Calculus: The Language of Change

Introduction

   1MmaNoteBooks.nb
   2MmaNumerics.nb
   3MmaSymbolics.nb
   4MmaGraphics.nb
   5MmaLists.nb

Chapter 1

Calculus with Computers
        1.1 Previews of Coming Attractions
        1.2 An Introduction to Computing
        1.3 Linearity in Local Coordinates
        1.4 Derivatives for Explicit Formulas
        1.5 Review of High School Math
        1.6 Free Advice

   Linear Functions with CAS

   Derivatives as Limits by CAS

   LinearIntro.nb or .mws

   DerivLimit.nb or .mws

Chapter 2

Using Calculus to Model Epidemics
        2.1 The First Model
        2.2 Shortening the Time Steps
        2.3 The Continuous Variable Model
        2.4 Analysis of Change
        2.5 Long-Term Change
        2.6 Calculus and the S-I-R Invariant
        2.7 Chapter Summary
        2.8 Projects

Chapter 3

Linearity vs. Local Linearity
        3.1 Approximation of Ox-bows
        3.2 Graphical Increments
        3.3 Algebra of Microscopes
        3.4 Symbolic Increments
        3.5 Projects

Chapter 4

Differential Equations and Derivatives
        4.1 The Cool Canary
        4.2 Instantaneous Rates of Change
        4.3 Projects

Chapter 5

Symbolic Increments
        5.1 The Gap for Power Functions
        5.2 Moving the Microscope
        5.3 Trigonometric Derivatives
        5.4 Derivatives of Log and Exp
        5.5 Continuity and the Derivative
        5.6 Projects and Theory

Chapter 6

Symbolic Differentiation
        6.1 Rules for Special Functions
        6.2 The Superposition Rule
        6.3 The Product Rule
        6.4 The Chain Rule
        6.5 General Exponentials
        6.6 Derivative of the Natural Log
        6.7 Combined Symbolic Rules
        6.8 Review - Inside the Microscope
        6.9 Projects

Chapter 7

Related Rates and Implicit Derivatives
        7.1 Differentiation with Parameters
        7.2 Implicit Differentiation
        7.3 Implicit Tangents and Derivatives
        7.4 Related Rates
        7.5 Implicitly Linked Variables
        7.6 Projects

Chapter 8

The Natural Log and Exponential
        8.1 The Official Natural Exponential
        8.2 e as a "Natural" Base
        8.3 Growth of Log, Exp, and Powers
        8.4 Official Properties
        8.5 Projects

   Exact Solution of y'=k y
   Exponentials and Percentage
   Orders of Infinity
   Rapid Exponential Growth
   Slow Logarithmic Growth

   ExpEquns.nb or .mws
   PercentGth.nb or .mws
   Infinities.nb or .mws
   ExpGth.nb or .mws
   LogGth.nb or .mws

Chapter 9

Graphs and the Derivative
        9.1 Graphs from Formulas
        9.2 Graphs Without Formulas
        9.3 Ups and Downs of the Derivative
        9.4 Bending & the Second Derivative
        9.5 Graphing Differential Equations
        9.6 Projects

Chapter 10

Velocity, Acceleration, and Calculus
        10.1 Acceleration
        10.2 Galileo's Law of Gravity
        10.3 Projects

Chapter 11

Maxima and Minima in One Variable
        11.1 A Graphical Minimum
        11.2 Critical Points
        11.3 Max - min with Endpoints
        11.4 Max - min without Endpoints
        11.5 Supply and Demand
        11.6 Constrained Max-Min
        11.7 Max-min with Parameters
        11.8 Projects

Chapter 12

Basic Integration
        12.1 Slice Approximations
        12.2 Distance When Speed Varies
        12.3 The Definite Integral
        12.4 Computer Summation
        12.5 The Algebra of Summation
        12.6 The Algebra of Integration
        12.7 Fundamental Theorem, Part 1
        12.8 Fundamental Theorem, Part 2

   Sums with CASs
   The Disk Method
   Numerical Approximation

Chapter 13

Symbolic Integration
        13.1 Indefinite Integrals
        13.2 Specific Integral Formulas
        13.3 Superposition of Antiderivatives
        13.4 "Substitution" for Integrals
        13.5 Change of Limits of Integration
        13.6 Trig Substitutions
        13.7 Integration by Parts
        13.8 Impossible Integrals

Chapter 14

Applications of Integration
        14.1 The Length of a Curve
        14.2 Duhamel's Principle
        14.3 Geometric Integrals
        14.4 Improper Integrals

Chapter 15

Basic Vector Geometry
        15.1 Cartesian Coordinates
        15.2 Position Vectors
        15.3 Geometry of Vector Addition
        15.4 Scalar Multiplication
        15.5 Differences and Displacements
        15.6 Angles and Projections
        15.7 Cross Product in 3 Dimensions
        15.8 Geometry and Algebra Lexicon

Chapter 16

Parametric Curves
        16.1 The Vector Parametric Line
        16.2 Parametric Circles
        16.3 Polar Curves
        16.4 3-D Parametric Curves
        16.5 Tangents and Velocity Vectors
        16.6 Projects

Chapter 17

Graphs in Several Variables
        17.1 The Expicit Plane in 3-D
        17.2 Vertical Slices and Chickenwire
        17.3 Implicit Linear Equations
        17.4 Gradients and Contour Plots
        17.5 Horizontal Slices and Contours
        17.6 Explicit, Implicit, Parametric
        17.7 Lines, Curves, & Planes

Chapter 18

Differentiation in Several Variables
        18.1 Partial and Total Derivatives
        18.2 Partial Differentiation Examples
        18.3 Differential Approximations
        18.4 Geometry of the Differential
        18.5 The Meaning of the Gradient
        18.6 Implicit Differentiation (Again)
        18.7 Review Exercises

Chapter 19

Several Variable Optimization
        19.1 Critical Points to Investigate
        19.2 Existence of Extreme Values
        19.3 Compact Regions
        19.4 Implicit Constraints
        19.5 Noncompact Extrema in 2-D
        19.6 Projects on Max - min

Chapter 20

Discrete Dynamical Systems
        20.1 Models for Price Adjustment
        20.2 Cobwebs
        20.3 The Linear System
        20.4 Logistic Growth
        20.5 Calculus and Nonlinearity
        20.6 Projects

Chapter 21

Continuous Dynamical Systems in 1-D
        21.1 Exponential Growth and Decay
        21.2 Logistic Growth Laws
        21.3 Some Helpful Theory
        21.4 Separation of Variables
        21.5 Projects

Chapter 22

Continuous Dynamical Systems in 2-D
        22.1 Basic Theory in 2-D
        22.2 Geometric Solution in 2-D
        22.3 Flows vs. Explicit Solutions
        22.4 Flow Analysis of Models
        22.5 Projects

Chapter 23

Linear Dynamical Systems
        23.1 The Shock Absorber Equation
        23.2 Constant Coefficient Systems
        23.3 Symbolic Exponential Solutions
        23.4 Rotation and Euler's Formula
        23.5 Basic Solutions
        23.6 Superposition and All Solutions
        23.7 Second-Order IVPs
        23.8 Projects

Chapter 25

Geometric Series
        25.1 Geometric Series: Convergence
        25.2 Convergence by Comparison
        25.3 Compound Interest

Chapter 26

Power Series
        26.1 Computation of Power Series
        26.2 The Ratio Test
        26.3 Integration of Series
        26.4 Differentiation of Power Series

Chapter 27

The Edge of Convergence
        27.1 Alternating Series
        27.2 Telescoping Series
        27.3 Integrals Compared to Series
        27.4 Limit Comparisons
        27.5 Fourier Series

Chapter 28

High School Review with Computing
        28.1 Linear Functions
        28.2 Polynomials
        28.3 Functions from Formulas
        28.4 Power Functions
        28.5 Trig Functions
        28.6 Logs and Exponentials
        28.7 Chaining or Composition
        28.8 Parameters
        28.9 Functional Identities

Chapter 29

Complex Numbers
        29.1 Algebra of Complex Numbers
        29.2 Geometry of Numbers