Calculus: The Language of Change

Text and Software Table of Chapter Contents

Homework solutions & more problems (as Mathematica NoteBooks)






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

   The First S-I-R Epidemic Model
   A Second Computation with Lists
   Euler's Recursion vs. CAS's
   CAS Solution of S-I-R Equations
   Limits for S-I-R & the Invariant

   FirstSIR.nb or .mws
   SecondSIR.nb or .mws
   ThirdSIR.nb or .mws
   SIRsolver.nb or .mws
   EpidemicRoots.nb or .mws

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

   Microscopic Zooming
   Zooming into the Tangent Gap
   Weierstrass' Function
   Zoom.nb or .mws
   SecantGapZ.nb or .mws
   Weierstrass.nb or .mws

Chapter 4

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

   Euler's Approximation for IVPs   EulerApprox.nb or .mws

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

   The Symbolic Microscope in 1-D
   Symbolic Increments of Functions
   Function Limits for Derivatives
   Exponentials and Percent Growth
   Micro1D.nb or .mws
   SymbIncr.nb or .mws
   DfctLimit.nb or .mws
   PercentGth.nb or .mws

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

   Computing with Your Own Rules
   Symbolic Differentiation with CAS
   Check Differentiation with CAS
   The Symbolic Microscope in 1-D
   The Tangent to the Graph

   DiffRules.nb or .mws
   DfDx.nb or .mws
   HomeWkCk.nb or .mws
   Micro1D.nb or .mws
   Tangents.nb or .mws

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

   Symbolic Differentiation with CAS
   Implicit Derivative and Tangent

   DfDx.nb or .mws
   ImplicitDeriv.nb or .mws

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

  Derivatives, Shape Tables, and Computer Plots    Graph2D.nb or .mws

Chapter 10

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

   Acceleration and Gravity    Gravity.nb or .mws

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

   Proof of The Extreme Value Thm
   Exact and Approximate Solution

   ExtremeValue.nb or .mws
   SolveEqns.nb or .mws

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

   Sums.nb or .mws
   ConeVol.nb or .mws
   IntegrAprx.nb or .mws

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

   Symbolic Integration with CAS    SymbolicIntegr.nb or .mws

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

   Arc Length of Curves
   Surface Area of a Torus
   Area Between Curves
   Area of a Parabolic Antenna

   ArcLength.nb or .mws
   DonutIcing.nb or .mws
   AreaBetween.nb or .mws
   AntennaArea.nb or .mws

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

   Position Vectors and Length
   Vector Sums, Tips to Tails
   Scalar Multiplication of Vectors
   Free Vectors, Displacements
   Angles, Lengths
   The Vector Cross Product
   An Algebraic-Geometric Lexicon

   PositVec.nb or .mws
   VectSum.nb or .mws
   ScalarMult.nb or .mws
   DisplVect.nb or .mws
   AnglPerp.nb or .mws
   CrossProd.nb or .mws
   AlgGeoLx.nb or .mws

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

   Parametric Equations for a Line
   Parametric Equations for a Circle
   Classic Parametric Curves in 2-D
   Curves in Polar Coordinates
   Parametric Curves in 3-D
   Velocity and Tangents

   ParamLine.nb or .mws
   Circles.nb or .mws
   ClassicCrvs.nb or .mws
   PolarCrvs.nb or .mws
   Param3D.nb or .mws
   PrmVelTan.nb or .mws

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

   Slices of Surfaces
   Basic Graphs in 3-D
   Graphs of Surfaces by Slices
   Explicit, Implicit Lines and Planes
   Plane Through Three Points
   Contour Plots of Linear Functions
   Contours and Horizontal Slices
   Explicit Plots for Problem 17.6
   Contour Plots for Problem 17.6

   SurfaceFlyBy.nb or .mws
   BasicGrfs3D.nb or .mws
   SurfSlices.nb or .mws
   PlaneLines.nb or .mws
   ThreePoints.nb or .mws
   LinearContours.nb or .mws
   ContourSlices.nb or .mws
   ExplicitSurfaces.nb or .mws
   ContourPlots.nb or .mws

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

   A Microscope for 3D Graphs
   Zooming to a Tangent
   Partial and Total Derivatives
   The Normal and Tangent
   The Gradient of a Function
   Review Uses of Partial Derivatives

   Micro3D.nb or .mws
   Zoom3D.nb or .mws
   PartialD.nb or .mws
   SurfNorm.nb or .mws
   Grad.nb or .mws
   DiffReview.nb or .mws

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

   Partial Information for Max-Min
   Max-min Examples using CAS

   MxMnThy.nb or .mws
   SvVarMxMn.nb or .mws

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

   A First Look at Discrete Systems
   Dynamical Systems and Cobwebs

   FstDscrDySy.nb or .mws
   CobWeb.nb or .mws

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

   Exact Solution of Percentage
   Rapid Exponential Growth
   Accurate Numerical Solution
   One Dimensional Flow
   Euler's Approximation
   Euler's Approximation Compared

   ExpEquns.nb or .mws
   ExpGth.nb or .mws
   AccDEsol.nb or .mws
   Flow1D.nb or .mws
   EulerApprox.nb or .mws
   EULERexact.nb or .mws

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

   Direction Fields in 2-D
   Flows in 2-D
   Views of Solutions of 2-D ODEs
   Accurate Numerical Solution
   Flows in 3-D
   DirField.nb or .mws
   Flow2D.nb or .mws
   SolnViews.nb or .mws
   AccDEsol.nb or .mws
   Flow3D.nb or .mws

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

   Equilibria of Linear ODEs in 2-D
   Computation of Local Stability
   Linear Flows in 2-D

   LinearEquilibria.nb or .mws
   LocalStability.nb or .mws
   Flow2D.nb or .mws

Chapter 25

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

   Basic Examples of Series
   The Geometric Series
   Series for the Classical Functions
   Power, Fourier, and Other Series
   Weierstrass' Function

   BasicSeries.nb or .mws
   GeomSeries.nb or .mws
   ClassicalSeries.nb or .mws
   FourierSeries.nb or .mws
   Weierstrass.nb or .mws

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

   Series for the Classical Functions
   A Series for the Absolute Value
   Formal Taylor Series
   The Binomial Series
   Specific Infinite Sums
   Series for Hyperbolic Functions
   Series for Bessel Functions
   ClassicalSeries.nb or .mws
   AbsSeries.nb or .mws
   FormalTSeries.nb or .mws
   BinomialSeries.nb or .mws
   SomeSums.nb or .mws
   CoshSeries.nb or .mws
   BesselSeries.nb or .mws

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

   Slowly Convergent Series    SlowSeries.nb or .mws

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

   Polynomials with Mathematica
   Defining Functions with CSAs
   Laws of Exponents
   Rapid Exponential Growth
   Slow Logarithmic Growth
   Translation and Scaling a Parabola

   Polynomials.nb or .mws
   Functions.nb or .mws
   ExpRules.nb or .mws
   ExpGth.nb or .mws
   LogGth.nb or .mws
   SlideSquash.nb or .mws

Chapter 29

Complex Numbers
        29.1 Algebra of Complex Numbers
        29.2 Geometry of Numbers

   Complex Arithmetic with CASs    CmplxNrs.nb or .mws

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