Elementos de Cálculo Diferencial
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Contenido Prefacio ................................................................................................................................................ 8
Contenido .......................................................................................................................................... 10
Primera Parte: Historia del Cálculo.................................................................................................... 22
Capítulo 1 : Entre inconmensurables y paradojas ............................................................................. 23
1.1 Thales y Pitágoras .................................................................................................................... 24
1.2 Continuidad, infinito e inconmesurables ................................................................................ 28
1.3 Zenón y las paradojas .............................................................................................................. 35
1.4 Ejercicios .................................................................................................................................. 38
Capítulo 2: Cálculo de áreas en la Grecia Antigua ............................................................................. 41
2.1 Eudoxo yArquímedes ............................................................................................................... 42
2.2 El método de Exhausción......................................................................................................... 43
2.3 Ejercicios .................................................................................................................................. 52
Capítulo 3: Medievo, ciencias y matemáticas ................................................................................... 55
3.1 De los romanos a la Escolástica. .............................................................................................. 56
3.2 Los árabes y las matemáticas .................................................................................................. 60
3.3 Técnicas, ideología y sociedad ................................................................................................. 64
3.4 Ejercicios .................................................................................................................................. 65
Capítulo 4 :El renacimiento : Un nuevo punto de partida ................................................................. 67
4.1 Renacimiento y humanismo .................................................................................................... 68
4.2 Navegación y ciencias matemáticas ........................................................................................ 71
4.3 Un poco de matemáticas ......................................................................................................... 72
4.4 Ejercicios .................................................................................................................................. 76
Capítulo 5: Revolución en la cosmología ........................................................................................... 77
5.1 Copérnico: el heliocentrismo ................................................................................................... 79
5.2 Kepler : la geometría celeste ................................................................................................... 82
5.3 Galileo y la nueva cosmología ................................................................................................. 85
5.4 Ejercicios .................................................................................................................................. 92
Capítulo 6: Los métodos de la nueva ciencia .................................................................................... 95
6.1 Galileo y la nueva ciencia......................................................................................................... 97
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6.2 Otros profetas de la nueva ciencia: Bacon y Descartes ......................................................... 102
6.3 Las sociedades científicas ...................................................................................................... 107
6.4 Ejercicios ................................................................................................................................ 109
Capítulo 7: Problemas para las matemáticas del siglo XVII ............................................................. 111
7.1 Los límites de la matemática antigua .................................................................................... 113
7.2 Cuatro problemas fundamentales ......................................................................................... 115
7.3 Las matemáticas del siglo XVII ............................................................................................... 118
7.4 Ejercicios ................................................................................................................................ 120
Capítulo 8: Geometría analítica y cálculo de tangentes .................................................................. 121
8.1 La geometría analítica ........................................................................................................... 122
8.2 Cálculo de tangentes, máximos y mínimos ........................................................................... 131
8.3 Ejercicios ................................................................................................................................ 139
Capítulo 9: Longitudes, áreas y volúmenes ..................................................................................... 141
9.1 Los métodos de Kepler .......................................................................................................... 142
9.2 Galileo y los conjuntos infinitos ............................................................................................. 146
9.3 El área bajo una curva ........................................................................................................... 148
9.4 Ejercicios ................................................................................................................................ 153
Capítulo 10 : Newton ....................................................................................................................... 154
10.1 La obra científica de Newton ............................................................................................... 156
10.2 El cálculo .............................................................................................................................. 158
10.3 Cálculo, series infinitas y publicaciones............................................................................... 165
10.4 Ejercicios .......................................................................................................................... 168
Capítulo 11: Leibniz ......................................................................................................................... 170
11.2 El cálculo de Leibniz ............................................................................................................. 172
11.3 Diferencias entre Newton y Leibniz ..................................................................................... 176
11.4 Ejercicios .............................................................................................................................. 178
Capítulo 12: Ciencia y sociedad en el siglo XVII ............................................................................... 180
12.1 La revolución industrial ....................................................................................................... 181
12.2 La química ............................................................................................................................ 183
12.3 Ejercicios .............................................................................................................................. 186
Capítulo 13: El análisis y las matemáticas del siglo XVII .................................................................. 188
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13.1 El carácter de las matemáticas ............................................................................................ 189
13.2 La época de Euler ................................................................................................................. 190
13.3 La Edad Heroica ................................................................................................................... 197
13.4 Ejercicios .............................................................................................................................. 203
Capítulo 14: Panorámica de las matemáticas del siglo XIX ............................................................. 205
14.1 La preocupación por el rigor ................................................................................................ 206
14.2 Gauss y las geometrías ........................................................................................................ 210
14.3 La teoría de grupos y el álgebra moderna ........................................................................... 214
14.4 Ejercicios .............................................................................................................................. 216
Capítulo 15: La aritmetización del análisis ...................................................................................... 218
15.1 el rigor a través del límite .................................................................................................... 220
15.2 Weierstrass .......................................................................................................................... 224
15.3 La construcción de los números reales ............................................................................... 228
15.4 Un balance ........................................................................................................................... 231
15.5 Ejercicios .............................................................................................................................. 233
Capítulo 16: El análisis no – standard y la naturaleza de las matemáticas ..................................... 235
16.1 Las matemáticas del siglo XIX : un balance general ............................................................ 236
16.2 Los infinitesimales y el análisis No -Standard ...................................................................... 237
16.3 Una síntesis final .................................................................................................................. 244
16.4 Ejercicios .............................................................................................................................. 246
Segunda Parte: Ejercicios Resueltos ............................................................................................ 248
Presentación ................................................................................................................................ 249
Capítulo 1: Las razones de cambio y la derivada ............................................................................. 251
Capítulo 2: Límites ........................................................................................................................... 259
Capítulo 3: Límites laterales y continuidad ..................................................................................... 269
Capítulo 4: Límites infinitos y al infinito .......................................................................................... 276
Capítulo 5: La derivada .................................................................................................................... 284
Capítulo 6 : Las funciones trigonométricas y el cálculo ................................................................... 298
Capítulo 7: Las funciones logarítmicas y exponenciales y el cálculo ............................................... 306
Capítulo 8: Algunas aplicaciones ..................................................................................................... 312
Capítulo 9: Temas adicionales: una introducción ............................................................................ 319
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Capítulo 10: Definiciones y métodos formales ................................................................................ 325
Bibliografía general .......................................................................................................................... 329
Indice de recuadros teóricos ........................................................................................................... 333
Indice analítico ................................................................................................................................. 334
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Prefacio
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Contenido
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Primera Parte: Historia del Cálculo
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Capítulo 1 : Entre inconmensurables y paradojas
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1.1 Thales y Pitágoras
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1.2 Continuidad, infinito e inconmesurables
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1.3 Zenón y las paradojas
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1.4 Ejercicios
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Capítulo 2: Cálculo de áreas en la Grecia Antigua
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2.1 Eudoxo yArquímedes
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2.2 El método de Exhausción
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2.3 Ejercicios
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Capítulo 3: Medievo, ciencias y matemáticas
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3.1 De los romanos a la Escolástica.
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3.2 Los árabes y las matemáticas
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3.3 Técnicas, ideología y sociedad
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3.4 Ejercicios
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Capítulo 4 :El renacimiento : Un nuevo punto de partida
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4.1 Renacimiento y humanismo
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4.2 Navegación y ciencias matemáticas
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4.3 Un poco de matemáticas
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4.4 Ejercicios
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Capítulo 5: Revolución en la cosmología
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5.1 Copérnico: el heliocentrismo
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5.2 Kepler : la geometría celeste
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5.3 Galileo y la nueva cosmología
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5.4 Ejercicios
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Capítulo 6: Los métodos de la nueva ciencia
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6.1 Galileo y la nueva ciencia
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6.2 Otros profetas de la nueva ciencia: Bacon y Descartes
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6.3 Las sociedades científicas
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6.4 Ejercicios
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Capítulo 7: Problemas para las matemáticas del siglo XVII
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7.1 Los límites de la matemática antigua
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7.2 Cuatro problemas fundamentales
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7.3 Las matemáticas del siglo XVII
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7.4 Ejercicios
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Capítulo 8: Geometría analítica y cálculo de tangentes
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8.1 La geometría analítica
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8.2 Cálculo de tangentes, máximos y mínimos
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8.3 Ejercicios
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Capítulo 9: Longitudes, áreas y volúmenes
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9.1 Los métodos de Kepler
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9.2 Galileo y los conjuntos infinitos
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9.3 El área bajo una curva
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9.4 Ejercicios
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Capítulo 10 : Newton
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10.1 La obra científica de Newton
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10.2 El cálculo
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10.3 Cálculo, series infinitas y publicaciones
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10.4 Ejercicios
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Capítulo 11: Leibniz
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11.1 Una mente universal
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11.2 El cálculo de Leibniz
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11.3 Diferencias entre Newton y Leibniz
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11.4 Ejercicios
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Capítulo 12: Ciencia y sociedad en el siglo XVII
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12.1 La revolución industrial
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12.2 La química
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12.3 Ejercicios
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Capítulo 13: El análisis y las matemáticas del siglo XVII
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13.1 El carácter de las matemáticas
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13.2 La época de Euler
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13.3 La Edad Heroica
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13.4 Ejercicios
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Capítulo 14: Panorámica de las matemáticas del siglo XIX
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14.1 La preocupación por el rigor
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14.2 Gauss y las geometrías
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14.3 La teoría de grupos y el álgebra moderna
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14.4 Ejercicios
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Capítulo 15: La aritmetización del análisis
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15.1 el rigor a través del límite
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15.2 Weierstrass
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15.3 La construcción de los números reales
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15.4 Un balance
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15.5 Ejercicios
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Capítulo 16: El análisis no – standard y la naturaleza de las matemáticas
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16.1 Las matemáticas del siglo XIX : un balance general
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16.2 Los infinitesimales y el análisis No -Standard
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16.3 Una síntesis final
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16.4 Ejercicios
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Segunda Parte: Ejercicios Resueltos
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Presentación
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Capítulo 1: Las razones de cambio y la derivada
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Capítulo 2: Límites
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Capítulo 3: Límites laterales y continuidad
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Capítulo 4: Límites infinitos y al infinito
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Capítulo 5: La derivada
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Capítulo 6 : Las funciones trigonométricas y el cálculo
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Capítulo 7: Las funciones logarítmicas y exponenciales y el cálculo
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Capítulo 8: Algunas aplicaciones
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Capítulo 9: Temas adicionales: una introducción
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Capítulo 10: Definiciones y métodos formales
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Bibliografía general
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Indice de recuadros teóricos
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Indice analítico
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