Question 1

Evaluate the following limit.

\[\lim_{x \to 3} \frac{1}{(x - 3)^2}\]

# install.packages("Ryacas")
library(Ryacas)
## Warning: package 'Ryacas' was built under R version 4.5.2
## 
## Attaching package: 'Ryacas'
## The following object is masked from 'package:stats':
## 
##     integrate
## The following objects are masked from 'package:base':
## 
##     %*%, det, diag, diag<-, lower.tri, upper.tri
x <- ysym("x")
f <- 1 / (x - 3)^2
result <- lim(f,x,3)
result
## y: Infinity

Question 2

Which one of the following is odd?

A. \(42 \times 45 \times 48\)

B. \(43 \times 45 \times 46\)

C. \(44 \times 46 \times 48\)

D. \(45 \times 47 \times 49\)

# 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()
## ✖ purrr::simplify() masks Ryacas::simplify()
## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors
q2_data <- data.frame(Choice = LETTERS[1:4],
                      Number = c(42*45*48,43*45*46,44*46*48,45*47*49))
correct_answer <- q2_data %>%
  mutate(Odd = Number %% 2 == 1) %>%
  filter(Odd == TRUE) %>%
  pull(Choice)
cat("The correct answer is:",correct_answer,"\n")
## The correct answer is: D

Question 3

What is \(\sqrt[3]{-512}\)?

# install.packages("pracma")
library(pracma)
## 
## Attaching package: 'pracma'
## The following object is masked from 'package:purrr':
## 
##     cross
value <- nthroot(x = -512,n = 3) # x = -512 (our value), n = 3 (cube root)
cat("Answer:",value,"\n")
## Answer: -8

Question 4

What is \(\int 4x^7 dx\)?

# 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
g <- makeFun(4 * x^7 ~ x)
anti_g <- antiD(g(x) ~ x)
anti_g
## function (x, C = 0) 
## x^8/2 + C