MAC 2242 — Life Sciences Calculus II — Syllabus
Prerequisites: C (2.0) or better in MAC 2241, or C (2.0) or better in
MAC 2281, or C (2.0) or better in MAC 2311.
Course Description: The course has two different formats: daytime
sections have three 50-minute lectures per week and two 50-minute help sessions;
evening sections have two 110-minute lectures per week. Successful completion of
the course merits 4 semester hours of credit. The schedule outlined below allows
time for three midterm exams plus a cumulative final exam, which are the norms
for this course. Students in this course are encouraged to submit projects that
explore mathematical topics in the sciences.
Text: Calculus for Biology and Medicine, 2nd Edition, by C.
Neuhauser
Course Content
Chapter 7: Integration Techniques and Computational Methods (4-5 weeks)
7.1 The Substitution Rule
7.1.1 Indefinite Integrals
7.1.2 Definite Integrals
7.2 Integration by Parts
7.3 Practicing Integration and Partial Fractions
7.3.1 Practicing Integration
7.3.2 Rational Functions and Partial Fractions
7.4 Improper Integrals
7.4.1 Type 1: Unbounded Intervals
7.4.2 Type 2: Unbounded Integrand
7.4.3 A Comparison Result for Improper Integrals
7.5 Numerical Integration
7.5.1 The Midpoint Rule
7.5.2 The Trapezoidal Rule
7.6 Numerical Integration (omit)
7.7 The Taylor Approximation
7.7.1 Taylor Polynomials
7.7.2 The Taylor Polynomial about x = a
7.7.3 How Accurate is the Approximation? (Optional)
Chapter 8: Differential Equations (2 weeks)
8.1 Solving Differential Equations
8.1.1 Pure time Differential Equations
8.1.2 Autonomous Differential Equations
8.1.3 Allometric Growth
8.2 Equilibria and the Stability
8.2.1 A First Look at Stability
8.2.2 Single Compartment or Pool
8.2.3 The Levins Model
8.2.4 The Allee Effect
8.3 Systems of Autonomous Equations (omit)
8.3.1 A Simple Epidemic Model
8.3.2 A Compartment Model
8.3.3 A Hierarchical Competition Model
Chapter 9: Linear Algebra and Analytic Geometry (2 weeks)
9.1 Linear Systems
9.1.1 Graphical Solution
9.1.2 Solving Systems of Linear Equations
9.2 Matrices
9.2.1 Basic Matrix Operations
9.2.2 Matrix Multiplication
9.2.3 Inverse Matrices
9.2.4 Computing Inverse Matrices (Optional)
9.2.5 An Application: The Leslie Matrix
9.3 Linear Maps, Eigenvectors, and Eigenvalues
9.3.1 Graphical Representation
9.3.2 Eigenvalues and Eigenvectors
9.3.3 Iterated Maps (Needed for Section 10.7)
9.4 Analytic Geometry
9.4.1 Points and Vectors in Higher Dimensions
9.4.2 The Dot Product
9.4.3 Parametric Equations of Lines
Chapter 10: Multivariable Calculus (3 weeks)
10.1 Functions of Two or More Independent Variables
10.2 Limits and Continuity
10.2.1 Informal Definition of Limits
10.2.2 Formal Definition of Limits (Optional)
10.2.3 Continuity
10.3 Partial Derivatives
10.3.1 Functions of Two Variables
10.3.2 Functions of More Than Two Variables
10.3.3 Higher-Order Partial Derivatives
10.4 Tangent Planes, Differentiability, and Linearization
10.4.1 Functions of Two Variables
10.4.2 Vector-Valued Functions
10.5 More about Derivatives (omit)
10.5.1 The Chain Rule for Functions of Two Variables
10.5.2 Implicit Differentiation
10.5.3 Directional Derivatives and Gradient Vectors
10.6 Applications (omit)
10.6.1 Maxima and Minima
10.6.2 Extrema with Constraints
10.6.3 Diffusion
10.7 Systems of Difference Equations (omit)
10.7.1 A Biological Example
10.7.2 Equilibria and Stability in Systems of Linear Difference Equations
10.7.3 Equilibria and Stability of Nonlinear Systems of Difference
Equations
Chapter 11: Systems of Differential Equations (omit)
11.1 Linear Systems: Theory
11.1.1 The Direction Field
11.1.2 Solving Linear Systems
11.1.3 Equilibria and Stability
11.2 Linear Systems: Applications
11.2.1 Compartment Models
11.2.2 The Harmonic Oscillator (Optional)
11.3 Nonlinear Autonomous Systems: Theory
11.3.1 Analytical Approach
11.3.2 Graphical Approach for 2 × 2 Systems
11.4 Nonlinear Systems: Applications
11.4.1 The Lotka-Volterra Model of Interspecific Competition
11.4.2 A Predator-Prey Model
11.4.3 The Community Matrix
11.4.4 A Mathematical Model for Neuron Activity
11.4.5 A Mathematical Model for Enzymatic Reactions
Chapter 12: Probability and Statistics (2 weeks)
12.1 Counting
12.1.1 The Multiplication Principle
12.1.2 Permutations
12.1.3 Combinations
12.1.4 Combining the Counting Principles
12.2 What is Probability?
12.2.1 Basic Definitions
12.2.2 Equally Likely Outcomes
12.3 Conditional Probability and Independence
12.3.1 Conditional Probability
12.3.2 The Law of Total Probability
12.3.3 Independence
12.3.4 The Bayes Formula
12.4 Discrete Random Variables and Discrete Distribution
12.4.1 Discrete Distributions
12.4.2 Mean and Variance
12.4.3 The Binomial Distribution
12.4.4 The Multinomial Distribution
12.4.5 Geometric Distribution
12.4.6 The Poisson Distribution
12.5 Continuous Distributions
12.5.1 Density Functions
12.5.2 The Normal Distribution
12.5.3 The Uniform Distribution
12.5.4 The Exponential Distribution
12.6 Limit Theorems
12.6.1 The Law of Large Numbers
12.6.2 The Central Limit Theorem
12.7 Statistical Tools
12.7.1 Collecting and Describing Data
12.7.2 Estimating Means and Proportions
12.7.3 Linear Regression
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