U N S A
Universidad Nacional de San Agustín
Aplicaciones de la Derivada e Integral en Ingeniería Ambiental Alumno: Luigui Valdivia Guzman Docente: Mg. Oscar Parisaca Z – Matemática Matemática II Arequipa- Perú,2016
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Introducción………………………………………………………………………………………………………………..…………………1 1. Derivadas…………………………………………………………………………………………………………………………………2 ………………………………………………………………………………………………………………2 1.1. La derivada en sí ………………………………………………………………………………………………………………2 1.1.1.Tangente 1.1.1.Tangente a una curva…………………………………………………………………………………………..……2 1.1.2.Razón 1.1.2.Razón de Cambio puntual y velocidad instantánea……………………………………………………2 1.1.3.Definición 1.1.3.Definición……………………………………………………………………………………………………………….…3 1.2. Aplicaciones en general ……………………………………………………………………………………..……………3 1.2.1.Ejercicios 1.2.1.Ejercicios de razón de cambio…………………………………………………………………………………..3 1.2.2.Ejercicio 1.2.2.Ejercicio de tangente…………………………………………………………………………………………………4 1.2.3.Optimización 1.2.3.Optimización de funciones……………………………………………………………………..…………………5 1.2.3.1. Definición de extremos absolutos ………………………………………………..……………6 1.2.3.2. Definición de punto crítico……………………………………………………………..…………6 1.2.3.3. Máximos y mínimos absolutos en un intervalo cerrado…………….………………7 1.2.3.4. Intervalos no cerrados. ……………………………………………………………………….……8 1.3. La derivada y la Ingeniería Ambiental……………………………………………………………………….……11 2. Integrales………………………………………………………………………………………………………………………….……16 2.1. La integral…………………………………………………………………………………………………………………..…16 2.2. Aplicaciones de Integrales definidas en general ……………………………………………………….……16 2.2.1.Movimiento 2.2.1.Movimiento rectilíneo ………………………………………………………………………………….…………16 2.2.2.Áreas 2.2.2.Áreas de figuras planas……………………………………………………………………………………………17 2.2.3.Volúmenes 2.2.3.Volúmenes………………………………………………………………………………………………………..……20 2.2.3.1. Volúmenes De Solidos De Sección Conocida……………………………………………20 2.2.3.2. Volúmenes De Solidos De Revolución. ……………………………………………………21 2.2.3.2.1. Método De Los Discos…………………………………………………………………21 2.2.3.2.2. Método De Los Tubos. ……………………………………………………….………23 2.2.4.Cálculo 2.2.4.Cálculo de trabajo con ayuda de la integral definida………………………….……………………27 2.2.5.Presión 2.2.5.Presión y fuerza del fluido …………………………………………………………………………………….…29 …………………………………………………………………………..…32 2.3. La integral y la Ingeniería Ambiental …………………………………………………………………………..…32 3. Extremos de funciones de varias variables……………………………………………………………………………..34 4. Integrales Dobles…………………………………………………………………………………………………………..……….42 5. Integrales Triples……………………………………………………………………………………………………………..…….44 6. Conclusiones…………………………………………………………………………………………………………………..………49 7. Bibliografía………………………………………………………………………………………………………………….…………50
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De antemano agradezco al Mg. Oscar Parisaca Z por enseñar el curso de Matemática Matemática II en Ing. Ambiental, Ambiental, al hacer claros y sencillos los temas; a mi familia por el apoyo y sobre todo, a mi Padre Domingo Valdivia Huamani por el gran apoyo económico económico en los gastos diarios. Que Dios les bendiga y espero disfruten del tema que investigue y doy a conocer.
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Introducción. Este trabajo abarcara los temas aplicados con algunas definiciones de apoyo tanto para las derivadas como para las integrales; desarrollando en si sus aplicaciones y poniéndolos en práctica a través de ejercicios y ejemplos resueltos. Por lo que los objetivos del presente son:
Conocer las aplicaciones de la derivada, las más usadas y comunes Presentar como es que ayuda la derivada en el campo de la Ingeniería Ambiental mediante algunos ejemplos básicos Conocer las aplicaciones de las Integrales, las más usadas y comunes Presentar como es que ayuda la integral en el campo de la Ingeniería Ambiental mediante algunos ejemplos básicos
Cada capítulo estará resaltado en un enfoque teórico practico en donde cada tema tendrá un ejemplo de aplicación además de tener su respectivo procedimiento. Y un apartado para problemas que contengan temas ambientales y de interés de la carrera.
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Derivadas 1.1La derivada en sí. La derivada de una función puede interpretarse geométricamente como la pendiente de una curva, y físicamente como una razón “instantánea” de cambio. 1.1.1Tangente 1.1.1Tangente a una curva.
Supongamos que queremos hallar la tangente a una curva de ecuación cartesiana y D f .x/ en el punto .a; f .a//. La estrategia, usada primero por Pierre de Fermat y más tarde por Newton, consiste en aproximar la tangente por rectas secantes cuyas pendientes sí pueden calcularse directamente. En particular, consideremos la recta que une el punto .a; f .a// con un punto cercano, .x; f .x//, de la gráfica de f . Esta recta se llama una secante (recta que corta a la curva, pero no es tangente a la curva). La pendiente de esta secante es:
Observa que una secante es una buena aproximación de la tangente, siempre que el punto (x, f(x)) esté próximo a (a, f(a)). Estas consideraciones llevan a definir la tangente a la gráfica de f en el punto (a, f(a)) como la recta que pasa por dicho punto y cuya pendiente es igual al límite:
1.1.2 Razón de cambio puntual y velocidad instantánea.
Muchas leyes de la Física, la Química, la Biología o la Economía, son funciones que relacionan una variable “dependiente” y con otra variable “independiente” x, lo que suele escribirse en la forma y
=f(x). Si la variable independiente independiente cambia de un valor inicial a a otro x, la variable y lo hace de f (a) a f (x). La razón de cambio promedio de y= f(x) con respecto a x en el intervalo [a,x] es:
Con frecuencia interesa considerar la razón de cambio en intervalos cada vez más pequeños. Esto lleva a definir lo que podemos llamar “razón de cambio puntual de y Df .x/ con respecto a x en el punto a” como:
El ejemplo más conocido de esto que decimos es el de un móvil que se mueve a lo largo de una recta sobre la cual hemos elegido un origen. Sea s.t/ la posición del móvil en el tiempo t, es decir, la distancia con signo del móvil al origen en el tiempo t. La razón de cambio promedio tiene en este caso una interpretación física natural:
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Es la velocidad media del móvil en el intervalo de tiempo comprendido entre a y a C h. Parece intuitivo que, en cada instante, el móvil se mueve con una determinada velocidad instantánea. Pero no hay manera de medir directamente una velocidad instantánea; un instante quiere decir una posición en la recta: la velocidad instantánea del móvil para t D a es la velocidad que tiene cuando está en la posición s.a/. La velocidad instantánea es una abstracción de una característica física del movimiento, pero no es una magnitud que podamos observar directamente. La única definición razonable de velocidad instantánea es como la razón de cambio puntual:
1.1.3Definiciòn
Se dice que una función
es derivable en un punto a€ I , si existe el límite:
Dicho número L se llama derivada de f en a y lo representaremos por f’(a) (notación debida a
Lagrange).
1.2. Aplicaciones en general. 1.2.1 . Ejercicios de razón de cambio
Se empezará con unas aplicaciones sencillas sencillas donde se aplican las derivadas; basándose en calcular la tasa de variación de una magnitud cuando se conoce la tasa de variación de otra magnitud relacionada con ella. En este tipo de ejercicios la “tasa de variación” se interpreta como una derivada
y, en la mayoría de los casos, basta usar la regla de la cadena para obtener lo que se pide. Hay que elegir las unidades de acuerdo con los datos del problema; por ejemplo, si un volumen se mide en litros tendremos que medir longitudes con decímetros. Ejemplos: a) Se está llenando un depósito cónico apoyado en su vértice a razón de 9 litros por segundo. Sabiendo que la altura del depósito es de 10 metros y el radio de la tapadera de 5 metros, ¿con qué rapidez se eleva el nivel del agua cuando ha alcanzado una profundidad de 6 metros? V (t) Solución. Expresaremos todas las medidas en metros. Si V(t) = 1/3* πr(t)2 h(t) donde h(t) es la altura, medida desde el vértice, alcanzada por el agua en el tiempo t y r(t) es el radio de la sección transversal del cono a la distancia h(t) desde el
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Luego
ℎ , y
b) Una bola esférica de hielo se está derritiendo de forma uniforme en toda la superficie, a razón de 50 cm3 por minuto. ¿Con qué velocidad está disminuyendo el radio de la bola cuando este mide 15 cm?
1.2.2 . Ejercicio de tangente
El siguiente ejercicio es de cálculo de derivada y de tangente y normal. Cuando en un ejercicio intervienen parámetros, debes debes expresar las soluciones de la forma más sencilla posible. posible. a) Calcula las ecuaciones de las rectas tangente y normal a una hipérbola de ecuación cartesiana y 2 x 2 D 1, en un punto genérico (u,v) de la misma
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Estos problemas llamados de optimización, desde el punto de vista matemático se reducen a problemas de determinación de máximos y mínimos absolutos de funciones de una variable real en determinados intervalos Para resolver estos ejercicios deberás entonces extremar una función. Tu primer paso será entonces individualizar con claridad cuál es la función a la que debes hallarle el máximo y / o mínimo absoluto. En ocasiones el enunciado del ejercicio te proporciona la expresión analítica de esa función. En otros, en cambio, tú deberás conseguir esa expresión analítica utilizando
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1.2.3.2Definición 1.2.3.2Definición de punto crítico
Un punto crítico de una función es entonces un punto del dominio donde siendo continua la función, su derivada en nula o no existe (punto singular). Las figuras siguientes indican posibles andamientos de la función en un punto crítico.
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1.2.3.3Máximos 1.2.3.3Máximos y mínimos absolutos en un intervalo cerrado.
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En este punto es importante que recuerdes el teorema de Weierstrass: Teorema “ Una función continua en un intervalo cerrado tiene Máximo y Mínimo absolutos” Para
encontrar los extremos en un intervalo [ a , b ] bastará que: 1ro) Encuentres los puntos críticos pertenecientes al intervalo ( a ,b ). 2do) Calcules los valores funcionales en cada uno de ellos. 3ro) Calcules los valores funcionales en los extremos del intervalo , f(a) y f(b) . 4to) Compara todos esos valores funcionales . El mayor de ellos será el máximo absoluto y el menor , el mínimo absoluto. 1.2.3.4Intervalos 1.2.3.4Intervalos no cerrados.
En caso de que el intervalo no sea cerrado deberás clasificar los puntos críticos y auxiliarte con alguna información adicional para encontrar los extremos absolutos, como ser la variación de la función en todo el intervalo de estudio utilizando el signo de la derivada 1ra., límites en extremos abiertos o valores funcionales en extremos cerrados. En los ejercicios encontrarás muy comúnmente un único punto crítico en el intervalo de estudio. Si lo clasificas como máximo relativo, dada la continuidad de la función, podrás asegurar sin más que es el máximo absoluto de la función. La clasificación puedes hacerla tanto usando el criterio de la derivada 1ra. como el criterio de la derivada 2da. visto antes. Ejemplos. a) Utilizando una de las paredes laterales de un galpón (como indica la figura) se desea construir un corral para alimentar libremente a un grupo de terneros. El corral tendrá sección rectangular y se utilizarán tres tiros de alambre, el más bajo de los cuales se colocará a una altura tal que permita la entrada de los pequeños animales impidiendo la de los demás.
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Siendo el área de 100 m2 se cumplirá que: x.y = 100 (2)
.
El punto crítico corresponde pues al mínimo absoluto de la función en el
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b) Se desea construir un silo de forma cilíndrica rematado por una bóveda semiesférica. El costo de construcción por m 2 es doble en la bóveda que en la parte cilíndrica. Encuentra las dimensiones h y φ del silo de volumen V dado, de forma que el costo de construcción sea
mínimo. Solución. Sea R el radio de la base y h la altura de la
parte cilíndrica. Tendremos: Sup. lateral = 2π R h Sup. bóveda = 2 π R2
Costo de superficie lateral: A ( $ / m2 ) Costo de bóveda : 2 A ( $ / m2 m2 ) Costo total: CT = 2π R h A + 2 π R2 .2 A
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1.3. La derivada y la Ingeniería Ambiental Uno de los objetivos de un Ingeniero Ambiental, es implementar y contribuir con la sostenibilidad ambiental de los sectores productivos; implementando programas prioritarios, tales como, Producción Más Limpia y Procesos para Mercados Verdes, donde las razones de cambio son vitales para la evaluación del funcionamiento y eficacia de dichos programas. El Cálculo Diferencial cumple un papel fundamental en esa búsqueda de soluciones a los fenómenos de la naturaleza que nos afectan. El Cálculo aplica en muchos los fenómenos naturales. Un Ingeniero Ambiental debe conocer y aplicar conceptos numéricos para la realización de proyectos ambientales, debe interpretar los fenómenos de la naturaleza por medio de expresiones o modelos matemáticos, físicos y/o químicos relacionados con el ámbito ambiental. Un Ejemplo, el Ingeniero Ambiental aplica la derivada para operar sistemas de tratamientos de aguas residuales, para sistemas de recolección y tratamiento tr atamiento de residuos, para hacer estudios de contaminación, diagnósticos, evaluación y monitoreo de ecosistemas, entre muchas otras aplicaciones. aplicaciones. Por eso, el cálculo está considerado como una Ciencia Básica para un Ingeniero Ambiental, si no sabemos cálculo, no somos Ingenieros, es así de simple. sim ple.
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2) Estudios realizados han permitido determinar que el nivel medio diario C de monóxido de carbono CO2 en el aire , en partes por millón (ppm) , en una ciudad , está relacionado con la población p expresada en miles de habitantes por la siguiente expresión :
Solución. Como la concentración C es función de la población p y ésta es función del tiempo
t, resulta ser C función compuesta de t. Debes calcular la derivada de la concentración respecto del tiempo, para lo cual podemos previamente hallar la función compuesta y luego derivar. Tendremos entonces:
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Derivando (1) y (2) respecto de t obtienes:
Para = 3 : p = 4 ,
Sustituyendo estos valores en (3) reencontramos:
3) Una empresa de agua potable desea saber cuánta cantidad de agua necesita para proveer
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) 2, ∇ ∇ℎ, ℎ, ( , 2ℎ 4 4 2 2 ℎ 2,2ℎ Derivada de G Res Res ecto ecto a h
Teniendo los gradientes aplicamos LaGrange así:
∇ ∇, , ∇ Donde α es el multiplicador de LaGrange
Derivada de G Res Res ecto a r
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Integrales. 2.1La integral La integración es una de las herramientas más versátiles del Cálculo, sus aplicaciones no se limitan a calcular áreas de regiones planas o volúmenes de sólidos, también se utiliza para calcular longitudes de curvas, centros de masas, momentos de inercia, áreas de superficies, para representar magnitudes físicas como el trabajo, la fuerza ejercida por una presión, o la energía potencial en un campo de fuerzas. Una función f definida en [a, b] se dice integrable en [a, b] si existe el límite de las sumas de Riemann de f (cuando la norma de P tiende a 0) y denotamos este límite mediante
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Ejemplo: Una pelota de tenis se lanza vertical mente hacia abajo desde una altura de 54 pies con una velocidad inicial de 8pies/s ¿Cuál es la velocidad de impacto si la pelota golpea en la cabeza a una persona de 6 pies de estatura? Solución. En este caso a(t) = -32, s(0) = 54 y, puesto que la pelota
se lanza hacia abajo, v(0) = -8. Luego,
AI usar la velocidad inicial v(0) = -8 encontramos C1 = - 8. En consecuencia, v(t) = -32t - 8. AI continuar encontramos
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En particular, si la función esta expresada en forma paramétrica x = x(t), y = y(t), el área viene
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Si la región está limitada por dos curvas y = f(x), y = g(x) entre dos rectas horizontales y = c e y = d, consideramos las funciones inversas e integramos respecto a la variable y. El área se expresa entonces como:
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∈
2) Calcular el área comprendida entre las curvas y = 1/ cos 2 x, y = sen6 x para x [0, π/4]. 2 6 Solución. En este caso también la curva y = 1/ cos x queda por encima de y = sen x. Bastará pues integrar la resta de ambas funciones en el intervalo indicado
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En particular, si las secciones son perpendiculares perpendiculares al eje OX entre los valores x0 y x1,
Así, en el ejemplo de la figura tenemos una pirámide de base
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Si el eje de giro es el eje OX
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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The world’s largest digital library
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