Calculus off the Beaten Path : a journey through its fundamental ideas / Ignacio Zalduendo
Editor: Cham : Springer, 2022Descripción: xvi, 213 p. : il. , 24cmISBN: 978-3-031-15764-6Otra clasificación: 26-01, 26A06, 40-01, 26Dxx1 The Real Numbers [1] -- The Rational Line [1] -- Density of Q [2] -- Some Basic Notions [3] -- Irrationality of x/2 [4] -- From Eudoxus to Dedekind [5] -- The Real Line [8] -- Dyadic Series—A Construction of R [11] -- The Scarcity of Q [13] -- The Completeness of R [14] -- Cardinality [16] -- Exercises [19] -- 2 Sequences and Series [21] -- Sequences [21] -- Limits of Sequences [22] -- Cantor’s Nested Intervals Theorem [24] -- Subsequences [25] -- Series [28] -- The Harmonic Series [29] -- Series of Positive Terms [30] -- Series with Positive and Negative Terms [32] -- The Riemann Series Theorem [34] -- Absolute and Unconditional Convergence [35] -- Exercises [35] -- 3 Functions [39] -- The Elementary Functions [39] -- Polynomials [40] -- Circular Functions [40] -- The Exponential Function: Bernoulli’s Inequality [43] -- Irrationality of e [49] -- Convergence of [50] -- Hyperbolic Functions [51] -- Injectivity and Inverse Functions [52] -- Curves in the Plane: Parametrized Curves [55] -- The Cycloid [55] -- Pythagorean Triples [57] -- Continuity [59] -- Bolzano and Weierstrass [60] -- Limits [62] -- Limits in Ancient Greece: The Area of a Circle [62] -- Three Important Limits [65] -- Exercises [67] -- 4 The Derivative [71] -- Derivative [71] -- Tangents [72] -- Newton-Raphson [75] -- Derivatives of the Elementary Functions [77] -- The Chain Rule [79] -- Derivative of the Inverse Function [80] -- The Derivative of a Parametrized Curve [82] -- First Derivative, Tangent Line, and Growth [84] -- The Mean Value Theorems [84] -- L’Hopital’s Rule [87] -- Snell’s Law [89] -- The Brachistochrone [92] -- Exercises [94] -- 5 The Integral [99] -- Measure and Integral [99] -- The Fundamental Theorem of Calculus [105] -- A Pause for Comments [108] -- Buffon’s Needle [111] -- Irrationality of it [113] -- Improper Integrals [114] -- Integration and Sums: Linearity of the Integral [118] -- Uniform Convergence—The Weierstrass [118] -- Gregory’s Series [123] -- Integration and Products: Integration by Parts [124] -- Stirling’s Formula [125] -- Integration and Composition: Integration by Substitution [126] -- A Note on Notation [127] -- Length of Curves. The Catenary [128] -- Area Enclosed by a Simple Closed Curve [131] -- Exercises [133] -- 6 More Derivatives [137] -- Second Derivative, Best-Fitting Parabola, and Curvature -- The Thylor Polynomial of Order Two [138] -- The Thylor Series [147] -- Exercises [153] -- 7 Convexity and the Isoperimetric Inequality [155] -- The Arithmetic-Geometric Inequality [155] -- Convexity [156] -- Young, Holder, Jensen, Cauchy-Schwarz [159] -- The Isoperimetric Inequality [164] -- Exercises [168] -- 8 More Integrals [171] -- Volume [171] -- Double Integrals [174] -- The Basel Problem [175] -- Solids of Revolution [178] -- Density Functions, Barycenter, and Expectation [183] -- Center of Mass or Barycenter [184] -- Pappus’ Theorem [188] -- The Method [190] -- Surface Area [192] -- Normal Distribution. Gauss, Laplace, and Stirling [194] -- Exercises [198] -- 9 The Gamma Function [201] -- The Gamma Function [201] -- Weierstrass’ Formula [202] -- Growth of the Harmonic Series, Again [205] -- Exercises [206] -- Bibliography [207] -- Index [209] --
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Instituto de Matemática, CONICET-UNS | Últimas adquisiciones | 26 Z22 (Browse shelf) | Available | A-9422 |
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1 The Real Numbers [1] --
The Rational Line [1] --
Density of Q [2] --
Some Basic Notions [3] --
Irrationality of x/2 [4] --
From Eudoxus to Dedekind [5] --
The Real Line [8] --
Dyadic Series—A Construction of R [11] --
The Scarcity of Q [13] --
The Completeness of R [14] --
Cardinality [16] --
Exercises [19] --
2 Sequences and Series [21] --
Sequences [21] --
Limits of Sequences [22] --
Cantor’s Nested Intervals Theorem [24] --
Subsequences [25] --
Series [28] --
The Harmonic Series [29] --
Series of Positive Terms [30] --
Series with Positive and Negative Terms [32] --
The Riemann Series Theorem [34] --
Absolute and Unconditional Convergence [35] --
Exercises [35] --
3 Functions [39] --
The Elementary Functions [39] --
Polynomials [40] --
Circular Functions [40] --
The Exponential Function: Bernoulli’s Inequality [43] --
Irrationality of e [49] --
Convergence of [50] --
Hyperbolic Functions [51] --
Injectivity and Inverse Functions [52] --
Curves in the Plane: Parametrized Curves [55] --
The Cycloid [55] --
Pythagorean Triples [57] --
Continuity [59] --
Bolzano and Weierstrass [60] --
Limits [62] --
Limits in Ancient Greece: The Area of a Circle [62] --
Three Important Limits [65] --
Exercises [67] --
4 The Derivative [71] --
Derivative [71] --
Tangents [72] --
Newton-Raphson [75] --
Derivatives of the Elementary Functions [77] --
The Chain Rule [79] --
Derivative of the Inverse Function [80] --
The Derivative of a Parametrized Curve [82] --
First Derivative, Tangent Line, and Growth [84] --
The Mean Value Theorems [84] --
L’Hopital’s Rule [87] --
Snell’s Law [89] --
The Brachistochrone [92] --
Exercises [94] --
5 The Integral [99] --
Measure and Integral [99] --
The Fundamental Theorem of Calculus [105] --
A Pause for Comments [108] --
Buffon’s Needle [111] --
Irrationality of it [113] --
Improper Integrals [114] --
Integration and Sums: Linearity of the Integral [118] --
Uniform Convergence—The Weierstrass [118] --
Gregory’s Series [123] --
Integration and Products: Integration by Parts [124] --
Stirling’s Formula [125] --
Integration and Composition: Integration by Substitution [126] --
A Note on Notation [127] --
Length of Curves. The Catenary [128] --
Area Enclosed by a Simple Closed Curve [131] --
Exercises [133] --
6 More Derivatives [137] --
Second Derivative, Best-Fitting Parabola, and Curvature --
The Thylor Polynomial of Order Two [138] --
The Thylor Series [147] --
Exercises [153] --
7 Convexity and the Isoperimetric Inequality [155] --
The Arithmetic-Geometric Inequality [155] --
Convexity [156] --
Young, Holder, Jensen, Cauchy-Schwarz [159] --
The Isoperimetric Inequality [164] --
Exercises [168] --
8 More Integrals [171] --
Volume [171] --
Double Integrals [174] --
The Basel Problem [175] --
Solids of Revolution [178] --
Density Functions, Barycenter, and Expectation [183] --
Center of Mass or Barycenter [184] --
Pappus’ Theorem [188] --
The Method [190] --
Surface Area [192] --
Normal Distribution. Gauss, Laplace, and Stirling [194] --
Exercises [198] --
9 The Gamma Function [201] --
The Gamma Function [201] --
Weierstrass’ Formula [202] --
Growth of the Harmonic Series, Again [205] --
Exercises [206] --
Bibliography [207] --
Index [209] --
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