Math 237 COURSE INFORMATION

Prerequisite:
Math 173 (Precalculus) with a grade of 'C' or higher or
Math 171 (College Algebra) and Math 172 (Trigonometry) both with grades of 'C' or higher or
equivalent.

Textbook:
Calculus for Biology & Medicine, Claudia Neuhauser, Prentice Hall, 2003, 2^nd edition.

Supplies:
A scientific calculator is required, graphing capabilities are desirable.

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 utilized. 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 population biology.

Course Objectives:
After successfully completing this course, the student will 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 or a table.
2. Evaluate a limit of a function at infinity using algebraic techniques or a table.
3. Evaluate a limit using a graph.
4. Evaluate left- and right-handed limits using algebraic techniques or a graph.
5. Evaluate limits using L'Hôpital's Rule.

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 Newtonís 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 funcn 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. Calculate approximate solutions to systems of differential equations using Euler's method.
4. Determine the qualitative behavior of solutions of systems of differential equations in the phase plane.
5. 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.