República Bolivariana de Venezuela Ministerio del Poder Popular para Educación Universitaria Fundación Misión Sucre Región Zulia Eje Costa Oriental del Lago Aldea Rafael Urdaneta Bachaquero - Estado Zulia
Fundamentos Matemáticos para las Ingenierías, Carreras Administrativas y otras carreras afines Preparado por el Ingeniero Shado Viñas, Docente Colaborador
Bachaquero, Febrero de 2.015
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2
«No puede existir un lenguaje más universal y simple, más carente de errores y oscuridades, y por lo tanto más apto para expresar las relaciones invariables de las cosas naturales [...]. Las matemáticas parecen constituir una facultad de la mente humana destinada a compensar la brevedad de la vida y la imperfección de los sentidos» Joseph Fourier. Físico y matemático
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3
Las Matemáticas están presentes en todo, absolutamente todo. No existe un único elemento, ni la más mínima partícula en el complejísimo Universo que no esté regida por alguna forma matemática. Sólo debes ser lo suficientemente atento a todo cuanto te rodea, para reconocer esta afirmación en tu vida! Observa detenidamente, presta muchísima atención. Cada partícula de tu ser, de todo cuanto te rodea, es una armoniosa combinación numérica la que lo define. Desde una simple molécula de agua, pasando por la laboriosa labor de las abejas en su colmena o la espiral de un caracol, hasta el más vasto arreglo de una constelación o un sistema planetario... todo está regido por leyes numéricas. El gran Maestro de la Creación nos habla a través de esta ciencia, que es Su Gran Ciencia, y nos prodiga Su Eterno e Infinito Amor!!! Lamentablemente, Lamentablemente, y esto es una gran realidad, re alidad, las matemáticas no suelen enseñarse como debiesen. En mis años como Profesor de Matemáticas, Física y Química, logré desarrollar métodos eficaces para la enseñanza de la ciencia a través de principios tangibles, cotidianos, sencillos, prácticos... pero más aún, divertidos. ¡La ciencia más exacta es también el lenguaje más tierno y sencillo de todo cuanto pueda existir! Te invito a que te sumerjas s umerjas en esta nueva aventura. Prometo ser paciente con tus inquietudes!!! Ánimo, tú puedes!!!
No existe un único elemento, ni la más mínima partícula partícula en el complejísimo c omplejísimo Universo que no esté regida por alguna forma matemática.
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De acuerdo con lo poco que he leído acerca de esta noble ciencia (y arte): Las Ciencias Matemáticas , me parece que una de las literaturas más ingeniosas en cuanto a la manera de presentar la misma ―como un maravilloso y enigmático encanto ‖, es la magistral obra de Júlio César de Mello Souza , mejor conocido como Malba Tahán. Digna obra que lleva por título: El Hombre que Calculaba. Para iniciar con pie firme nuestro Curso, nos permitiremos, sin ánimos de usurpar ningún derecho o propiedad intelectual, parafrasear algunas breves palabras citadas por tan prodigioso Autor: "Cuando miramos hacia el cielo, en las noches límpidas y calmas, sentimos que nuestra inteligencia es pequeña para concebir las obras maravillosas del Creador. Delante de nuestra mirada sorprendida, las estrellas son una caravana luminosa que desfila por el desierto insondable del infinito; las nebulosas inmensas y los planetas giran según leyes eternas por los abismos del espacio. Una noción surge, entretanto, bien nítida, en nuestro espíritu: la noción de número. Vivió otrora, en Grecia, cuando ese país era dominado por el paganismo, un filósofo notable llamado Platón. (Alah es, sin embargo, más sabio). Consultado por un discípulo sobre las fuerzas dominantes de los destinos del hombre, el gran sabio respondió: " Los números gobiernan el mundo". Realmente es así. El pensamiento más simple no puede ser formulado sin que en él se involucre, bajo múltiples aspectos, el concepto fundamental de número. El beduino que en medio del desierto, en el momento de la oración, murmura el nombre de Dios, tiene su espíritu dominado por un número: ¡La Unidad! Sí; Dios, según las verdades escritas en el Libro Santo y repetidas por el Profeta, es UNO, eterno e inmutable. Luego, el número UNO aparece en el cuadro de nuestra inteligencia como el símbolo del propio Creador. Creador. Del número que es la base de la razón y del entendimiento, surge otra noción de indiscutible importancia: la noción de medida. Medir es comparar. Por lo tanto, solo son susceptibles de medirse las magnitudes que admiten un elemento como base de comparación. ¿Será posible medir la extensión del espacio? De ningún modo. El espacio es infinito y, siendo así, no admite término de comparación. ¿Será posible avaluar la eternidad? De ninguna manera. Dentro de las posibilidades humanas, el tiempo es siempre finito, y en el cálculo de la Eternidad no puede lo efímero servir de unidad de avaluación. En muchos casos, sin embargo, embargo, nos es posible representar una magnitud que no se adapte a los sistemas de medida, por otra que pueda ser avaluada con exactitud. Ese cambio de magnitudes, tendiente a simplificar los procesos de medidas, constituye el objeto principal de una ciencia, que los hombres denominan Matemática".
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A modo de cierre de ésta parte introductoria, plasmaré algunas citas célebres de algunos matemáticos destacados ó de algún otro pensador que, al igual que un ingenioso matemático, ha dejado en la literatura universal su huella imborrable con algún planteamiento ideológico sobresaliente. Es importante destacar que el orden en que dichas citas aparecen, obedecen única y e xclusivamente un orden cronológico, y ningún otro criterio de relevada importancia. En definitiva, todas éstas citas, estoy convencido, servirán suficientemente como fuente motivacional para emprender con buen pie nuestro Curso. ―Comprender las cosas que nos rodean es la mejor preparación para comprender las cosas que hay más allá ‖. Hipatia de Alejandría (355 ó 370–415) Filósofa y maestra neoplatónica griega. ―La matemática es llave y puerta de la ciencia c iencia ‖. Roger Bacon (1214-1294) Filósofo, científico, y teólogo inglés. ―Ninguna investigación humana puede ser denominada ciencia si no pasa a través de pruebas matemáticas‖ . Leonardo Da Vinci (1452-1519) Pintor, anatomista, arquitecto, artista, botánico, botánico, científico, escritor, escultor, filósofo, ingeniero, inventor, músico, poeta y urbanista italiano. ―Las matemáticas son el alfabeto con el cual Dios ha escrito el Universo ‖. Galileo Galilei (1564-1642) Físico y astrónomo italiano. italiano. ―La matemática es la ciencia del orden y la medida, de bellas cadenas de razonamientos, todos sencillos y fáciles‖ . René Descartes (1596-1650) Filósofo y matemático francés. ―Hay que tratar de unir lentamente en la instrucción del niño el saber y el poder. Las matemáticas matemáticas parecen ser, e ntre todas las ciencias, el único medio de satisfacer este fin‖. Immanuel Kant (1724–1804) Filósofo alemán; considerado como uno de los pensadores más influyentes de la filosofía universal. ―Los encantos de esta ciencia sublime, las matemáticas, sólo se le revelan a aquellos que tienen el valor de profundizar en ella ‖. Carl Friedrich Gauss (1777–1855) Matemático, astrónomo, geodesta, y físico alemán. ―No hay rama de la matemática, por lo abstracta que sea, que no pueda aplicarse algún día a los fenómenos del mundo real‖. Nikolái Lobachevski (17921856) Matemático ruso. ―Las matemáticas parecen dotar a uno de un nuevo sentido se ntido ‖. Charles Darwin (1809–1882) Naturalista inglés. ―Un matemático que no es también algo de poeta nunca será un matemático completo‖ . Karl Weierstrass (1815-1897) Matemático alemán. ―No puedo imaginar a las matemáticas como algo difícil y aburrido ‖. William Thomson Kelvin (1824-1907) Físico y matemático británico. ―El arte es la ciencia de la belleza, las matemáticas son la ciencia de la verdad‖ . Oscar Wilde (1854-1900) Escritor, poeta y dramaturgo irlandés. ―En la actualidad, no sólo nuestros reyes ignoran la matemática; también nuestros filósofos, y para ir más lejos, ni nuestros matemáticos saben matemáticas‖. Julius Robert Oppenheimer (1904-1967) Físico judío estadounidense. es tadounidense. ―Para aquellos que no conocen las matemáticas, es difícil sentir la belleza, la profunda belleza de la naturaleza... Si quieres aprender sobre la naturaleza, apreciar la naturaleza, es necesario aprender el lenguaje en el que habla‖. Richard Phillips Feynman (1918-1988) Físico estadounidense. es tadounidense. ―La riqueza de un hombre no se encuentra en la cantidad de dinero que posee, sino en la calidad de su conocimiento y educación ‖. Javier Herrera (1960actualidad) Diplomático español.
Aclaratoria Aprovecharé este breve espacio para hacerte conocer el Código Cromático que he implementado a lo largo de ésta guía, el cual debes tener muy presente para identificar, de manera rápida y eficaz, cada uno de los elementos destacados en el presente trabajo, a saber:
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Contenido
Capítulo 1: Capítulo 2: Capítulo 3: Capítulo 4: Capítulo 5: Capítulo 6: Trabajo 1: Modelo 1:
La Historia de los Números y de las Operaciones Matemáticas Teoría de Números Propiedades Numéricas y Operacionales Artificios Matemáticos. Parámetros Sobrentendidos y Dilates Valor Absoluto y Valor Posicional Algunas curiosidades sobre el sabio Eratóstenes y los Números Primos Primer Contenido a ser evaluado en forma de Trabajo Escrito (5%) Modelo de Práctica para el Primer Evaluativo tipo Examen (10%)
Capítulo 7: Capítulo 8: Capítulo 9: Capítulo 10: Capítulo 11: Trabajo 2: Modelo 2:
La Suma y la Resta. Sumatoria. Suma Algebraica. Naturaleza y Dualidad La Multiplicación ó Producto. Productoria. Factorial. Primorial. Combinatoria La División ó Cociente. Divisibilidad. Criterios de Divisibilidad Descomposición Factorial. Mínimo Común Múltiplo y Máximo Común Divisor Las Potencias y Las Raíces Segundo Contenido a ser evaluado en forma de Trabajo Escrito (5%) Modelo de Práctica para el Segundo Evaluativo tipo Examen (10%)
Capítulo 12: Capítulo 13: Capítulo 14: Capítu Cap ítulo lo 15: Trabajo 3: Modelo 3:
Ecuaciones ó Igualdades. Sistemas de Ecuaciones. Métodos de Resolución Inecuaciones ó Desigualdades. Algunas Aplicaciones Importantes Matrices y Determinantes. Aplicaciones Matriciales Vectores en el Plano y en el Espacio. Operaciones Vectoriales. Vectoriales. Aplicaciones ones Vectoriales Vectoriales Tercer Contenido a ser evaluado en forma de Trabajo Escrito (5%) Modelo de Práctica para el Tercer Evaluativo tipo Examen (10%)
Capítulo 16: Capítulo 17: Capítulo 18: Trabajo 4: Modelo 4:
Límites y Continuidad Derivadas por Definición y por Tablas Aplicaciones de la Derivada. Antiderivada. Una introducción al Cálculo Integral Cuarto Contenido a ser evaluado en forma de Trabajo Escrito (5%) Modelo de Práctica para el Cuarto Evaluativo tipo Examen (10%)
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7
Capítulo 1: La Historia de los Números y de las Operaciones Matemáticas Matemáticas Como bien sabemos, las Operaciones Operaciones Matemáticas elementales son: Adición, Sustracción, Producto y Cociente (más popularizado como: Suma, Resta, Multiplicación y División); las cuales, van de la mano con otras cuatro, a saber: la Potenciación, la Radicación, la Logaritmación y la Exponenciación. En ese justo orden, el hombre comenzó a desarrollar sus planteamientos ideológicos, basados principalmente en necesidades. Es, de esta forma, como surge la necesidad de contar y la idea de número. Así, se originan, inicialmente, los Números Naturales ( ), iniciando, lógicamente por el concepto de Unidad (UNO). (Debemos acá recordar que el cero lo inventaron los indios (de la India) por el año 500 de nuestra era, ellos denominaron a éste símbolo ―sunya‖, que quiere dec ir "vacío". Éste símbolo de la nada fue recogido por los árabes hacia el siglo VIII, quienes lo denominaron ― céfer‖, que en su propio idioma quería decir también " vacío". Esta última expresión dio origen a las palabras castellanas ―cero‖ y ―cifra‖. Con mucha lentitud llegaron llegaron los números arábigos a occidente y reemplazaron a los números romanos, que estos habían esparcido por todo su imperio). Ahora bien, como decíamos, la repetición sucesiva de esta unidad ( UNO) produce la generación de los siguientes números naturales; es decir: 2 = 1+1; 3 = 1+1+1 ó, lo que es igual, 3 = 1+2 1+2; y así subsiguientemente. Es así como surgió también el concepto de Suma. No obstante, al comenzar a contar y sumar, el hombre se dio cuenta que los números Naturales ― ℕ‖ ya no eran suficientes para resolver todos los problemas que se planteara; si se deseaba, por ejemplo: ―Encontrar un número que Sumado con 11 resulte 7‖ (Simbólicamente se escribe: x + 11 = 7 ). Por lo tanto, se dio apertura a un nuevo Conjunto de números que más tarde sería conocido como Enteros ( ); e introduciendo el nuevo concepto de Resta. (Hoy en día sabemos que el número ―x‖ buscado anteriormente anteriormente es el -4, es decir x = -4; y se lee equis es igual a ―menos cuatro‖).
Con el tiempo, el pastor de los rebaños, el sembrador del campo, así como el cazador y el artesano, necesitaron de otras operaciones operaciones numéricas, aparte de la Suma y de la Resta, para poder contar más fácilmente sus posesiones y verificar que, al final de la jornada, todo estuviese ―contado‖. Fue así como surgió el concepto de Multiplicación. Ahora bien, resultó que al plantear problemas del tipo, por ejemplo: ―Hallar un número que Multiplicado con 7 resulte 11‖ (Simbólicamente : x*7 = 11), ya el Conjunto de Enteros ( ) no era suficiente. Por tanto, se dio apertura a un nuevo Conjunto de números que más tarde sería conocido como Racionales ( ); e introduciendo el nuevo sabemos que el número ―x‖ es el 11/7, 11/7, es decir: x = 11/7 –se lee: once séptimos–). La representación Matemática del Conjunto de los concepto de División. (Hoy en día sabemos Números Racionales es: {n ε Q n = a/b a, b ε Z; b≠0}.
Más tarde, cuando de figuras geométricas se trató, surgió la necesidad de extrapolar y transmutar las Sumas en Productos y los Productos en Potencias. Ello trajo a camente: x 7 = 11), ya el su vez una nueva polémica, al plantear problemas del tipo, por ejemplo: ―Encontrar un número que Elevado a la Potencia 7 resulte 11‖ (Simbólicamente: Conjunto de Racionales ( ) no era suficiente. Por lo tanto, se dio apertura a un nuevo Conjunto de números que más tarde sería conocido como Irracionales (I); e busc ado anteriormente anteriormente es el e l 7√11, es decir x = 7√ 11 –se lee: raíz séptima introduciendo endo el nuevo concepto de Radicación. (Hoy en día sabemos que el número ―x‖ buscado s éptima de once-). En el conjunto de los Números Irracionales están incluidos todos los radicales ó raíces, es decir, aquellos números que no tienen raíz exacta, por ejemplo: √2 = 1,41421; √3 = 1,73205. También están incluidos todos los números decimales inconmensurables (que no pueden medirse) ó trascendentes como lo son: φ 1,6180339
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astronómicos. Estas tres necesidades pueden ser relacionadas en cierta forma a la subdivisión amplia de la matemática en el estudio de la estructura, el espacio y el cambio. La razón para que actualmente se utilice un sistema decimal, se deriva principalmente de que el ser humano necesitó hacer una representación simbólica del conteo con su propio cuerpo, y para ello ssee valió básicamente de los 10 dedos de las manos y aunque éste no fue el único sistema s istema utilizado zado por la humanidad sí fue el más difundido. difundido. En 1.614 John Napier, llamado Nepper o Neperius, inventó los logaritmos (del griego logos = razón y arithmos = número). Un logaritmo es un número que indica la potencia a la que hay que elevar otro dado para que resulte un tercero también conocido. El matemático inglés John Wallis (1.616-1.703) fue el que consiguió dar sentido a los números Imaginarios (número que se inventa y se le asigna el símbolo ― i‖) en 1.685, así como los números complejos. Luego, en 1.744 el matemático suizo Leonhard Euler (1.707-1.783) descubrió los números trascendentes, que son los que jamás constituirán una solución a cualquier ecuación algebraica que pueda escribirse. En 1.845 el am Rowan Hamilton (1.815-1.865) comenzó a trabajar con números hipercomplejos, o como él los llamó: Cuaternios. Un campo importante en matemático irlandés William matemática aplicada es la probabilidad y la estadística, que permiten la descripción, el análisis y la predicción de fenómenos que tienen variables aleatorias y que se usan en todas las ciencias. El Análisis Numérico investiga los métodos para realizar los cálculos en computado c omputadoras. ras. De acuerdo a mis modestas apreciaciones, yo creo que podríamos afirmar, sin temor a equivocarnos -puesto que las matemáticas no se desarrollaron en un día-, que su infancia aún se encuentra muy lejos de nosotros. Tal conclusión me conlleva a cuestionarme lo siguiente: ¿Qué tiene de extraño entonces, puesto que los hombres han tardado tantos miles de años en representar los números y sus s us operaciones, que un estudiante encuentre algunas dificultades dificultades en éste és te tema?
Capítulo 2: Teoría de Números Bien sabemos que un número es una entidad abstracta que representa una cantidad o una magnitud (todo aquello que pueda medirse). Por contraposición: ―si algo no puede medirse, entonces no puede ser representado a través de un número‖. El símbolo de un número recibe el nombre de numeral o cifra (también se lo suele llamar ― guarismo‖). Los números se usan en la vida diaria como etiquetas, números telefónicos, identificcaciones aciones personales, numeración de carreteras, casas; también como indicadores de orden, posicionadores, números de serie o también como códigos. Los números más conocidos son los Números Naturales, que se usan para contar y básicamente están en todas las cosas. Éstos, conjuntamente con los números negativos, conforman el conjunto de los Enteros. Cocientes de números enteros generan los números Racionales. Si se incluyen incluyen todos los números que pueden expresarse con decimales pero no con fracciones de enteros (Irracionales), se habla entonces de los números Reales; si a éstos se les añade los números Imaginarios, se obtendrán todos los números necesarios necesarios para resolver cualquier ecuación algebraica. algebraica. A esta serie pueden añadirse también los números infinitos, los hiperreales y los transfinitos. Entre los reales, existen números que no son soluciones de una ecuación polinomial o algebraica, que reciben el nombre de transcendentes. Ejemplos famosos de estos números son el número φ = 1,6180339 (que es el número Áureo, y se lee ―fi‖), el número π (relación entre el Perímetro y el Diámetro de la circunferencia) y el número e (éste último es la base de los logaritmos naturales es o Neperianos). Los dos últimos números ( π y e) están relacionados entre sí por la Identidad de Euler. Resulta significativo e interesante que los números, todos sin excepción, poseen características y peculiaridades, por demás, atractivísimas; las cuales, implican una condición de por sí ―delirante y meritoria‖ conocer. Por ejemplo: Los Números Naturales pueden ser clasificados (aparte del UNO ―1‖) en Números Primos y Números Compuestos. Un Número Primo es un Número Natural que tiene exactamente dos divisores naturales distintos: él mismo y el 1; en otras palabras, un Número Primo sólo e nteros consecutivos son es divisible por él mismo y por la unidad (los primeros Números Primos son: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31...). Dato de Interés: ―Dos números enteros ―coprimos‖ ó primos entre sí, es decir, no tienen divisores comunes distintos de la unidad‖. Por su parte, un Número Compuesto tiene uno o más divisores distintos a 1 y a
8
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suman 284. Por su parte, los divisores propios de 284 son 1, 2, 4, 71 y 142, que suman 220. Para los pitagóricos los Números Amigos tenían muchas propiedades místicas. Por su parte, un Número Perfecto es un número natural que es igual a la suma de sus divisores propios positivos, sin incluirse él mismo. Dicho de otra forma, un Número Perfecto es aquel que es amigo de sí mismo. Así, 6 es un número perfecto, porque sus divisores propios son 1, 2 y 3; y 6 = 1 + 2 + 3 . Los primeros Números Perfectos son: 6, 28, 496, 8.128, 33.550.336 y 8.589.869.056. Un Número Defectivo o Deficiente es un Número Natural que es mayor que la suma de sus divisores propios exceptuándose a sí mismo (en éste caso, los primeros números defectivos son: 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23…). Todos los números primos son defect ivos, y también lo son las potencias de los números primos y los divisores propios de los números defectivos y perfectos. Contrario a esto, si la suma es mayor, entonces se considera al número como Número Abundante. Un Número Abundante o Número Excesivo es un número ―n‖ el cual es menor que la suma de todos los divisores positivos de ―n‖, menores que él. Una definición equivalente es que los divisores propios del número (todos los divisores excepto el propio número) sumen más que dicho número (en éste caso, los primeros Números Abundantes son: 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56…). Resulta importante, además, saber que el Primorial de un número n se define como el producto de todos los números primos menores o iguales a él, y se indica como n#; de éste modo: 2# = 2, 3# = 6. (Los primeros Números Primoriales son: 2, 6, 30, 210, 2.310, 2.3 10, 30.030). Geométricamente hablando, los números representan también figuras en el plano y en el espacio. Estos números, en forma genérica, reciben el nombre de Números Poligonales. De éste modo: a) Un Número Triangular cumple con la siguiente relación numérica: ½n(n+1); de ésta manera logramos obtener los primeros Números Triangulares, los cuales son: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66... b) Un Número Cuadrado cumple con la relación numérica: ½n(2n); y los primeros Números Cuadrados, son: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 … c) Un Número Pentagonal cumple la relación numérica: ½n(3n-1); y los primeros Números Pentagonales son: 1, 5, 12, 22, 35, 51, 70, 92, 117 … d) Un Número Hexagonal cumple la relación: ½n(4n-2); y los primeros Números Hexagonales son: 1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231 … e) Un Número Heptagonal cumple la relación: ½n(5n-3); y los primeros Números Heptagonales son: 1, 7, 18, 34, 55, 81, 112, 148, 189, 235 … f) Un Número Octogonal cumple la relación: ½n(6n-4); y los primeros Números Octogonales son: 1, 8, 21, 40, 65, 96, 133, 176, 225, 280 … g) Un Número Nonagonal cumple la relación: ½n(7n-5); y los primeros Números Nonagonales son : 1, 9, 24, 46, 75, 111, 154, 204, 261, 325 … h) Finalmente, un Número Decagonal cumple la relación: ½n(8n-6); y los primeros Números Decagonales son: 1, 10, 27, 52, 85, 126, 175 … Una de las mayores curiosidades que he estudiado en mi vida, relacionada con los números, son los Números Narcicistas, los Números Omirpes, los Números Vampiros y los Números Capicúas. Un Número Narcisista es un Número de ―n‖ dígitos que resulta ser igual a la suma de las potencias de orden ―n‖ de sus dígitos. Ejemplo: 153 = 1³ + 5³ + 3³ . Un Número Omirp es un Número Primo que al invertir sus dígitos da otro Número Primo. Ejemplos: 1597 y 7951 son primos. Un Número Vampiro es un Número ―Flagrante‖ el cual se consigue a partir del producto de dos números ―obtenidos‖ a partir de sus dígitos. Ejemplo: 2187 = 27 x 81. Un Número capicúa es aquel que puede leerse igual ―al derecho‖ que ―al revés‖. Ejemplos : 1, 11, 121, 2332, 51215, 9137319, 35785958753. A manera de Dato Final, compartiré con ustedes los nombres de algunos de los dignos hombres y mujeres que dedicaron su vida al estudio de los números y sus propiedades; para ello, daré preferencia al orden alfabético y ningún otro criterio: 1. Adrien-Marie Legendre 21. 21. Bernhard Riemann 41. 41. Ernst Kummer 61. 61. Hermann Minkowski 2. Alan Baker 22. Bill Gosper 42. Ernst Steinitz 62. Hermann Weyl 3. Alan Turing 23. Blaise Pascal 43. Étienne Bézout 63 Herón de Alejandría
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81. 81. Joseph-Louis de LaGrange 82. Julius Wilhelm Richard Dedekind 83. Karl Rubin 84. 84. Karl Weierstrass 85. Klaus Roth 86. Kurt Gödel 87. 87. Leonardo de Pisa 88. Leonardo Fibonacci 89. Luca Pacioli 90. Leonhard Euler 91. 91. Leopold Kronecker
92. Luitzen Egbertus Jan Brouwer 93. 93. Marin Mersenne 94. 94. Martin Davis 95. 95. Michael Stifel 96. 96. Nicolo Tartaglia 97. 97. Niels Henrik Abel 98. 98. Nikolái Chebotariov 99. 99. Nikolái Ivánovich Lobachevski 100. Omar Jayam 101. Pafnuti Lvóvich Chebyshov 102. Paul Erdös
103. Peter Gustav Lejeune Dirichlet 104. Pierre de Fermat 105. Pierre Deligne 106. Pierre-Simon Laplace 107. Pitágoras Pitágoras de Samos 108. Qin Jiushao 109. Rafael Bombelli 110. René Descartes 111. Richard Dedekind 112. Richard Guy 113. Robert Daniel Carmichael
114. Robert Langlands 115. Serge Lang 116. Sofia Vasílievna Kovalévskaya 117. Sophie Germain 118. Srinivasa Aiyangar Ramanujan 119. Stefan Banach 120. Tales de Mileto 121. Teiji Takagi 122. Thabit ibn Qurra 123. William Timothy Gowers.
propiedades… Honor, admiración admiración y respeto; a todos esos hombres y mujeres que dedicaron su vida al estudio de los números y sus propiedades…
Capítulo 3: Propiedades Numéricas Numéricas y Operacionales Toda vez que iniciemos con pie firme nuestro curso (recordemos que estamos en los preliminares), necesitaremos de una serie de pautas y reglas básicas por las cuales debemos regirnos y resulta importante, además de necesario, que logremos conocer y manejar hábilmente dichas reglas. No es menester aprendernos nada de memoria, para eso están las tablas, gráficos e imágenes; pero sí es fundamental que logremos conocer su existencia y en parte su significado, además de saber cómo utilizarlas as en el instante propicio. En esta serie, te presento apenas 27 de estas propiedades y leyes (de las miles existentes); las cuales son las más básicas y regulares, para todo estudio es tudio matemático co que se desee emprender. Pasaré de inmediato a dar una breve exposición de cada una de las propiedades que ahora te traigo: 1. El orden de los Sumandos NO altera el resultado de la suma (Propiedad Conmutativa): Conmutativa): Ejemplo: 2+3 = 3+2 2. Dados tres números cualesquiera, pueden ser sumados de a dos, sin que se altere el resultado final (Propiedad Asociativa): Asociativa): Ejemplo: 2+3+4 = (2+3) +4 = (2+4) (2+4) +3 = 2 +(3+4) = 9 2+3+4 = (5) +4 = (6) +3 = 2 +(7) = 9 3. Todo número tiene un opuesto sumativo, el cual lo SUPRIME (Elemento Opuesto): Ejemplo: 3+(-3) = 0 4. El cero no altera la suma, pero sí el producto (OJO): Ejemplo: 3+0 = 3 pero 3 *0 = 0 (muy pendiente con con esto) 5. El orden de los factores NO altera el producto (Propiedad Conmutativa): Ejemplo: 2*3 = 3*2
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11. Si tomamos un número base y lo elevamos a un logaritmo de la misma base, el resultado siempre será el ARGUMENTO del Logaritmo: log log 3 log 5 Ejemplos: 2 2 = 3 y 7 7 = 5 12. Una raíz puede ser escrita como potencia y viceversa: Ejemplos: √3 = 3 ½ ; y también: 52/3 = 3√52 13. La Potencia de un Producto es igual al Producto de Potencias: Ejemplo: (3 * 4)2 = 32 * 42 ; en efecto: (12) 2 = 9 * 16 = 144 14. Si dos potencias tienen la misma base y se Multiplican, entonces esto puede escribirse como la base elevada a la Suma S uma de los exponentes: 2 3 (2 + 3) Ejemplo: (3 ) * (3 ) = 3 ; en efecto: 9 * 27 = 35 = 243 15. Si dos potencias tienen la misma base y se Dividen, entonces esto puede escribirse como la base elevada a la Resta de los exponentes: Ejemplo: (45) / (43) = 4(5 - 3); en efecto: 1.024 / 64 = 42 = 16 16. Si una Potencia P otencia está elevada a otra Potencia, entonces esto puede escribirse como la base elevada a la Multiplicación de los exponentes: Ejemplo: (42)3 = 4(2*3); en efecto: 163 = 46 = 4.096 17. Todo elemento es igual a sí mismo (Propiedad Reflexiva): Ejemplos: 2 = 2 -123 = -123 18. Si un número es igual a otro, entonces el otro es igual al primero (Propiedad Simétrica): Ejemplo: Si √25 √25 = 5; entonces 5 = √25 19. Si dos números son iguales a un tercero, entonces son iguales entre sí (Propiedad Transitiva): Ejemplo: Si √81 = 9 y 32 = 9 entonces √81 = 3 2 20. Al sumar o restar un mismo número a ambos a mbos miembros de una igualdad, ésta no varía: Ejemplo: Si 3 = √9 entonces 3 + 2 = √9 + 2 21. Dos igualdades pueden ser sumadas (o multiplicadas) miembro a miembro: Ejemplo: Si √25 = 5 y √9 = 3; Entonces: √25 + √9= 5 + 3 y también √25*√9 = 5*3 22. Si suprimimos el mismo término a ambos miembros de una igualdad, ésta no varía: Ejemplo: Si √9 + 7 = 3 + 7; entonces √9 = 3 23. Si multiplicamos ó dividimos ambos miembros de una igualdad por el mismo término, ésta no varía:
11
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Capítulo 4: Artificios Matemáticos A partir de ahora, conocerás (sólo algunos) de los trucos que los matemáticos utilizamos utilizamos para lograr desarrollar y resolver complicadísimos problemas problemas matemáticos; problemas que, sin la ayuda de estas fastuosas ―tretas‖, serían más que imposibles resolver. Estos famosos ―trucos‖ de los cuales te hablo, reciben muy amablemente el título de ―ARTIFICIOS ‖; aunque algunos autores prefieren también el uso del término ―Artilugio‖ (Ambos términos muy apropiados) para referirse a estos ―mecanismos‖. ―mecanismos‖. Lo importante importante de todo es que a partir de hoy logres conocer algunos de esos es os truquitos. Afortunadamente, Afortunadamente, quienes enseñan matemáticas verdaderamente, verdaderamente, han tenido el interés de ilustrar a sus discípulos os en el manejo de estas formidables ―estrategias‖ y es un deber moral e inte lectual, para todos quienes deseamos formar a otros en esta ciencia, brindar la mayor parte de estas ―herramientas‖ a quienes se encuentran en la etapa de adiestramiento; con el firme propósito de hacerles más fácil el camino que les conllevará a su propio éxito. é xito. He allí la gran diferencia entre quienes le ―suavizan‖ el trabajo a sus discentes y quienes retienen para sí mismos estas her ramientas, con el fin de mantener en ascuas a sus interesados alumnos; quienes ciertamente desarrollarían, de esta manera, un clima de aversión frente a los números. Con estas repugnantes prácticas, quienes pretenden enseñar, sin ser verdaderos Ayos, lo único que logran es demostrar un destacado espíritu egoísta; además de pretender crear un escenario de oscurantismo entre los pueblos de la tierra: ― La Luz es para compartirla‖. (Si es de tu agrado y deseas conocer un poco más sobre mi posición crítica frente a éste punto, te invito a que leas el poema de mi autoría titulado “Innumeración “Innumeración”, ”, al final de esta esta guía de estudio, en el Apéndice 4. 4. Pero recuerda que sólo si es de tu agrado, ya que de lo contrario, puedes ignorarlo completamente, ya que no forma parte de nuestro estudio matemático ).
En otras palabras, como yo mismo lo diría: ―De aquí en adelante, la cosa se pone buena‖ . Y realmente es así. De modo que aprovecha al máximo cada una de las claves que ahora te traigo. Disfrútalas, hazlas parte de ti.
De acuerdo a mi poca experiencia durante mis años de docencia e investigación, procuraré clasificarlas en dos grandes grupos que yo mismo he concertado en llamarlos: ―Parámetros Sobreentendidos‖ y ―Dilates‖. Los primeros ―Artificios‖ pretenden hacerte ver muchas de las propiedades que poseen los números, aún cuando tus ―ojos naturales‖ así no las vean; es preciso que tus ― ojos mentales‖ sí logren reconocerlos de ahora en adelante. Los segundos ―Artificios‖ son lo que podríamos podríamos llamar ―Escapatorias‖ ante una situación de gran enredo. Veamos en detalle de qué se trata toda esta locura. Recuerda que denotaremos con las letras ― m‖ y ―n‖ un par de números cualesquiera, es decir, ― m‖ y ―n‖ representan cualquier número que puedas imaginar (sean éstos un número natural, par, impar, primo, compuesto, positivo, negativo, entero, fraccionario, radical, trascendente, imaginario, complejo, e inclusive el cero…).
4.1. Parámetros Sobreentendidos
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4.1.7. El cero puede ser se r multiplicado cado o dividido por cualquier número, sin que ello lo altere: 0=0*n y 0=0/n (OJO: Es el CERO el que se divide; pero NO es el número que divide a otro). En consecuencia: 0 = 0*3, 0 = 0/1, 0 = 0 *7, 0 = 0/17, 0 = 0*√5, 0 = 0/Ln27, 0 = 0*3xy, 0 = 0*i 4.1.8. El cero puede ser elevado a cualquier potencia y extraérsele cualquier raíz, sin que ello lo altere: 0 = 0n y 0=n√0 En consecuencia: 0 = 03, 0 = √0, 0 = 07, 0 = √0, 0 = 0√5, 0 = 3√0Ln27, 0 = 03xy, 0 = 0i 4.1.9. Todo número multiplicado por la unidad (1) es igual a SÍ MISMO, pero multiplicado por cero se hace CERO. n*1 = n; pero: n*0 = 0 4.1.10. Todo número elevado a la CERO, siempre es igual a 1. (Esta no debes olvidarla jamás). n0 = 1 Estos son sólo las más elementales; sin embargo, por ahora es suficiente con estas. Si ahora intentas aplicar, al menos, estas diez nuevas propiedades a un número cualquiera, podrás visualizar inmediatamente que ya no podrás volver a ver los números del mismo modo en que te los presentaron en otrora, cuando eras apenas un chiquillo o un mozuelo. Veamos un par de ejemplos. Con lo visto hasta ahora, podremos establecer las siguientes construcciones: Ejemplo 1: 3 = { 3√ [x√+(1*3/1)x] + [ y√+((7*0)/1234)y]3 }1 (Tal vez nunca lo imaginaste).
Ejemplo 2:
7 = 23√ { [ 7√(7 √(7*987650)7] + [123√(1234*0/4321) 234 + (987*0/2345)] } 23 (Tal vez te parezca una locura).
Y, aunque parezca mentira, la expresión 23√ { [ 7√(7*987650)7] + [123√(1234*0/4321)234 +(987*0/2345)]} 0/2345)] }23 realmente es igual a 7; el mismo 7 que hemos propuesto como valor inicial en la identidad. Te invito a que lo verifiques. Ahora bien, con una idea más clara (tal vez), podrás deleitarte entonces con los siguiente s iguientess Artificios. Veamos.
4.2.Dilates 4.2. Dilates
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Ejemplo:
√ √
√ √
{ }
4.2.4. 2.4. Identidad Valiosísima: La siguiente igualdad simbólica es a lo que yo llamo una Identidad Valiosísima, puesto que la misma es una de las tantas que puedes utilizar para convertir un simple cociente en sumas y restas cuadráticas. Espero que siempre la tengas presente a la hora de resolver problemas matemáticos que involucren fracciones, estúdiala con atención y mantenla siempre como tu As bajo la manga.
4.2.5. Fracciones Parciales: Éste método consiste en descomponer un cociente un cociente de polinomios de polinomios en una suma de fracciones de fracciones de polinomios de menor grado. menor grado. El requisito más importante es que el grado del polinomio del denominador sea estrictamente mayor mayor que el del numerador. Se distinguen 4 casos; a saber: s aber: a) Factores lineales distintos c) Factores cuadráticos distintos b) Factores lineales repetidos d) Factores cuadráticos repetidos Por la simple razón de que no es taremos implementando implementando éste artificio durante el presente curso, no daré mayor detalle. No obstante, te invito a que logres profundizar un poco más sobre el tema, ya que será de gran utilidad para cursos posteriores.
Sin ánimos de fastidiarte con mi exposición, te invito a que compruebes las siguientes identidades. Deseando, de todo corazón, que te diviertas un rato.
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Capítulo 5: Valor Absoluto Absoluto y Valor Posicional Posicional Te haré una pregunta de entrada, pero no deseo que la respondas. Es sólo es para hacerte reflexionar: ¿Sabías tú que los antiguos mayas tuvieron un sistema de numeración posicional el cual, aún cuando ya no se usa, era un sistema matemático perfecto? El Sistema de Numeración Decimal, considerado como universal en la utilización más corriente, es un sistema posicional con base en 10 elementos o cifras (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), que adquieren un valor posicional a la hora de determinar la cantidad o magnitud que puede componerse con dichas cifras. En éste sentido, se ntido, las posiciones se inician por la derecha: La primera posición o lugar indica las unidades; la segunda las decenas; la tercera las centenas; la cuarta el millar; siendo cada cifra a la izquierda tantas unidades de la potencia de 10 que corresponda al número de la posición. Por ejemplo: El número 3.476: Se lee: Tres mil cuatrocientos setenta y seis. Y consta de 3 unidades de mil (3 unidades de millar; millar = 10 ³), 4 unidades de cien (4 centenas; centena = 10 ²), 7 unidades de 10 (7 decenas; decena = 10 1 ) y 6 unidades de 1 (6 unidades; unidad = 100). Si contamos desde 0, incrementando incrementando una unidad cada vez, al llegar a 9 unidades, hemos agotado los símbolos disponibles, y si queremos seguir contando no disponemos de un nuevo símbolo para representar la cantidad que hemos contado. Por tanto añadimos una nueva columna a la izquierda del número, reutilizamos los símbolos de que disponemos, y decimos que tenemos una unidad de segundo orden (decena), reiniciam reiniciamos os a cero las unidades, y seguimos s eguimos contando. De igual forma, cuando contamos hasta 99, hemos agotado los símbolos disponibles para las dos columnas; por tanto si contamos (sumamos) una unidad más, debemos reiniciar a cero la columna de la derecha y sumar 1 a la de la izquierda (decenas). Pero la columna de la izquierda ya ha agotado los símbolos disponibles, así que también la reiniciamos a cero, y sumamos 1 a la siguiente columna (centena). (centena). Como resultado nos queda que 99+1 = 100. El cuentakilómetros mecánico, al util izar el sistema de numeración posicional decimal, nos muestra lo anterior: va sumando ― 1‖ a la columna de la derecha y cuando la rueda de esa columna ha completado una vuelta (se agotan los símbolos), se reinicia a cero y se añade una unidad a la siguiente s iguiente columna columna de la izquierda. Pero estamos tan habituados a contar usando el sistema decimal, que no somos conscientes de éste comportamiento, y damos por hecho que 99+1 = 100, sin detenernos a pensar en el significado que encierra esa expresión. Tal es la costumbre de calcular en decimal que la mayoría de la población ni siquiera se imagina que puedan existir otros sistemas de numeración diferentes al de base 10, y tan válidos y útiles (y, tal vez, mucho más prácticos) como éste. Entre esos sistemas s istemas se encuentran el Sistema de base 2, 2 , llamado Sistema Binario (el cual se compone exclusivamente de las cifras 0 y 1), el Sistema de base 8 ó Sistema Octal y el Sistema de base 16, conocido como Sistema Hexadecimal. También los antiguos mayas tuvieron un Sistema de Numeración Posicional que fue, por más, un Sistema matemático perfecto.
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5.3. Otras propiedades del Valor Absoluto | – a | = | a | | a – b | = 0 a = b | a – b | ≤ | a – c | + | c – b | | a – b | ≥ | | a | – |b | b | | | a/ a/ b | = |a |a | / | b | b≠ b≠0
⇔
⇔⇔
Otras dos propiedades útiles de Valor Absoluto son: |a| ≤ b – b ≤ a ≤ b |a| ≥ b a ≥ b ˅ a ≤ – b
Simetría Identidad de indiscernibles es Desigualdad triangular (equivalente (equivalente a la propiedad aditiva) Preservación de la división (equivalente a la propiedad multiplicativa) Desigualdad Menor o igual que Desigualdad Mayor o igual que
Estas dos últimas son de gran utilidad para la resolución de inecuaciones, inecuaciones, como por ejemplo: – 9 – 9 ≤ x – 3 3 ≤ 9 |x -3| ≤ 9 – 6 – 6 ≤ x ≤ 12 12
5.4. Valor Posicional El sentido más amplio de Valor Posicional está referido, precisamente, a la posición que ocupa (en el sentido de derecha a izquierda) una cifra en determinado número. Por ejemplo, en el número 1.234; la cifra ó dígito 4 ocupa la Primera Posición ó Posición 1; seguidamente, la cifra ó dígito 3 ocupa la Segunda Posición ó Posición 2; luego, la cifra ó dígito 2 ocupa la Tercera Posición ó Posición 3; y, finalmente, la cifra ó dígito 1 ocupa la Cuarta Posición ó Posición 4. La notación posicional es un modo de escritura numérica en el cual, cada dígito posee un valor diferente que depende de su posición relativa. El modo que utilizamos habitualmente mente es el sistema s istema decimal (base 10), necesitándose necesitándose diez dígitos diferentes, cuyo valor en orden creciente es: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Utilizando la Notación Posicional, el mismo dígito 5 toma diferente valor en los números 5, 50 y 500. Esto es una consecuencia de la descomposición de números en múltiplos múltiplos de factores bn (donde b es la base y n cualquier número entero). entero).
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Eratóstenes era hijo de Aglaos. Estudió en Alejandría y, durante algún tiempo en Atenas. Sus profesores incluían al erudito Lisanias de Cirene y al filósofo Aristón de Chios quien había estudiado con Zenón de Citio, el fundador de la escuela Estoica de filosofía. Eratóstenes también estudió con el poeta y erudito Calímaco, quien también había nacido en Cirene. Después Eratóstenes pasó algunos años estudiando en Atenas, y también gran amigo de Arquímedes. En el año 236 antes de Cristo, Ptolomeo III de Egipto le llamó para que se hiciera cargo de la Biblioteca de Alejandría; puesto que ocupó hasta el fin de sus días. Suidas afirma que, tras perder la vista, se dejó morir de hambre a la edad de ochenta años; sin embargo, Luciano afirma que llegó a la edad de ochenta y dos, y Censorino sostiene que falleció cuando tenía ochenta y dos. Eratóstenes poseía una gran variedad de conocimientos y aptitudes para el estudio. Suidas afirma que también era conocido como el segundo Platón, y diversos autores dicen que se le daba el sobrenombre de Beta, por la segunda s egunda letra del alfabeto griego, porque porque ocupó el segundo s egundo lugar en todas las ramas de la ciencia que cultivó. En matemáticas, la Criba de Eratóstenes es un simple y antiguo algoritmo para encontrar todos los Números Primos hasta un número entero especificado. Se trabaja eficientemente para los primos más pequeños. Esta fue creada por Eratóstenes, del cual ya hemos hablado ampliamente. Para encontrar todos los Números Primos menores o iguales a un entero ― n‖ se sigue la propuesta del ―método de Eratóstenes‖: Se forma una tabla con todos los números naturales comprendidos entre 2 y n, y se van tachando los números que no son primos de la siguiente manera: cuando se encuentra un número entero que no ha sido tachado, ese número es declarado primo, primo, y se s e procede a tachar todos sus múltiplos. El proceso termina cuando el cuadrado del mayor número confirmado como primo es mayor que n. Te invito ahora a que te diviertas sobremanera, checando checando un par de aplicaciones que te demostrarán en detalle de qué se trata todo lo expuesto e xpuesto con antelación:
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2. El Máximo Común Divisor de dos o más números es el e l mayor número natural que divide a todos ellos. Es igual al producto de los factores comunes con su mínimo exponente. En el ejemplo anterior, el máximo común divisor de 10 y 12 es 2. 3. Finalmente, dos o más números son Coprimos, o primos entre sí, si no tienen ningún factor primo común; es decir, si su máximo común divisor es 1. Un Número Primo es, así, Coprimo con cualquier número natural que no sea múltiplo de él mismo. Ejemplos: 21 y 13 son Coprimos, pero también lo son 45 y 22, aún cuando estos últimos no sean primos ninguno de los dos. Por su parte, todo número natural no primo, a excepción del 1, se denomina Compuesto, es decir, tiene uno o más divisores distintos a 1 y a sí mismo. También se utiliza el término DIVISIBLE para referirse estos números. Los 20 primeros números compuestos son: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30 y 32. Una característica de los Números Compuestos es que pueden escribirse como producto de dos enteros positivos menores que él. Así, el número 20 es compuesto porque puede expresarse como 4 x 5; y también el 87 ya que se expresa como 3 x 29. Sin embargo, no es posible hacer lo mismo con el 17 ó el 23 porque son Números Primos. Cada Número Compuesto se puede expresar como multiplicación cación de dos (o más) números primos específicos, cuyo proceso se conoce como Factorización.
6.2.2. Otras Curiosidades: 1. En el Sistema Decimal, todos los números primos salvo el 2 y el 5 acaban en 1, 3, 7 ó 9. 2. El Teorema de Wilson establece que: ―Un número natural n>1 es primo si y solo si: (n-1)! + 1 es divisible por n. Asimismo, un número natural n>4 es compuesto si y sólo si (n-1)! es divisible por n‖.
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Trabajo I (5%- 01 punto) 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
Qué significa, para usted, el Número 1 y cuál es su Importancia? Qué significa, para usted, el Número 0 y cuál es su Importancia? Cómo convertiría usted, un Producto en una Suma? Por qué razón todo Número multiplicado por 0, resulta 0? Qué es el Sistema S istema Binario y porqué es tan Importante éste Sistema? Muestre, con al menos 3 ejemplos diferentes, la Conversión de un Número en Base Decimal a Base Binaria. Muestre, con al menos 3 ejemplos diferentes, la Conversión de un Número en Base Binaria a Base Decimal. Escriba las Operaciones Básicas en el Sistema Binario: Suma Binaria y Multiplicación Binaria. Explique brevemente qué es un Logaritmo? Escriba las Propiedades de Logaritmo: Logaritmo de la Unidad, Logaritmo de la Base, Suma de Logaritmos, Diferencia de Logaritmos, Logaritmo de un Producto, Logaritmo de un Cociente, Exponente de un Logaritmo y Raíz de un Logaritmo. Anote en su Cuaderno, ó Carpeta de Trabajo,o, 5 ejemplos de entidades o parámetros que no pueden representarse numéricamente. numéricamente. Investigue la Identidad de Euler y el Teorema de De Moivre, anótelo en su cuaderno c uaderno.. Investigue sobre el proceso de Factorización, tanto para Números Enteros, como para Expresiones Polinómicas. Polinómicas. Defectivos ó Deficientes menores que 100. Complete la Tabla de Números Defectivos
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Modelo I (10%- 02 puntos) INSTRUCCIONES a) Mantenga su Celular o Teléfono Completamente Apagado (NO en Silencio ni en Modo Vibrar) y no intente establecer ningún tipo de comunicación con alguno de sus compañeros. b) Dispone íntegramente íntegramente de sus s us dos (02) horas para desarrollar su Test; aprovéchelas. NO pierda tiempo!!! c) Cada ítem tiene un valor de cuatro (04) puntos; es decir: 03 puntos por procedimiento y 01 punto por resultado correcto. d) Utilice el FORMULARIO aportado en caso de que lo requiera.
FORMULARIO Propiedades Numéricas y Operacionales Propiedad Asociativa a + b + c = (a + b) + c = (a + b) + c = a + (b +c) a * b * c = (a * b) * c = (a * b) * c = a * (b * c) Propiedad Distributiva a * (b + c) = (a * b) + (a * c) Potencia de un Producto (a * b)c = ac * bc
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Bloque II – Aritmética Capítulo 7: La Suma y la Resta. Sumatoria. Suma Algebraica. Naturaleza y Dualidad Con lo que has visto hasta ahora, tal vez te has reanimado un poco para continuar avanzando. Espero que sí. De hecho, si estás leyendo estas líneas es porque has elegido correctamente. De entrada, te haré ―reflexionar‖ con un par de preguntas; recuerda que no debes contestar de inmediato, sólo es para que reflexiones: 1. ¿A qué crees que se refiere el Término Sumatoria? 2. ¿Cómo crees que debe estructurar es tructurarse se una Sumatoria?
7.1. Suma. Sumatoria La Suma o Adición es la operación básica de composición; la cual consiste en combinar dos números por agregación, o añadir a un primer número uno o más números, para obtener una cantidad final o total. La Suma también ilustra el proceso de juntar dos colecciones de objetos con el fin de obtener una sola s ola colección. Por otro lado, la acción repetitiva de Sumar Uno es la forma más básica de contar. En términos más formales, la suma es una operación aritmética definida sobre conjuntos de
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7.3. Naturaleza-Dualidad
A lo largo de estos breves capítulos, cuenta te has podido dar de la relación fundamental entre 2 grandes grupos de números Reales, a saber: Los Positivos y los Negativos. Independientemente que éstos sean Enteros Racionales Irracionales. Es decir; podemos verificar, efectivamente, que tenemos:
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3. P-N = 295 +(-731)= 295-731 = -436. (En Sumas y Restas se coloca el signo del número con mayor Valor Absoluto; pero NUNCA en Multiplicaciones y Divisiones; por favor, no lo olvides).
Ejemplo 3: Determina, para la siguient s iguientee operación, su SUMA S UMA ALGEBRAICA: ALGEBRAICA: 3 -√2 +¼ -3 +8 -½ +¾ +√¼ +8 -15 -3 +6 -√2 +3 +7√2 = ? 1.
P = 3+¼+8+¾+√¼+8+6+3+7√2 3+¼+8+¾+√¼+8+6+3+7√2 = 29 +½ +7√2. +7√2. (Esto se debe a que que √¼ = ½, y ¼+¾ = 1).
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∏ ∏
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Los Primoriales son números definidos en la demostración de la infinitud de los números primos de Euclides. La demostración consiste en suponer un conjunto finito de números primos. Si se toma el Producto de todos ellos y se añade uno, ese número debe tener algún factor primo que no está en el conjunto. Aplicando el procedimiento infinitas veces, surgen infinitos números primos que no estaban en el conjunto inicial. La sucesión de los Primoriales crece muy rápidamente. He aquí los quince primeros Números Primos y sus Primoriales: 2# = 2 13# = 30030 31# = 200560490130 3# = 6 17# = 510510 37# = 7420738134810
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variaciones con repetición de m elementos tomados de n en n (m ≥ n); son s on los diferentes grupos formados por n elementos. En ellos: Las variacionescon 1. Sí entran entran todos los elementos del conjunto dado; siempre y cuando m ≤ n. Si, por el e l contrario m > n, entonces no entran todos los elementos. 2. Sí importa el orden en que se coloquen los elementos. 3. Sí pueden pueden repetirse uno ó más de los elementos dados.
Su notación es de la forma mVRn, ó de la manera más común VR ( ). Su cálculo se determina a través de la relación:
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después de chocar es porque existen fuerzas que actúan sobre cada una de las bolas, ya que las 2 modifican sus estados de reposo y movimiento respectivamente. Lo anterior significa que: ―Un sólo objeto no puede experimentar ni ejercer, por sí mismo, ninguna fuerza. Como dato final, es preciso conocer que esta acción mutua no siempre se ejerce entre 2 objetos en contacto. En muchas ocasiones parece tener lugar ―a distancia‖; éste es el caso de un objeto atraído por la Tierra, y viceversa, con una fuerza que es el peso del objeto. Entonces se habla de campos de fuerzas, y en el caso concreto del objeto atraído por la Tierra se habla del campo gravitatorio gravitatorio terrestre; las cargas eléctricas se atraen o se repelen debido a la presencia de un campo eléctrico. Por su parte, el momento de una fuerza es la medida del efecto de rotación (tendencia al giro) causado por una fuerza. Se le calcula como el producto de una fuerza por la distancia, en forma perpendicular, de su línea de acción a un determinado punto ó eje de giro. Cuando se aplica una fuerza a una puerta pesada para abrirla, la fuerza se ejerce perpendicularmente a la puerta y a la máxima distancia de las bisagras. Así se logra un momento máximo. Si se empujara la puerta con la misma fuerza en un punto situado a medio camino entre el tirador y las bisagras, la magnitud del momento sería la mitad. Si la fuerza se aplicara de forma paralela a la puerta (es decir, de canto), el momento sería nulo. Para que un objeto esté en equilibrio, los momentos dextrógiros u horarios (a derechas) en torno a todo eje deben cancelarse con los momentos levógiros ó antihorarios (a izquierdas) en torno a ese eje. Puede demostrarse que si los momentos se cancelan para un eje determinado, se cancelan para todos los ejes. En lugar de describir la dinámica de rotación en función de los momentos de las fuerzas, se puede hacer en función de pares de fuerzas. Un par de fuerzas es un conjunto de 2 fuerzas iguales y de sentido contrario aplicadas en puntos distintos. El momento del par de fuerzas o torque se representa por un vector perpendicular al plano del par, cuyo módulo es igual al producto de la intensidad común de las fuerzas por la distancia entre sus rectas soporte, y cuyo sentido está ligado al sentido de rotación del par por la 'regla del sacacorchos'.
15.3. Vectores Vectores En Matemáticas y Física, un vector (también llamado Vector Euclidiano o Vector Geométrico) es una magnitud una magnitud física definida por un punto del espacio es pacio donde se mide dicha magnitud, además de un módulo (o magnitud), su dirección (u orientación) (u orientación) y y su sentido (que distingue el origen del extremo). Sin embargo, vale recordar que, en inglés, la palabra " direction" indica tanto la dirección como el sentido del vector, con lo que se define el vector con solo dos características: módulo y dirección. Por otro lado, como todas las fuerzas aplicadas sobre una misma partícula tienen el mismo punto de aplicación, cada vector fuerza queda completamente definido únicamente por su magnitud ó módulo y por su dirección. En resumidas cuentas: Todo vector quedará completamente definido por su módulo y por su dirección. Algunos ejemplos de magnitu de magnitudes des físicas que son magnitudes vectoriales son: la velocidad con que se desplaza un móvil, ya que no queda definida tan sólo por su módulo (lo que marca el velocímetro, en el caso de un automóvil), automóvil), sino que se requiere indicar la dirección y eell sentido (hacia donde se dirige); la fuerza que actúa sobre un objeto, ya que su efecto depende, además de su intensidad o módulo, de la dirección en la que actúa; también, el desplazamiento de un objeto; y también la aceleración. No se debe confundir el desplazamiento con la distancia; ya que el desplazamiento está indicado por un cambio en la posición del objeto mediante una magnitud y un ángulo o dirección (es decir, el desplazamiento es un vector), mientras que la distancia es una cantidad escalar. La definición clásica de vector es la siguiente: Un vector es un elemento de un espacio vectorial. Esta noción es muy abstracta, y para muchos espacios vectoriales no es posible representar sus vectores mediante su módulo y dirección. En particular, ar, los espacios infinitos sin producto escalar no son representables de ese modo. Un vector también puede verse desde el punto de vista geométrico como un vector geométrico; es decir, como un segmento de recta orientado ó ―flecha‖ (y para ello, se hace uso del plano R2 ó del espacio R3); es decir, los vectores se pueden representar geométricamente como segmentos de recta dirigidos «flechas», en el plano cartesiano R2 o en el espacio euclidiano R3. Así que, otro modo particular ar de definir a un vector sería el siguiente: Un vector es una tupla una tupla de 3 números reales (que se llaman 2 componentes del vector). Así, un vector perteneciente al plano cartesiano R se representa como: y, un 3 vector perteneciente al espacio euclidiano R se representa como: .
⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗
15.4. Notación Vectorial al vectoriales se representan en los textos impresos por letras en negrita, para diferenciarlas de las magnitudes escalares que se representan en letra Las magnitudes vectoriales normal –sin el uso de negritas–. También, suele usarse frecuentemente una flecha sobre la letra, para diferenciarlo completamente de su módulo ó magnitud (el cual es un escalar) un escalar).. Por ejemplo: . 15.5. Definiciones Básicas 1. Vector: Es todo segmento de recta dirigido en el espacio. Lo denotaremos por . v
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6. Sentido: Se indica mediante una punta de flecha situada en el extremo del vector, indicando hacia qué lado de la línea de acción se dirige. 7. Magnitudes Escalares: Denominamos Magnitudes Escalares a aquellas en las cuales las medidas quedan correctamente expresadas por medio de un número y la correspondiente correspondiente unidad. Ejemplo de ello son: la masa, la temperatura, el volumen, la presión, la densidad. densidad. ales: Las magnitudes vectoriales son magnitudes que para estar determinadas precisan de un valor numérico y una dirección. una dirección. Ejemplo de ello 8. Magnitudes Vectoriales: son: el desplazamiento, la velocidad, la aceleración, la fuerza, el peso. el sistema de referencia de los vectores, los vectores, que que estará formado por un origen y tres ejes perpendiculares. ares. Éste 9. Sistema de Referencia: Hay que tener muy en cuenta el sistema sistema de referencia permite fijar la posición de un punto cualquiera con exactitud. El sistema de referencia que usaremos, como norma general, es el Plano Cartesiano ó el Espacio Euclidiano.
Para poder representar cada vector en nuestro Sistema de Referencia, haremos uso de 3 vectores unitarios. Estos vectores unitarios, son 10. 10. Vector Unitario: Para poder
11.11. 12. 12. 13. 13. 14. 14. 15. 16. 16. 17.
unidimensionales y tienen módulo 1, son perpendiculares entre sí y corresponderán a cada uno de los ejes del sistema de referencia. El unitario i para el eje x; el unitario j para el eje y; y el unitario k para el eje z. Vectores Iguales: Dos vectores son iguales cuando tienen el mismo módulo y la misma dirección. Vector Libre: Un vector libre queda caracterizado por su módulo y dirección. El vector libre es independiente del lugar en el que se encuentra ubicado en el espacio. Los vectores libres no están aplicados en ningún punto en particular. Vector Fijo ó Vector Ligado: Contrario al vector libre, éste vector está aplicado en un punto en particular. particular. Vector Deslizante: Es aquel vector cuyo punto de aplicación se puede deslizar a lo largo de su línea de acción. Vectores Opuestos: Son vectores de igual magnitud, pero dirección contraria. En éste caso, la dirección del vector vec tor opuesto se consigue sumando, a la dirección del vector de referencia, 1 ángulo llano; es decir, 2 ángulos rectos ó 180° sexagesimales. Vectores Coplanares ó Vectores Vectores Coplanarios: Son vectores que se encuentran o actúan dentro de un mismo plano en el espacio. Vectores Colineales o Vectores Vectores Paralelos: Son aquellos que actúan en una misma línea de acción. Ejemplos: En los instrumentos de cuerda, el punto donde está atada la cuerda (puente) se puede representar a la fuerza de tensión en un sentido s entido y al punto donde se afina la cuerda (llave) será otra fuerza en sentido contrario. Otro ejemplo puede ser cuando se levanta un objeto con una cuerda, la fuerza que representa la tensión de la cuerda va hacia arriba y, la fuerza que representa al peso del objeto, va hacia abajo.
Vectores Angulares: Son aquellos que parten de un mismo punto de aplicación; es decir, sus direcciones o líneas de acción pasan por 18. Vectores Concurrentes o Vectores un mismo punto. También se les suele llamar angulares porque forman un ángulo entre ellos. Ejemplos: Cuando dos aviones salen de un mismo lugar, cuando dos o más cuerdas tiran del mismo punto o levantan un objeto del mismo punto.
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20. 20. Vector Equilibrante: Es un vector igual en magnitud y dirección al vector resultante pero en sentido contrario; es decir, a 180°. 21. 21. Vectores Axiales ó Pseudovectores: Cuando consideramos 2 magnitudes físicas vectoriales , su producto vectorial es otra mangitud física ―aparentemente‖ vectorial que tiene un extraño comportamiento respecto a los cambios de sistema de referencia. Los vectores que presentan esas anomalías se llaman pseudovectores llaman pseudovectores o vectores axiales. Esas anomalías se deben a que no todo ente formado de 3 componentes c omponentes es un vector un vector físico.
15.6. Magnitudes vectoriales ―versus‖ magnitudes escalares Frente a aquellas magnitudes físicas, tales como la masa, la masa, la la presión, el volumen, la la energía, la la temperatura; que emperatura; que quedan completamente definidas por un número y las unidades utilizadas en su medida, aparecen otras, tales como el e l desplazamiento, la velocidad, la aceleración, la fuerza, el peso; que no quedan completamente definidas dando un dato numérico, sino que llevan asociadas una dirección. Estas últimas magnitudes son llamadas vectoriales en contraposición a las primeras, las cuales son llamadas escalares. Las magnitudes vectoriales quedan representadas por un ente matemático que recibe el nombre de vector. En un espacio un espacio euclidiano, euclidiano, de no más de 3 dimensiones, un vector se representa por un segmento de recta orientado. Así, un vector queda caracterizado por los siguientes elementos: su longitud o módulo, siempre positivo por definición, y su dirección, la cual puede ser representada mediante la suma de sus componentes sus componentes vectoriales ortogonales, paralelas a los ejes de coordenadas; o mediante mediante coordenadas polares, que determinan el ángulo que forma el vector con los ejes positivos de coordenadas. No cualquier n-tupla de funciones o números reales constituye un vector físico. Para que una n-tupla represente un vector físico, los valores numéricos de sus componentes, medidos por diferentes diferentes observadores, deben transformarse de acuerdo con ciertas relaciones fijas. En mecánica newtoniana generalmente se utilizan vectores genuinos, llamados a veces vectores polares, junto con pseudovectores , llamados vectores axiales. El momento angular, el campo magnético y todas las vectores axiales. magnitudes magnitudes que en cuya definición interviene el e l producto vectorial son en realidad pseudovectores o vectores 15.7. Módulo, Magnitud ó Norma de un Vector Se define como la longitud del vector en el espacio métrico considerado. La interpretación geométrica del módulo, magnitud ó norma de un vector es: la distancia que existe entre su origen y su s u extremo. Se calcula a través del producto interno ó producto escalar del vector consigo mismo.
|⃗| ⃗ ⃗ |⃗⃗| ⃗⃗ ⃗ ⃗ |⃗| ⃗⃗ ()⃗
Por componentes, tomando la base la base canónica en R3, formada por los vectores unitarios {i, j j, k}: Efectuado el producto escalar, es calar, tenemos:
15.8. Componentes de un vector Un vector se puede definir por sus sus c oordenadas, oordenadas, si si el vector está en el plano cartesiano xy, se representa: siendo sus coordenadas: vx y vy. Y, por lo tanto, el vector es la suma vectorial de sus coordenadas:
⃗⃗ (⃗ ) ⃗
Si un vector es de 3 dimensiones reales, entonces se puede representar sobre los ejes x, y, z:
siendo sus coordenadas: vx, vx y vz. Y, por lo tanto, el vector es la suma vectorial de sus coordenadas:
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⃗
El lema de Zorn, consecuencia Zorn, consecuencia del del axioma de elección, permite establecer que todo espacio vectorial admite una base una base vectorial, por lo que todo vector es representable como el producto de unas componentes respecto a dicha base. Dado un vector sólo existen un número finito de componentes diferentes de cero. Expuesto lo anterior, podemos decir que las componentes vectoriales tienen la siguiente constitución algebraica: 1. En 2 dimensiones (R2): Consideremos un vector F situado en un plano cualquiera. Ahora bien, éste vector puede ser descompuesto sobre un sistema de ejes componentes Fx, Fy . En los ejes cartesianos se definen 2 vectores unitarios que son: i, j, (respectivamente en las direcciones del eje x y cartesianos, en forma de sus 2 componentesFx del eje y, como se muestra en la figura adjunta).
Vectorialmente esta descomposición puede expresarse: F = Fx i + Fy j. (Siendo Fx, Fy los módulos respectivos de Fx, Fy). Por otra parte, si F es el módulo de F se verifica que: Fx = F.cos α y Fy = F.cos β = F.sen α. Estas cantidades Fx, Fy se llaman llaman "proyecciones" del vector F sobre los ejes x e y; mientras que vistos como vectores sobre los ejes, Fx, Fy, se llaman "componentes" "componentes" del vector F. Además, mediante la relación re lación trigonométrica trigonométrica tangente del ángulo (tg a) se obtiene: tg α = Fy/Fx. Resulta interesante saber que se da la siguient s iguientee relación entre F (módulo de F) y sus proyecciones (debido do al llamado Teorema de Pitágoras): Por lo que, despejando a F nos queda:
2. En 3 dimensiones (R3): Lo que hemos dicho para el plano cartesiano, se puede fácilmente generalizar al espacio euclidiano. En tal sentido, consideremos un vector F en el espacio, éste vector puede ser descompuesto sobre un sistema de ejes cartesianos xyz , tal como se aprecia en la figura adjunta, donde i, j y k son los vectores unitarios en cada una de las direcciones de los ejes:
Observa en la figura con mucha atención cada uno de los ángulos: Así, el ángulo α es el ángulo que forma F con el eje x, el ángulo β es el ángulo que forma F con el eje y, y el ángulo γ es el ángulo que forma F con el eje z. Las 3 componentes componentes de F son: Fx, Fy, Fz, cada una en su respectiva dirección de eje. Las proyecciones se obtienen F.cos β, Fz = F.cosγ F.cosγ; y, por último, el módulo de F es mediante: Fx = F.cosα , Fy = F.cosβ es::
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15.9. Operaciones con Vectores Vectores Las operaciones que se pueden hacer con Vectores provienen de sus aplicaciones, sobre todo de las aplicaciones en Álgebra Vectorial. Las Operaciones que están completamente definidas para los Vectores son: a. La Suma o Adición y la Resta o Sustracción. b. El Producto de un Escalar por un Vector (ó Producto de un Vector por un Escalar). c. El Producto Punto ó Producto P roducto Escalar. d. El Producto Cruz ó Producto P roducto Vectorial. e. El Producto Triple. f. El Producto Mixto. Nota de Sumo Interés : La División de Vectores Vectores no está definida. a. Suma y Resta de Vectores vectores en el orden u+v produce otro vector, al cual podemos llamar vector w. Sumar 2 vectores es como encadenar un vector u y luego La definición suma de 2 vectores un vector v. Diremos que u+v se simplifica como un vector w. La resta de 2 vectores consiste en la Suma del primer vector con el opuesto del segundo vector. La suma y/o resta de vectores está bien definida si ambos vectores pertenecen al mismo espacio vectorial. En física, para que 2 vectores puedan ser sumados deben estar aplicados en el mismo punto; es decir, deben ser s er coplanares y concurrentes. La composición de fuerzas sobre un sólido un sólido rígido cuando los puntos de aplicación no coinciden lleva a la noción de momento de la fuerza, dadas fuerza, dadas 2 fuerzas F1 y F2, con puntos de aplicación P1 y P2 diferentes. w
v
u
vectores goza de las siguient La operación suma de vectores s iguientes es propiedades: 1. Decir que u+v = v+u, es exigir que las 2 sumas simplifiquen en el mismo vector (el coloreado en negro). 2. Decir que u+(v+w) = (u+v)+w, es exigir que exista la asociatividad de de sumas de vectores. 3. Decir que existe un vector cero ó vector nulo (elemento neutro) tal neutro) tal que u+0 = u, equivale a exigir que exista un vector, el cual es incapaz de efectuar, mediante mediante la suma, modificación alguna a cualquier otro vector. un elemento opuesto, -u, que sumado a u simplifique en un vector cero. 4. Decir que u +(-u) = 0, es exigir la existencia de un elemento La suma y/o resta de vectores la podemos realizar realizar de 2 maneras diferentes: gráficamente ó analíticamente. a.1. Procedimiento Gráfico a.1.1. Método del paralelogramo La suma de 2 vectores libres es otro vector libre que se determina de la siguiente forma: Se sitúa el punto de aplicación de uno de ellos sobre el extremo del otro; el vector suma es el vector que tiene su s u origen en el origen del primero y su extremo en el extremo del segundo. Por tanto, el vector suma de dos vectores coincide con una de las diagonales del paralelogramo que puede formarse con los vectores que se suman; la otra diagonal representa la resta de dichos vectores. Un paralelogramo es una vectores concurrentes, los cuales después de dibujarse a figura geométrica de cuatro lados, con sus lados opuestos paralelos dos a dos. En éste método se nos dan 2 vectores escala en un sistema s istema de ejes cartesianos, se les dibujarán arán otros 2 vectores auxiliares, ares, con ayuda de 2 escuadras; siendo la resultante del sistema la diagonal que parte del origen y llega al punto donde se intersecan los vectores auxiliares y siendo la diferencia de los vectores la otra diagonal del paralelogramo. paralelogramo. v
u+
u
v
u-
u
Éste método permite solamente sumar vectores de dos en dos. Para efectuar sumas o restas de 3 o más vectores, el proceso es idéntico. Basta con aplicar la la propiedad asociativa. Al vector que se obtiene al sumar o restar varios vectores se le denomina Vector Resultante. Pero hay que tener muy presente lo siguiente:
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u +v
v
u
a.2. Procedimiento Analítico a.2.1. Método Algebraico (Sumatoria de Componentes) Hagamos un par de ejemplos. Dados 2 vectores libres, u y v:
⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗⃗ ⃗( ⃗ ⃗)⃗ ( ⃗ ⃗⃗ ) ⃗ ⃗ ⃗ ⃗ ⃗⃗ ⃗⃗ (( ⃗⃗ ⃗⃗ )) ((⃗⃗ ⃗⃗ )) ⃗ ⃗ ⃗ ⃗ ⃗⃗ ⃗ ⃗
El resultado de su suma se expresa en la forma:
y sumando, una a una, sus componentes, nos queda:
Por otro lado, la diferencia viene dada por:
y restando, una a una, sus componentes, nos queda:
a.2.2. Método Trigonométrico
En éste caso, se toman cada uno de los vectores a sumar como a y b, y su resultante se toma como c. De ese modo: El teorema del coseno se aplica a los lados y ángulos de triángulos cualesquiera y relaciona los tres lados con uno de los ángulos. Con dicho Teorema, se calcula el módulo del Vector Resultante.
–
El teorema del seno se aplica a los lados y ángulos de un triángulo cualquiera y relaciona cada dos lados con sus ángulos opuestos. Con dicho Teorema, se calcula la dirección del Vector Resultante
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1. Si k es positivo, NO altera el sentido; y cuando k es negativo, SI altera el sentido. 2. Para valores de k, mayores que 1, el vector sufre una ampliación; y para para valores de k menores que 1, sufre una reducción. 3. Si k es nulo (es decir, 0), entonces se origina el vector 0, cuyas coordenadas en R2 son: 0i + 0 j; y en R3 son: 0i + 0 j + 0k. Partiendo de la representación gráfica del vector, sobre la misma línea de su dirección tomamos tantas veces el módulo de vector como indica el escalar. Sean k un escalar y v un vector; entonces el producto de k por v se representa kv y se s e desarrolla multiplicando cando cada una de las componentes del vector v por el escalar k; esto es:
Ejemplo: Sean k = 3 y
⃗ ⃗ ⃗ ⃗ ⃗ ⃗⃗ ⃗⃗ ⃗ ⃗ ⃗ ⃗⃗ , entonces:
El Producto de un escalar por un vector goza de las siguientes propiedades (debes recordar que a y b representan magnitudes escalares y que u y v representan magnitudes vectoriales ales): a. Decir que a(bu) = (ab)u, es exigir que los productos encadenados a(b(u)) pueden simplificarse como uno, c=ab, luego (ab) u queda como cu. b. Decir que existe el escalar 1 tal que 1*u = u, equivale a decir exista un escalar incapaz de efectuar, mediante producto, modificación cación alguna a todos los vectores. c. Decir que a(u+v) = au+av, es exigir la propiedad distributiva respecto la suma vectorial. d. Decir que (a+b)u = au+bu, es exigir la propiedad distributiva respecto la suma escalar.
c. Producto Punto ó Producto Escalar El Producto Punto, también conocido como Producto Interno, Producto Interior ó Producto Escalar, es una operación una operación binaria definida sobre 2 vectores 2 vectores de un mismo espacio mismo espacio euclídeo. El euclídeo. El resultado de esta operación es un número ó escalar. ó escalar. El Producto Punto ó Producto Escalar de 2 vectores 2 vectores es una función que toma como argumentos dos elementos de un espacio vectorial, y devuelve un elemento del cuerpo k ó escalar. Esta definición de carácter geométrico es independiente independiente del sistema de coordenadas elegido y por lo tanto de la base la base del espacio vectorial escogida. El Producto Punto ó Producto Escalar de 2 vectores en un espacio un espacio euclídeo se define como el producto de sus módulos sus módulos por elel coseno del ángulo θ que forman entre ellos los vectores u y v. En el el espacio euclidiano, la notación notación usual de producto escalar es u • v.
⃗ ⃗ |⃗| |⃗| ⃗⃗ ⃗⃗ ⃗⃗
Expresión Analítica del producto escalar El Producto Punto ó Producto Escalar es una operación entre dos vectores u y v, que da como resultado un número (es decir, un escalar), por lo que también se le conoce como Producto Escalar. El Producto Punto ó Producto Escalar de 2 vectores, expresado analíticamente como u · v, se obtiene de la suma de los productos formados por las componentes de uno y otro vector. Si los vectores u y v se expresan en función de sus componentes cartesianas rectangulares, tomando la base la base 3 canónicaen canónica en R formada por los vectores unitarios {i, j j, k},
y, teniendo en cuenta que los productos escalares de los vectores: i · i = j · j j = k · k = 1, y los productos escalares de los vectores i • j = i • k = j • k = 0, entonces el Producto Punto ó Producto Escalar está definido como:
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Ángulo entre 2 vectores La expresión geométrica del Producto Punto ó Producto Escalar permite calcular el coseno del ángulo existente entre 2 vectores, mediante la siguiente definición formal que nos dice que: la multiplicación de un escalar denominado ―k‖ tiene que ser diferente de cero. El E l ángulo determinado por las direcciones de dos vectores u y v viene dado por la siguiente relación:
|⃗⃗| ⃗|⃗|
El cos θ dará siempre entre 0 y 1 (sea positivo ó negativo). El cos θ nos dice si los vectores son paralelos paralelos o perpendiculares. Si cos θ = 0, entonces los vectores son perpendiculares. Si cos θ = 1, entonces los vectores son paralelos. Si cos θ es positivo, entonces el ángulo entre los vectores es agudo; pero, si cos θ es negativo, entonces el ángulo es obtuso. Vectores Ortogonales y Vectores Paralelos (ó en una misma línea de acción) 1. Dos vectores son son ortogonales o perpendiculares perpendiculares cuando forman ángulo recto entre sí. Si el producto escalar de dos vectores es cero, ambos vectores son ortogonales. 2. Dos vectores son paralelos son paralelos o son colineales si el ángulo que forman es de 0 radianes 0 radianes (0°) ó de π radianes (180°). Cuando 2 vectores forman un ángulo cero, el valor del coseno es la unidad, por lo tanto el producto de los módulos vale lo mismo que el producto escalar.
⃗ ⃗ |⃗| |⃗|
d. Producto Cruz ó Producto Vectorial al Cuando los vectores son tridimensionales (esto es, son vectores de R3) es posible definir otra multiplicación de vectores cuyo resultado sea también un vector; dicha operación se denomina Producto Cruz o Producto Vectorial, definido mediante el determinante: determinante:
⃗ ⃗ ⃗
Donde i, j j, k son los vectores unitarios en la dirección de los tres ejes x, y, z. El producto u x v corresponde a un vector perpendicular ar a u y v, cuya norma o módulo es: donde nuevamente, θ es el ángulo entre los vectores.
⃗ ⃗
|⃗ ⃗ | |⃗| |⃗|
Del resultado anterior se derivan 2 deducciones: 1. El valor de es es igual al área del paralelogramo determinado determinado por u y v. 2. Los vectores u y v son paralelos (colineales) si y sólo si u x v = 0 (Observemos la similitud similitud entre éste és te criterio y el de perpendicularidad perpendicularidad para el producto punto). El producto vectorial entre u y v da como resultado un nuevo vector, nuevo vector, w. El Producto Vectorial entre u y v se denota mediante u x v, por ello se lo llama también Producto Cruz. La dirección del vector Producto Cruz está dada por la regla de la mano derecha y θ es, es, como antes, el ángulo entre u y v. A la regla de la mano derecha se la llama a menudo también regla del sacacorchos. u
x
v
v
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Puede verificarse fácilmente que el vector w es ortogonal a los vectores u y v, efectuando el producto escalar y verificando que éste es nulo (condición de perpendicularidad de perpendicularidad de vectores). Propiedades del Producto Cruz Cualesquiera era que sean s ean los vectores u, v y w: (u x v) v), (anticonmutatividad) 1. u x v = – (u 2. u • (u x v) = 0, cancelación por ortogonalidad. dad. 3. Si u x v = 0, con u≠0 y v≠0, entonces u y v son paralelos; esto es, la anulación del producto vectorial proporciona proporciona la condición la condición de paralelismo entre dos direcciones. 4. (u + v) x w = u x w + v x w. w) – w (u • v) v), conocida como Regla de la Expulsión. 5. u x (v x w) = v (u • w) de Jacobi. 6. u x (v x w) + w x (u x w) + v x (w x u) = 0, conocida como Identidad de Jacobi senθ, en la expresión del término de la derecha, sería el módulo de los vectores u y v, siendo θ, el ángulo menor entre los vectores u y v; esta 7. | u x v | = |u| |v| senθ expresión relaciona al producto vectorial con el área del paralelogramo del paralelogramo que definen ambos vectores. 8. El módulo de u x v puede calcularse fácilmente sin efectuar el producto vectorial: | u x v | = (|u|2 |v|2 - (u · v)2 ) ½
Bases ortonormales y producto vectorial Sea un sistema un sistema de referencia S = {0, i, j, k}, en en el espacio vectorial R3. Se dice que S es una Base Ortonormal Derecha si cumple c umple con las siguientes condiciones: vectores unitarios. 1. |i| = j |j| = |k| = 1; es decir, los 3 vectores son vectores 2. i · j = j · k = k · i = 0; es decir, los 3 vectores unitarios son son ortogonales entre sí y; por lo tanto, dada la propiedad anterior, anterior, son ortonormales. es. 3. i x j = k, j x k = i, k x i = j; es decir, cumplen la regla de la mano derecha. e. Producto Triple Llamamos Producto Triple (ó también conocido como Triple Producto Vectorial) de 3 vectores a la expresión u x (v x w) ó (u x v) x w; esto es, elel producto vectorial de 2 vectores se multiplica vectorialmente por un tercer vector. Para calcular el doble producto vectorial se utiliza la siguient s iguientee fórmula: u x (v x w) = v (u • w) – w (u • v) Propiedades del Producto Triple 1. De acuerdo a la fórmula, el vector que se obtiene en el Producto Triple, es un vector contenido en el plano definido por los vectores v y w. 2. Evidentemente, el producto vectorial no tiene la propiedad asociativa, ya que es antisimétrico (o anticonmutativo). 3. (u x v) x w = –w x (u x v). 4. Identidad de Jacobi: de Jacobi: u x (v x w) + w x (u x v) v) + v x (w x u) = 0. f. Producto Mixto Llamamos Producto Mixto (o también conocido como triple producto escalar) de 3 vectores, a la operación que combina el el producto escalar con el producto el producto vectorial para obtener como resultado un escalar. un escalar. Los triples productos aparecen cuando se desea definir multiplicaciones entre 3 vectores. Una expresión de la forma ( u · v · w) no tiene mucho sentido; porque el resultado del primer producto es un escalar y no es posible calcular el Producto Punto entre un número (escalar) y un vector. Sin embargo, cuando los vectores son
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La interpretación geométrica anterior proporciona un tercer criterio geométrico de estilo similar a los señalados para los otros productos: Tres vectores u, v y w, son coplanares son coplanares si y sólo si: u · (v x w) = 0. Lo anterior se sigue de que el volumen del paralelepípedo elepípedo tendrá volumen cero si y sólo si los vectores que los definen están en un mismo plano. Ejemplo: Sean 3 vectores u, v y w en el espacio el espacio vectorial R3: , y . Entonces, el Producto Triple combinado de los 3 vectores se calcula desarrollando ando el determinante de la siguiente matriz:
⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⟨ ⟨ ⟨ ⟨ ⟨ ⟨ ⃗ ⃗ ⃗
Si se emplea el método de los menores y cofactores, se tiene:
Dando como resultado:
Si tomamos el valor absoluto de |u · (v x w)| = |-12|, resulta 12; el cual es el volumen del paralelepípedo formado formado por los 3 vectores en el espacio euclidiano.
15.10. Cálculo Vectorial Vectorial El cálculo vectorial es un campo de las matemáticas referidas al análisis real multivariable de vectores en 2 o más dimensiones. Consiste en una serie de fórmulas y técnicas para solucionar problemas muy útiles para la la ingeniería y la física. Consideramos los campos vectoriales, que asocian un vector a cada punto en el espacio, y campos escalares, que asocian un escalar a cada punto en el espacio. Por ejemplo, la la temperatura de una piscina es un campo escalar: a cada punto asociamos un valor escalar de temperatura. El flujo de l agua en la misma piscina es un campo vectorial: a cada punto asociamos un vector de de velocidad. Cinco operaciones son importantes en el Cálculo Vectorial: 1. El Gradiente: mide la tasa y la dirección del cambio en un campo escalar; el gradiente de un campo escalar es calar es un campo vectorial. 2. El Rotor o Rotacional: mide la tendencia de un campo vectorial a rotar alrededor de un punto; el rotor de un campo vectorial es otro campo (seudo) vectorial. 3. La Divergencia: mide la tendencia de un campo vectorial a ―originarse en‖ ciertos puntos, ó a ―converger‖ hacia ciertos puntos; la divergenci a de un campo vectorial es un campo escalar.
jacobiano ano o al determinante de la matriz jacobiana. Tanto la matriz 4. El Jacobiano: En el cálculo vectorial, se llama determinante jacobiano ó simplemente jacobi
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hidráulica en todos sus aspectos, la aeronáutica, el trabajo no conservativo, la fricción, la conservación de la energía, la transferencia de calor, la entalpía ó energía de transporte, los campos electromagnéticos, cos, y muchos otros conceptos ingenieriles; se resuelven de manera fácil y sencilla mediante el Álgebra Vectorial.
15.11. Aplicaciones Ingenieriles del Álgebra Vectorial al 15.11.1. Análisis de fuerzas en estructuras Sabemos que la fuerza es una magnitud vectorial, puesto que el momento lineal lo es, y esto significa que tiene módulo, dirección y sentido. Al conjunto de fuerzas que actúan sobre un cuerpo se le llama sistema de fuerzas. Si las fuerzas tienen el mismo punto de aplicación se habla de fuerzas concurrentes. Si son paralelas paralelas y tienen distinto punto de aplicación se habla de fuerzas paralelas. Cuando sobre un objeto actúan varias fuerzas, éstas se suman vectorialmente vectorialmente para dar lugar a una fuerza total o resultante. Si la fuerza resultante es nula, el objeto no se acelerará: seguirá rá parado o detenido o continuará moviéndose con velocidad velocidad constante. Esto quiere decir que todo cuerpo permanece en estado de reposo o de movimiento rectilíneo y uniforme mientras no actúe sobre él una fuerza resultante no nula. Por ejemplo, si una persona empuja un triciclo con una fuerza de magnitud igual a la fuerza de rozamiento que se opone al movimiento del triciclo, las fuerzas se compensarán, produciendo una fuerza resultante nula. Eso hace que se mueva con velocidad constante. Si la persona deja de empujar el triciclo, la única fuerza que actúa sobre el artefacto es la fuerza de rozamiento, rozamiento, la cual le imprime una desaceleración, y su velocidad va disminuyendo hasta hacerse cero; es decir, hasta detenerse. Con frecuencia, sobre un cuerpo actúan simultáneamente simultáneamente varias fuerzas. Puede resultar muy complejo calcular por separado el efecto de cada una de las fuerzas; sin embargo, las fuerzas son vectores y se pueden sumar para formar una única fuerza total o resultante (R) que permite determinar el comportamiento del cuerpo. Si un objeto está en equilibrio, la fuerza total ejercida sobre él debe ser igual a cero. Un libro colocado sobre una mesa es atraído hacia abajo por la atracción gravitacional de la Tierra y es empujado hacia arriba por la repulsión molecular de la mesa. En éste caso, la suma de las 2 fuerzas es igual cero; por lo tanto, el libro está en equilibrio. Para calcular la fuerza total, hay que sumar s umar las fuerzas como vectores. En ingeniería se necesita saber cómo responden los materiales sólidos a fuerzas externas como la tensión, la compresión, la torsión, la flexión o la cizalladura. Los materiales sólidos responden a dichas fuerzas con una deformación elástica (en la que el material vuelve a su tamaño y forma originales cuando se elimina la fuerza externa), una deformación deformación permanente o una fractura. Los efectos de una fuerza externa dependientes del tiempo son la plastodeformación y la fatiga. a. La tensión es una fuerza de estiramiento ó elongación; por ejemplo, la fuerza que actúa sobre un cable que sostiene un peso. Bajo tensión, un material suele estirarse, y recupera su longitud original si la fuerza no supera el límite elástico del material. Bajo tensiones mayores, el material no vuelve completamente a su
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El conocimiento del esfuerzo de tensión, los límites elásticos y la resistencia de los materiales a la plastodeformación y la fatiga son extremadamente importantes en ingeniería.
15.11.2. Aplicaciones al cálculo de velocidad velocidad Si despreciamos la resistencia del aire, una pelota lanzada formando ángulo describe una parábola. La velocidad de la pelota ( v) tiene una componente vertical (vv) y otra horizontal (vh); la componente horizontal no cambia en ningún momento, mientras que la vertical, la única afectada por la gravedad, cambia de forma continua. 15.11.3. Aplicaciones en cálculo de movimiento movimiento bidimensional Utilizando vectores, se puede resolver gráficamente cualquier problema relacionado relacionado con el movimiento de un objeto bajo la influencia de varias fuerzas. El uso sencillo de los vectores así como los cálculos utilizando vectores quedan ilustrados en éste diagrama, que muestra el movimiento de una barca para atravesar una corriente de agua. El vector a, (ó, lo que es lo mismo, vector ), indica el movimiento de la barca durante un determinado periodo de tiempo si estuviera navegando en aguas tranquilas; el vector b, ó vector , representa el desvío o empuje de la corriente durante el mismo periodo de tiempo. El recorrido real de la barca, bajo la influencia de su propia propulsión y de la corriente, se representa con el vector c, ó vector . B c O
b a
A
La magnitud del vector puede graficarse de una manera proporcional a su valor real siempre y cuando hagamos uso de una escala adecuada, que puede ser un cierto número de centímetros (ó milímetros) por cada kilómetro. En el dibujo anterior, la velocidad al remar es de 3 km/h, el tiempo transcurrido es 1 hora y la escala es 1 cm por cada km. Por tanto, el vector vector mide 3 cm y representa 3 km. km. La velocidad de la corriente del río es de 4 km/h, km/h, y se representa con el vector vector que mide 4 cm, cm, lo que indica que la corriente recorre una distancia de 4 km en 1 hora. Éste segundo vector se dibuja con su origen en el e xtremo del vector a y en dirección paralela al movimiento de la corriente. El punto B, extremo del segundo vector, es la posición real de la barca después de una hora de viaje, y la distancia recorrida es la longitud del vector c, u
73
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Trabajo 3 (5%- 01 punto) 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
74
Qué es un Sistema S istema de Coordenadas? Cite, cuando menos, tres situaciones reales en e n las cuales se s e utilizan los Sistemas de Coordenadas. Coordenadas. Qué es una Escala? Qué es el Plano P lano Cartesiano? Cartesiano? A quién se debe el origen de ese es e nombre? Cómo se determina la Distancia entre dos puntos cualesquiera de un Plano Cartesiano? Qué es una Proporción y cómo se calcula? Dé al menos 3 ejemplos. Qué es el Porcentaje P orcentaje y cómo se calcula? Dé al menos 3 ejemplos. Qué es la Pendiente de una recta y cómo se calcula? Dé al menos 3 ejemplos. Qué es una Regla de Correspondencia? Qué es un Producto Cartesiano y cómo se define el Producto Cartesiano de 2 Conjuntos? Qué es el Dominio de una Función? Ilústrelo con 4 ejemplos sencillo e ncillos.s. Qué es el Rango o Recorrido de una Función? Ilústrelo con 4 ejemplos sencillos. Cómo se determina el Dominio y el Rango de una función? Brinde, cuando menos, 3 ejemplos para cada uno. Qué es una Función definida "A Trozos"? Defina la Función "Valor Absoluto". Qué es una Sucesión? Dé 3 ejemplos. Qué es una Progresión? Cuántos tipos de Progresiones existen? Defina cada una de ellas y brinde 3 ejemplos de cada una. diferente Orden: Dadas las siguientes Matrices, de diferenteOrden
A=
0 1
D=
E=
0 1
G=
* +
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19. Grafique en papel milimetrado, en cada uno de los casos, la terna de puntos dados P1(x1, y1), P2(x2, y2), y P3(x3, y3) y calcule el área A del triángulo P1P2P3 (ignorando siempre el signo algebraico) en base a la Aplicación Geométrica de Matrices (punto ―e.2‖ de la Guía de Matrices). Compare la solución obtenida por medio de la fórmula conocida para el Área de un Triángulo ( A = ½ (b.h) ó por medio del Método de ― conteo‖): P1(1, 2), P2(5, 2), y P3(5, 5) P1(3, 3), P2(8, 3), y P3(12, 8) P1(3, 4), P2(6, 7), y P3(14, 7) P1(1, 2), P2(6, 2), y P3(6, 7) P1(3, 2), P2(5, 7), y P3(13, 7) P1(3, 4), P2(3, 9), y P3(7, 9)
20. Demuestre que las siguientes ternas de puntos P1(x1, y1), P2(x2, y2), y P3(x3, y3), son colineales; es decir, que pertenecen a la misma recta y Grafique en papel milimetrado: P1(1, 2), P2(2, 4), y P3(4, 8) P1(0, 1), P2(1, 2), y P3(2, 3) P1(3, 0), P2(4, 1), y P3(5, 2) P1(1, 1), P2(2, 3), y P3(3, 5) P1(3, 3), P2(2, 2), y P3(1, 1) P1(0, 3), P2(1, 4), y P3(3, 6)
21. Resuelva, utilizando la Regla de Cramer, los siguientes Sistemas de Ecuaciones Lineales : a) b) c) d) e) f)
g)
h)
i)
j)
k)
l)
m)
n)
75
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⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗⃗ ⃗
⃗⃗ ⃗ ⃗⃗ ⃗ ⃗ ⃗
g) Calcule los Productos Mixtos y el volumen del paralelepípedo que forman los vectores dados, en cada caso: h) Demuestre la Identidad de Jacobi con los vectores dados, en cada caso:
⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗⃗⃗
i) Utilice los resultados obtenidos en la Pregunta Número 19; y compruebe que, efectivamente, es posible determinar el área A del triángulo dado a través de Matrices y Determinantes, así como a través de Vectores. Ayuda: Es importante recordar que, para desarrollar la Parte ―i‖ de la Pregunta Número 22, es preciso preciso que primeramente primeramente logres convertir, un par de lados vectoriales; es decir: cualesquiera del triángulo dado, en vectores y, luego, implementar cualquiera de las siguientes 2 relaciones de productos vectoriales -1 |a|.|b|.senα (donde α = cos {(a.b)/|a|.|b|}). e) Si utilizas el Producto Punto, la relación será: A = ½ |a|.|b|.senα f) Si utilizas el Producto Cruz, la relación será: A = ½ |a x b|. Ejemplo: Para la primera terna de puntos: P1(1, 2), P2(5, 2), y P3(5, 5). Bien podemos escoger como nuestros vectores a =P = P1P2 y b = P1P3. De ese modo: a = (x(x2-x1)i +(y2-y1) j j = (5-1)i +(2-2) j j = 4i +0 j b = (x(x3-x1)i +(y3-y1) j j = (5-1)i +(5-2) j j = 4i +3 j y Por lo que: |a| = a = √ [(4)2 + (0) 2] = √16 = 4; |b| = b = √ [(4)2 + (3) 2] = √25 = 5 = 20 y |a|.|b| = 4*5 =20
{16//20 20}} = Por otro lado, tenemos que: a.b = (4)(4) + (0)(3) = 16 + 0 = 16; con lo cual, obtenemos el ángulo entre los vectores así: α = cos-1 {(a.b)/|a|.|b|} = cos-1 {16 36,87°; entonces: sen α = 0,6. Finalmente: A = ½ |a|.|b|.sen
½ (20 (20)(0,6) )(0,6) = 6
76
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Modelo 3 (10%- 02 puntos)
77
INSTRUCCIONES a) Mantenga su Celular o Teléfono Completamente Apagado (NO en Silencio ni en Modo Vibrar) y no intente establecer ningún tipo de comunicación con alguno de sus compañeros. b) Dispone íntegramente íntegramente de sus s us dos (02) horas para desarrollar su Test; aprovéchelas. NO pierda tiempo!!! c) Cada ítem tiene un valor de cinco (04) puntos; es decir: 04 puntos por procedimiento y 01 punto por resultado correcto. d) Para éste evaluativo, usted debe estar en la capacidad de preparar un FORMULARIO adecuado, en caso de que lo requiera.
PLANTEAMIENTOS
1. Dadas las siguientes Matrices Cuadradas, de Orden 3:
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Bloque IV – Cálculo Infinitesimal Capítulo 16: Límites y Continuidad. Propiedades de los Límites. Álgebra de Límites Sir Isaac Newton fue un físico, matemático, astrónomo, alquimista, filósofo y alto funcionario administrativo inglés. Independientemente de Leibniz, fundó el Cálculo Infinitesimal y realizó importantes aportes al álgebra; dedujo la Ley de Gravitación Universal y realizó experimentos sobre la Naturaleza de la Luz y el Color. En Matemática, el Método de Newton lleva su nombre y en física, la Mecánica Newtoniana, con ayuda de la cual, entre otras cosas, se pudieron derivar matemáticamente las leyes de Kepler. En el año 1.665, Newton trabaja en su Teorema Fundamental del Cálculo y desarrolla su versión del Cálculo Infinitesimal, con el cual le brinda a la humanidad la herramienta más poderosa hasta hoy conocida dentro del campo de las Matemáticas. La obra de Isaac Newton representa una de las mayores contribuciones a la ciencia realizadas nunca por un solo individuo. Durante el siglo XVII tuvieron lugar los más importantes avances en las Matemáticas desde la era de Arquímedes y Apolonio. La base teórica que permitió a Sir Isaac Newton establecer sus leyes está también precisada en su magistral obra: Principios Matemáticos de la Filosofía de la Naturaleza. En dicho trabajo, el primer concepto que maneja es el de masa, que identifica con «cantidad de materia». La importancia de esta precisión está en que permite
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x f(x)
–0,9
–0,99
–0,999
–0,9999
–0,9999
3,51
3,9501
3,995001
3,99950001
3,99950001
... ...
–1
4
Por otro lado, si nos acercamos lo suficiente al valor x = –1, ahora por el lado derecho, vemos cómo la función f(x) ―tiende‖, de igual modo, al valor 4. En resumidas cuentas; los valores de la función están próximos a 4 para valores de x suficientemente cercanos a –1. Éste comportamiento de la función puede observarse gráficamente: gráficamente:
79
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Esto se representa matemáticamente: matemáticamente:
80
lí m f (x) (x ) = L
x→ a
16.2. Definición formal de Límite Para llegar a una definición formal del concepto de límite retomaremos el ejemplo anterior; en el cual dada la función f: R → R, definida por: f(x) = (3x2 – 3)/(x – 1); con dominio Dom f = {xε {xεR/ x ≠ 1}, se obtuvo que el límite, cuando x tiende a 1, es igual a 6. Para profundizar el significado de la expresión: f(x) tiende a 6 cuando x tiende a 1, se estudiará el comportamiento de las distancias entre ― x y 1‖ y entre ―f(x) y 6‖. Se agregan a las tablas confeccionadas confeccionadas anteriormente 2 filas ó renglones adicionales, las cuales estarán encabezadas por |x – 1 | y | f(x) – 6|.
x f(x) |x–1| |f(x) – 6|
0,9 5,7 0,1 0,3
x
1,1
0,99 5,97 0,01 0,03
0,999 5,997 0,001 0,003
0,9999 5,9997 0,0001 0,0003
1,01
1,001
1,0001
0,99999 5,99997 0,00001 0,00003 1,00001
... ... … …
1 6 1/10n 3/10n 1
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De éste modo, los valores de la función y = f(x) se aproximan al límite L, conforme x lo hace al número a. El valor absoluto de la diferencia entre f(x) y L puede hacerse tan pequeña como se desee, tomando x suficientemente cerca de ―a‖, pero nunca igual al valor a. su representación gráfica es la siguiente:
16.3. Propiedades 16.3. Propiedades de de Límites a. Si f es la función identidad f(x) = x, para cualquier valor de a, entonces el límite es igual al valor a.
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82
4)
lí m [ f(x) m g(x g( x ) = L1 * L2 ] = l ím f ( x ) * l í * g(x)
x→ a
x→ a
x→ a
5) l í m g(x g( x ) = L / m f ( x / ) l í m [ f(x) / ] = lí L2 ; con L2 ≠ 0. / g(x) 1 L x→ a
x→ a
x→ a
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3.Economía: 3.Economía: Un banco ofrece una tarjeta de crédito. Por datos obtenidos a lo largo del tiempo, han determinado que el porcentaje de cobranza de las que se otorgan en un mes cualquiera es función del tiempo transcurrido después de concederlas. Esta función es P = f(t) = 0,9(1 —3—0,08t), donde P es el porcentaje de cuentas por cobrar t meses después de otorgar la tarjeta. a)¿Qué a)¿Qué porcentaje se espera cobrar luego de 2 y 5 meses? (R: 14,51 % y 32 % ) . b)Si b)Si el número de meses transcurridos desde el otorgamiento ento de la tarjeta crece indefinidamente, damente, determine el % de las mismas que se espera es pera cobrar. (R: 90% ) . 4.Ingeniería 4.Ingeniería Industrial: Los ingenieros industriales han estudiado un trabajo particular en una línea de montaje. La función y = f(t) = 120 —80e—0,3t es la función de la
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10. Bacteriología: Matemáticamente, se ha logrado determinar que un cultivo de ciertas bacterias crece siguiendo la ley y = (1,25) / (1 + 0,25e –0,4t), donde el tiempo t ≥ 0 se mide en horas y el peso del cultivo en gramos. a)Determine a)Determine el peso del cultivo transcurridos transcurridos 60 minutos. (R: 1,07 g) . b)¿Cuál b)¿Cuál será el peso del mismo cuando el número de horas crece indefinidamente? (R: 1,25 g) . 11. 11. Estadística: En una academia de mecanografía, el número medio de palabras N por minuto escritas luego de t semanas de lecciones prácticas, está dado por N(t) = (157) / (1 + 5,4e –0,12t).
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Capítulo 17: Derivadas por Definición Definición y por Tablas En Matemáticas, todo concepto, toda idea y toda operación, tiene siempre su nombre preciso y conciso. Muchos de esos términos son adoptados del lenguaje cotidiano por sus similitudes o parecidos con un concepto tradicional y c omún. En algunos otros casos, términos y conceptos exclusivamente matemáticos se ―cuelan‖ al lenguaje cotidiano, a tal grado que con el paso del tiempo y su frecuente uso, las personas llegan a olvidar que el verdadero origen de dichos términos se encuentra, precisamente, en el mundo de las Matemáticas. Tal es el caso del Concepto de Derivada. Desde un punto de vista puramente geométrico, la Derivada de una función en un punto, es la pendiente de la recta tangente a la función, la cual pasa por dicho punto.
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y así sucesivamente.
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Se lee «derivada de f de x (ó de y) con respecto a x». Esta notación tiene la ventaja de sugerir a la derivada de una función con respecto res pecto a otra como un cociente de diferenciales. de diferenciales. La La notación de Leibniz es muy útil, por cuanto permite especificar la variable de diferenciación (en el denominador); lo cual es pertinente en caso de diferenciación de diferenciación parcial. También parcial. También facilita recordar la Regla de la Cadena , porque los términos « d» ―parecen‖ cancelarse simbólicamente:
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