Question 1

If \(f(x) = \sqrt{x}\), what is \(f''(x)\)?

# install.packages("Deriv")
library(Deriv)
f <- function(x) {
  sqrt(x)
}
f_prime <- Deriv(f)
f_double_prime <- Deriv(f_prime)
f_double_prime
## function (x) 
## -(0.25/(x * sqrt(x)))

Question 2

Write 7,420,206 in words.

# install.packages("english")
library(english)
## Warning: package 'english' was built under R version 4.5.2
words(7420206)
## [1] "seven million four hundred twenty thousand two hundred six"

Question 3

Find the roots of \(g(x) = 20x^3 - 16x^2 - 35x + 28\) when \(g(x) = 0\).

# install.packages("rootSolve")
library(rootSolve)
g <- function(x) {
  20 * x^3 - 16 * x^2 - 35 * x + 28
}
roots <- uniroot.all(g,c(-10,10))
for (x in seq_along(roots)) {
  cat("Root",x,":",roots[x],"\n")
}
## Root 1 : -1.322876 
## Root 2 : 0.8 
## Root 3 : 1.322714