Ejercicio 1.
Para cada una de las siguientes funciones realiza la respectiva gráfica en el intervalo dado. Compara las gráficas de las derivadas aproximadas (con dos tamaños de paso \(h\neq 0\) y \(h=0\)) y la derivada exacta en tal intervalo.
- \(f(x)=e^{2x}-cos 2x\), \(x\in [0,2]\)
f<- function(x){exp(2*x)-cos(2*x)}
x<- seq(from=0, to=2, length.out=100)
y<- f(x)
graf_1a<- ggplot()+
geom_line(aes(x, y), color="yellow", size=.5)+
theme_bw()
ggplotly(graf_1a)der_exacta <- function(x){2*exp(2*x)+2*sin(2*x)}
der_h1.25 <- fderiv(f, x, n=1, h=1.25, method = "central")
der_h1 <- fderiv(f, x, n=1, h=1, method = "central")
der_h.75 <- fderiv(f, x, n=1, h=0.75, method = "central")
der_h0 <- fderiv(f, x, n=1, h=0, method = "central")
graf_der_1a <- ggplot()+
geom_line(aes(x, der_exacta(x)), color= "red", size=1.2) +
geom_line(aes(x, der_h1.25), color= "blue", size=1) +
geom_line(aes(x, der_h1), color= "pink") +
geom_line(aes(x, der_h.75), color= "green") +
geom_line(aes(x, der_h0), color= "purple") +
theme_bw()
ggplotly(graf_der_1a)Se puede observar que la curva purpura se aproxima más a la función de \(f(x)=e^{2x}-cos 2x\) que es donde h equivale a un tamaño de paso de 0
error_h1.25 <-abs (der_exacta(x)-der_h1.25)
ggplot()+
geom_line(aes(x, error_h1.25), color="brown")+
theme_bw()
error_h0 <-abs (der_exacta(x)-der_h0)
ggplot()+
geom_line(aes(x, error_h0), color="brown", size=1.2)+
theme_bw()
- \(f(x)=log(x+2)-(x+1)^2\), \(x\in [0,5]\)
f<- function(x){log(x+2)-(x+1)^2}
x<- seq(from=0, to=5, by=.005)
y<- f(x)
graf_1b<- ggplot()+
geom_line(aes(x, y), color="red", size=.5)+
theme_bw()
ggplotly(graf_1b)der_exacta <- function(x){1/(x+2)-2*(x+1)}
der_h1.5 <- fderiv(f, x, n=1, h=1.5, method = "central")
der_h1 <- fderiv(f, x, n=1, h=1, method = "central")
der_h.5 <- fderiv(f, x, n=1, h=0.5, method = "central")
der_h0 <- fderiv(f, x, n=1, h=0, method = "central")
graf_der_1b <- ggplot()+
geom_line(aes(x, der_exacta(x)), color= "red", size=1.25) +
geom_line(aes(x, der_h1.5), color= "gold", size=1) +
geom_line(aes(x, der_h1), color= "pink") +
geom_line(aes(x, der_h.5), color= "green") +
geom_line(aes(x, der_h0), color= "purple") +
theme_bw()
ggplotly(graf_der_1b)La línea amarilla con una reducción en un tamaño de paso de 1.5, sin embargo, es solamente una línea paralela a las de otros tamaños por lo que a paser de las diferentes reducciones la aproximación será bastante buena
error_h1.5 <-abs (der_exacta(x)-der_h1.5)
ggplot()+
geom_line(aes(x, error_h1.5), color="brown")+
theme_bw()
error_h0 <-abs (der_exacta(x)-der_h0)
ggplot()+
geom_line(aes(x, error_h0), color="brown", size=1.2)+
theme_bw()
- \(f(x)=x\, sen\,x+x^2cos\,x\), \(x\in [0,\pi]\)
f<- function(x){x*sin(x)+x^2*cos(x)}
x<- seq(from=0, to=pi, length.out=100)
y<- f(x)
graf_1c<- ggplot()+
geom_line(aes(x, y), color="grey", size=.5)+
theme_bw()
ggplotly(graf_1c)der_exacta <- function(x){3*x*cos(x)+(1-x^2)*sin(x)}
der_h_1 <- fderiv(f, x, n=1, h=1, method = "central")
der_h_0.5 <- fderiv(f, x, n=1, h=0.5, method = "central")
der_h_0 <- fderiv(f, x, n=1, h=0, method = "central")
graf_der_1c <- ggplot()+
geom_line(aes(x, der_exacta(x)), color= "red", size=1.2) +
geom_line(aes(x, der_h_1), color= "gold") +
geom_line(aes(x, der_h_0.5), color= "pink") +
geom_line(aes(x, der_h_0), color= "brown") +
theme_bw()
ggplotly(graf_der_1c)Si reducimos el tamaño de paso a 0 vemos que la aproximación es practicamente total para la función \(f(x)=x\, sen\,x+x^2cos\,x\)
error_h1 <-abs (der_exacta(x)-der_h_1)
ggplot()+
geom_line(aes(x, error_h1), color="brown")+
theme_bw()
error_h0 <-abs (der_exacta(x)-der_h_0)
ggplot()+
geom_line(aes(x, error_h0), color="brown", size=1.2)+
theme_bw()
- \(f(x)=(cos\,3x)^2-e^{2x}\), \(x\in [0,\pi/2]\)
f<- function(x){cos(3*x)^2-exp(2*x)}
x<- seq(from=0, to=pi/2, length.out=100)
y<- f(x)
graf_1d<- ggplot()+
geom_line(aes(x, y), color="blue", size=.5)+
theme_bw()
ggplotly(graf_1d)der_exacta <- function(x){-3*sin(6*x)-exp(2*x)*2}
der_h_1 <- fderiv(f, x, n=1, h=1, method = "central")
der_h_0.5 <- fderiv(f, x, n=1, h=0.5, method = "central")
der_h_0 <- fderiv(f, x, n=1, h=0, method = "central")
graf_der_1d <- ggplot()+
geom_line(aes(x, der_exacta(x)), color= "red", size=1.2) +
geom_line(aes(x, der_h_1), color= "blue") +
geom_line(aes(x, der_h_0.5), color= "pink") +
geom_line(aes(x, der_h_0), color= "green") +
theme_bw()
ggplotly(graf_der_1d)Se observa tanto en la gráfica superior como en la de error que la aproximación de las rectas no es tan buena para la función \(f(x)=(cos\,3x)^2-e^{2x}\) a excepción de cuando h vale 0
error_h1 <-abs (der_exacta(x)-der_h_1)
ggplot()+
geom_line(aes(x, error_h1), color="brown")+
theme_bw()
error_h0 <-abs (der_exacta(x)-der_h_0)
ggplot()+
geom_line(aes(x, error_h0), color="brown", size=1.2)+
theme_bw()
Ejercicio 2
Da el valor aproximado (por medio de las funciones integral y cotes, del package pracma) y exacto (en caso de ser posible) de las siguientes integrales (realiza la respectiva gráfica).
\[\begin{equation} \int_{0.5}^1 x^4 dx \end{equation}\]
f<-function(x){x^4}
x<- seq(from=.5, to=1, length.out=100)
y<- f(x)
graf_2a <-ggplot()+
geom_line(aes(x, y), color="red")+
geom_area(aes(x, y), fill="blue", alpha=.5)+
theme_bw()
ggplotly(graf_2a)Función integral
Método: Kronrod
pracma::integral(f, 0.5, 1, method= "Kronrod")## [1] 0.19375Método: Clenshaw
pracma::integral(f, 0.5, 1, method= "Clenshaw")## [1] 0.19375Método: Simpson
pracma::integral(f, .5, 1, method= "Simpson")## [1] 0.19375Método: Newton-Cotes
2 nodos
cotes(f, .5, 1)## [1] 0.19375\[\begin{equation} \int_{0}^{0.5} \frac{2}{x-4} dx \end{equation}\]
f<-function(x){2/(x-4)}
x<- seq(from=0, to=.5, length.out=100)
y<- f(x)
graf_2b <-ggplot()+
geom_line(aes(x, y), color="red")+
geom_area(aes(x, y), fill="blue", alpha=.5)+
theme_bw()
ggplotly(graf_2b)Función integral
Método: Kronrod
pracma::integral(f, 0, .5, method= "Kronrod")## [1] -0.2670628Método: Clenshaw
pracma::integral(f, 0, .5, method= "Clenshaw")## [1] -0.2670628Método: Simpson
pracma::integral(f, 0, .5, method= "Simpson")## [1] -0.2670628Método: Newton-Cotes
2 nodos
cotes(f, 0, .5)## [1] -0.2670628\[\begin{equation} \int_{1}^{1.5} x^2\, log(x) dx \end{equation}\]
f<-function(x){x^2*log(x)}
x<- seq(from=1, to=1.5, length.out=100)
y<- f(x)
graf_2c <-ggplot()+
geom_line(aes(x, y), color="green")+
geom_area(aes(x, y), fill="brown", alpha=.5)+
theme_bw()
ggplotly(graf_2c)Función integral
Método: Kronrod
pracma::integral(f, 1, 1.5, method= "Kronrod")## [1] 0.1922594Método: Clenshaw
pracma::integral(f, 1, 1.5, method= "Clenshaw")## [1] 0.1922594Método: Simpson
pracma::integral(f, 1, 1.5, method= "Simpson")## [1] 0.1922594Método: Newton-Cotes
2 nodos
cotes(f, 1, 1.5)## [1] 0.1922594\[\begin{equation} \int_{0}^{1} x^2 e^{-x}\, log(x) dx \end{equation}\]
f<-function(x){x^2*exp(-x)*log(x)}
x<- seq(from=0, to=1, length.out=100)
y<- f(x)
graf_2d <-ggplot()+
geom_line(aes(x, y), color="red")+
geom_area(aes(x, y), fill="blue", alpha=.5)+
theme_bw()
ggplotly(graf_2d)## Warning: Removed 1 rows containing missing values (position_stack).Función integral
Método: Kronrod
pracma::integral(f, 0, 1, method= "Kronrod")## [1] -0.06471696Método: Clenshaw
pracma::integral(f, 0, 1, method= "Clenshaw")## [1] -0.06471696Método: Simpson
pracma::integral(f, 0, 1, method= "Simpson")## [1] -0.06471697Método: Newton-Cotes
2 nodos
cotes(f, 0, 1)## [1] NaN\[\begin{equation} \int_{1}^{1.6} \frac{2x}{x^2-4} dx \end{equation}\]
f<-function(x){(2*x)/x^2-4}
x<- seq(from=1, to=1.6, length.out=100)
y<- f(x)
graf_2e <-ggplot()+
geom_line(aes(x, y), color="gold")+
geom_area(aes(x, y), fill="black", alpha=.5)+
theme_bw()
ggplotly(graf_2e)Función integral
Método: Kronrod
pracma::integral(f, 1, 1.6, method= "Kronrod")## [1] -1.459993Método: Clenshaw
pracma::integral(f, 1, 1.6, method= "Clenshaw")## [1] -1.459993Método: Simpson
pracma::integral(f, 1, 1.6, method= "Simpson")## [1] -1.459993Método: Newton-Cotes
2 nodos
cotes(f, 1, 1.6)## [1] -1.459993\[\begin{equation} \int_{0}^{\pi/4} e^{3x}sin(2x) dx \end{equation}\]
f<-function(x){exp(3*x)*sin(2*x)}
x<- seq(from=0, to=pi/4, length.out=100)
y<- f(x)
graf_2f <-ggplot()+
geom_line(aes(x, y), color="green")+
geom_area(aes(x, y), fill="purple", alpha=.3)+
theme_bw()
ggplotly(graf_2f)Función integral
Método: Kronrod
pracma::integral(f, 0, pi/4, method= "Kronrod")## [1] 2.588629Método: Clenshaw
pracma::integral(f, 0, pi/4, method= "Clenshaw")## [1] 2.588629Método: Simpson
pracma::integral(f, 0, pi/4, method= "Simpson")## [1] 2.588629Método: Newton-Cotes
2 nodos
cotes(f, 0, pi/4)## [1] 2.588629