Question 1

A function is \(f(x) = \sqrt{\frac{x^2 + 1}{x^2 - 1}}\). What is \(f'(x)\)?

# install.packages("Deriv")
library(Deriv)
f <- function(x) {
  sqrt((x^2 + 1) / (x^2 - 1))
}
f_prime <- Deriv(f)
f_prime
## function (x) 
## {
##     .e1 <- x^2
##     .e2 <- .e1 - 1
##     .e3 <- (1 + .e1)/.e2
##     0.5 * (x * (2 - 2 * .e3)/(sqrt(.e3) * .e2))
## }

Question 2

A tumor size satisfies \(\frac{dS}{dt} = 0.7 S\) with \(S(0) = 3\). What is \(S(5)\)?

# 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
s <- makeFun(0.7 * t ~ t)
S <- antiD(s(t) ~ t)
answer <- S(t = 5,C = 3)
cat("S(5) =",answer,"\n")
## S(5) = 11.75

Question 3

Write 208 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(208)
## [1] "two hundred eight"