Description:

This course focuses on the study and mathematical modeling of biological systems. Through a host of biological and medical applications, the rudiments of calculus are developed. Concepts include measuring the slope of a curve, writing equations of tangent lines, maximizing and minimizing a function, determining the rate of change of a function, and measuring the area under a curve. Solution techniques, both analytic and numeric, for difference and differential equations are used. Modeling activities are heavily emphasized. Qualitative analysis of solutions of differential equations is incorporated in modeling activities. Application areas include mathematical physiology, pharmacology, cell biology and populations biology. 5 hrs. lecture/wk.

Supplies: Refer to the instructor's course syllabus for details about any supplies that may be required.

Textbook(s): For information see - http://bookstore.jccc.net

Course Fees: NONE

Course Objectives:

Upon successful completion of this course the student should be able to:

1. Model biological and medical phenomena using difference equations.
2. Evaluate limits of functions using graphs, tables, and algebraic methods.
3. Demonstrate the use of limits to determine continuity of a function at a point.
4. Determine differentiability of a function at a point using limits and graphs.
5. Demonstrate the use of the limit definition to find the derivative.
6. Differentiate algebraic, exponential, logarithmic, and trigonometric functions.
7. Produce equations of tangent lines.
8. Demonstrate the use of derivatives to describe the behavior of a function.
9. Apply derivatives in biological and medical applications.
10. Antidifferentiate algebraic, exponential, and trigonometric functions.
11. Apply the Fundamental Theorem of Calculus to find the area under a curve and between two curves.
12. Solve differential equations.
13. Solve systems of differential equations.
14. Interpret solutions of differential equations.
15. Model biological and medical phenomena using the concepts of calculus and algebra.
16. Analyze mathematical models for some select biological and medical phenomena.
17. Compare and contrast competing models for a biological/medical scenario.

Content Outline & Competencies:

I. Difference Equations
   A. Utilize updating functions in the context of biological and medical
applications.
   B. Review units and dimensions in describing physical phenomena.
   C. Review algebraic, exponential, and trigonometric functions.
   D. Examine equilibria and stability issues.

II. Limits 
   A. Evaluate Limits.
      1. Evaluate a limit at a point using algebraic techniques and
tables.
      2. Evaluate a limit of a function at infinity using algebraic
techniques and tables.
      3. Evaluate a limit using a graph.
      4. Evaluate left- and right-handed limits using algebraic techniques
and tables.
      5. Evaluate limits using L’Hôpital’s Rule.
   B. Use Limits.
      1. Use the limit to determine continuity of a function at a point.
      2. Use limits for stability analysis in mathematical models.
      3. Use a limit to determine differentiability of a function.
      4. Use the limit definition of the derivative to determine
differentiability of a function and to find the derivative of the
function.

III. Derivatives 
   A. Find and estimate derivatives.
      1. Find the derivatives of algebraic, trigonometric, exponential,
and logarithmic functions using the power rule, product rule, quotient
rule, and chain rule.
      2. Find derivatives using implicit differentiation.
      3. Use a graph to estimate the intervals over which the first
derivative is positive or negative.
      4. Use a graph to estimate the intervals over which the second
derivative is positive or negative.
      5. Find the equation of both the tangent and normal line to a curve
at a given point.
   B. Apply derivative techniques to curve sketching.
      1. Use the derivative to find critical points.
      2. Determine the behavior of a function using the first derivative.
      3. Use the second derivative to find inflection points.
      4. Determine the concavity of a function using the second
derivative.
      5. Sketch a function using information gathered from the first and
second derivatives.
      6. Utilize Newton’s Method to approximate the zeros of functions.
   C. Apply derivative techniques to applied problems in biology,
medicine, and physics.
      1. Use derivatives to predict rates of change.
      2. Use derivatives to determine the maxima/minima (optimization).
      3. Use derivatives to determine outcomes in related rates problems.
      4. Use derivatives to find and explain rates of change for position
functions, including the relationship between position, velocity, and
acceleration.
      5. Find extrema of functions with restricted domains.

V. Integrals
   A. Identify the antiderivative for a given function using elementary
techniques.
   B. Identify the antiderivative for a given function using
u-substitution.
   C. Apply the Fundamental Theorem of Calculus.
      1. Evaluate a definite integral using elementary techniques.
      2. Evaluate a definite integral using u-substitution.
      3. Calculate the area between two curves.
      4. Calculate the area under a curve numerically using sums.

VI. Differential equations
   A. Analyze a single differential equation.
      1. Identify differential equations.
      2. Solve pure-time differential equations.
      3. Solve differential equations using the method of separation of
variables.
      4. Calculate approximate solutions to differential equations using
Euler’s method.
      5. Determine the qualitative behavior of solutions of differential
equations.
      6. Classify equilibrium solutions as to their stability.
   B. Analyze a system of differential equations.
      1. Identify systems of differential equations.
      2. Solve systems of differential equations.
      3. Solve differential equations using the method of separation of
variables.
      4. Calculate approximate solutions to differential equations using
Euler’s method.
      5. Determine the qualitative behavior of solutions of systems of
differential equations in the phase plane.
      6. Classify equilibrium solutions as to their stability.

VII. Modeling techniques in biology and medicine 
   A. Analyze documented biological and medical mathematical models.
      1. Analyze allometric models.
      2. Analyze models in cell diffusion.
      3. Analyze models in population growth models.
      4. Analyze models in population biology for interacting species.
      5. Analyze models for respiration and control of respiration.
      6. Analyze models for cardiac dynamics and control of heart
rhythms.
      7. Analyze models for neuron dynamics.
      8. Analyze models in pharmacology.
      9. Utilize a computer-algebra system in a model’s analysis.
   B. Analyze proposed biological and medical mathematical models.
      1. Identify a proposed model for a measurable phenomena.
      2. Utilize a computer-algebra system in the model’s design and/or
analysis.
      3. Critically evaluate a proposed model for a measurable
phenomena.

Methods of Evaluation of Competencies:

Evaluation of student mastery of course competencies will be accomplished using the following methods:

Unit Exams, Unit Papers and/or Unit Projects   40% - 80%
Homework, Quizzes and/or Small Projects         0% - 50%
Final Exam**                                   10% - 40%

**The final exam must count at least as much as any unit exam, unit paper
or unit project. In any course where unit exams are not proctored, the
instructor may require that the student score at least a 70% on the final
exam to earn a ‘C’ for the course. At the instructor's discretion, the
grade on all or any part of the final exam may replace any lower test
score.

Caveats:

1. The majority of mathematics courses are sequential. Students must earn a grade of C or higher in a prerequisite mathematics course to progress to its subsequent mathematics course.
2. In accordance with the assertion made on your billing statement, during the first two weeks of the semester, if a student is found not to have successfully fulfilled the prerequisite(s) for this course, the student will be dropped from the course. He/she will be allowed to enroll in the appropriate lower level course on a space-available basis with an even exchange of tuition. After the first two weeks, students who have not met the prerequisite(s) will be dropped from the course with no refund of tuition.

Disabilities:

If you are a student with a disability, and if you will be requesting accommodations, it is your responsibility to contact Access Services. Access Services will recommend any appropriate accommodations to your professor and his/her director. The professor and director will identify for you which accommodations will be arranged.

JCCC provides a range of services to allow persons with disabilities to participate in educational programs and activities. If you desire support services, contact the office of Access Services for Students With Disabilities (913) 469-8500, ext. 3521 or TDD (913) 469-3885. The Access Services office is located in the Success Center on the second floor of the Student Center.