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Calculus: The Language of Change

Chapter 1: Introduction to the Scientific Projects

What good is it?
How Much Work Are They?
How to Write Up a Project
Help on the World Wide Web

Chapter 2: Epidemiological Applications

Review of the S-I-R Model
Basic Assumptions
Derivation of the Equations of Change

Project 3: Vaccination for Herd Immunity

Herd Immunity
The Contact Number Data
Project Issues
Vaccine Failures

Project 4: S-I-S Diseases and The Endemic Limit

Basic Assumptions
The Continuous S-I-S Variables
Parameters for The SIS Model
The Importance of the Contact Ratio}
Conjectures
Conclusions

Chapter 3: The Role of Rules for Derivatives

Project 7: The Expanding House

Volume Expansion Explained by Calculus

Chapter 4: Applications of The Increment Approximation

Project 10: Functional Identities

Differential Equations from Increment Geometry}{55}

Project 12: The Isochrone

Conservation of Energy

Project 13: The Catenary

The Catenary Hypotheses
Parameters
Variables
The Equation for Tension
Optimizing Length and Strength

Chapter 5: Log and Exponential Functions

Project 15: Drug Concentration and ``Bi-Exponential'' Functions

Primary Variables of the Model
Parameters of the Model
The Formulas for Concentration
Comparison with Mythical Data
Comparison with Real Data

Project 16: Measurement of Kidney Function by Drug Concentration

Variables and Parameters
Overview of the Project
Drug Data

Chapter 6: Theory of Derivatives

Project 20: The Mean Value Math Police

The Mean Value Theorem for Regular Derivatives
The Theorem of Bolzano
The Mean Value Theorem for Pointwise Derivatives
Overall Speed IS an Average

Project 21: Inverse Functions and Their Derivatives

Graphical Representation of the Inverse
The Derivative of the Inverse
Non-Elementary Inversion

Project 22: Taylor's Formula

The Increment Equation and Increasing
Taylor's Formula and Bending
Symmetric Differences and Taylor's Formula
Direct Computation of Second Derivatives
Direct Interpretation of Higher Order Derivatives

Chapter 7: Applications to Physics

Project 24: Falling with Air Resistance: Data and A Linear Model

Terminal Velocity
Comparison with The Symbolic Solution

Project 25: Bungee Diving

Forces Acting on the Jumper Before the Cord is Stretched
Forces Acting on the Jumper After He Falls L Feet
Modeling the Jump
The Derivation of Planck's Law of Radiation
Wavelength Form and First Plots
Maximum Intensity in Terms of a Parameter

Project 27: Fermat's Principle Implies Snell's Law

Reflection off a Curved Mirror
Computation of Reflection Angles

Chapter 8: Applications in Economics

Project 28: Monopoly Pricing

Going Into Politics

Project 29: Discrete Dynamics of Price Adjustment

The Story
The Basic Linear Model
Taxation in the Linear Economy (Optional)
A Nonlinear Economy (Optional)
Taxation in the Nonlinear Economy

Chapter 9: Advanced Max - min Problems

Project 31: Geometric Optimization Projects

Distance Between Lines
Distance between Curves
An Implicit-Parametric Approach
Distance from a Curve to a Surface

Project 32: Least Squares Fit and Max-Min

Introductory Example
The Critical Point
The General Critical Equations

Project 33: Local Max-Min and Stability of Equilibria

Steepest Ascent
The Second Derivative Test in Two Variables

Chapter 10: Applications of Linear Differential Equations

Project 34: Lanchester's Combat Models

The Principle of Concentration
The Square Law
Guerrilla Combat
Operational Losses (Optional)

Project 35: Drug Dynamics and Pharmacokinetics

Derivation of the Equations of Change
Where do we go from here?
Periodic Intravenous Injections (Optional Project Conclusion)
Intramuscular Injection - a Third Compartment

Chapter 11: Forced Linear Equations

Project 36: Forced Vibration - Non-autonomous Equations

Solution of the Autonomous Linear Equation
Transients - Limiting Behavior
Superposition for the Spring System
Equations Forced by Gravity
Equations Forced by Sinusoids
Non-homogeneous I. V. P.s

Project 37: Resonance - Maximal Response to Forcing

Some Useful Trig
Resonance in Forced Linear Oscillators
An Electrical Circuit Experiment
Nonlinear Damping

Project 38: A Notch Filter - Minimal Response to Forcing

The Laws of Kirchoff, Ohm, and Coulomb
A Check on \$a[t]^2\$
Where's the Min?

Chapter 12: Applications in Ecology

Project 39: Logistic Growth with Hunting

Basic Fertility
Logistic Growth
Voodoo Discovers the Mice

Project 40: Predator - Prey Interactions

Bunny Island
Rabbit Island

Project 41: Competition and Cooperation Between Species

Biological Niches
Cooperation between Species

Project 42: Sustained Harvest of Sei Whales

Carrying Capacity, Environmental and Mathematical
The Actual Carrying Capacity

Chapter 13: Derivations with Vectors

Project 43: Wheels Rolling on Wheels

Epicycloids
Cycloids
Hypocycloids

Project 44: The Perfecto Skier

The Mountain's Contribution
Gravity and the Mountain
The Pendulum as Constrained Motion
The Explicit Surface Case

Project 45: Low Level Bombing

Significance of Vector Air Resistance

Project 46: The Pendulum

Derivation of the Pendulum Equation
Numerical Solutions of the Pendulum Equation
Linear Approximation to the Pendulum Equation
Friction in The Pendulum (Optional)
The Spring Pendulum (Optional)

Project 47: Using Jupiter as a Slingshot

Setting up the Problem: Scaling and Units
Newton's Law of Gravity
Newton's     F = m  a Law
Numerical Flights Out of the Solar System

Chapter 14: Chemical Reactions

Project 48: Stability of a Tank Reaction

Mass Balance
Arrhenius' Law
Heat Balance
Stability of Equilibria
Forced Cooling (Optional)

Project 49: Beer, Coke \& Vitamin C

Enzyme-mediated Reactions
Molar Concentration and Reaction Rates
The Briggs-Haldane Dynamics Approximation
The Michaelis-Menten Dynamics Approximation
Blood Ethanol
Blood CO2