Question 1

Evaluate the following limit.

\[\lim_{x \to 0} \frac{\sqrt{x + 1} - 1}{x}\]

# 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 <- (sqrt(x + 1) - 1) / x
result <- lim(f,x,0)
result
## y: 1/2

Question 2

How many even numbers are there in the set \([2,13,27,32,22,42,19,23,29,3,25,17,5,12,15]\)?

# install.packages("comprehenr")
library(comprehenr)
## Warning: package 'comprehenr' was built under R version 4.5.2
even_nums <- to_vec(for (x in c(2,13,27,32,22,42,19,23,29,3,25,17,5,12,15)) if (x %% 2 == 0) x)
cat("There are",length(even_nums),"even numbers in the set of numbers above.","\n")
## There are 5 even numbers in the set of numbers above.

Question 3

Graph the corresponding polygon given the point data below.

# 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
q3_data <- data.frame(X = c(-5,-5,5,7,7),
                      Y = c(10,7,2,7,10))
ggplot(q3_data,aes(x = X,y = Y)) +
  geom_polygon(fill = "yellow",color = "black",lwd = 1.25) +
  geom_point(size = 3) +
  coord_equal() +
  theme_minimal()

Question 4

Solve the system of equations below.

\[3x - 4y = 12 \\ 5x + 3y = 32\]

q4_data <- data.frame(x = c(3,5),
                      y = c(-4,3),
                      constants = c(12,32))
q4_model <- lm(constants ~ . - 1,data = q4_data)
coef(q4_model)
##        x        y 
## 5.655172 1.241379

Question 5

What is \(\int \frac{\ln(x)}{x^3} dx\)?

# 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(log(x) / (x^3) ~ x)
anti_g <- antiD(g(x) ~ x)
anti_g
## function (x, C = 0) 
## {
##     F <- makeF(log(x)/(x^3))
##     evalFun(F, x = x, .const = C)
## }
## <environment: 0x000001e8f13b0a18>