Question 1

Find the derivative of \(f(x) = x^3 + 4x^2 + x\).

# install.packages("Deriv")
library(Deriv)
f <- function(x) {
  x^3 + 4 * x^2 + x
}
f_prime <- Deriv(f)
f_prime
## function (x) 
## 1 + x * (3 * x + 8)

Question 2

Evaluate: \(\frac{d}{dx} \lbrace 2 \sin(x) \cos(x) \rbrace\).

# install.packages("Deriv")
library(Deriv)
g <- function(x) {
  2 * sin(x) * cos(x)
}
g_prime <- Deriv(g)
g_prime
## function (x) 
## 2 * (cos(x)^2 - sin(x)^2)

Question 3

Calculate descriptive statistics for \([1,2,5,7,7,9,15,15,15,22,23,24,26]\).

# install.packages("summarytools")
library(summarytools)
nums <- c(1,2,5,7,7,9,15,15,15,22,23,24,26)
descr(nums)
## Descriptive Statistics  
## nums  
## N: 13  
## 
##                       nums
## ----------------- --------
##              Mean    13.15
##           Std.Dev     8.66
##               Min     1.00
##                Q1     7.00
##            Median    15.00
##                Q3    22.00
##               Max    26.00
##               MAD    11.86
##               IQR    15.00
##                CV     0.66
##          Skewness     0.09
##       SE.Skewness     0.62
##          Kurtosis    -1.61
##           N.Valid    13.00
##                 N    13.00
##         Pct.Valid   100.00

Question 4

Find the norm of the vector \(\vec{v} = (-1,3,2,2)\).

# install.packages("pracma")
library(pracma)
v <- c(-1,3,2,2)
norm_v <- Norm(v)
cat("The norm of vector v is:",norm_v,"\n")
## The norm of vector v is: 4.242641

Question 5

You are given the data frame below with several objects and their weights. Determine which object has the highest weight.

  1. Define the data frame.

Note: Weight is defined in pounds (lb.) except where there are asterisks (*). These weights are defined in tons. Keep in mind that 1 ton is 2000 pounds.

q5_data <- data.frame(Object = c("Kayak","Truck","Boat","Scooter","Plane*","Racecar","Motorcycle","Car","Cruise Ship*","Bike"),
                      Weight = c(40,16000,1600,27,400,1600,350,3300,100000,15))
q5_data
##          Object Weight
## 1         Kayak     40
## 2         Truck  16000
## 3          Boat   1600
## 4       Scooter     27
## 5        Plane*    400
## 6       Racecar   1600
## 7    Motorcycle    350
## 8           Car   3300
## 9  Cruise Ship* 100000
## 10         Bike     15
  1. Determine which object has the highest weight.
# 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() ──
## ✖ purrr::cross()  masks pracma::cross()
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag()    masks stats::lag()
## ✖ tibble::view()  masks summarytools::view()
## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors
heaviest_object <- q5_data %>%
  mutate(Weight = if_else(
    str_detect(Object,"\\*"), # finding the asterisk "*"
    Weight * 2000, # if an asterisk is present, multiply value by 2000 (1 ton = 2000 lbs.)
    Weight # keeping value if no asterisk at all
  )) %>%
  slice_max(Weight,n = 1) %>% # getting the maximum weight
  pull(Object) %>% # extracting the object with the maximum weight
  str_remove("\\*") # removing asterisk from final result
cat("The object with the highest weight is:",heaviest_object,"\n")
## The object with the highest weight is: Cruise Ship