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Calculus II final project

The goal of the final project is for each student to have the time and space to investigate a topic of interest to them, and to apply mathematical thinking and tools germane to the course to that topic.


You may work in a group or share a project topic with another course.  This will allow your investigation to be more thorough.   You must use the tools of the course to address the topic at hand.  This may be through a novel idea or data set that you would like to model or you may take an established and published project and reproduce a portion of the results.


1. Please communicate with me regularly about your project idea.  

2. Once you have informal approval, then begin searching for literature that will serve to provide context for both the research question and the mathematical tools.

3. Form a research question and write a proposal.  

  • This proposal will be due Wednesday, April 16th along with a short presentation to the class..  

  • Provide a working title, abstract, and the names of any partners or any other class with which you are doing the research.  

  • Cite at least 5 peer-reviewed resources (books and journal articles) both in a references section as well as incorporated into an introduction.  It may help you to organize your thoughts to add a summary of some important quotes from each article after each reference in the literature cited section.  The introduction must provide context and motivation for your research question.  

  • The research question must be very clear, with one or two sentences of how you intend to address it.  The scope of your project should reflect whether you are investigating individually or with others, whether you are using as a joint projected with other courses, etc. and will be scaled in expectations accordingly.  

  • Declare by what method you will formally present your results (poster or oral/ppt).

  • NOTE: If you are either a math minor or an honors student, I expect that you will use Matlab as a tool investigating your research question.

4. Once the proposal is approved, work on the final project.  You are expected to have both formal oral and written presentation of results.  We will reserve class time for practicing the oral presentation, but you should submit your abstract as soon as it is approved to student conference (write it at the level of Calc I students).  The formal research paper will be in lab report style like the other reports in class.  This is an opportunity to present (to me) in-depth evidence of your work, your methods, and the context in which it is interpreted.

5. The exposition of the work must encompass:

  • title (a rephrasing of the research question)

  • introduction (contextual motivation for the research question)

  • methods (a description about the mathematical tools used)

  • results (the results derived from the use of the mathematical tools)

  • discussion (an interpretation of the results and their significance)

  • literature cited

6. All components of the final project will be due by the final exam period.

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Average Value of weighted functions

This project applies a lot of integration techniques:

Using the mathematics discussed in class: finding average value of a function with probabilistic inputs.  You are to come up with an extension to the wind the wind turbine problem done in class.  You may either find wind data for a new site or find a new power curve .  Or you might look for data for other kinds of energy production.  Weighted functions are not unique to power generation though.  The average of any function, f, with probabilistic input , pdf p, is


http://t.co/MCz9LEKf has a quick summary of weighted functions in computing energy as a result of wind power generation.  Here is our data from the activity from class. More info


If input cannot be found as a probability, (if the area under the input curve is not 1), then w(t) is a weight, and we calculate the average as



MIT Open Course has a tutorial on general weighted functions with a “non-power” related application.  Other applications can be found in center of mass and mean phenotype as a result of artificial selection.


You may work in groups to research your idea if you have a partner with similar interests, but each person must write their own report with their own scientific questions.  Your idea with some preliminary background research is due as a ppt next class.  If you are working in a group, each member’s subproject should be clear.  The class will discuss and critique and the each member will write his/her scope of work as a lab report due the Wednesday after break.  


Category                                                   Points  

HW Part & Action plan                                 13% (10 points)

(Present in ppt form to class Wed before break, needs references, detailed data)    

Lab report draft due Sunday (to peer) Wednesday after break            20% (12 points)

    Peer review feedback due Friday, Plan for revision due Sunday        17% (8 points)

Lab report, due following Friday                         50% (30 points)

Background Research                      

Distribution Curve                  To fit a distribution curve, you may need this JMP tutorial        

Power Curve                            For these curves, you may need this JMP tutorial                 


    Meaning and significance   


The lab research report will be graded using the Center for Biodiversity Research report rubric.

Lab Report Format   Lab Report Rubric   Lab Report Common Mistakes

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Programming Project: Trapezoid Rule

This project is to familiarize students with Matlab after they already know how to use the trapezoid rule without programming:

Student Programming Project: Trapezoid Rule Project


We have investigated integrals in two ways: by explicitly using antiderivatives and

by approximation using sum of rectangles. For example, we can calculate



This integral is easy to calculate since the antiderivative of its integrand, -3/x,

can be found exactly. But what if we want to allow the upper limit to be infinity?



Also some important functions do not have explicit antiderivatives.

Consider this function, which is used in representing normal distributions:

p(x)=e-x2/2/(2 ).

We do not have the tools to calculate the explicit antiderivative for this function, so we must approximate its integral numerically.


We will use MatLab to calculate integrals numerically with the trapezoid rule which we developed in class.  The code can be found in the BGL book.


1. Estimate using the trapezoid rule the area under the curve of


on the interval [1, 10] using 100 subintervals.  You will have to modify “function” in the f.m file in order to do this.  Then run it and put in the appropriate values for a, b and n.


2. Now calculate the approximate integrals for the b=xmax 100 and then 1000.  Note: you may have to modify n in order to get good accuracy, so when you state your results, mention what n you used.  You may want to think about what n gives you the same xabove.


3. Calculate the exact value of the integrals in #2 using antiderivatives.  Compare 1-3.  What do you think is the value of 1(3x-2)dx?


4. Estimate using the trapezoid rule the area under the curve of

p(x)=e-x2/2/(2 )

over the interval [-1, 1] using an appropriate number of subintervals (you choose n based on what gives you enough accuracy).  Hint: e^x is actually exp(x) in matlab, is just pi, and the square root function is sqrt(x).


5.  Now calculate the same integral over [-2,2] for the same number of subintervals.  Then try 10 times as many intervals.


6. Repeat, but for [-5,5], then [-10,10].  Conjecture the exact value of this integral from (-infinity, infinity).


7. Write up a research report, with a title, results, and conclusions/discussion.  Also discuss the accuracy of your results as well as comparing and contrasting results.   Are the results what you expected or different?  Include a copy of your modified function files as an appendix.

There will be two phases of research report

Phase I.  Due Thursday 2/26.  You must upload your document to canvas or submit a GoogleDoc url (make sure you change settings to share by link).  


Phase II.  You will be assigned 2 reports to read of peers and give feedback.  Please review by Thursday 3/3.


Follow the peer review guidelines.

Your grade will be a combination of completion, effort in draft, insight and proper conclusions, (all assessed by the biodiversity lab rubric here) and peer review quality of comments you make for other people. No introduction or methods are required.

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Breaking it down

This project helps students with the concept of using the rectangle method to determine the area under a function:


Work in Groups to find a solution for finding the area under a curve.

Group 1 - use rectangles as “pieces,” with the left side of each interval to determine the height

Group 2 - use rectangles as “pieces,” with the right side of each interval to determine the height

Group 3 - use rectangles as “pieces,” with the midpoint side of each interval to determine the height

Group 4 - use trapezoids as “pieces” (rectangles + triangles)


Course Outcomes addressed:

  • Understand integrals visually as area under a function (definite integral) and as a process of reversing derivatives (antiderivative).

  • Calculate integrals of elementary functions and compositions of elementary functions.

  • Recognize integrals in the practical and professional world around them, particularly in environmental and life science.

  • Work in groups … and communicate the results of mathematical investigations ...


Suggested Process/Methods:

  1. First use the curve f(x) = x^2 from 0 to 1 for #2-6 below, then consider the same 2-5 for f(x) = exp(x^2) from 0 to 1.

  2. Find the exact area under the curve using WolframAlpha.com (you can query in natural language).

  3. Draw a picture of the curve, and break it into 4 pieces.  Find the formula that yields an approximation to the area under the curve.  Compute and compare this to the value that you achieved in part 1.  What is the error?

  4. Repeat for 8 pieces and compare the value to the true value.  What is the error?

  5. Write a formula that breaks up the area into n pieces.  What is the width of the each piece when you use n pieces?

  6. Now consider the limit as n -> infinity.  What is the limit of the width of each piece as n-> infinity?  Can you compute the limit of the area equation as n-> infinity?

  7. Finally consider any function, f(x), from x=a to x=b broken into n intervals.  Simplify as much as possible.  Write the formula that best approximates the area under this curve.


Communicating the results:

Group presentation to class on in one week, showing your peers the answers to the above questions and the process to get there.  The presentation may be a board presentation, but you are expected to provide a handout.  Alternatively, you may present a short powerpoint.  Each group should aim for a presentation in the neighborhood of 7 minutes, with each member contributing equally to each part of the presentation.  The grading rubric is on Canvas.

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Presentation Description

Journal Article Presentation for the final project in Calc. I of Dr. Carrie Diaz Eaton's syllabus:

Journal Article Presentation

Calculus I

1) The last day of class will be a Q&A review.  Presentations will be the full week prior.


2)  Use the Unity College library online resources to find a current primary literature article in a peer-reviewed journal that is related to the current class discussion on mathematical techniques and related to your field of interest by Friday, two weeks before your presentation.  Let me know your interests, and I will pick one for you if your article does not meet the criteria.


3) You/your group must meet with me at least one week in advance of presenting to discuss your article choice.  Make an appt when you submit your article.


3)  When reading the article, try to think about the following questions:

  • The issue being explored

  • The benefits and limitations of the mathematics presented

  • The type of mathematics, and how it relates to class discussion

  • What the model means

  • The major results of the article

  • The contribution of the article to its area


4) You are to prepare a 5 minute brief Powerpoint review of the article with the questions above in mind, explaining it to your classmates.  


6) The original .pdf of the article (not the link!), the presentation must be uploaded to CAMS by the day before the presentation.  


Grading Rubric for Presentation:

E-mailing me article on time, meeting with me during week specified    10 pts

Connection of Article to Class Mathematics, Model Explanation    20 pts

Discussion of Major Results in Context – Benefits and Limitations    10 pts

Overall Presentation Quality:                        10 pts

Total                                    50 pts

Let me know if you need any help or have questions!

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Sustainable Harvesting Project

This project is to go along with differential equations:

Learning Objectives addressed:

  1. Understand sequences, limits, and continuity algebraically, numerically, visually, and verbally.

3.   Be able to model simple scenarios of change through either difference equations or differential equations.

5.   Recognize limits and derivatives in the practical and professional world, particularly in environmental and life science.

6.   Be able to use a computer algebra system and spreadsheet system to investigate or evaluate given problems.

7.   Work in groups to investigate problems and communicate solutions on an introductory level.


Project methods:

A small lake trust in Maine is undergoing an investigation that has both environmental and economic impacts.  The exact numbers presented here have been simplified for ease of analysis.  

To build up a population of trout in a small lake, 200 young trout are added each year. In addition, the population increases its own numbers by 20% each year. Let xn denote the size of the population after n years.

a) If x0 = 1200, determine the largest n such that when xn <2800.

b) Once xn =2800 the lake is no longer stocked and fishermen will catch 600 fish per year. What is the fate of the population?


Now extend this problem.

c) Suppose you have only have enough money to stock the lake for the 3 years it takes to reach 2800.  What level of fishing could be sustained when the population reaches 2800?

d)  Suppose you want to harvest 600, and that is your main goal.  How long should you stock the lake before switching to allowing permitted fishing?

e) Take the recommendation from part d.  What if you accidentally had overestimated your reproduction rate, and it is really only 15%?  What happens to your population long-term following the guideline recommended in part d?


Communicating the solution:

Consider the above, and write a 2-3 page, group report, outlining the management problem and your solution.  In so doing, take into consideration all of the mathematical work and discussion from above. Write the report as if you are writing a recommendation to the lake trust board, using summarizing graphs as necessary.  Submit through Canvas as a url (GoogleDocs).


Components of a “lite” management report for this lake (see rubric on Canvas):

  1. Title describing the charge of the report

  2. Background on the known information prior to the study.  Statement of intent for the report.

  3. Investigation under financial constraints: (a-c)

  4. Investigation under ecotourism constraints: (d)

  5. Possible pitfalls, violation of assumptions (e)

  6. Overall recommendation

  7. Appendix which includes a summary of work (may be a separate file)  This part will not be graded, however, if there is a mistake in any of the calculations, this gives me the opportunity to go back and see how the mistake was made and determine appropriate partial credit.

Team work:  If you do not contribute in pre-defined, agreed upon ways, you will not receive credit.  Regardless of any individual’s participation, the entire team is responsible for handing in a full report. Please acknowledge all team members on the final report, and do the participation quiz after submitting the report.

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