Question 1

Compute \(\int (8x + 6) \sqrt{4x^2 - 6x + 9} dx\) for a physical potential function.

# install.packages(c("ggformula","mosaicCalc"))
library(ggformula)
## Warning: package 'ggformula' was built under R version 4.5.2
## Loading required package: ggplot2
## Loading required package: scales
## Loading required package: ggiraph
## Warning: package 'ggiraph' was built under R version 4.5.2
## Loading required package: ggridges
## Warning: package 'ggridges' was built under R version 4.5.2
## 
## New to ggformula?  Try the tutorials: 
##  learnr::run_tutorial("introduction", package = "ggformula")
##  learnr::run_tutorial("refining", package = "ggformula")
library(mosaicCalc)
## Warning: package 'mosaicCalc' was built under R version 4.5.2
## Registered S3 method overwritten by 'mosaic':
##   method                           from   
##   fortify.SpatialPolygonsDataFrame ggplot2
## 
## Attaching package: 'mosaicCalc'
## The following object is masked from 'package:stats':
## 
##     D
f <- makeFun((8 * x + 6) * sqrt(4 * x^2 - 6 * x + 9) ~ x)
anti_f <- antiD(f(x) ~ x)
anti_f
## function (x, C = 0) 
## {
##     F <- makeF((8 * x + 6) * sqrt(4 * x^2 - 6 * x + 9))
##     evalFun(F, x = x, .const = C)
## }
## <environment: 0x0000017da2674b28>

Question 2

Write 5,390,052 in words.

# install.packages("english")
library(english)
## Warning: package 'english' was built under R version 4.5.2
## 
## Attaching package: 'english'
## The following object is masked from 'package:scales':
## 
##     ordinal
words(5390052)
## [1] "five million three hundred ninety thousand fifty-two"

Question 3

Two fair dice are thrown. What is the probability that the score on the first die is 4 or the score on the second die is odd?

die <- 1:6
counter <- 0
N <- 1e5
for (i in 1:N) {
  roll <- sample(x = die,size = 2,replace = T)
  if (roll[1] == 4 || roll[2] %% 2 == 1) {
    counter <- counter + 1
  }
}
probability <- counter / N
cat("The probability is:",probability,"\n")
## The probability is: 0.58361