This is not a project and/or assignment for marks, as this is a textbook question, and will not be copying the tutors work. This question is not a academic integrity nor a honor code violation, since the purpose of this is to increase my knowledge and understanding in the topic, and to aid me a bit in the viewing of how they are connected, I will not be submitting any of the work done by tutor as this question just helps me to get a better and bigger graps of content. Follow the instructions and guidelines to end the activity. Please share the demos link when done, all references are in the images and show all of your work.

A consortium of math afficionados have decided to open an math-themed amusement park north of Toronto. The crown jewel in the park's design will be a roller coaster called "Leibniz's Dragon". Named for the 17^th-century German philosopher and mathematician, Gottfried Leibniz, the Dragon needs to be as fun, exciting, and scary as calculus.

The consortium has decided to open the design contest to the public, and you have decided to enter a record-breaking design to the contest.

Task

You will design and sell a roller coaster to the judging committee.

Submission Guidelines

Your design must consist of two parts:

1. You must use desmos to generatea piecewise function consisting of at least six parts that will define h(t), height in terms of time. You also need to generate a graph of its derivative. In order to guarantee an exciting ride, you must use each of the following types of functions at least once: quadratic, cubic, sinusoidal, and exponential/logarithmic.

a. Note that your roller coaster must end with a full circuit every time it runs, and as such, it needs to start and finish at the same height.

b. Also note that the function, h(t), must be continuous, and appear to be differentiable at a glance.

c. As with any roller coaster, once it's going, it should stay going without needing to be hauled up to the top of a hill. For this reason, your tallest hill should be the first one the coaster reaches.

2. You must generatea presentation board to sell your design, as well as an accompanying technical document.

a. The presentation board should highlight the main features of your design and is designed to catch the eye.

b. You will have to display your presentation board and model in an area that is 1 square metre in size.

c. The technical document is where your mathematical justification actually goes. For example, if one of the features that you use to sell your coaster is its maximum speed, then the calculations that you do to find the maximum speed should be included in the technical document.

d. Your technical document must include a minimum list of features of your coaster: Consult the attached Checklist

Checklist

Piecewise Function

Proof of the function's continuity

Derivatives of each piece of the function

Second derivatives of each piece of the function

Maximum and minimum height of each piece

Overall maximum and minimum height

Maximum and minimum velocity of each piece

Overall maximum and minimum velocity