Calculus: Understanding Its Concepts and Methods

Cover of Calculus: Understanding Its 
      Concepts and Methods Darel Hardy, Fred Richman, Carol Walker, and Robert Wisner
©2005
Publisher: MacKichan Software, Inc.
ISBN: 0-9766806-9-6

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Table of Contents

Preface

Introduction

Review Chapter: Functions and Their Graphs

  0.0 Introduction

  0.1 Ways to define a function

  0.2 Polynomials and rational functions

  0.3 Transcendental functions

  0.4 Plotting equations

  0.5 Parametric curves

  0.6 Shifting, scaling, and combining functions

Chapter 1: Tangents and Derivatives

  1.0 Introduction

  1.2 Local linearity

  1.3 Limits

  1.4 Continuity

  1.5 The derivative at a point

  1.6 Derivatives as functions

  1.7 Asymptotic behavior

Chapter 2: Differentiation Rules and Properties

  2.0 Introduction

  2.1 Product and quotient rules

  2.2 Chain rule

  2.3 Derivatives of trigonometric functions

  2.4 Differentiating implicit functions

  2.5 Higher derivatives

  2.6 Differentials

  2.7 Parametric curves

Chapter 3: Applications of Derivatives

  3.0 Introduction

  3.1 Extreme values

  3.2 Mean value theorem

  3.3 Shape of a graph

  3.4 Optimization

  3.5 Newton's method

  3.6 Indeterminate forms

  3.7 Related rates

Chapter 4: Integrals

  4.0 Introduction

  4.1 Area function

  4.2 Antiderivatives

  4.3 Definite integrals

  4.4 Fundamental theorem of calculus

  4.5 Change of variable

Chapter 5: Applications of Integration

  5.0 Introduction

  5.1 Velocity and acceleration

  5.2 Area between curves

  5.3 Volume by cross sections

  5.4 Volume by cylindrical shells

  5.5 Average value of a function

Chapter 6: Calculus of Transcendental Functions

  6.0 Introduction

  6.1 Inverse functions

  6.2 Natural logarithm

  6.3 Exponential functions

  6.4 Inverse trigonometric functions

  6.5 Hyperbolic functions

Chapter 7: Techniques of Integration

  7.0 Introduction

  7.1 Integration by parts

  7.2 Trigonometric functions

  7.3 Trigonometric substitution

  7.4 Partial fractions

  7.5 Tables of integrals and further substitutions

  7.6 Improper integrals

Chapter 8: Further Applications of Integration

  8.0 Introduction

  8.1 Polar coordinates

  8.2 Arc length

  8.3 Surface of revolution

  8.4 Exponential growth and decay

  8.5 Moments and center of mass

Chapter 9: Function Approximations

  9.0 Introduction

  9.1 Taylor polynomials

  9.2 Polynomial interpolation

  9.3 Splines

  9.4 Bézier curves

  9.5 Rational functions

  9.6 Trigonometric functions

Chapter 10: Infinite Series

  10.0 Introduction

  10.1 Sequences

  10.2 Series

  10.3 Convergence tests

  10.4 Power series

  10.5 Maclaurin and Taylor series

  10.6 Complex functions

Chapter 11: Numerical Integration

  11.0 Introduction

  11.1 Riemann sums

  11.2 Simpson's rule

  11.3 Taylor polynomials

  11.4 Other numerical integration methods

  11.5 Euler's method

Chapter 12: Vectors in Two and Three Dimensions

  12.0 Introduction

  12.1 Vectors in the plane

  12.2 Vectors in space

  12.3 Inner products and projections

  12.4 Cross product

  12.5 Lines and planes

  12.6 Cylindrical and spherical coordinate systems

  12.7 Surfaces

Chapter 13: Partial Derivatives

  13.0 Introduction

  13.1 Functions of several variables

  13.2 Partial derivatives

  13.3 Rules for partial derivatives

  13.4 Local linearity

  13.5 Directional derivatives and the gradient

  13.6 Normals and the tangent plane

  13.7 Extrema

  13.8 Lagrange multipliers

Chapter 14: Multiple Integrals

  14.0 Introduction

  14.1 Double integrals

  14.2 Iterated integrals

  14.3 Double integrals in polar coordinates

  14.4 Surface area

  14.5 Triple integrals

  14.6 Cylindrical and spherical coordinates

Chapter 15: Vector-Valued Functions

  15.0 Introduction

  15.1 Space curves

  15.2 Derivatives and integrals

  15.3 Arc length and curvature

  15.4 Velocity and acceleration

Chapter 16: Vector Calculus

  16.0 Introduction

  16.1 Vector fields

  16.2 Line integrals

  16.3 Green's theorem

  16.4 Surface integrals

  16.6 Divergence theorem

Chapter 17: Differential Equations

  17.0 Introduction

  17.1 Solutions to differential equations

  17.2 Differential equations with separable variables

  17.3 Homogeneous differential equations

  17.4 Exact differential equations

  17.5 Exactness from integrating factors

Appendix A: Animations

  A0: Introduction

  A1: 2D Animations

  A2: 3D Plots

  A3: 3D Animations

Appendix B: Business Examples

  B.1 Marginal analysis

  B.2 Interest

  B.3 Consumer and producer surplus

  B.4 Probability

  B.5 Expected value

Appendix C: Complex Numbers

  C.1 Complex numbers

Appendix D: Matrices and Determinants

  D0: Introduction

  D1: Matrices and vectors

  D2: Determinants

  D3: Geometric transformations in two dimensions

  D4: Geometric transformations in three dimensions

Appendix E: Engineering Examples

  E.1 Work

  E.2 Pressure

  E.3 Moments of inertia

Index

Examples

Explorations



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