Question 1

Let \(I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) and \(\quad J = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\). What is \((I + J)^2\)?

# install.packages("matrixcalc")
library(matrixcalc)
## Warning: package 'matrixcalc' was built under R version 4.5.2
I <- diag(2) # identity matrix of 2x2
J <- matrix(data = c(0,1,-1,0),nrow = 2,ncol = 2,byrow = T)
answer <- matrix.power(x = I + J,k = 2) # x = I + J (our value), k = 2 (square it)
answer
##      [,1] [,2]
## [1,]    0    2
## [2,]   -2    0

Question 2

Use the sort() method and the tidyverse package to order the values \([0.35,0.035,3.05,0.305]\) from least to greatest.

A. Use the sort() function.

values <- c(0.35,0.035,3.05,0.305)
sort(values,decreasing = F) # decreasing = F (least to greatest)
## [1] 0.035 0.305 0.350 3.050

B. Use the tidyverse package.

# install.packages("tidyverse")
library(tidyverse)
## Warning: package 'lubridate' was built under R version 4.5.2
## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
## ✔ dplyr     1.1.4     ✔ readr     2.1.5
## ✔ forcats   1.0.1     ✔ stringr   1.5.2
## ✔ ggplot2   4.0.0     ✔ tibble    3.3.0
## ✔ lubridate 1.9.4     ✔ tidyr     1.3.1
## ✔ purrr     1.1.0     
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag()    masks stats::lag()
## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors
q2_data <- data.frame(Values = c(0.35,0.035,3.05,0.305))
q2_data %>%
  arrange(Values) # arrange(): least to greatest values
##   Values
## 1  0.035
## 2  0.305
## 3  0.350
## 4  3.050

Question 3

Solve the system of equations below.

\[3x + 5y = 9 \\ x - y = -5\]

q3_data <- data.frame(X = c(3,1),
                      Y = c(5,-1),
                      Constants = c(9,-5))
q3_model <- lm(Constants ~ . - 1,data = q3_data) # "." means all other variables in the dataset, -1 to exclude intercept
coef(q3_model)
##  X  Y 
## -2  3

Question 4

Differentiate \(f(x) = \tan(2x)\).

# install.packages("Deriv")
library(Deriv)
f <- function(x) {
  tan(2 * x)
}
f_prime <- Deriv(f)
f_prime
## function (x) 
## 2/cos(2 * x)^2