Calculus and its Origins

(Spectrum)
David Perkins

The Mathematical Association of America | 2012 | 180 páginas | pdf | 3,7 Mb

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Calculus answers questions that had been explored for centuries before calculus was born. Calculus and Its Origins begins with these ancient questions and details the remarkable story of how subsequent scholars wove these inquiries into a unified theory. This book does not presuppose knowledge of calculus, it requires only a basic knowledge of geometry and algebra (similar triangles, polynomials, factoring). Inside you will find the accounts of how Archimedes discovered the area of a parabolic segment, ibn Al-Haytham calculated the volume of a revolved area, Jyesthadeva explained the infinite series for sine and cosine, Wallis deduced the link between hyperbolas and logarithms, Newton generalized the binomial theorem, Leibniz discovered integration by parts, and much more. Each chapter ends with further results, in the form of exercises, by such luminaries as Pascal, Maclaurin, Barrow, Cauchy and Euler.

Contents
Preface ix
1 The Ancients 1
1.1 Zeno holds a mirror to the infinite . . 2
1.2 The ‘infinitely small’  . . 4
1.3 Archimedes exhausts a parabolic segment . . 5
1.4 Patterns . . 7
1.5 The evolution of notation  . . 10
1.6 Furthermore  . . 10
2 East of Greece 15
2.1 Ibn al-Haytham sums the fourth powers . . 15
2.2 Ibn al-Haytham’s parabolic volume . . 17
2.3 Jyesthadeva expands 1/(1+x)  . . 20
2.4 Jyesthadeva expresses as a series . . 23
2.5 Furthermore. . 26
3 Curves 29
3.1 Oresme invents a precursor to a coordinate system . . 29
3.2 Fermat studies the maximums of curves. . 32
3.3 Fermat extends his method to tangent lines . . 33
3.4 Descartes proposes a geometric method . . 35
3.5 Furthermore . . 37
4 Indivisibles 43
4.1 Cavalieri’s quadrature of the parabola . . 43
4.2 Roberval’s quadrature of the cycloid  . . 47
4.3 Worry over indivisibles. . 50
4.4 Furthermore. . 51
5 Quadrature 57
5.1 Gregory studies hyperbolas . . 57
5.2 De Sarasa invokes logarithms . . 59
5.3 Brouncker finds a quadrature of a hyperbola. . 61
5.4 Mercator andWallis finish the task . . 63
5.5 Furthermore  . . 65
6 The Fundamental Theorem of Calculus 77
6.1 Newton links quadrature to rate of change  . . 77
6.2 Newton reverses the link  . . 79
6.3 Leibniz discovers the transmutation theorem  . . 81
6.4 Leibniz attains Jyesthadeva’s series for  pi. . 83
6.5 Furthermore  . . 85
7 Notation 95
7.1 Leibniz describes differentials. . 95
7.2 The fundamental theorem with new notation  . . 98
7.3 Leibniz integrates the cycloid . . 100
7.4 Furthermore . . 102
8 Chords 109
8.1 Preliminary results known to the Greeks. . 109
8.2 Jyesthadeva finds series for sine and cosine. . 110
8.3 Newton derives a series for arcsine . . 116
8.4 Furthermore . . 118
9 Zero over zero 123
9.1 D’Alembert and the convergence of series  . . 123
9.2 Lagrange defines the ‘derived function’. . 126
9.3 Taylor approximates functions. . 128
9.4 Bolzano and Cauchy define convergence  . . 130
9.5 Furthermore . . 133
10 Rigor 137
10.1 Cauchy defines continuity . . 137
10.2 Bolzano invents a peculiar function . . 140
10.3 Weierstrass investigates the convergence of functions. . 145
10.4 Dirichlet’s nowhere-continuous function . . 147
10.5 A few final words about the infinite. . 149
10.6 Furthermore . . 150

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