Table of Contents:
| Preface | i | |||
| 1 | Complete Mathematical Induction | 1 | ||
| 1.1 | The Principle of Complete Induction | 1 | ||
| 1.2 | Bernoulli's Inequality | 3 | ||
| 1.3 | Binomial Coefficients | 4 | ||
| 1.4 | Combinatorial Significance of Binomial Coefficients | 6 | ||
| 1.5 | The Sum Sign | 7 | ||
| 1.6 | Binomial Theorem | 8 | ||
| 2 | Inequalities and Absolute Value | 11 | ||
| 2.1 | The Absolute Value | 12 | ||
| 2.2 | The Concept of Monotony | 16 | ||
| 2.3 | The Completeness of \(\mathbb{R}\) | 18 | ||
| 2.4 | The Monotone Convergence Criterion for Sequences | 18 | ||
| 3 | Limit Values | 21 | ||
| 3.1 | Definition of a Limit | 21 | ||
| 3.2 | Differentiation | 27 | ||
| 3.3 | Basic Rules of Differentiation | 29 | ||
| 4 | Important Functions | 33 | ||
| 4.1 | The Exponential Function in the Real | 33 | ||
| 4.2 | The Natural Logarithm | 36 | ||
| 4.3 | The General Power Function in the Real | 40 | ||
| 4.4 | Polynomials | 42 | ||
| 4.4.1 | Identity Theorem for Polynomials | 45 | ||
| 5 | Existence Theorems | 47 | ||
| 5.1 | The Nested Interval Theorem | 47 | ||
| 5.2 | The Bolzano-Weierstraß Theorem | 47 | ||
| 5.3 | The Extreme Value Theorem | 49 | ||
| 5.4 | The Intermediate Value Theorem | 50 | ||
| 5.5 | The Inverse Function | 51 | ||
| 5.6 | Local Extrema | 54 | ||
| 5.7 | Rolle's Theorem | 55 | ||
| 5.8 | The Mean Value Theorem | 55 | ||
| 6 | Applications of the Mean Value Theorem | 57 | ||
| 6.1 | Monotony, Local Extrema, Convexity | 57 | ||
| 6.1.1 | The Monotony Criterion of Differential Calculus | 57 | ||
| 6.1.2 | Criterion for Strict Local Extrema | 58 | ||
| 6.1.3 | Convexity Criterion | 59 | ||
| 6.2 | Hölder's Inequality | 60 | ||
| 6.3 | Newton's Method | 62 | ||
| 6.4 | The Extended Mean Value Theorem | 64 | ||
| 6.5 | The Rule of de l'Hospital | 65 | ||
| 6.6 | Taylor's Formula | 66 | ||
| 6.6.1 | The Exponential Series | 68 | ||
| 6.6.2 | The Logarithm Series | 69 | ||
| 6.6.3 | The Binomial Series | 71 | ||
| 7 | Oscillation Equation and Trigonometric Functions | 73 | ||
| 7.1 | Intermediate Section on Power Series | 76 | ||
| 7.1.1 | Leibniz' Convergence Criterion | 77 | ||
| 7.1.2 | Theorem on the Radius of Convergence | 78 | ||
| 7.1.3 | A Criterion for Determining the Radius of Convergence | 79 | ||
| 7.1.4 | The Derivative of a Power Series | 80 | ||
| 7.2 | Inverse Functions | 84 | ||
| 7.2.1 | Geometric Significance of the Angular Functions Sine and Cosine | 86 | ||
| 8 | The Euclidean Plane and its Affine Mappings | 89 | ||
| 8.1 | Rotations Around the Origin | 91 | ||
| 8.2 | Basic Geometric Sine and Cosine Laws | 92 | ||
| 8.3 | The Scalar Product | 93 | ||
| 8.4 | Hessian Normal Form of a Straight Line | 96 | ||
| 9 | Complex Numbers | 101 | ||
| 9.1 | The Cauchy Criterion Applies in \(\mathbb{C}\) | 106 | ||
| 9.1.1 | Moivre's Theorem | 107 | ||
| 9.2 | The Fundamental Theorem of Algebra in \(\mathbb{C}\) | 108 | ||
| 9.3 | Partial Fraction Decomposition | 111 | ||