Calculus: The Language of Change

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Table of Contents

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

Project 1: Linear Approximation of CO2 Data

Chapter 2: Epidemiological Applications

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

Project 2: The 1968-69 New York Hong Kong Flu Epidemic

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}

Project 5: Max-min in S-I-R Epidemics

Chapter 3: The Role of Rules for Derivatives

Project 6: The Expanding Economy

Project 7: The Expanding House

        Volume Expansion Explained by Calculus

Chapter 4: Applications of The Increment Approximation

Project 8: A Derivation of Hubble's Law

Project 9: Functional Linearity

Project 10: Functional Identities

        Additive Functions
        Differential Equations from Increment Geometry}{55}

Project 11: The Tractrix

Project 12: The Isochrone

        Conservation of Energy

Project 13: The Catenary

        The Catenary Hypotheses
        The Equation for Tension
        Optimizing Length and Strength

Chapter 5: Log and Exponential Functions

Project 14: The Canary Resurrected

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

Project 17: Numerical Derivatives of Exponentials

Project 18: Repeated Exponents

Project 19: Solve dx = r[t]  x[t]\ dt + f[t]

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 23: The Falling Ladder (or Dad's Disaster)

        Air Resistance on Dad (Optional)

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 Business
        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
        Why Trade?}{169}

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)
        Steady Intravenous Flow
        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
        Steady State Solution
        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


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

Chapter 15: More Mathematical Projects

Project 50: Rearrangement of Conditionally Convergent Series

Project 51: Computation of Fourier Series

Project 52: The Big Bite of the Subtraction Bug

Chapter 16: Additional Project References

        Lowering the Water Table
        The Light Speed Lighthouse
        Horizontal, Vertical, and Slant Asymptotes

C:TLC Projects

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