Question 1

The cost function is \(C(x) = 12x^4 - 5x^3 + 2x + 11\). What is \(C'(x)\)?

# install.packages("Deriv")
library(Deriv)
C <- function(x) {
  12 * x^4 - 5 * x^3 + 2 * x + 11 
}
C_prime <- Deriv(C)
C_prime

## function (x) 
## 2 + x^2 * (2 * (12 * x - 5) + 24 * x - 5)

Question 2

Solve the system of equations below.

\[2x - 3y = 7 \\ 4x + 5y = 1\]

q2_data <- data.frame(X = c(2,4),
                      Y = c(-3,5),
                      Constants = c(7,1))
q2_model <- lm(Constants ~ . - 1,data = q2_data) # ".": all other variables except response variable, -1 excludes intercept
coef(q2_model)

##         X         Y 
##  1.727273 -1.181818

Question 3

Calculate the eigenvalues and eigenvectors of \(E = \begin{pmatrix} 2 & -1 & 1 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \end{pmatrix}\).

E <- matrix(data = c(2,-1,1,0,1,-1,0,0,1),nrow = 3,ncol = 3,byrow = T)
eigen(E)

## eigen() decomposition
## $values
## [1] 2 1 1
## 
## $vectors
##      [,1]      [,2]         [,3]
## [1,]    1 0.7071068 7.071068e-01
## [2,]    0 0.7071068 7.071068e-01
## [3,]    0 0.0000000 1.570092e-16

Question 4

Use the data frame below to construct a bar graph.

First, define the data frame.

q4_data <- data.frame(Color = c("Red","Yellow","Blue","Green","Gray","Purple","Orange"),
                      Frequency = c(8,5,6,3,5,2,1))
q4_data

##    Color Frequency
## 1    Red         8
## 2 Yellow         5
## 3   Blue         6
## 4  Green         3
## 5   Gray         5
## 6 Purple         2
## 7 Orange         1

Construct a bar graph.

# 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

ggplot(q4_data,aes(x = factor(Color),y = Frequency)) +
  geom_col() +
  labs(title = "Colors of Cars Passing the School Gate During a Ten-Minute Period",
       x = "Color",
       y = "Frequency") +
  theme_gray()

Question 5

Write the fractions below in order from largest to smallest.

\[\text{Fractions} = [\frac{1}{4},\frac{1}{10},\frac{1}{8},\frac{1}{6}]\]

# install.packages("tidyverse")
library(tidyverse)
q5_data <- data.frame(Fractions = c(1/4,1/10,1/8,1/6))
q5_data %>%
  arrange(desc(Fractions))

##   Fractions
## 1 0.2500000
## 2 0.1666667
## 3 0.1250000
## 4 0.1000000

Question 6

What is the least common multiple of 9 and 24?

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

## 
## Attaching package: 'pracma'

## The following object is masked from 'package:purrr':
## 
##     cross

Lcm(a = 9,b = 24)

## [1] 72

Math Problems 9/30/26

Gavin Gunder

2026-09-30