Master Math: Calculus: Calculus Book

Introduction 1

Chapter 1 Functions 3

1.1 Functions: Types, Properties, and Definitions 3

1.2 Exponents and Logarithms 8

1.3 Trigonometric Functions 10

1.4 Circular Motion 20

1.5 Relationship Between Trigonometric and Exponential Functions 25

1.6 Hyperbolic Functions 26

1.7 Polynomical Functions 28

1.8 Functions of More Than One Variable and Contour Diagrams 29

1.9 Coordinate Systems 33

1.10 Complex Numbers 36

1.11 Parabolas, Circles, Ellipses, and Hyperbolas 38

Chapter 2 The Derivative 47

2.1 The Limit 47

2.2 Continuity 50

2.3 Differentiability 52

2.4 The Definition of the Derivative and Rate of Change 53

2.5 (delta) Notation and the Definition of the Derivative 57

2.6 Slope of a Tangent Line and the Definition of the Derivative 58

2.7 Velocity, Distance, Slope, Area, and the Definition of the Derivative 60

2.8 Evaluating Derivatives of Constants and Linear Functions 63

2.9 Evaluating Derivatives Using the Derivative Formula 64

2.10 The Derivatives of a Variable, a Constant with a Variable, a Constant with a Function, and a Variable Raised to a Power 66

2.11 Examples of Differentiating Using the Derivative Formula 68

2.12 Derivatives of Powers of Functions 69

2.13 Derivatives of a^x, e^x, and ln x 71

2.14 Applications of Exponential Equations 77

2.15 Differentiating Sums, Differences, and Polynomials 80

2.16 Taking Second Derivatives 81

2.17 Derivatives of Products: The Product Rule 82

2.18 Derivatives of Quotients: The Quotient Rule 85

2.19 The Chain Rule for Differentiating Complicated Functions 86

2.20 Rate Problem Examples 90

2.21 Differentiating Trigonometric Functions 91

2.22Inverse Functions and Inverse Trigonometric Functions and Their Derivatives 95

2.23 Differentiating Hyperbolic Functions 99

2.24 Differentiating Multivariable Functions 101

2.25 Differentiation of Implicit Vs. Explicit Functions 101

2.26 Selected Rules of Differentiation 102

2.27 Minimum, Maximum, and the First and Second Derivatives 103

2.28 Notes on Local Linearity, Approximating Slope of Curve, and Numerical Methods 109

Chapter 3 The Integral 113

3.1 Introduction 113

3.2 Sums and Sigma Notation 114

3.3 The Antiderivative or Indefinite Integral and the Integral Formula 117

3.4 The Definite Integral and the Fundamental Theorem of Calculus 120

3.5 Improper Integrals 122

3.6 The Integral and the Area Under a Curve 124

3.7 Estimating Integrals Using Sums and the Associated Error 128

3.8 The Integral and the Average Value 131

3.9 Area Below the X-axis, Even and Odd Functions, and The Integrals 131

3.10 Integrating a Function and a Constant, the Sum of Functions, a Polynomial, and Properties of Integrals 134

3.11 Multiple Integrals 136

3.12 Examples of Common Integrals 138

3.13 Integrals Describing Length 139

3.14 Integrals Describing Area 140

3.15 Integrals Describing Volume 145

3.16 Changing Coordinates and Variables 152

3.17 Applications of the Integral 157

3.18 Evaluating Integrals Using Integration by Parts 162

3.19 Evaluating Integrals Using Substitution 164

3.20 Evaluating Integrals Using Partial Fractions 172

3.21 Evaluating Integrals Using Tables 177

Chapter 4 Series and Approximation 179

4.1 Sequences, Progressions, and Series 179

4.2 Infinite Series and Tests for Convergence 183

4.3 Expanding Functions Into Series, the Power Series, Taylor Series, Maclaurin Series, and the Binomial Expansion 188

Chapter 5 Vectors, Matrices, Curves, Surfaces, and Motion 195

5.1 Introduction to Vectors 195

5.2 Introduction to Matrices 202

5.3 Multiplication of Vectors and Matrices 205

5.4 Dot or Scalar Products 208

5.5 Vector or Cross Product 211

5.6 Summary of Determinants 215

5.7 Matrices and Linear Algebra 217

5.8 The Position Vector Parametric Equations, Curves, and Surfaces 224

5.9 Motion, Velocity, and Acceleration 230

Chapter 6 Partial Derivatives 243

6.1 Partial Derivatives: Representation and Evaluation 243

6.2 The Chain Rule 246

6.3 Representation on a Graph 247

6.4 Local Linearity, Linear Approximations, Quadratic Approximations, and Differentials 250

6.5 Directional Derivative and Gradient 255

6.6 Minima, Maxima, and Optimization 259

Chapter 7 Vector Calculus 267

7.1 Summary of Scalars, Vectors, the Directional Derivative, and the Gradient 267

7.2 Vector Fields and Field Lines 271

7.3 Line Integrals and Conservative Vector Fields 276

7.4 Green's Theorem: Tangent and Normal (Flux) Forms 282

7.5 Surface Integrals and Flux 287

7.6 Divergence 295

7.7 Curl 300

7.8 Stokes' Theorem 304

Chapter 8 Introduction to Differential Equations 307

8.1 First-Order Differential Equations 308

8.2 Second-Order Linear Differential Equations 312

8.3 Higher-Order Linear Differential Equations 315

8.4 Series Solutions to Differential Equations 317

8.5 Systems of Differential Equations 319

8.6 Laplace Transform Method 321

8.7 Numerical Methods for Solving Differential Equations 322

8.8 Partial Differential Equations 324