Difference between revisions of "Calculus (Book)"

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Calculus (Book) (view source)

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| <nowiki>*</nowiki>I 1.4 || Exercises || 8
| <nowiki>*</nowiki>I 1.4 || Exercises || 8
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| I 1.5 || Rational numbers || 8
| I 1.5 || A critical analysis of the Archimedes' method || 8
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| I 1.6 || Multiplicative inverses || 10
| I 1.6 || The approach to calculus to be used in this book || 10
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! colspan="3" | Chapter 2: Linear Equations
! colspan="3" | Part 2: Some Basic Concepts of the Theory of Sets
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| 1 || Equations in two unknowns || 53
| I 2.1 || Introduction to set theory || 11
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| 2 || Equations in three unknowns || 57
| I 2.2 || Notations for designating sets || 12
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! colspan="3" | Chapter 3: Real Numbers
| I 2.3 || Subsets || 12
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| 1 || Addition and multiplication || 61
| I 2.4 || Unions, intersections, complements || 13
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| 2 || Real numbers: positivity || 64
| I 2.5 || Exercises || 15
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| 3 || Powers and roots || 70
! colspan="3" | Part 3: A set of Axioms for the Real-Number System
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| 4 || Inequalities || 75
| I 3.1 || Introduction || 17
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! colspan="3" | Chapter 4: Quadratic Equations
| I 3.2 || The field axioms || 17
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| <nowiki>*</nowiki>I 3.3 || Exercises || 19
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| I 3.4 || The order axioms || 19
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| <nowiki>*</nowiki>I 3.5 || Exercises || 21
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| I 3.6 || Integers and rational numbers || 21
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| I 3.7 || Geometric interpretation of real numbers as points on a line || 22
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| I 3.8 || Upper bound of a set, maximum element, least upper bound (supremum) || 23
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| I 3.9 || The least-Upper-bound axiom (completeness axiom) || 25
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| I 3.10 || The Archimedean property of the real-number system || 25
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| I 3.11 || Fundamental properties of the supremum and infimum || 26
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| <nowiki>*</nowiki>I 3.12 || Exercises || 28
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| <nowiki>*</nowiki>I 3.13 || Existence of square roots of nonnegative real numbers || 29
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| <nowiki>*</nowiki>I 3.14 || Roots of higher order. Rational powers || 30
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| <nowiki>*</nowiki>I 3.15 || Representation of real numbers by decimals || 30
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! colspan="3" | Part 4: Mathematical Induction, Summation Notation, and Related Topics
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| I 4.1 || An example of a proof by mathematical induction || 32
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| I 4.2 || The principle of mathematical induction || 34
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| <nowiki>*</nowiki>I 4.3 || The well-ordering principle || 34
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| I 4.4 || Exercises || 35
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| <nowiki>*</nowiki>I 4.5 || Proof of the well-ordering principle || 37
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| I 4.6 || The summation notation || 37
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| I 4.7 || Exercises || 39
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| I 4.8 || Absolute values and the triangle inequality || 41
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| I 4.9 || Exercises || 43
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| <nowiki>*</nowiki>I 4.10 || Miscellaneous exercises involving induction || 44
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! colspan="3" | Interlude: On Logic and Mathematical Expressions
! colspan="3" | Interlude: On Logic and Mathematical Expressions
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