Biocalculus II

MATH 221A- Spring 2008

Benedictine University


Basic Information


Attendance and Tardiness

Course Description


Academic Honesty

Course Objectives

Big Project

Academic Accommodations For Religious Obligations (AAFRO)

Learner Outcomes

In-Class Work

Other Information


Daily Questions

Homework Assignments

Technology Requirement


Dr. Tim Comar's Homepage 

Basic Information:

Instructor: Dr. Timothy D. Comar

Location:  Monday, Wednesday, Friday: TBA

Office: Birck 128

Phone: 829 - 6555

Time: MWF: 9:00 a.m.- 9:50 a.m.



Web Site:

Blackboard (WebCT) login:

Office Hours:


12:30 p.m. - 1:30 p.m.


12:30 p.m. - 2:00 p.m.


10:00 a.m. - 11:30 a.m.


by appointment

Textbooks: C. Neuhauser, Calculus for Biology and Medicine, 2e, Prentice Hall, 2004

G. de Vries, et al. A Course in Mathematical Biology, SIAM, 2006

Calculator: TI-83 or TI-84 series recommended

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Course Description:

This is the second course in a two-semester sequence in calculus with biological applications. There is a strong emphasis on biological models and examples using real biological data. Topics include applications of the definite integral, methods of integration, differential equations, systems of linear equations, matrices, eigenvalues and eigenvectors, analytic geometry, functions of several variables, partial derivatives, differentiability, tangent planes and linearization, systems of difference equations, systems of linear and nonlinear differential equations, equilibria and stability, and an introduction to probability. Applications may include allometric growth, age-structured population matrix models, epidemic models, competition models, host-parasitoid models, and models for neuron activity. The course uses the computer algebra system MATLAB and the modeling program Berkeley Madonna to explore calculus concepts and biological models

We will approach material using the Rule of Four: Symbolically, Graphically, Numerically, and Verbally. We will emphasize the technical aspects of the course material as well as effective communication of the mathematics. We will use technology including graphing calculators, the computer algebra system, MATLAB and the modeling program Berkeley Madonna to solve problems when appropriate.

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Course Objectives:

We would like to develop proficient understanding of the course material and the ability to use the course material in further course work as well as outside the classroom. To serve these ends, we will emphasize critical thinking and effective communication skills, both verbal and written. Success in this course will be dependent upon your ability to communicate your technical understanding of course material to your peers as well as to the instructor.  You will also be expected to successfully work collaboratively with others.

Basic skills/topics include: (ISBE Math Content Area Standards in bold)

1.      Techniques of Integration (Substitution, Integration by Parts, Partial Fractions) and Improper Integrals (8C5, 8C6)

2.      Applications of Integration to Biological Problems (4A, 4B, 4D, 4E, 6C6, 8C5, 8C6, 8G2, 8G3, 9E3)

3.      Introduction to Differential Equations (solutions, geometric and numerical solutions, equilibria and stability) (8C5, 8C6)

  1. Basic topics in linear algebra: systems of linear equations, matrices, eigenvalues and eigenvectors, analytic geometry
  2. Functions of several variables, partial derivatives, differentiability, tangent planes and linearization
  3. Systems of difference equations and applications to biological models
  4. Systems of linear and nonlinear differential equations and applications of biological models
  5. Using multivariable calculuus and linear algebra to analyze systems of difference and differential equations (linearization, equilibria, and stability)
  6. Using Calculus and Computational Methods to Solve Biologicallly Oriented Problems (2A, 2C, 3A, 3B, 3C, 4A, 4B, 4D, 4E, 7A3, 7A5, 7B5, 7C7, 8C6, 8E9, 8E10, 8F5, 8G2, 8G3, 9E6, 10A3, 10A4)
  7. Introduction to Probabiliity and Statistics
  8. Communicating Mathematics Accurately and Effectively (1C)
  9. Using Computational Software to Investigate and Solve Biologically Oriented Problems (2A, 2B, 2C, 3B, 5A, 5B, 7C8, 10E4; Core Technolgy: 5G)
  10. Working Collaboratively with Peers

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Learner Outcomes: (ISBE Math Content Area Standards in bold)

To successfully complete this course, the student will:

  1. Demonstrate ability to successfully approach mathematics problems four different ways: geometrically, algebraically, numerically, and verbally (oral and written forms); this will be achieved throughhomework exercises and exams.
  2. Evidence mastery of differentiation and antidifferentiation rules through homework assignments and exams.
  3. Evidence understanding of many applications of differentiation and applications of integration, particularly those related to biology, through problem solving in homework exercises and exams..
  4. Evidence the ability to read, represent, and interpret data in numerical and graphical formats through homework exercises and exams..
  5. Evidence the ability to interpret and apply biological models expressed as basic difference and differential equations through homework exercises and exams..
  6. Evidence the ability to analyze quantitative biological data using sophisticated mathematical computational software through homework exercises.
  7. Evidence the ability to apply the concepts of rate of change and total change to biological problems through homework exercises and exams..
  8. Evidence the ability to interpret biological mathematical models and the ability to formulate a biological mathematical model from a verbal description through homework exercises and exams..
  9. Evidence the ability to interpret and apply biological models expressed as systems of difference equations, systems of differential equations, and matrix models through homework exercises and exams.
  10. Evidence the ability to apply basic concepts and techniques in linear algebra and multivariable calculus to biological models expressed as systems of difference equations or systems of differential equations through homework exercises and exams..
  11. Evidence the ability to compute probabilities and expectations of continuous random variables through exams.
  12. Evidence the ability to work collaboratively through collaborative homework exercises.

IDEA Objectives:

      1. Gaining factual knowledge (terminology, classifications, methods, trends). (Essential)
      2. Learning fundamental principles, generalizations, or theories. (Essential)
      3. Learning to apply course material (to improve thinking, problem solving, and decisions). (Important)

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This course is fast-paced and demanding.  It is expected that you will study at least two hours for each class hour.  You should devote at least eight hours of study to this class per week (including the lab Math 208).    You are expected to read the required section in the text and attempt the assigned problems from the section before the material is either summarized or expanded upon in class.  Your notes from studying should include the following: the title of the section, a list of key concepts from the section, a brief summary of the ideas and techniques presented, solutions to the problems you have solved and a list of questions and problems you have not solved.  Ask questions!  If there is material with which you are not fully comfortable, you are expected to ask questions either during class, online, or during office hours. 

We are a community of learners working together to achieve our course goals. As such, it is incumbent upon all class members to show appropriate respect for each other. Each class member has something important to contribute to the class and should feel comfortable sharing with the class. Cell phones and pagers must be turned off. Inappropriate and disrespectful behavior including cell phone usage will result in dismissal from the class for the remainder of the class period.

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Technology Requirement:

Students are expected to use WebCT for all course communications, accessing notes and course information, and the completion of certain assignments as indicated in this syllabus. Students are expected to be familar with graphing calculuator and are expected to know how to use the computer algebra system Derive from the MATH 207 course. Derive will not be taught in this class. In this course, Berkeley Madonna, Excel, and MATLAB will be the primary software tools.

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Daily Questions


Group Homework


Big Project


Exam 1


Exam 2


Exam 3


Final Exam



The grading scale is 90% for A, 80% for B, 70% for C, and 60% for a D.  It is the student’s responsibility to seek clarification of the course requirements and evaluation policy.

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The homework assignment sheet lists the sections that will be covered in each class and suggested homework problems for you to use to reinforce the text's concepts. It is recommended that you attempt at least seventy percent of the suggested problems listed. These problems may be the content of quizzes. It is your responsibility either to know how to solve all assigned problems or to ask for assistance.  Your homework assignments also require daily questions. (Read on.)

There will be 6-7 Group Homework Problem sets assigned throughout the term. You will collaboratively with 2-3 other class members and submit one solution set for a grade. The instructor will indicated which problems will be collected approximately one week in advance. Any of the group homework problems may appear on pop quizzes or on exams. Questions for a particular assignment will not be addressed on the due date of the assignment. No late papers will be accepted.

Studying mathematics is a social process. Much benefit can be gained by sharing insights and by struggling through problems with your peers.  You are strongly encouraged to study and work with other class members.  You are also strongly encouraged to consult Dr. Comar outside of the class periods either during office hours or via e-mail at

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Big Project:

You will be required to analyze a single model a greater detail than in homework problems and lab projects. This may require some outside reading and computer programming. You will be required to implement the model electronically and explain the behavior of the model from mathematical and biological perspectives. You will be required to present your work EXTERNALLY at the ISMAA Annual Meeting on April 4-5 at Eastern Illinois University or at the ACCA Student Symposium on April 12 at Lewis University. The final written portion of the the project will be due on Tuesday, 5/6 at the begining of the Lab Final.

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In-Class Work:

Cooperative-learning exercises will take place on a regular basis. Learn to work with each other and learn from each other. Some activities may require follow-up work and re-writing outside of class. Some exercises will be graded for accuracy, and others will be granted credit for participation.

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Daily Questions:

Reading a mathematics textbook is very different from reading a novel and is often difficult.  To help you gain practice reading mathematics, you will be required to read the assigned sections, answer a question based on the reading, and submit at least two important questions of your own related to each assigned section.  The questions you pose may be significant questions that the text answers for you.  (In this case, provide a brief answer.)  Other important questions may arise from concepts that are unclear to you or from issues or extensions of concepts that the text does not discuss.

Questions should be significant and should indicate that you have thought carefully about what you have read.  Questions should not be of the form "What was Section 7.3 about?", "Does anybody really care about Section 7.3", "Can you do problem 46?", or "What does "homeomorphism" mean?" You still encouraged to ask about specific homework problems or examples in the text as long as you clearly indicate your issue with such problems or examples.  You may find that by identifying your difficultly and looking back in the text may enable you to answer your own question--a job well done!  All questions will be answered either in class, outside of class, or in written form.  Moral: ask questions!

Many of the basic concepts in the text will not be addressed explicitly in class.  Your questions will help direct discussion to important, yet difficult, issues and leave time for applied or exploratory activities.

Your questions and responses are required prior to each non-lab session and should be submitted to Dr. Comar via WebCT in the following manner.

Responses to Instructor’s Study Questions:  Mail tool

Your Questions:                                           Discussion tool

These are due by to 7:00 a.m. prior to the next class meeting.  Questions not submitted in by this time will not receive credit and may not be addressed in the next day’s class. Your postings will be graded as a participation grade.  Credit is earned by submitting your questions and by seriously attempting to answer to the study questions--right or wrong.  Extra credit of one half a participation score may be earned (once each submission day) by correctly responding to a fellow’s student question before class discussion of the question.  The instructor reserves the right to post questions with responses to the class discussion board on WebCT

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There will three in-class exams and a two-hour comprehensive final exam. The in-class exam dates are 2/6/08, 3/9/08, and 4/9/08.  The comprehensive final exam will take place on 5/9/08, 3:15 p.m. – 5:15 p.m.  Exams and quizzes cannot be made up or retaken.  If you miss an exam, your total exam score will be based on your performance on the other exams including the final.  Use of graphing calculators is strongly recommended on tests and quizzes.  Calculators with computer algebra systems including the TI-89, TI-92, and the TI-92+ are not permitted on exams.  The instructor reserves the right to delete all calculator memory prior to an exam.

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Absence and Tardiness:

Absence due to documented illness, participation in Benedictine University athletic activities, religious observance, or other extenuating circumstances will be excused.  It is your responsibility to inform Dr. Comar in the event of such absences.  Class attendance is very important.  Others will depend on you to be to participate in group exercises.  It is incumbent upon you to obtain class notes and updated assignments for missed classes. Tardiness will interfere with your time to complete homework quizzes and exams.  No student shall be admitted fifteen minutes after the scheduled classtime.

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Academic Honesty:

The search for truth and the dissemination of knowledge are the central missions of a university. Benedictine University pursues these missions in an environment guided by our Roman Catholic tradition and our Benedictine heritage. Integrity and honesty are therefore expected of all members of the University community, including students, faculty members, administration, and staff. Actions such as cheating, plagiarism, collusion, fabrication, forgery, falsification, destruction, multiple submission, solicitation, and misrepresentation, are violations of these expectations and constitute unacceptable behavior in the University community. The penalties for such actions can range from a private verbal warning, all the way to expulsion from the University. The University's Academic Honesty Policy is available at , and students are expected to read it. Acts of any sort of academic dishonesty will not be tolerated.  All instances will be pursued.  The first case of any academic dishonesty will result in a grade of zero for the assignment.  A second case will result in failure of the course. Any incident of academic honesty on the final exam will result in failure of the course.

Your name should appear on all of your submissions of your work.  If collaboration is allowed, you must state with whom you have collaborated.  You are responsible for understanding any authorized collaboriation policies on specific assignments. You must also properly reference any other print, electronic, or human resource that you consult.

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Academic Accommodations For Religious Obligations (AAFRO)

A student whose religious obligation conflicts with a course requirement may request an academic accommodation from the instructor. Students must make such requests in writing by the end of the first week of the class.

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Other Information:

If you have a documented learning, psychological, or physical disability, you may be eligible for reasonable academic accommodations or services.  To request accommodations or services, contact Tina Sonderby in the the Academic Resource Center, 249 Kindlon Hall, 630-829-6512.  All students are expected to fulfill essential course requirements.  The University will not waive any essential skill or requirement of a course or degree program.

Final Drop Date: April 13, 2008

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This syllabus is subject to change.  Any changes will be communicated to all class members electronically.

Contact Dr. Comar:

 Dr. Tim Comar's Homepage 

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