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.

Associated Costs: These are additional (out-of-pocket) expense considerations that students should expect in addition to the course tuition, fees, and textbooks. $0 to $100.

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 the limits of functions.
3. State whether a function is continuous or discontinuous based on both the graph and the definition of continuity.
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. Compute definite integrals using the Fundamental Theorem of Calculus, by numerical techniques, and by substitution.
12. Solve differential equations.
13. Interpret solutions of differential equations.
14. Use integration results to calculate areas, volumes, and average values.
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 the limit of a function at a point both algebraically
and graphically.
      2. Evaluate the limit of a function at infinity both algebraically
and graphically.
      3. Use the definition of a limit to verify a value of the limit of a
function.
   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 limiting process to find the derivative of the function.
 
III. Derivatives 
   A. Finding derivatives.
      1. Find derivatives involving powers, exponents, and sums.
      2. Find derivatives involving products and quotients.
      3. Find derivatives involving the chain rule.
      4. Find derivatives involving exponential and logarithmic
functions.
      5. Find derivatives involving trigonometric and inverse
trigonometric functions.
      6. Find derivatives involving implicit differentiation.
      7. Use the derivative to find velocity, acceleration, and other
rates of change.
      8. Use the derivative to find the equation of a line tangent and a
line normal to a curve at a given point.
   B. Apply derivative techniques to curve sketching.
      1. Use the derivative to find critical points.
      2. Apply the Mean-Value Theorem for derivatives.
      3. Determine the behavior of a function using the first derivative.
      4. Use the second derivative to find inflection points.
      5. Determine the concavity of a function using the second
derivative.
      6. Sketch the graph of the function using information gathered from
the first and second derivatives.
      7. Interpret graphs of functions.
   C. Apply derivative techniques to applied problems in biology,
medicine, and physics.
      1. Use optimization techniques in the life sciences, physical
sciences, and geometry.
      2. Solve related rates problems.
      3. Utilize Newton’s Method.
      4. Utilize differentials to estimate change.
      5. Evaluate limits using L’Hôpital’s Rule.
 
IV. Integration
   A. Finding Integrals
      1. Find area using Riemann sums.
      2. Express the limit of a Riemann sum as a definite integral.
      3. Evaluate the definite integral using geometry.
      4. Integrate algebraic, exponential, and trigonometric functions.
      5. Evaluate definite integrals using the Fundamental Theorem of
Calculus.
      6. Interpret definite integrals both in terms of area and cumulative
change.
      7. Integrate indefinite integrals.
      8. Integrate using substitution.
      9. Integrate using integration by parts.
     10. Integrate using numerical techniques.
     11. Determine whether an improper integral converges or diverges.
     12. Evaluate a convergent improper integral.
    B. Using the Integral
      1. Calculate the average value of a function on a closed interval.
      2. Calculate the area between curves using integration techniques.
      3. Calculate the volume of a solid by the cross sections method.
      4. Calculate the volume of a solid of revolution by the washer
method.
      5. Calculate the volume of a solid of revolution by the cylindrical
shells method.
      6. Calculate the arc length and surface area using integration
techniques.
      7. Apply integration techniques to the life sciences, the physical
sciences, and geometry.

V. 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.

VI. Modeling techniques in biology and medicine 
   A. Analyze documented mathematical models in biology and medicine.
   B. Analyze proposed mathematical models in biology and medicine.

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.