| Preface | p. xi |
| Acknowledgments | p. xiii |
| Introduction | p. 1 |
| What Is Computable Calculus? | p. 1 |
| What Are the Advantages of Computable Calculus, If Any? | p. 3 |
| A Brief Description of Computable Calculus | p. 4 |
| The Real Numbers | p. 7 |
| Definition of a Real Number | p. 7 |
| An Ordering of Intervals | p. 10 |
| Interval Arithmetic | p. 14 |
| The Real Number Operations +, -, x, [divide] | p. 15 |
| The Absolute Value of a Real Number | p. 18 |
| Solvable Problems and Nonsolvable Problems | p. 21 |
| Introduction | p. 21 |
| Turing's Resolution of the Halting Problem | p. 22 |
| A Certain Computation Problem | p. 24 |
| Deciding Whether a Number Is Zero | p. 26 |
| A List of Nonsolvable Problems | p. 27 |
| Solvable Problems | p. 28 |
| Key Nonsolvable Problems | p. 30 |
| Deciding Whether a Number Is Rational or Irrational | p. 31 |
| Deciding Which of Two Real Numbers Is Larger | p. 32 |
| Sequences and Functions | p. 35 |
| Sequences of Real Numbers | p. 35 |
| The Cantor Counting Theory | p. 36 |
| Functions | p. 37 |
| The nth Root of a Real Number | p. 40 |
| An Algebra of Functions | p. 41 |
| The Function sgn(x) | p. 42 |
| Functions Defined on Intervals | p. 44 |
| Semifunctions | p. 46 |
| Other Calculus Concepts | p. 47 |
| The Ideal Computer | p. 51 |
| The Goal of This Chapter | p. 51 |
| The Various Methods of Proof | p. 52 |
| Definition of the Ideal Computer | p. 54 |
| The Ideal Computer Steps | p. 55 |
| Viewing, Compiling, and Executing Ideal Computer Programs | p. 59 |
| More Ideal Computer Details | p. 62 |
| Programming the Ideal Computer to Add Natural Integers | p. 64 |
| The Addition of Two General Integers | p. 68 |
| The Subtraction, Multiplication, and Division of Integers | p. 70 |
| Rational Number Arithmetic | p. 71 |
| Interval Arithmetic | p. 73 |
| The Retrieval of a Program from Disk Memory | p. 74 |
| The Real Number [pi] | p. 76 |
| Changing an Approximation Algorithm | p. 77 |
| Rational Numbers Converted to Real Numbers | p. 79 |
| nth Roots of Real Numbers | p. 79 |
| Arithmetic Operations on Real Numbers | p. 80 |
| Rational Sequence Arithmetic | p. 81 |
| Function Programs | p. 82 |
| The Exponential Function e[superscript x] | p. 83 |
| Two Semifunction Examples | p. 84 |
| The Beginning of the Nonsolvability Proof for Problem 3.1 | p. 86 |
| Programs That Decide Whether a Real Number Is Zero | p. 88 |
| Limits | p. 93 |
| Limit of a Sequence | p. 93 |
| Monotone Sequences | p. 95 |
| The Specker Theorem | p. 97 |
| Consequences of Specker's Theorem | p. 99 |
| Limit of a Function | p. 99 |
| Using Limits to Extend a Function's Domain | p. 105 |
| A Standard Result for Functions | p. 107 |
| Unbounded Continuous Functions on [a, b] | p. 110 |
| Limits of Sequences of Functions | p. 112 |
| Limits of Functions with More Than One Variable | p. 114 |
| Uniformly Continuous Functions | p. 119 |
| Introduction | p. 119 |
| Bounds of Uniformly Continuous Functions | p. 124 |
| The Derivative | p. 131 |
| A Difficulty with the Derivative Definition | p. 131 |
| Rules of Differentiation | p. 135 |
| A Computation Problem | p. 137 |
| The Mean Value Theorem | p. 138 |
| The Riemann Integral | p. 143 |
| Riemann Sums | p. 143 |
| The Integration of Uniformly Continuous Functions | p. 144 |
| Properties of Integrals | p. 146 |
| Defining a Function by Means of an Integral | p. 148 |
| A Mean Value Theorem for Integrals | p. 150 |
| Functions of Two Variables | p. 151 |
| Partial Derivatives | p. 151 |
| The Chain Rule | p. 154 |
| Equality of Cross Derivatives | p. 157 |
| The Differential Equation y' = f(x,y) | p. 163 |
| Introduction | p. 163 |
| The Lipschitz Condition | p. 165 |
| The Possibility of No Solution | p. 170 |
| Ideal Computer Simulation | p. 175 |
| The Extended Ideal Computer | p. 175 |
| Call Loops | p. 176 |
| Composing and Deleting a Program | p. 178 |
| Managing the Ideal Computer's Input and Output | p. 180 |
| Possible Projects | p. 182 |
| References | p. 185 |
| About the CD... | p. 189 |
| Index | p. 191 |
| Table of Contents provided by Rittenhouse. All Rights Reserved. |
![Computable Calculus with CDROM [With CDROM] - Oliver Aberth](https://www.booktopia.com.au/covers/big/9780120417520/0000/computable-calculus-with-cdrom-with-cdrom-.jpg)
