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Ap_CalculoDiferencialUnaVariable.wxm
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Ap_CalculoDiferencialUnaVariable.wxm
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/* [wxMaxima batch file version 1] [ DO NOT EDIT BY HAND! ]*/
/* [ Created with wxMaxima version 22.04.0 ] */
/* [wxMaxima: title start ]
Ejemplos de aplicaciones del cálculo diferencial
a funciones de una variable
[wxMaxima: title end ] */
/* [wxMaxima: section start ]
Continuidad, derivabilidad, máximos y mínimos
[wxMaxima: section end ] */
/* [wxMaxima: caption start ]
EJERCICIO 1 SOBRE CONTINUIDAD, DERIVABILIDAD Y EXTREMOS
[wxMaxima: caption end ] */
/* [wxMaxima: image start ]
png
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
[wxMaxima: image end ] */
/* [wxMaxima: comment start ]
Comenzaremos dibujando la función.
IMPORTANTE: Los comandos draw2d y draw3d (más potentes que plot2d y plot3d) requieren cargar el paquete draw.
Sóla hay que hacerlo una vez, salvo que reiniciemos Maxima. Hay dos posibilidades para realizar la carga:
1. Manualmente mediante el comando load(draw)$
2. Que Maxima se ocupe de cargarlo automáticamente si lo necesita. Esta posibilidad está explicada en el "manualico"
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
wxdraw2d(proportional_axes=xy,
explicit( sqrt(1+x), x,-1,0),
explicit( sin(x)/x, x,0,2),
yrange=[-1,2])$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Aparentemente f es continua. Es clara la continuidad en [-1,0) y en (0,2] por tratarse de fórmulas
en las que aparecen funciones continuas. El punto 0 es el problemático porque en él
se "empalman" dos funciones continuas: ¿es f continua en x=0?
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
g(x):= sqrt(1+x); G(x):= sin(x)/x;
limit(g(x),x,0,minus);
tlimit(G(x),x,0,plus);
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
La función f es continua porque los límites
por la izquierda y por la derecha en 0 coinciden y valen 1.
Al igual que antes la derivabilidad esta garantizada, salvo el algún punto
que hay que analizar con más cuidado, y el valor de la derivada se
calcula fácilmente
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
diff(g(x),x);
diff(G(x),x);
wxdraw2d(explicit( diff(g(x),x), x,-0.9,0),
explicit(diff(G(x),x),x,0,2))$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
A la vista de las fórmulas (y de las gráficas) los puntos problemáticos son x=-1 y x=0.
Los analizamos por separado
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
limit( (g(0+h)-1)/h, h, 0, minus);
limit( (G(0+h)-1)/h, h, 0, plus);
print("Los límites laterales existen, pero no coinciden: f no es derivable en 0")$
limit( (g(-1+h)-1)/h, h, 0, plus);
print("En x=-1 f no es derivable")$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Como f es continua en el intervalo [-1,2] que es compacto, tiene extremos relativos.
En x=0 hay máximo absoluto y el mínimo absoluto está en x=-1. En x=2 hay un mínimo relativo.
[wxMaxima: comment end ] */
/* [wxMaxima: caption start ]
[wxMaxima: caption end ] */
/* [wxMaxima: image start ]
png
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
[wxMaxima: image end ] */
/* [wxMaxima: comment start ]
Comenzamos por dibujar la función en diferentes intervalos para hacernos
una idea sobre sus extremos en base al grafismo.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
f(x):=exp(x)+exp(-x)+2*cos(x);
wxdraw2d( proportional_axes=xy,
explicit(f(x), x,-3,3));
wxplot2d(f(x),[x,-3,3]);
print("Esta gráfica no está adecuadamente dimensionada: despista")$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
A la vista de las gráficas diríase que f tiene un mínimo absoluto en x=0 y que
no tiene máximo absoluto.
La figura muestra la simetría respecto al eje 0Y: evidente, pues cambiar x por -x en
la fórmula produce el mismo resultado.
Si nos facilita el estudio podemos suponer que x > = 0.
Que f no está acotada superiormente es evidente porque la exponencial
no está acotada superiormente.
Su desarrollo de Taylor limitado afianza la conjetura del mínimo relativo está el origen
y que su valor es 4.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
taylor(f(x),x,0,20);
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Los posibles extremos de f (función infinitamente derivable) son puntos críticos.
Le pedimos a Maxima que derive y resuelva la ecuación
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
diff(f(x),x);
solve(%,x);
print("Maxima parece no saber resolverla.
Es una ecuación difícil de resolver.
Por lo menos hay una solución sencilla x=0, ")$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Vamos a hacer más derivadas, a ver si la convexidad, Taylor... nos dan alguna idea
para abordar analíticamente el problema.
A nivel visual la función parece estrictamente creciente, y no sólo eso,
la pendiente parece ser cada vez mayor
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
f(x);
Df: diff(f(x),x);
diff(f(x),x,2);
diff(f(x),x,3);
diff(f(x),x,4);
print("Curioso, f coincide con su cuarta derivada")$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
La función f es la suma de dos exponenciales (> 0) y un coseno que en [0, %pi/2]
es positivo y por tanto también f, a partir de %pi/2 empieza a restar pero
exp(%pi/2) ya supera a 1 que es lo máximo que puede restar el coseno (y sigue creciendo).
Consecuencia f(x)> 0 para todo x ( podemos suponer x>0 por la simetría).
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
exp(%pi/2),numer;
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Luego diff(f(x),x,4)>0 para todo x.
En consecuencia diff(f(x),x,3) es estrictamente creciente. Y en x=0 vale 0.
En consecuencia diff(f(x),x,3)>0 en (0, inf)
Repitiendo el razonamiento se obtiene que diff(f(x),x,2)>0 en (0, inf). Y en x=0 vale 0.
Volviendo a repetir el razonamiento se obtiene que diff(f(x),x)>0 en (0, inf). Y en x=0 vale 0.
En resumen f alcanza en 0 su mínimo absoluto (relativo, en particular)
y en los demás puntos es estrictamente creciente, no existiendo máximo alguno
(ni absoluto, ni relativo).
[wxMaxima: comment end ] */
/* [wxMaxima: section start ]
Analizando la veracidad de algunas desigualdades
[wxMaxima: section end ] */
/* [wxMaxima: caption start ]
EJERCICIO SOBRE DESIGUALDADES
[wxMaxima: caption end ] */
/* [wxMaxima: image start ]
png
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
[wxMaxima: image end ] */
/* [wxMaxima: comment start ]
Comenzaremos haciendo un dibujo, por si nos resultara útil en los razonamientos.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
f(x):=log(1+x^2)$
g(x):=log(2)*x$
wxdraw2d( key_pos=top_left, /* Posición de la leyenda log(1+x^2)...*/
color=blue,
key="log(1+x^2)", /* Etiqueta de la leyenda */
explicit(f(x), x,0,4),
key="x*log(2)",
color=red,
explicit(g(x), x,0,4))$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
El dibujo confirma lo que deseamos demostrar.
Vamos a dibujar las funciones en un intervalo más grande.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
wxdraw2d( key_pos=top_left,
color=blue,
key="log(1+x^2)",
explicit(f(x), x,-1,6),
key="x*log(2)",
color=red,
explicit(g(x), x,-1,6))$
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
/* Los punto de corte de f y g */
solve(f(x)=g(x),x);
print("Para un humano es claro que x=0 y x=1 son soluciones.
Pero hay más, la gráfica lo muestra.
Maxima no se atreve a dar soluciones de la ecuación.
La gráfica nos permite ayudarle a obtener, al menos, soluciones aproximadas")$
find_root(f(x)=g(x),x,-1,0.5);
find_root(f(x)=g(x),x, 0.5,1.5);
find_root(f(x)=g(x),x, 2, 5);
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Visualmente la función f parece cóncava para x>1 y convexa para x<1.
De ser así las desigualdades estarían claras:
(1) porque la secante entre dos puntos de la curva se sitúa por encima
de ella cuando hay convexidad
(2) porque la secante se sitúa por debajo en caso de concavidad
[wxMaxima: comment end ] */
/* [wxMaxima: comment start ]
Calculemos la derivada segunda de f y sus ceros a fin de analizar la convexidad
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
print("La derivada segunda de f es")$
D2f: diff(f(x),x,2);
print("que, operando, se transforma en")$
D2f,fullratsimp;
/* El denominador siempre es positivo, pero el numerador cambia de signo,
¿donde se anula el numerador y por tanto la derivada segunda de f?
La respuesta es muy fácil para usted (y para Maxima) */
solve(%,x);
print("son los ceros de la derivada segunda de f")$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Resulta entonces claro de lo anterior que:
A) la segunda derivada de f es positiva en [0,1], siendo f convexa en dicho intervalo,
B) la segunda derivada de f es negativa en [1,4], siendo f cóncava en dicho intervalo.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
/* Pedimos a Maxima que lo compruebe, sólo por ponerlo a prueba */
assume(0 <x, x<1)$ is (D2f >0);
forget(0 <x, x<1)$ assume(x>1,x<4)$ is (D2f <0);
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
La función g corresponde a la secante que une los puntos de coordenadas (0,0) y (1,log(2))
y siendo f convexa en dicho intervalo se tiene demostrada la primera desigualdad
por las propiedades de las funciones convexas.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
wxdraw2d( key_pos=top_left,
color=blue,
key="log(1+x^2)",
explicit(f(x), x,0,1),
key="x*log(2)",
color=red,
explicit(g(x), x,0,1))$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
La función g puede ser vista también como una secante que une los puntos
(1,log(2)) y (c,log(1+c^2)) para cierto punto c situado visualmente en el dibujo
entre 4 y 5 y que corresponde, como antes hemos visto a 4.257461914447933
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
wxdraw2d( key_pos=top_left,
color=blue,
key="log(1+x^2)",
explicit(f(x), x,1,4.5),
key="x*log(2)",
color=red,
explicit(g(x), x,1,4.5))$
/* [wxMaxima: input end ] */
/* [wxMaxima: caption start ]
Otro ejercicio sobre desigualdades
[wxMaxima: caption end ] */
/* [wxMaxima: image start ]
png
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
[wxMaxima: image end ] */
/* [wxMaxima: comment start ]
Probar esta desigualdad es equivalente a probar que la función
f(x):=tan(x) * sin(x) - x^2
verifica f(x)>0 en (0,%pi/2) y eso es lo que probaremos.
[wxMaxima: comment end ] */
/* [wxMaxima: subsect start ]
Una forma de abordar el problema
[wxMaxima: subsect end ] */
/* [wxMaxima: input start ] */
f(x):=tan(x) * sin(x) - x^2;
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Están claros los límites laterales de f en 0 y %pi/2
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
f(0);
limit(f(x),x,%pi/2,minus);
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
A tenor de lo anterior dibujamos la gráfica de f para visualizar si
la desigualdad es razonable. La gráfica no deja lugar a dudas
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
wxdraw2d( explicit(f(x), x,0,%pi/2-0.2 )); /* algo menos de %pi/2 para evitar la asíntota */;
wxdraw2d( explicit(f(x), x,0,0.2 )); /* cerca de 0 */;
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Parece que f es estrictamente creciente. Si así fuera,
como en f(0)= 0, habríamos acabado.
Calculemos entonces f'(x) para ver si lo conseguimos.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
diff(f(x),x);
/* a ver si simplificando fuera más fácil verlo */
trigsimp(%);
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
No está claro que sea f'>0 en (0, %pi/2), ¿cierto?
1. Parece convexa: ¿lo será?
2. Si lo fuera ¿podemos garantizar que f(x)> 0 en (0, %pi/2)?
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
diff(f(x),x,2);
trigsimp(%);
factor(%);
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
A la vista de las cuentas que acabamos de realizar
(porque visualmente no hay dudas)
¿podemos responder a las dos preguntas anteriores?
SU TURNO
[wxMaxima: comment end ] */
/* [wxMaxima: subsect start ]
¿Disponemos de más herramientas?
[wxMaxima: subsect end ] */
/* [wxMaxima: comment start ]
Las funciones trigonométricas que figuran en la definición f son las que
están creando las dificultades para poder responder a las preguntas
(naturales a la luz de la gráfica) que nos hemos ido planteando.
Si f hubiera sido un polinomio, quizá hubiera sido más sencillo
¿Polinomio, polinomios...? ¡¡Se me ocurre usar otra herramienta!!
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
taylor(f(x), x, 0, 7 ); /* desarrollo de Taylor en 0 hasta grado 7 */
/* observe la etiqueta en la output */;
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Pero eso significa que f(x)=x^4/6 + R siendo R el resto de Lagrange
de orden 4 en un punto c situado entre 0 y x.
Vamos a calcularlo.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
diff(f(x),x,4);
print("¡Socorro! ¿Alguien puede prestar ayuda?")$
D4: trigsimp( diff(f(x),x,4) );
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
R4: ev( D4 , x=c)*x^4/4! ;
print("No podemos evaluar R, pero es claro que R > 0
¿por qué? ")$
print("Resumiendo: f(x)>0 en (0,%pi/2).
La desigualdad era cierta (la gráfica permitía visualizarlo) ")$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Hacer el desarrollo de Taylor hasta el orden 3, ¿serviría?
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
trigsimp( diff(f(x),x,3) );
D3: ev(%, x=c); /* derivada tercera en c */
print("El valor de f(x) coincide con: ")$
taylor(f(x),x,0,3) + D3*x^3/3!;
print("Claramente esa expresión es > 0 cuando x y c pertenecen a (0, %pi/2)")$
/* [wxMaxima: input end ] */
/* Old versions of Maxima abort on loading files that end in a comment. */
"Created with wxMaxima 22.04.0"$