Tarea 6. Diferenciación e integración numérica.

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.

1. \(f(x)=e^{2x}-cos 2x\), \(x\in [0,2]\)

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="brown", 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="purple")+
  geom_line(aes(x,der_h_0.05), color="pink")+
  geom_line(aes(x,der_h_0), color="yellow")+
  theme_bw()
ggplotly(graf_der_1a)

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="red", size=1.2)+
  theme_bw()

2. \(f(x)=log(x+2)-(x+1)^2\), \(x\in [0,5]\)

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="firebrick")+
  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="green")+
  geom_line(aes(x,der_h_1.5), color="orange")+
  geom_line(aes(x,der_h_1), color="blue")+
  geom_line(aes(x,der_h_0.05), color="pink")+
  geom_line(aes(x,der_h_0), color="yellow")+
  theme_bw()
ggplotly(graf_der_1b)

error_h1 <- abs(der_exacta(x)-der_h_1.5)

ggplot()+
  geom_line(aes(x, error_h1), color="purple")+
  theme_bw()

3. \(f(x)=x\, sen\,x+x^2cos\,x\), \(x\in [0,\pi]\)

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="forestgreen", 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="brown")+
  geom_line(aes(x,der_h_1), color="red")+
  geom_line(aes(x,der_h_0.05), color="purple")+
  geom_line(aes(x,der_h_0), color="blue")+
  theme_bw()
ggplotly(graf_der_c)

error_h1 <- abs(der_exacta(x)-der_h_1)

ggplot()+
  geom_line(aes(x, error_h1), color="#235ADE")+
  theme_bw()

error_h0 <- abs(der_exacta(x)-der_h_0)

ggplot()+
  geom_line(aes(x, error_h0), color="pink", size=1.2)+
  theme_bw()

4. \(f(x)=(cos\,3x)^2-e^{2x}\), \(x\in [0,\pi/2]\)

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="firebrick", size=1)+
  theme_bw()
ggplotly(graf_1d)

error_h1 <- abs(der_exacta(x)-der_h_1)

ggplot()+
  geom_line(aes(x, error_h1), color="blue")+
  theme_bw()

error_h0 <- abs(der_exacta(x)-der_h_0)

ggplot()+
  geom_line(aes(x, error_h0), color="red", size=1)+
  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=1.5, length.out=100)
y <- f(x)

graf_2a <- ggplot()+
  geom_line(aes(x,y), color="gold", size=1)+
  geom_area(aes(x,y), fill="purple", alpha=0.5)+
  theme_bw()
ggplotly(graf_2a)

Metodo clenshaw

pracma::integral(f, 0.5,1, method="Clenshaw")

## [1] 0.19375

Metodo simpson

pracma::integral(f, 0.5,1, method="Simpson")

## [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=1, length.out=100)
y <- f(x)

graf_2b <- ggplot()+
  geom_line(aes(x,y), color="blue", size=1.5)+
  geom_area(aes(x,y), fill="pink", alpha=0.5)+
  theme_bw()
ggplotly(graf_2b)

Metodo clenshaw

pracma::integral(f, 0,0.5, method="Clenshaw")

## [1] -0.2670628

Metodo simpson

pracma::integral(f, 0,0.5, method="Simpson")

## [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="purple", size=1)+
  geom_area(aes(x,y), fill="red", alpha=0.5)+
  theme_bw()
ggplotly(graf_2c)

Metodo clenshaw

pracma::integral(f, 1,1.5, method="Clenshaw")

## [1] 0.1922594

Metodo simpson

pracma::integral(f, 1,1.5, method="Simpson")

## [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="yellow", size=1)+
  geom_area(aes(x,y), fill="pink", alpha=0.5)+
  theme_bw()
ggplotly(graf_2d)

## Warning: Removed 1 rows containing missing values (position_stack).

\[\begin{equation} \int_{1}^{1.6} \frac{2x}{x^2-4} dx \end{equation}\]

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="purple", size=1)+
  geom_area(aes(x,y), fill="yellow", alpha=0.5)+
  theme_bw()
ggplotly(graf_2e)

\[\begin{equation} \int_{0}^{\pi/4} e^{3x}sin(2x) dx \end{equation}\]

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="red", size=1)+
  geom_area(aes(x,y), fill="pink", alpha=0.5)+
  theme_bw()
ggplotly(graf_2f)