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="#235ADE", size=1)+
theme_bw()
ggplotly(graf_1a)der_exacta <- function (x){2*exp(2*x)+2*sin(2*x)}
der_h_1<- fderiv(f, x, n=1, h=1, method="central")
der_h_0.05<- fderiv(f, x, n=1, h=0.5, method="central")
der_h_0<- fderiv(f, x, n=1, h=0, method="central")
graf_der_1a <- ggplot()+
geom_line(aes(x,der_exacta(x)), color="black")+
geom_line(aes(x,der_h_1), color="orange")+
geom_line(aes(x,der_h_0.05), color="pink")+
geom_line(aes(x,der_h_0), color="green")+
theme_bw()
ggplotly(graf_der_1a)error_h1 <- abs(der_exacta(x)-der_h_1)
ggplot()+
geom_line(aes(x, error_h1), color="forestgreen")+
theme_bw()
error_h0 <- abs(der_exacta(x)-der_h_0)
ggplot()+
geom_line(aes(x, error_h0), color="firebrick", 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, length.out=100)
y <- f(x)
graf_1b <- ggplot()+
geom_line(aes(x,y), color="#235ADE")+
theme_bw()
ggplotly(graf_1b)der_exacta <- function(x){1/(x+2)-2*(x+1)}
der_h_1.5<- fderiv(f, x, n=1, h=1.5, method="central")
der_h_1<- fderiv(f, x, n=1, h=1, method="central")
der_h_0.05<- fderiv(f, x, n=1, h=0.5, method="central")
der_h_0<- fderiv(f, x, n=1, h=0, method="central")
graf_der_1b <- ggplot()+
geom_line(aes(x,der_exacta(x)), color="black")+
geom_line(aes(x,der_h_1.5), color="red")+
geom_line(aes(x,der_h_1), color="orange")+
geom_line(aes(x,der_h_0.05), color="pink")+
geom_line(aes(x,der_h_0), color="green")+
theme_bw()
ggplotly(graf_der_1b)error_h1 <- abs(der_exacta(x)-der_h_1.5)
ggplot()+
geom_line(aes(x, error_h1), color="forestgreen")+
theme_bw()
error_h0 <- abs(der_exacta(x)-der_h_0)
ggplot()+
geom_line(aes(x, error_h0), color="firebrick", 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="#235ADE", size=1)+
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_0.05<- fderiv(f, x, n=1, h=0.5, method="central")
der_h_0<- fderiv(f, x, n=1, h=0, method="central")
graf_der_c <- ggplot()+
geom_line(aes(x,der_exacta(x)), color="black")+
geom_line(aes(x,der_h_1), color="orange")+
geom_line(aes(x,der_h_0.05), color="pink")+
geom_line(aes(x,der_h_0), color="green")+
theme_bw()
ggplotly(graf_der_c)error_h1 <- abs(der_exacta(x)-der_h_1)
ggplot()+
geom_line(aes(x, error_h1), color="forestgreen")+
theme_bw()
error_h0 <- abs(der_exacta(x)-der_h_0)
ggplot()+
geom_line(aes(x, error_h0), color="firebrick", 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="#235ADE", size=1)+
theme_bw()
ggplotly(graf_1d)der_exacta <- function (x){-3*sin(6*x)-2*exp(2*x)}
der_h_1<- fderiv(f, x, n=1, h=1, method="central")
der_h_0.05<- 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="black")+
geom_line(aes(x,der_h_1), color="orange")+
geom_line(aes(x,der_h_0.05), color="pink")+
geom_line(aes(x,der_h_0), color="green")+
theme_bw()
ggplotly(graf_der_1d)error_h1 <- abs(der_exacta(x)-der_h_1)
ggplot()+
geom_line(aes(x, error_h1), color="forestgreen")+
theme_bw()
error_h0 <- abs(der_exacta(x)-der_h_0)
ggplot()+
geom_line(aes(x, error_h0), color="firebrick", 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=0.5, to=1, length.out=100)
y <- f(x)
graf_2a <- ggplot()+
geom_line(aes(x,y), color="#235ADE", size=1)+
geom_area(aes(x,y), fill="red", alpha=0.5)+
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, 0.5,1, method="Simpson")## [1] 0.19375Función Newton-Cotes
Nodos=8
cotes(f, 0.5, 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=0.5, length.out=100)
y <- f(x)
graf_2b <- ggplot()+
geom_line(aes(x,y), color="#235ADE", size=1)+
geom_area(aes(x,y), fill="red", alpha=0.5)+
theme_bw()
ggplotly(graf_2b)Función integral
Método: Kronrod
pracma::integral(f, 0,0.5, method="Kronrod")## [1] -0.2670628Método: Clenshaw
pracma::integral(f, 0,0.5, method="Clenshaw")## [1] -0.2670628Método: Simpson
pracma::integral(f, 0,0.5, method="Simpson")## [1] -0.2670628Función Newton-Cotes
Nodos=8
cotes(f, 0, 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="#235ADE", size=1)+
geom_area(aes(x,y), fill="red", alpha=0.5)+
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] 0.1922594Función Newton-Cotes
Nodos=8
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="#235ADE", size=1)+
geom_area(aes(x,y), fill="red", alpha=0.5)+
theme_bw()
ggplotly(graf_2d)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] -0.06471697Función Newton-Cotes
Nodos=8
cotes(f, 0, 1)## [1] NaN##Este método no ha funcionado para calcular esta integral debido a la forma en la que ha sido programado.
\[\begin{equation} \int_{1}^{1.6} \frac{2x}{x^2-4} dx \end{equation}\]
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="#235ADE", size=1)+
geom_area(aes(x,y), fill="red", alpha=0.5)+
theme_bw()
ggplotly(graf_2e)Función integral
Método: Kronrod
pracma::integral(f, 1,1.6, method="Kronrod")## [1] -0.7339692Método: Clenshaw
pracma::integral(f, 1,1.6, method="Clenshaw")## [1] -0.7339692Método: Simpson
pracma::integral(f, 1,1.6, method="Simpson")## [1] -0.7339692Función Newton-Cotes
Nodos=8
cotes(f, 1, 1.6)## [1] -0.7339692\[\begin{equation} \int_{0}^{\pi/4} e^{3x}sin(2x) dx \end{equation}\]
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="#235ADE", size=1)+
geom_area(aes(x,y), fill="red", alpha=0.5)+
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] 2.588629Función Newton-Cotes
Nodos=8
cotes(f, 0, pi/4)## [1] 2.588629