Math Problems 9/19/26

Question 1

Compute \(\int (3x^2- 4x) (x^3 - 2x^2 + 7)^5 \space dx\) from a composite response function.

# install.packages(c("ggformula","mosaicCalc"))
library(ggformula)

## Warning: package 'ggformula' was built under R version 4.5.2

## Loading required package: ggplot2

## Loading required package: scales

## 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((3 * x^2 - 4 * x) * (x^3 - 2 * x^2 + 7)^5 ~ x)
anti_f <- antiD(f(x) ~ x)
anti_f

## function (x, C = 0) 
## (x^18 - 12 * x^17 + 60 * x^16 - 118 * x^15 - 180 * x^14 + 1488 * 
##     x^13 - 2561 * x^12 - 2520 * x^11 + 16296 * x^10 - 16660 * 
##     x^9 - 29400 * x^8 + 82320 * x^7 - 18865 * x^6 - 144060 * 
##     x^5 + 144060 * x^4 + 100842 * x^3 - 201684 * x^2)/6 + C

Question 2

Carl spins the spinner twice and adds the two scores together. He wins if he gets a total score of 9. What is the probability of winning?

spinner <- c(3,1,5,1,5,2,1,5)
N <- 1e5
counter <- 0
for (i in 1:N) {
  spin1 <- sample(x = spinner,size = 1,replace = T)
  spin2 <- sample(x = spinner,size = 1,replace = T)
  if (spin1 + spin2 >= 9) {
    counter <- counter + 1
  }
}
probability <- counter / N
cat("The probability of winning is:",probability,"\n")

## The probability of winning is: 0.14159

Question 3

Solve and graph the following definite integral.

\[\int_{-6}^{-2} -x^2 - 5x + 6 \space dx\]

# install.packages("tidyverse")
library(tidyverse)

## Warning: package 'lubridate' was built under R version 4.5.2

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## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors

g <- function(x) {
  -x^2 - 5 * x + 6
}
answer <- integrate(f = g,lower = -6,upper = -2)$value
x_values <- seq(-6.01,-1.99,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 = 2) +
  geom_ribbon(data = subset(q3_data,x >= -6 & x <= -2),
              aes(ymin = 0,ymax = y),
              fill = "steelblue") +
  labs(title = "Graph of g(x) = -x^2 - 5x + 6",
       caption = paste("Answer:",round(answer,4)),
       x = "x",
       y = "y") +
  theme_gray()