Here is a link to a Reading assignment in mathematical biology:

http://journals.plos.org/plosbiology/article?id=10.1371/journal.pbio.1002017

This is an example of The applications of integrals.

Here is the link: https://www.wyzant.com/resources/lessons/math/statistics_and_probability/expected_value/variance

Here is a video on Partial Fraction integration:

A video on Integration by parts:

Educational video on U substitution:

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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

110(3x-2)dx=3x-1-1|110=-3x|110=(-0.3)-(-3)=2.7

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?

1(3x-2)dx

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

f(x)=3x-2

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.

A table of integrals to help make computing easier: 

Link to pdf:http://www.physics.umd.edu/hep/drew/IntegralTable.pdf

Here is a link to some useful Matlab files:

http://mathematicsforthelifesciences.com/MatlabFiles.html

This an example of a biology reading used in Dr. Carrie Eatons Calculus 2 class. 

Here is a link:http://skepchick.org/2013/11/why-biology-students-have-to-learn-calculus/

Matlab is a coding program that helps compute many things using m files:

here is a link to the tutorials: https://sites.google.com/site/profbodine/tutorials

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

Goal:

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.

An Educational video on first and second derivatives:

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!

Here is some reading involving calculus and biology concepts. Link to pdf:

https://drive.google.com/a/unity.edu/file/d/0B_Vlmt-dGhKdMWc1NXNwTFpsMlk/view

Here is a link to the pdf:

https://drive.google.com/a/unity.edu/file/d/0B_Vlmt-dGhKddEhyLUlzWlhLSzQ/edit

Video on Determining derivatives using the power rule:

Here is a link to the pdf:

https://drive.google.com/a/unity.edu/file/d/0B2k1BmkveWWxc2pqNGhtdFVRVUtSRVNBTjhFdzBiUQ/view

A worksheet for derivative calculation:

Lab 3 Part I, Calculus I

Unity College, Carrie Diaz Eaton

Consider an environmental indicator which can be described by the following equation:

P(t)=et

where t is in years.

1. Sketch a graph of P(t) and from that, sketch what you expect P’(t) to look like.

2. Write an equation for P(t+∆t)

3. Try to calculate the derivative algebraically using the limit definition of derivative

4. Create a spreadsheet that calculates the derivative numerically at 21 different points of the graph:  t = 0, .5, 1, 1.5, … 10

(Hint: Create a table for with 21 rows (t = 0, .5, 1, 1.5, … 10) and 8 columns: t,P(t),  then for∆t=1 P(t+∆t), P/∆t=[P(t+∆t)-P(t)]/∆t, then for∆t=.1 P(t+∆t), P/∆t=[P(t+∆t)-P(t)]/∆t, and for∆t=.01 P(t+∆t), P/∆t=[P(t+∆t)-P(t)]/∆t.)

5. Results Graph a scatterplot of P(t) and your estimates of P’(t) based on the numerical work performed in #4.

6. Discussion Do the visual results based on visual, algebraic and/or numerical work agree?

7. Discussion Contrast the initial and long-term value of the instantaneous rate of change.  

Discussion Suppose a different function was hypothesized for P(t), such as t^2 or e^(-t).  How would you modify your spreadsheet to approximate the graph of the derivative?

Educational video on how to get a tangent line equation:

This educational video is on Taking derivatives :

Another video on more advanced derivatives:

video on the basics of derivatives:

This Khan Academy video is to help better understand the equation for a derivative determining the slope at any given time: