Math Problems 10/7/26

Question 1

For \(p(x) = \ln(\sin(2x) + 3)\), what is \(p'(x)\)?

# install.packages("Deriv")
library(Deriv)
p <- function(x) {
  log(sin(2 * x) + 3)
}
p_prime <- Deriv(p)
p_prime

## function (x) 
## {
##     .e1 <- 2 * x
##     2 * (cos(.e1)/(3 + sin(.e1)))
## }

Question 2

Write 4,037,203 in words.

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

## Warning: package 'english' was built under R version 4.5.2

words(4037203)

## [1] "four million thirty-seven thousand two hundred three"

Question 3

Find the inverse of the matrix \(Y = \begin{pmatrix} 6 & 8 \\ -2 & -3 \end{pmateix}\).

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

## Warning: package 'matrixcalc' was built under R version 4.5.2

Y <- matrix(data = c(6,8,-2,-3),nrow = 2,ncol = 2,byrow = T)
matrix.inverse(Y)

##      [,1] [,2]
## [1,]  1.5    4
## [2,] -1.0   -3