x̄ - > Basic integration & Basic differentiation

# Basic integration using integrate function result 

<- integrate(function(x) x^2, 0, 1) 

print(result$value)

# Integration with parameters 

a <- 1 b <- 2 

result 

<- integrate(function(x, a, b) a*x + b, lower = 0, upper = 1, a = a, b = b) 

print(result$value)

# Basic differentiation using diff function 

x <- seq(1, 10, by = 1) 

y <- x^2 derivative <- diff(y) / diff(x) 

print(derivative) # Differentiation with parameters 

a <- 2 

b <- 3 

x <- seq(1, 10, by = 1) 

y <- a*x + b derivative <- diff(y) / diff(x) 

print(derivative) 

# Install and load Deriv package 

# install.packages("Deriv") library(Deriv)

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# Define function 

f <- function(x) x^2 + 2*x + 1 

# Calculate derivative 

df_dx <- Deriv(f, "x")

# Evaluate derivative at a specific point result 

<- df_dx(2) print(result)

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f <- function(x) x^2 

result 

<- integrate(f, 0, 1) print(result)

g <- function(x, a) a * x^2 

result <- integrate(g, 0, 1, a = 2) print(result)

h <- function(x) x^2 + sin(x) 

result <- integrate(h, 0, 2) 

# Single integration print(result)

f <- function(x) x^3 

df_dx <- diff(f, "x") print(df_dx)

g <- function(x, a) a * x^2 

dg_dx <- D(g, "x") print(dg_dx)

h <- function(x, y) x^2 + y^3 

dh_dx <- D(h, "x")

dh_dy <- D(h, "y") 

print(dh_dx) print(dh_dy)

f <- function(x) x^2 + sin(x) 

result_int <- integrate(f, 0, 2) # Integration

df_dx <- D(f, "x") 

# Differentiation 

print(result_int) print(df_dx)

g <- function(x) x^2 + sin(x) 

result_int1 <- integrate(g, 0, 1) 

result_int2 <- integrate(g, 1, 2)

dg_dx <- D(g, "x") 

print(result_int1) 

print(result_int2) print(dg_dx)

h <- function(x) x^2 + sin(x) 

result_int <- integrate(h, 0, 1) 

dh_dx <- D(h, "x") 

print(result_int) print(dh_dx)

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