Question 1

A tank’s volume is \(V(t) = 12t^3 - 5t^2 + 4\). What is \(V'(t)\)?

# install.packages("Deriv")
library(Deriv)
V <- function(t) {
  12 * t^3 - 5 * t^2 + 4
}
V_prime <- Deriv(V)
V_prime
## function (t) 
## {
##     .e1 <- 12 * t
##     t * (.e1 + 2 * (.e1 - 5))
## }

Question 2

Your friend asks you to choose one card from the seven cards in his hand without looking. Which one are you least likely to choose?

set.seed(123)
cards <- c("8 Heart","6 Spade","2 Club","4 Heart","4 Diamond","6 Club","2 Spade")
simulation <- sample(x = cards,size = 1e5,replace = T)
probabilities <- prop.table(table(simulation))
probabilities
## simulation
##    2 Club   2 Spade 4 Diamond   4 Heart    6 Club   6 Spade   8 Heart 
##   0.14284   0.14294   0.14385   0.14148   0.14145   0.14428   0.14316

Question 3

Put \([5613,6531,3156,3516]\) in ascending order.

# 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
q3_data <- data.frame(Number = c(5613,6531,3156,3516))
q3_data %>%
  arrange(Number) # arrange(): ascending order
##   Number
## 1   3156
## 2   3516
## 3   5613
## 4   6531

Question 4

Plot the corresponding polygon to the given data below.

# install.packages("tidyverse")
library(tidyverse)
q4_data <- data.frame(X = c(-9,-9,-4,-4),
                      Y = c(-7,7,7,-7))
ggplot(q4_data,aes(x = X,y = Y)) +
  geom_polygon(col = "black",fill = "lightblue",lwd = 1.25) +
  geom_point(size = 4) +
  coord_equal() +
  theme_gray()

Question 5

Find \(\sqrt[3] {80}\) correct to 2 decimal places.

# install.packages("pracma")
library(pracma)
## 
## Attaching package: 'pracma'
## The following object is masked from 'package:purrr':
## 
##     cross
value <- nthroot(x = 80,n = 3) # x = 80 (our value),n = 3 (cube root)
cat("Value:",round(value,2),"\n")
## Value: 4.31

Question 6

What is \(\int \sqrt{x^3} \space dx\)?

# install.packages(c("ggformula","mosaicCalc"))
library(ggformula)
## Warning: package 'ggformula' was built under R version 4.5.2
## Loading required package: scales
## 
## Attaching package: 'scales'
## The following object is masked from 'package:purrr':
## 
##     discard
## The following object is masked from 'package:readr':
## 
##     col_factor
## 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
f <- makeFun(sqrt(x^3) ~ x)
anti_f <- antiD(f(x) ~ x)
anti_f
## function (x, C = 0) 
## {
##     F <- makeF(sqrt(x^3))
##     evalFun(F, x = x, .const = C)
## }
## <environment: 0x00000220774a2cb8>