Mathematical Biology Tools using webMathematica

– :: Textbook Articulation :: –

This resource page consists of links to pages that are utilized in a variety of undergraduate mathematical biology courses or those courses related to it. The pages are grouped around the organization of a textbook.

Tools developed for mathematical biology were most often collaboratively developed and attributions are given on each particular page; however, a majority of the tools were developed with Mike Martin and either Tim Comar or Glenn Ledder.  Steve Wilson and Drew Cousino have played roles in the project, too. The development and dissemination of some of these tools is supported for Ledder and Martin by National Science Foundation grant #0535924.  In addition, NSF grant #0633232 supports Comar in some of his contributions to this page.



This page is maintained by Mike Martin
who works collaboratively to develop the tools found here,
making use of them in the courses he teaches
as a Math Professor at Johnson County
Community College.  Developed with support from
the National Science Foundation.

Chapter 1 _ Preliminaries:  Functions & Models


Chapter 2 _ Difference Equations, Sequences, & Limits

Chapter 3 _ Limits & Continuity of Functions of a Real Variable

Chapter 4 _ The Derivative

  • Introduction to the Derivative:  Geometric & Numerical Intuition
  • The Definition of the Derivative
  • Interpretation:  From Difference Equations to Differential Equations
  • Techniques of Differentiation I:  Powers, Polynomials, Sums, & Differences
  • Techniques of Differentiation II:  Products & Quotients
  • Techniques of Differentiation III:  The Chain Rule
  • Trigonometric Derivatives
  • Derivatives of Exponential Functions
  • Implicit Differentiation
  • Derivatives of Inverses
  • Higher Order Derivatives
  • Slopes of Parametric Curves

Chapter 5 _ Applications of the Derivative

  • The Mean Value Theorem
  • Derivatives & the Geometry of Curves
  • Optimization
  • Models Using Differential Equations
  • Equilibria & Stability of Difference Equations; Cobwebbing
  • Linear Approximations, Differentials, & Relative Error
  • Taylor Polynomial Approximations
  • L'Hopital's Rule
  • Related Rates
  • Antiderivatives

Chapter 6 _ The Definite Integral

  • Riemann Sums
  • The Definite Integral
  • The First Fundamental Theorem of Calculus
  • Interpretation of the Definite Integral:  Total Change
  • Numerical Integration
  • The Average Value of a Function

Chapter 7 _ Integration Techniques

  • Substitution
  • Using Integration Tables & Computer Algebra Systems
  • Integration by Parts
  • Partial Fractions
  • Additional Techniques
  • Improper Integrals

Chapter 8 _ Applications of the Definite Integral

  • Volumes by Slicing
  • Arc Length
  • Probability & Integration

Chapter 9 _ Ordinary Differential Equations

  • Differential Equations
    • Differential Equations:  Definitions
    • Equilibria of Differential Equations
    • The Phase Portrait & Phase Line Analysis
  • Examples of Mathematical Models with Differential Equations
    • Simple Population Models
    • Other Models
  • Slope Fields & Euler's Method
    • Slope Fields
    • Euler's Method
  • Elementary Solution Techniques for Ordinary Differential Equations
    • Separable Differential Equations
    • Examples of Solutions to Separable Differential Equations
    • Solutions to First-Order Linear Differential Equations (Optional)
  • Equilibria & Stability of Ordinary Differential Equations
    • The Stability Criterion:  Analysis & Eigenvalues
    • Bifurcations
  • A Glimpse at Systems of Autonomous Differential Equations
    • The Lotka-Volterra Predator-Prey Model
    • The Kermack-McKendrick SIR Model

Chapter 10 _ Matrix Models & Techniques

  • Vectors in n-Dimensional Space
  • Representations of Lines & Planes
  • Systems of Linear Equations
  • Matrix Algebra
  • Linear Transformations:  Eigenvalues & Eigenvectors
  • Iteration of Linear Transformations & Matrix Exponentials
  • Structured Population Models
  • Markov Chains

Chapter 11 _ Differential Calculus of Functions of Several Variables

  • Functions of Several Variables
  • Limits & Continuity of Functions of Several Variables
  • Differentiability:  Partial Derivatives & the Jacobian
  • Tangent Spaces & Linearization
  • The Chain Rule
  • Implicit Differentiation
  • Directional Derivatives & the Gradients
  • Optimization
  • Diffusion

Chapter 12 _ Systems of Equations:  Difference & Differential Equations

  • Linear Systems of Differential Equations
  • The Direction Field & Phase Plane
  • Elementary Solution Techniques
  • Equilibria & Stability of Linear Systems of Differential Equations
    • Equilibria, Bifurcations, & Singular Perturbation Theory
  • Examples of Linear Systems of Differential Equations
    • General Linear Compartment Model
    • Simple Drug Absorption Model
    • A Nonautonomous Drug Absorption Model

Chapter 13 _ Nonlinear Systems of Difference Equations & Systems of Differential Equations

  • Systems of Nonlinear Difference Equations:  Equilibria & Stability
  • Examples of Nonlinear Difference Equations
  • Nonlinear Autonomous Systems of Differential Equations:  Equilibria & Stability
  • Examples of Autonomous Systems of Differential Equations

Chapter 14 _ Probability:  Applications & Constructive Modeling

  • Probability
    • Definitions
  • Counting
    • The Multiplication Principle
    • Permutations
    • Combinations
    • Equally Likely Outcomes
    • Hypergeometric & the Capture-Recapture Problem
  • Conditional Probability
    • Definitions
    • The Law of Total Probability
    • Bayes Theorem
  • Discrete Random Variables
    • Definitions & the Probability Mass Function
    • Bernoulli Trials
    • The Binomial Distribution
    • The Geometric Distribution
    • The Poisson Process & Distribution
    • The Poisson Approximation to the Binomial Distribution
  • Continuous Random Variables
    • Definitions & the Probability Density Function
    • The Exponential Distribution
    • The Gamma Distribution
    • The χ2 Distribution
    • The Normal Distribution
    • Functions of Random Variables
  • Limit Theorems
    • The Law of Large Numbers
    • The Central Limit Theorem
    • Convergence in Probability & Distribution
    • Using the Normal Approximation of Discrete Random Variables
  • Jukes-Cantor & Kimura Models

Chapter 15 _ Statistical Inference:  Applications & Models

  • Describing Data
    • Graphical Displays of Data
    • Measures of Central Tendencies
    • Measures of Spread
  • Estimating Means & Proportions via Intervals
    • Confidence Intervals for Means
    • Confidence Intervals for Proportions
  • Hypothesis Testing
    • Hypothesis Testing Paradigm
    • Hypothesis Testing for Means
    • Hypothesis Testing for Proportions
    • Goodness of Fit Tests ( χ2 )
  • Statistical Modeling
    • Linear Regression
    • Logarithmic Regression & Dose-Response Curves
    • Other Models Including and Introduction to Nonlinear Models
    • Maximum Likelihood Estimation
    • Model Comparison
  • Introduction to Stochastic Modeling
    • Markov Chains
    • Diffusion
    • An Epidemiological Model

please see a separate list that is organized by bioscience & medicine content or another, more general, webMathematica index of all JCCC pages.




Mike Martin & Steve Wilson received the 2004 International Conference on Technology in Collegiate Mathematics Award for Excellence and Innovation with the Use of Technology in Collegiate Mathematics for their development of a subset of these tools.  The award was presented at the conference in New Orleans in October of 2004.

This page is maintained by Mike Martin.
Last updated: 15 December 2011

mmartin@jccc.edu











Copyright 2002 Johnson County Community College