Math Problems 7/28/26

Question 1

A car’s position is \(\vec{r}(t) = \langle t^3,4t \rangle\) meters. What is its velocity vector at \(t = 2\)?

# install.packages("Deriv")
library(Deriv)
r <- function(t) {
  c(t^3,4 * t)
}
v <- Deriv(r) # velocity vector
answer <- v(t = 2)
cat("The velocity vector at t = 2 is:",answer,"\n")

## The velocity vector at t = 2 is: 12 4

Question 2

Write 751 in words.

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

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

words(751)

## [1] "seven hundred fifty-one"

Question 3

\(225^{\circ}\) is the same as _____ radians.

# install.packages("pracma")
library(pracma)
value <- deg2rad(deg = 225)
cat("225 degrees =",value,"radians","\n")

## 225 degrees = 3.926991 radians

Question 4

Find the coordinates of the point or points of intersection of the straight line \(f(x) = x - 4\) and the curve \(g(x) = \sqrt{2x - 5}\).

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

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

## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
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## ✖ purrr::cross()  masks pracma::cross()
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## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors

f <- function(x) {
  x - 4
}
g <- function(x) {
  sqrt(2 * x - 5)
}
x_values <- seq(5,10,length.out = 500)
f_values <- f(x_values)
g_values <- g(x_values)
q4_data <- data.frame(x = x_values,f = f_values,g = g_values)
result <- uniroot(function(x) f(x) - g(x),interval = c(5,10))$root
ggplot(q4_data,aes(x = x)) +
  geom_line(aes(y = f),col = "blue",lwd = 1.25) + # f(x)
  geom_line(aes(y = g),col = "red",lwd = 1.25) + # g(x)
  labs(title = "Graphs of f(x) and g(x)",
       caption = paste("The intersection is at: (",round(result,2),",",round(f(result),2),")"),
       x = "x",
       y = "y") +
  theme_gray(base_size = 14)

Question 5

What is \(\int 5 \sin^4(x) \cos(x) dx\)?

# 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

## The following object is masked from 'package:english':
## 
##     ordinal

## 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(5 * (sin(x))^4 * cos(x) ~ x)
anti_h <- antiD(h(x) ~ x)
anti_h

## function (x, C = 0) 
## (10 * sin(x) + 5 * sin(-3 * x) - sin(-5 * x))/16 + C