Question 1

Evaluate and graph the following definite integral.

\[\int_{4}^{5} x^3 + 2x + 5 \space dx\]

# 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
g <- function(x) {
  x^3 + 2 * x + 5
}
value <- integrate(g,lower = 4,upper = 5)$value
x_values <- seq(3,6,length.out = 500)
y_values <- g(x_values)
q1_data <- data.frame(x = x_values,y = y_values)
ggplot(q1_data,aes(x = x,y = y)) +
  geom_line(col = "black",lwd = 1.25) +
  geom_ribbon(data = subset(q1_data,x >= 4 & x <= 5),
              aes(ymin = 0,ymax = y),
              fill = "blue") +
  labs(title = "Graph of f(x) = x^3 + 2x + 5",
       caption = paste("Answer:",value),
       x = "x",
       y = "y") +
  theme_gray(base_size = 14)

Question 2

Find the first and second derivatives of the function \(f(x) = \sec(x)\).

# install.packages("Deriv")
library(Deriv)
f <- function(x) {
  1 / cos(x) # sec(x) = 1 / cos(x) - define it this way due to the Deriv package capability
}
f_prime <- Deriv(f) # f'(x)
f_double_prime <- Deriv(f_prime) # f''(x)
cat("f'(x) =","\n")
## f'(x) =
print(f_prime)
## function (x) 
## sin(x)/cos(x)^2
cat("f''(x) =","\n")
## f''(x) =
print(f_double_prime)
## function (x) 
## {
##     .e1 <- cos(x)
##     (1 + 2 * (sin(x)^2/.e1^2))/.e1
## }

Question 3

Calculate descriptive statistics for \([15,12,10,13]\).

# install.packages("summarytools")
library(summarytools)
## 
## Attaching package: 'summarytools'
## The following object is masked from 'package:tibble':
## 
##     view
nums <- c(15,12,10,13)
descr(nums)
## Descriptive Statistics  
## nums  
## N: 4  
## 
##                       nums
## ----------------- --------
##              Mean    12.50
##           Std.Dev     2.08
##               Min    10.00
##                Q1    11.00
##            Median    12.50
##                Q3    14.00
##               Max    15.00
##               MAD     2.22
##               IQR     2.00
##                CV     0.17
##          Skewness     0.00
##       SE.Skewness     1.01
##          Kurtosis    -1.96
##           N.Valid     4.00
##                 N     4.00
##         Pct.Valid   100.00

Question 4

Four friends - Dan, Fran, Ian, Jan - went strawberry picking last Saturday. The data below shows the number of punnets of strawberries each of them picked. How many more punnets of strawberries did Dan pick than Jan picked?

  1. Define the data frame.
q4_data <- data.frame(Friend = c("Dan","Fran","Ian","Jan"),
                      Number = c(35,20,30,25))
q4_data
##   Friend Number
## 1    Dan     35
## 2   Fran     20
## 3    Ian     30
## 4    Jan     25
  1. Determine how many more punnets of strawberries Dan picked than Jan.
# install.packages("tidyverse")
library(tidyverse)
dan <- q4_data %>%
  filter(Friend == "Dan") %>%
  pull(Number)
jan <- q4_data %>%
  filter(Friend == "Jan") %>%
  pull(Number)
cat("Dan picked",dan - jan,"more strawberries than Jan.","\n")
## Dan picked 10 more strawberries than Jan.

Question 5

Arrange these products and prices in order from greatest price to least price.

  1. Define the data frame.
q5_data <- data.frame(Product = c("Salt","Bread","Tissues","Toothbrush"),
                      Price = c(0.55,0.99,0.79,0.95))
q5_data
##      Product Price
## 1       Salt  0.55
## 2      Bread  0.99
## 3    Tissues  0.79
## 4 Toothbrush  0.95
  1. Order the data frame from greatest price to least price.
# install.packages("tidyverse")
library(tidyverse)
q5_data %>%
  arrange(desc(Price))
##      Product Price
## 1      Bread  0.99
## 2 Toothbrush  0.95
## 3    Tissues  0.79
## 4       Salt  0.55

Question 6

How many of the square numbers between 0 and 200 have one of their digits equal to 9?

# install.packages("comprehenr")
library(comprehenr)
## Warning: package 'comprehenr' was built under R version 4.5.2
valid_numbers <- to_vec(for (x in 0:200) if (sqrt(x) %% 1 == 0 & grepl("9",as.character(x))) x)
cat("There are",length(valid_numbers),"perfect square numbers that have one of their digits equal to 9.","\n")
## There are 4 perfect square numbers that have one of their digits equal to 9.