What Is Calculus?: From Simple Algebra To Deep Analysis

by R Michael Range

Overview

This unique book provides a new and well-motivated introduction to calculus and analysis, historically significant fundamental areas of mathematics that are widely used in many disciplines. It begins with familiar elementary high school geometry and algebra, and develops important concepts such as tangents and derivatives without using any advanced tools based on limits and infinite processes that dominate the traditional introductions to the subject. This simple algebraic method is a modern version of an idea that goes back to René Descartes and that has been largely forgotten. Moving beyond algebra, the need for new analytic concepts based on completeness, continuity, and limits becomes clearly visible to the reader while investigating exponential functions.The author carefully develops the necessary foundations while minimizing the use of technical language. He expertly guides the reader to deep fundamental analysis results, including completeness, key differential equations, definite integrals, Taylor series for standard functions, and the Euler identity. This pioneering book takes the sophisticated reader from simple familiar algebra to the heart of analysis. Furthermore, it should be of interest as a source of new ideas and as supplementary reading for high school teachers, and for students and instructors of calculus and analysis.

Product Details

ISBN-13:	9789814644488
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	10/06/2015
Pages:	372
Product dimensions:	6.00(w) x 8.90(h) x 0.90(d)

Table of Contents

Preface xv

Notes for Instructors xxiii

Prelude to Calculus 1

1 Introduction 1

2 Tangents to Circles 2

2.1 Exercises 8

3 Tangents to Parabolas 8

3.1 Exercises 10

4 Motion with Variable Speed 11

4.1 Exercises 15

5 Tangents to Graphs of Polynomials 16

5.1 Exercises 21

6 Rules for Differentiation 21

6.1 Elementary Rules 22

6.2 Inverse Function Rule 24

6.3 Product Rule 26

6.4 Quotient Rule 28

6.5 Exercises 30

7 More General Algebraic Functions 31

7.1 Exercises 35

8 Beyond Algebraic Functions 35

8.1 Exercises 42

I The Cast: Functions of a. Real Variable 43

I.1 Real Numbers 43

I.1.1 Rational Numbers 43

I.1.2 Order Properties 46

I.1.3 Irrational Numbers 47

I.1.4 Completeness of the Real Numbers 50

I.1.5 Intervals and Other Properties of K 53

I.1.6 Exercises 58

I.2 Functions 60

I.2.1 Functions of Real Variables 60

I.2.2 Graphs 61

I.2.3 Some Simple Examples 65

I.2.4 Linear Functions, Lines, and Slopes 66

I.2.5 Exercises 70

I.3 Simple Periodic Functions 71

I.3.1 The Basic Trigonometric Functions 72

I.3.2 Radian Measure 74

I.3.3 Simple Trigonometric Identities 75

I.3.4 Graphs 77

I.3.5 Exercises 78

I.4 Exponential Functions 79

I.4.1 Compound Interest 79

I.4.2 The Functional Equation 80

I.4.3 Definition of Exponential Functions for Rational Numbers 81

I.4.4 Properties of Exponential Functions 83

I.4.5 Exponential Functions for Real Numbers 86

I.4.6 Exercises 88

I.5 Natural Operations on Functions 89

I.5.1 Compositions 89

I.5.2 Inverse Functions 91

I.5.3 Logarithm Functions 93

I.5.4 Inverting Functions on Smaller Domains 97

I.5.5 Exercises 99

I.6 Algebraic Operations and Functions 101

I.6.1 Sums and Products of Functions 101

I.6.2 Simple Algebraic Functions 102

I.6.3 Local Boundedness of Algebraic Functions 104

I.6.4 Global Boundedness 107

I.6.5 Exercises 108

II Derivatives: How to Measure Change 111

II.1 Algebraic Derivatives by Approximation 112

II.1.1 From Factorization to Average Rates of Change 112

II.1.2 From Average to Instantaneous Rates of Change 120

II.1.3 Approximation of Algebraic Derivatives 122

II.1.4 Exercises 126

II.2 Derivatives of Exponential Functions 127

II.2.1 Tangents for y = 2^x 128

II.2.2 The Tangent to y = 2^x at x = 0 130

II.2.3 Other Exponential Functions 136

II.2.4 The "Natural" Exponential Function 137

II.2.5 The Natural Logarithm 139

II.2.6 The Derivative of ln x 140

II.2.7 The Differential Equation of Exponential Functions 141

II.2.8 Exercises 143

II.3 Differentiability and Local Linear Approximation 144

II.3.1 Limits 144

II.3.2 Continuous Functions 147

II.3.3 Differentiable Functions 150

II.3.4 Local Linear Approximation 152

II.3.5 Exercises 157

II.4 Properties of Continuous Functions 158

II.4.1 Rules for Limits 158

II.4.2 Rules for Continuous Functions 160

II.4.3 The Intermediate Value Theorem 161

II.4.4 Continuity and Boundedness 162

II.4.5 Exercises 164

II.5 Derivatives of Trigonometric Functions 165

II.5.1 Continuity of sine and cosine Functions 165

II.5.2 The Derivative of sine at t = 0 166

II.5.3 The Derivative of sin t 169

II.5.4 The cosine Function 172

II.5.5 A Differential Equation for sine and cosine 172

II.5.6 Exercises 173

II.6 Simple Differentiation Rules 175

II.6.1 Linearity 175

II.6.2 Chain Rule 175

II.6.3 Power Functions with Real Exponents 177

II.6.4 Inverse Functions 178

II.6.5 Inverse Trigonometric Functions 180

II.6.6 Exercises 182

II.7 Product and Quotient Rules 184

II.7.1 Statement of the Rules 184

II.7.2 Examples 185

II.7.3 Exercises 187

III Some Applications of Derivatives 189

III.1 Exponential Models 189

III.1.1 Growth and Decay Models 189

III.1.2 Radiocarbon Dating 190

III.1.3 Compound Interest 191

III.1.4 Exercises 193

III.2 The Inverse Problem and Antiderivatives 195

III.2.1 Functions with Zero Derivative 195

III.2.2 The Mean Value Inequality 197

III.2.3 Antiderivatives 198

III.2.4 Solutions of y' = ky 199

III.2.5 Initial Value Problems 200

III.2.6 Exercises 201

III.3 "Explosive Growth" Models 202

III.3.1 Beyond Exponential Growth 202

III.3.2 An Explicit Solution of y' = y² 203

III.3.3 Exercises 205

III.4 Acceleration and Motion with Constant Acceleration 206

III.4.1 Acceleration 206

III.4.2 Free Fall 207

III.4.3 Constant Deceleration 208

III.4.4 Exercises 209

III.5 Periodic Motions 210

III.5.1 A Model for a Bouncing Spring 210

III.5.2 The Solutions of y" + w²y = 0 213

III.5.3 The Motion of a Pendulum 214

III.5.4 Exercises 218

III.6 Geometric Properties of Graphs 218

III.6.1 Increasing and Decreasing Functions 218

III.6.2 Relationship with Derivatives 219

III.6.3 Local Extrema 221

III.6.4 Convexity 223

III.6.5 Points of Inflection 224

III.6.6 Graphing with Derivatives 226

III.6.7 Exercises 229

III.7 An Algorithm for Solving Equations 231

III.7.1 Newton's Method 231

III.7.2 Numerical Examples 233

III.7.3 "Chaotic" Behavior 235

III.4 Exercises 236

III.8 Applications to Optimization 237

III.8.1 Basic Principles 237

III.8.2 Some Applications 239

III.8.3 Exercises 241

III.9 Higher Order Approximations and Taylor Polynomials 242

III.9.1 Quadratic Approximations 242

III.9.2 Higher Order Taylor Polynomials 244

III.9.3 Taylor Approximations for the sine Function 246

III.9.4 The Natural Exponential Function 247

III.9.5 Exercises 250

IV The Definite Integral 251

IV.1 The Inverse Problem: Construction of Antiderivatives 251

IV.1.1 Antiderivatives and New Functions 251

IV.1.2 Finding Distance from Velocity 252

IV.1.3 Control of the Approximation 254

IV.1.4 A Geometric Construction of Antiderivatives 257

IV.1.5 A Simple Example 262

IV.1.6 The Definite Integral 263

IV.1.7 Exercises 265

IV.2 The Area Problem 267

IV.2.1 Approximation by Sums of Areas of Rectangles 267

IV.2.2 Rectangles and Triangles 268

IV.2.3 Area under a Parabola 271

IV.2.4 Area of a Disc 272

IV.2.5 Exercises 275

IV.3 More Applications of Definite Integrals 276

IV.3.1 Riemann Sums 276

IV.3.2 Areas Bounded by Graphs 277

IV.3.3 Volume of a Sphere 279

IV.3.4 Work of a Spring 280

IV.3.5 Length of a Curve 282

IV.3.6 Income Streams 283

IV.3.7 Probability Distributions 285

IV.3.8 Exercises 287

IV.4 Properties of Definite Integrals 288

IV.4.1 Riemann Integrable Functions 288

IV.4.2 Basic Rules for Integrals 291

IV.4.3 Examples 292

IV.4.4 Exercises 293

IV.5 The Fundamental Theorem of Calculus 294

IV.5.1 The Derivative of an Integral 294

IV.5.2 The Integral of a Derivative 296

IV.5.3 Some Examples 297

IV.5.4 Exercises 298

IV.6 Existence of Definite Integrals 299

IV.6.1 Monotonic Functions 299

IV.6.2 Functions with Bounded Derivatives 301

IV.6.3 Exercises 303

IV.7 Reversing the Chain Rule: Substitution 303

IV.7.1 Integrals that Fit the Chain Rule 303

IV.7.2 Examples 304

IV.7.3 Changing Integrals by Substitution 305

IV.7.4 Exercises 307

IV.8 Reversing the Product Rule: Integration by Parts 308

IV.8.1 Partial Integration 308

IV.8.2 Some Other Examples 310

IV.8.3 Partial Integration of Differentials 311

IV.8.4 Remarks on Integration Techniques 311

IV.8.5 A Word of Caution 312

IV.8.6 Exercises 313

IV.9 Higher Order Approximations, Part 2: Taylor's Theorem 313

IV.9.1 An Application of Integration by Parts 314

IV.9.2 Taylor Series of Elementary Transcendental Functions 316

IV.9.3 Power Series 319

IV.9.4 Analytic Functions 322

IV.9.5 Exercises 325

IV.10 Excursion into Complex Numbers and the Euler Identity 326

IV.10.1 Complex Numbers 327

IV.10.2 The Exponential Function for Complex Numbers 329

IV.10.3 The Euler Identity 331

IV.10.4 Exercises 333

Epilogue 335

Index 337