Math Problems 10/9/26

Question 1

If \(f(x) = \sin(x) \ln(x^2 + 1)\), what is \(f'(x)\)?

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

## function (x) 
## {
##     .e1 <- x^2
##     2 * (x * sin(x)/(1 + .e1)) + cos(x) * log1p(.e1)
## }

Question 2

Which one of the following four decimal numbers is the second least?

A. 386.09

B. 38.691

C. 389.06

D. 390.618

# 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

q2_data <- data.frame(Choice = LETTERS[1:4],
                      Value = c(386.09,38.691,389.06,390.618))
correct_answer <- q2_data %>%
  arrange(Value) %>%
  slice(2)
correct_answer

##   Choice  Value
## 1      A 386.09

Question 3

Find the roots and graph \(g(x) = x^2 - 11x + 24\).

# install.packages(c("rootSolve","tidyverse"))
library(rootSolve)
library(tidyverse)
g <- function(x) {
  x^2 - 11 * x + 24
}
roots <- uniroot.all(g,c(-10,10))
for (x in seq_along(roots)) {
  cat("Root",x,":",roots[x],"\n")
}

## Root 1 : 3 
## Root 2 : 8

x_values <- seq(2,9,length.out = 500)
y_values <- g(x_values)
q3_data <- data.frame(x = x_values,y = y_values)
ggplot(q3_data,aes(x = x,y = y)) +
  geom_line(col = "black",lwd = 1.25) +
  annotate("point",x = roots[1],y = g(roots[1]),col = "blue",size = 5) +
  annotate("point",x = roots[2],y = g(roots[2]),col = "blue",size = 5) +
  labs(title = "Graph of g(x) = x^2 - 11x + 24",
       caption = paste("Roots:",roots[1],"and",roots[2]),
       x = "x",
       y = "y") +
  theme_gray()

Question 4

What is \(\int \frac{4}{1 + 9x^2} \space dx\)?

# 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

h <- makeFun(4 / (1 + 9 * x^2) ~ x)
anti_h <- antiD(h(x) ~ x)
anti_h

## function (x, C = 0) 
## (4 * atan(3 * x))/3 + C