Foundations, Methods, and Algorithms: Computer Science Analysis

Table of contents :
Numbers................................................1
The Real Numbers......................................1
Order Relation and Arithmetic on R...........................5
Machine Numbers......................................8
Rounding...........................................10
Exercises...........................................11
Real-Valued Functions......................................13
Basic Notions........................................13
Some Elementary Functions...............................17
Exercises...........................................23
Trigonometry............................................27
Trigonometric Functions at the Triangle.......................27
Extension of the Trigonometric Functions to R..................31
Cyclometric Functions..................................33
Exercises...........................................35
Complex Numbers........................................39
The Notion of Complex Numbers...........................39
The Complex Exponential Function..........................42
Mapping Properties of Complex Functions.....................44
Exercises...........................................46
Sequences and Series.......................................49
The Notion of an Infinite Sequence..........................49
The Completeness of the Set of Real Numbers...................55
Infinite Series........................................58
Supplement: Accumulation Points of Sequences..................62
Exercises...........................................65
Limits and Continuity of Functions..............................69
The Notion of Continuity.................................69
Trigonometric Limits...................................74

ix

Zeros of Continuous Functions.............................75
Exercises...........................................78

The Derivative of a Function..................................81
Motivation..........................................81
The Derivative........................................83
Interpretations of the Derivative............................87
Differentiation Rules....................................90
Numerical Differentiation................................96
Exercises..........................................101
Applications of the Derivative................................105
Curve Sketching......................................105
Newton’s Method.....................................110
Regression Line Through the Origin.........................115
Exercises..........................................118
Fractals and L-systems.....................................123
Fractals............................................124
Mandelbrot Sets......................................130
Julia Sets...........................................131
Newton’s Method in C..................................132
L-systems..........................................134
Exercises..........................................138
Antiderivatives..........................................139
Indefinite Integrals....................................139
Integration Formulas...................................142
Exercises..........................................146
Definite Integrals.........................................149
The Riemann Integral..................................149
Fundamental Theorems of Calculus.........................155
Applications of the Definite Integral.........................158
Exercises..........................................161
Taylor Series...........................................165
Taylor’s Formula.....................................165
Taylor’s Theorem.....................................169
Applications of Taylor’s Formula..........................170
Exercises..........................................173
Numerical Integration.....................................175
Quadrature Formulas...................................175
Accuracy and Efficiency................................180
Exercises..........................................182
Curves...............................................185
Parametrised Curves in the Plane...........................185
Arc Length and Curvature...............................193
Plane Curves in Polar Coordinates..........................200
Parametrised Space Curves...............................202
Exercises..........................................204
Scalar-Valued Functions of Two Variables........................209
Graph and Partial Mappings..............................209
Continuity..........................................211
Partial Derivatives....................................212
The Fréchet Derivative.................................216
Directional Derivative and Gradient.........................221
The Taylor Formula in Two Variables.......................223
Local Maxima and Minima...............................224
Exercises..........................................228
Vector-Valued Functions of Two Variables.......................231
Vector Fields and the Jacobian............................231
Newton’s Method in Two Variables.........................233
Parametric Surfaces...................................236
Exercises..........................................238
Integration of Functions of Two Variables........................241
Double Integrals......................................241
Applications of the Double Integral.........................247
The Transformation Formula.............................249
Exercises..........................................253
Linear Regression........................................255
Simple Linear Regression................................255
Rudiments of the Analysis of Variance.......................261
Multiple Linear Regression...............................265
Model Fitting and Variable Selection........................267
Exercises..........................................271
Differential Equations.....................................275
Initial Value Problems..................................275
First-Order Linear Differential Equations.....................278
Existence and Uniqueness of the Solution.....................283
Method of Power Series.................................286
Qualitative Theory....................................288
Second-Order Problems.................................290
Exercises..........................................294
Systems of Differential Equations..............................297
Systems of Linear Differential Equations.....................297
Systems of Nonlinear Differential Equations...................308
The Pendulum Equation.................................312
Exercises..........................................317
Numerical Solution of Differential Equations......................321
The Explicit Euler Method...............................321
Stability and Stiff Problems..............................324
Systems of Differential Equations..........................327
Exercises..........................................328

Appendix A: Vector Algebra.................................331

Appendix B: Matrices.....................................343

Appendix C: Further Results on Continuity.......................353

Appendix D: Description of the Supplementary Software..............365

References.............................................367

Index................................................369

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