Tutorial Exercises Week 6

Question 1

Write an R function that calculates f(x) in the equation below:

f(x) = \frac{e^x}{1+e^x}

  • What is f(-2)?
  • What is f(0)?
  • What is f(2)?

In each case provide your answer rounded to 4 decimal places.

Question 2

Plot the function

f(x) = x^3 - 12x^2

in the interval [-6, 13].

At what values of x are the extreme points of this function? Both extreme points are integers (whole numbers).

Hint: Add the layers theme_minimal() + scale_x_continuous(breaks = -6:13) to add more line verticial breaks to help see where the extreme points are.

Question 3

Use R to find the minimum of the following function:

f(x)=x^2 - 3x - 12

  • What is the minimizing value of x?
  • What value does the function achieve at its minimum?

Question 4

Use R to find the maximum of the following function:

f(x)=100+2x-x^2

What value of x maximizes this function?

Question 5

Create a function in R with two arguments x and z which does the following:

f(x,z)= \begin{cases} x+z & \text{ if } x > z\\ x-z & \text{ if } x = z\\ xz & \text{ if } x < z\\ \end{cases}

What is the value of the function at the following values:

  • x=2 and z=3.
  • x=2 and z=2.
  • x=3 and z=2.

Question 6

Download and read in the file rotterdam-airbnb.csv.

Using the data create a factor variable called n_beds according to:

  • "1" if bedrooms = 1.
  • "2" if bedrooms = 2.
  • "3+ if bedrooms > 2.

Create a bar plot of the variable n_beds. Which describes the shape of the bar plot?

  • Most listings have 1 bedroom. Listings with 3+ bedrooms are relatively rare.
  • Most listings have 3+ bedrooms. Listings with 1 bedroom are relatively rare.
  • Most listings have 2 bedrooms. Listings with 1 bedroom are relatively rare.

Question 7

Using the same dataset as the previous question, create a factor variable called affordability which takes the following values:

  • "Cheap" if the price is below 120
  • "Expensive" if the price is above 250.
  • "Affordable" otherwise (between 120-250).

Set the levels of the factor to go "Cheap", then "Affordable", then "Expensive".

Create a bar plot of the variable n_beds from the previous question with colors filling the bars that represent the affordability.

Which category of n_beds contains the most listings labelled as "Expensive"?

  • 1
  • 2
  • 3+