hline
P(n)  & 500 &\qquad\qquad &\qquad\qquad & \qquad\qquad &\qquad\qquad &\qquad\qquad  \\
\hline
\end{array}
$$
\vskip 10pt
\item[C.]  What happens to the population in the long run as $n$ increase?  Does
it tend to a definite value?
\end{enumerate}
\vskip 10pt
\noindent
IV.  Answer the following questions with a few sentences each.
\begin{enumerate}
\item[A.]  If you are fitting an exponential model to a data set $(x_i,y_i)$ ``by hand,''
you start by transforming the data to what new form $(X_i,Y_i)$?.  If the best fit
regression line for the transformed data is $Y = m X + b$, what is the corresponding
exponential model?  (You may assume the logarithms have base 10 as we discussed in class.)
\item[B.]  If you are fitting an power function model to a data set $(x_i,y_i)$ ``by hand,''
you start by transforming the data to what new form $(X_i,Y_i)$?  If the best fit
regression line for the transformed data is $Y = m X + b$, what is the corresponding
power function model?  (You may assume any logarithms have used base 10 as we discussed in class.)
\item[C.]  What does the $R^2$ statistic in measure in linear regression?  How
did we use it?  Explain what it would mean, for instance if $R^2 = 1$.
\item[D.]   What type of chart (scatterplot, pie chart, bar chart, etc.) would be
most useful to describe the composition of a forest if there 5 different types of trees
present in different concentrations per acre?   Explain, and illustrate your answer
with a chart if a typical acre of forest contains 10 oaks, 12 maples, 5 pines, 2 hemlocks,
and 1 chestnut.
\item[E.]  What difference equation would model a population undergoing logistic growth
if the population was growing at about $4\%$ per year when the population is much smaller than
the carrying capacity $M = 400$ of the habitat?  
\item[F.]  What feature of the solutions of the Lotka-Volterra equations is considered a
confirmation that this model is capturing an important aspect of real-world predator-prey
interactions?
\end{enumerate}
\vskip 10pt
\noindent
Full solutions for all of these will be posted at the start of exam week so you can check your
work.  Some additional practice problems to look at are:
\vskip 10pt
\begin{enumerate}
\item Chapter 1/5, 11, 13  (Note: the conversion factors you need
are given in the text. I will supply any information like that you would need
for questions like these ones.)
\item Chapter 2/2, 7
\item Chapter 4/6, 8, 9  
\item Chapter 5/2, 4, 6, 7, 8, 9
\item Chapter 7/1, 4 (know the general solution for affine first-order
difference equations and how to determine equilibrium solutions algebraically; 
I will not give you that formula), 8 -- sketch
the solutions by hand in rough, qualitative terms
\item Chapter 8/3
\end{enumerate}
\vskip 10pt
\noindent
{\it Note:}  I could also ask you ``qualitative'' 
questions relative to the process of fitting a linear, exponential, or power function
model to data like one of the questions from the midterm, including
questions about the process of transforming the data via logarithms
for the exponential and power law cases.  (See IV A and B above for what I mean by this.)
I {\it can't really} ask you to compute a regression equation
for fitting a linear or exponential model to data,
of course, since apart from one simple linear example, we only did 
those computations with the spreadsheet.)  I could give you the equation of
a regression line, though, and ask you to give a model prediction
for an $x$-value not occurring in the data.  
\end{document}
