Math 1030-007
Extra Credit Project
Fall 2010
Expectations for Extra Credit Project
Summary: In this project you will have the opportunity to apply what you have learned in
Calculus to answer questions about real world problems. You will be expected to use the
principles of calculus to solve the problems mathematically, and you will also be expected to do
enough background research to collect the information necessary to solve the problems. In many
cases, you may even need to learn about a new topic in Calculus that we did not cover in this
course but is very closely related to topics that we have covered.
The project is a report on an investigation of a question. This means your report should
begin with an introductory paragraph in which you discuss the question or problem that you are
examining and introduce briefly how you approached your work, perhaps give a brief statement
of something that you discovered or a conclusion you will examine in more detail later in the
report. The next part of your report will be the main body in which you discuss your work, give
your analysis, and demonstrate your results. The last part of your report should be a nice
summary of your findings and conclusions backed up by the evidence you presented in the
middle part of your report.
Instructions:
-
You may work on this project individually or in a group of no more than three people.
Select one of the approved topics for your project.
Projects must be typewritten on 81/2 x 11 paper, double spaced, 12 point font.
Math formulas, equations, diagrams can be written in by hand only if done neatly.
Equations and calculations should be accompanied by explanations given in complete
sentences of what these calculations mean.
If tables or graphs are attached at the back, then the main body of the report should
include correct references to page numbers, figures, and appendices.
Sources used for information or data should be described and adequate references for
these sources given. Material taken from references must be summarized in one's own
words.
It is recommended that you edit your report at least once before submitting it for grading.
Things to look for in the editing process include typographical errors, grammar and
spelling errors, and awkward writing. If you are working in a group, all members of the
group should read the final draft of the project, approve it before it is submitted, and sign
their name on the title page. If there is group member who has made no contribution to
the project, the instructor should be informed and that student's name not included in the
report.
Due Date:
Monday, December 6, 2010
Grading:
-
Mechanics:
20%
-
Mathematics & Analysis:
80%
(spelling, grammar, punctuation, citations,
formatting, etc.)
(accurate calculations, correct use of concepts,
proper reasoning, logical flow of ideas etc.)
Math 1030-007
Extra Credit Project
Fall 2010
Option #1: (Startup Company)
In this course you learned that one of the most important
applications of the derivative is in optimization problems (see section 3.4). We went over many
cool examples in class but in this project you will have the opportunity to do a much more in
depth exploration real world optimization problems. You may need to do some research in order
to determine some of the formulas and numbers you will need in order to answer the following
questions. Be sure to cite your sources.
(a)
You wish to open a small business in Salt Lake City where you will offer tutoring
services for high school mathematics. You have done some market analysis and have found that
the demand function is
where p is the price you charge for one hour of tutoring and
D(p) is the combined number of tutoring hours that all of your customers want at the give price
each month. In this situation, what should you set your price at in order to maximize your profit
(ignore operating costs for the time being).
(b)
You have several tutors working under you, and the number of hours they are willing to
work each month depends on how much you pay them. Suppose your supply-side analysis
shows that the supply function for tutors is
where q is the price you pay each tutor
per hour and S(q) is the combined number of hours each tutor is willing to work at that price
each month. Using your result from part (a), determine the price to pay your tutors that will
maximize profit for your company. Use your common sense to make certain simplifying
assumptions if necessary.
(c)
You decide it would be a good idea to try and advertise your company to the public. Do
some research to figure out how much it would cost per month to run an ad on the local
television channel and to get a large ad in the local yellow pages. You do some market analysis
and determine that the function for returns for advertising is
where p is the price of
tutoring per hour that you advertise and A(p) is the number of additional hours demanded by
customers each month as a result of your advertisements. What is your revenue function in
terms of price when you place this ad on TV? What is your profit function, ignoring for the
moment the cost of paying your tutors. What price should you set to maximize your profit? At
this optimal price, what is the price that you should pay your tutors? Do you make more money
with or without advertising in this case?
(d)
Do some research to figure out how much it would cost each month to rent 1,500 square
feet of office space in Salt Lake City, include in this calculation the costs for utilities,
maintenance, insurance, and relevant licenses. Use your imagination to determine what sorts of
overhead costs you might have just for getting the company started, as well as operating costs
and variable costs. Do the necessary research to reasonably well approximate how much each
of these things would cost. Include all of this in your cost function and use this information to
write down your profit function. Then determine the price you would need to set to maximize
your profit, and use that to determine the price you would need to pay your tutors in order to
maximize profit. Is it worthwhile to start up this company? What would be your annual salary if
you kept 70% of the profits and invested the other 30% back into the company? Would this be
better than having an ordinary job, working for somebody else?
Math 1030-007
Extra Credit Project
Fall 2010
Option #2: (Google Ad Words)
There are many important algebraic equations that can arise
from important real world applications, many of these equations are not solvable using basic
techniques from algebra. In these situations it is often appropriate to approximate the solution
using a root finding method such as Newton's method. Read section 3.7 to become more
acquainted with Newton's Method. You may need to do some research in order to determine
some of the formulas and numbers you will need in order to answer the following questions. Be
sure to cite your sources.
(a)
The first part of this project is to write a computer program, in the programming language
of your choice (C, C++, Pearl, Java, JavaSript, Matlab, Mathematica, Maple, TI-83/89, Fortran
etc.), that will approximate the root of a given function, f(x), using Newton's method. Let
represent your initial guess of the root and let n represent the number of iterations you perform
using Newton's method. Your program should somehow take as input from the user the
quantities and n. It should output the approximate root of the function f(x) that is specified in
your program. Note that Newton's method requires you to be able to evaluate the derivative of
the function, f(x), at a given point. You should include the programming code in the appendix of
your report.
(b)
To test to make sure your code is working properly, you want to have it approximate the
root of a function for which you already know the answer (e.g.
). Choose
four completely different functions, all of which you know one or more of the roots, and test
your code on these functions. (Note: A function may have more than one root, for those
functions only approximate one of the roots of a given function) Construct a table to report the
output of your program in the way illustrated below In each cell you report the absolute error in
your program's approximation. Be sure to report the error to at least 4 decimal places. For each
function, graph the absolute error against the value of n on a logarithmic scale. Do you notice a
trend?
10
100
1,000
10,000
100,000
EXACT
(c)
You are the owner of an online company that buys and sells books and you have
purchased some Google Ad Words in order to help draw traffic to your website. You have
collected lots of data over the past year and have run several regressions to determine a function
that approximates the daily traffic to your site given the price you pay for the Ad Words. Let p
represent the Ad Words price (in dollars) and
represent the expected number of
visitors to your site at the give price. You have also discovered that approximately 3% of people
who visit your website will end up purchasing a book. Suppose that each book you sell, before
considering advertising costs, yields a net profit of 5 cents. What price should you pay Google
for the Ad Words in order to maximize your profit? (HINT: At some point you will need to use
your computer program) Assume the only customers you get are the ones that find you online
through Google.
Math 1030-007
Extra Credit Project
Fall 2010
Option #3: (Melting Polar Ice Caps):
In many practical problems from science,
engineering, and economics, you may have a reasonable understanding about how a quantity
relates to its own rate of change. For example, how the current population of the U.S. relates to
the current population growth rate. In such a situation you may be interested in knowing what
the population of the U.S. will be at some point in the future. Equations that relate the derivative
of a quantity to the quantity itself are called differential equations and there are entire courses
you can take to study certain types of differential equations. Based on what you've learned so far
this semester you are capable of solving many simple differential equations which you can then
use to answer many important questions. I would recommend reading section 3.9 of the
textbook and become familiar with the idea of differential equations. You may need to do some
research in order to determine some of the formulas and numbers you will need in order to
answer the following questions. Be sure to cite your sources.
(a)
The rate at which an ice cube melts is proportional to its surface area. Assume the ice
cube you are concerned about has the shape of a sphere. Let V stand for volume of the ice cube,
S stand for surface area. Write a differential equation that describes how the ice cube melts, in
terms of S and V.
(b)
Let r(t) denote the radius of the ice cube and denote the initial radius of the ice cube.
Use your differential equation from part (a) to solve for the function r(t), the radius of the ice
cube at time t. How long does it take for the ice cube to melt completely?
(c)
Repeat the above procedure for a cylindrical ice cube. Assume that the radius and height
of the cylinder remain proportional to each other as the ice cube melts (i.e.
where c
is a constant). Let and
denote the initial radius and height of the cylinder, respectively.
Use the differential equation from part (a), except this time surface area and volume are
calculated differently. Solve the differential equation for
.
(d)
The polar ice caps have the approximate shape of a very large cylinder with a large radius
(the radius of the area of earth's surface covered by the ice cap) and a smaller height (the depth of
the ice). Go online and use real world data to determine the dimensions of a cylinder that would
roughly approximate the shape of either polar ice cap.
(e)
Plug this data into your result from part (c) to describe how the polar ice caps will melt
over time. The only remaining unknown in your equation should be the constant of
proportionality between the rate of melting and surface area. This constant will change
depending on the temperature of the surrounding water, do some research and find what it should
be when the surrounding water is 4°C. Use this constant for your final equation.
(f) How long will it take for the polar ice caps to melt completely (assuming there are no cracks
in the ice)?
Math 1030-007
Extra Credit Project
Fall 2010
Option #4: (Numerical Integration)
This semester you have learned how to integrate in
order to find the area beneath a curve. The most difficult part of integration is determining the
anti-derivative of the integrand. As it turns out, there are many functions in which it is not
possible to express the anti-derivative in a concise mathematical form. In order to find the area
under these functions scientists often rely on approximation methods such as the Riemann sums
(adding up the area of many rectangles). This is known as numerical integration. I would
recommend reading section 4.6 of the textbook to guide you in this project. You may need to do
some research in order to determine some of the formulas and numbers you will need in order to
answer the following questions. Be sure to cite your sources.
(a)
The first part of this project is to write a computer program, in the programming language
of your choice (C, C++, Rex, Pearl, Java, JavaSript, Matlab, Mathematica, Maple, TI-83/89,
Fortran etc.), that will approximate the integral of a given function. Let n represent the number
of rectangles you will use to approximate the area, and let [a,b] denote the interval over which
you are integrating. Your program should somehow take as input from the user the quantities a,
b, and n. It should then return (output) the approximate area under a given function f(x) that is
specified in your program. Your approximation should use the midpoint rule. You should
include the programming code in the appendix of your report.
(b)
To test to make sure your code is working properly, you want to have it approximate the
integral of a function that you know how to integrate (e.g.
). Choose four completely
different functions, all of which you know how to integrate, and test your code on these
functions. Construct a table to report the output of your program in the way illustrated below In
each cell you report the absolute error in your program's approximation. Be sure to report the
error to at least 4 decimal places. For each function, graph the absolute error against the value of
n on a logarithmic scale. Do you notice a trend?
10
100
1,000
10,000
100,000
EXACT
(c)
In statistics, one makes heavy use of normal distributions in order to draw important
conclusions about samples of data. Students with no background in calculus are usually told
(without proof) that for normally distributed data, approximately 68% of results occur within one
standard deviation of the mean and approximately 95% of the data occurs within two standard
deviations of the mean. You will verify this fact for the standard normal distribution which has a
mean of 0 and standard deviation of 1. The standard normal distribution curve is described by
the function
. The probability of a given value of x landing between a and b is
simply the integral of
from a to b. So to calculate the probability of landing within one
standard deviation of the mean, you would integrate this function from
-1 to 1. The probability of landing within two standard deviations is the integral from -2 and 2.
Math 1030-007
Extra Credit Project
Fall 2010
Are these probabilities approximately 0.68 and 0.95 as reported? It may be helpful for you to do
some reading about normal distributions and their applications to statistics.