Calculus Coasters

Using continuity and differentiability in roller coaster design

Contributors: Shonda Kuiper DASIL

NikoNomad. (n.d.). iStock

Part 2A: Introduction

What makes one roller coaster more thrilling than others? In this lab, we will explore how to use mathematical models to create models of roller coasters and then analyze how calculus can help us design better models.

Part 2B: Connecting Calculus and Coasters

This activity will focus on two core concepts discussed in most calculus classes: continuity and differentiability.

Continuity: continuous functions do not have any jumps or gaps.

Differentiability: a function is differentiable at a point when it is continuous and changes in a smooth manner without sudden changes in direction. In other words, the function must not have any sharp turns (like a "V" shape) at any point. The instantaneous slope (i.e. the derivative) can be calculated at each point.

On a roller coaster track, continuity and differentiability would ensure that the path is smooth, without sharp angles or breaks. The coaster would move in a smooth manner that helps riders maintain a sense of speed and excitement while avoiding jarring movements or even crashes.

Part 2C: Designing a Coaster

When designing your first roller coaster, you will create a piecewise function that represents a continuous and differentiable roller coaster path. This means the track must flow smoothly without any breaks or sharp turns.

Goal: Create a coaster with the most vertical distance travelled.

Base Rules:

Enter at least one equation.
The coaster must start at (x=0, y=0) and end at (x = 50, any y).
The maximum y value is 500.
Equations need to be smoooth and connected.
For every equation, y must be 0 or higher.
Going up the first hill does not count towards your score.

Here are some key terms that will be recorded during the coaster game:

Player ID: You can use any name you like, but please note that it will be visible on the internet. Use a code name that you will remember, but do not use a name that will easily identify you.
Group ID: You can use any name you like. However, if you are playing this game for a course, use the exact Group ID provided by your instructor, which is the same for every person in the class.
Click the Play button
Select Level 1 and enter your roller coaster functions.
Part 2D will demonstrate sample equations.

It may be helpful to use a graphing tool, such as Desmos, to explore various functions that can be used in your roller coaster design.

NOTE: If you are unable to play the game, either try a different browser or go to the game website directly Here. The game may take time to load.

Part 2D: Example - Level 1 Coaster

We will step through entering a piecewise function into the coaster game in order to get a feel for how the game interacts with piecewise functions.

Typically, a coaster will have some type of conveyor to get you to the top of the coaster, and then gravity will carry you through the rest of the coaster. For the first part of our piecewise function, we create a lift ramp or conveyor. For example, we enter a linear equation:
f(x) = 0 + x for 0 < x < 20
Note that for our function to be continuous, the equation entries require that the ending x value of the first equation is the starting value of the following equation. This ensures that there are no horizontal gaps in the coaster (and the game does this automatically!)
Furthermore, to be continuous, the y value of the first function when x = 20 must equal the y value of the second function when x = 20. This ensures that there are no vertical gaps in the coaster. For example, we could enter the second part of our piecewise equation as
f(x) = 30 - 0.5x for 20 < x < 50.

"Graph"

Let's try instead making the second part of our piecewise function a quadratic, for example,
f(x) = -20 + 3x - 0.05x² for 20 < x < 50.

"Graph"

Part 2E: Exploring Level 1 Coasters

You'll notice that when we entered the equations in Part 2D, we received a score of 20. Your goal is now to create a model of a roller coaster with the highest possible thrill score.

As you experiment with coasters, answer the following questions:

What was your maximum coaster score, and the equations that achieved it?
What calculations need to be done to verify that your coaster is continuous and differentiable at every point?
What strategies did you use to design your coaster and why?
If you were to design a new roller coaster game, what restrictions would you have?
- Does it make sense to assume the function can't go below 0?
- Should we require that when x = 0 the height should be 0 as well?
- Discuss any other rules that should be implemented to model a real coaster more accurately.

The graph highlights models created by students in a previous course.

Part 2F: Challenge - Level 2 Coasters

As you were working on the Level 1 Coasters, you may have felt restricted and wanted to be able to use more parts of a piecewise function and/or higher degree polynomials. Well, you're in luck - now you can do that by trying Level 2 of the game!

Your goal is to create a piecewise function in level 2 that creates a valid coaster with a higher thrill score than you made in level 1. The rules for Level 2 are:

Enter one to five equations.
The coaster must start at the origin (x = 0, y = 0).
The coaster must end with (0 < x ≤ 100) and (0 < y < 1).
The maximum y value is 500.
Equations need to be smooth and connected.
For every equation, y must be 0 or higher.
The first hill must be the highest.
Going up the first hill does not count towards your score.

Revisit the questions in part 2E and see if any of your answers have changed. Additionally, you should think about:

What was easier and what was harder about this level?
Should the first hill of your coaster be higher than all the others? Why is this typically important?
Should we require that the beginning and end of the track be horizontal? Why or why not?

Part 2F: Challenge - Level 2 Coasters

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