design a roller coaster using at least 5 curves (min. piecewise functions with detail mathematical communication) that are continuous at all transition points...this means no breaks. Your coaster must also be differentiable at all transition points ...this means no cusps, jumps or angled turns. You need to show your detail calculation to find the turning points/ inflections points of the functions.

You need to use differentiation concepts to mathematically model the path of roller coasters and to analyse the path of these coasters with regard to peak and valley points (local extreme

points), points of steepest ascent and descent (inflection points), and "thrill."

Things to remember:

1. Your roller coaster must begin and end at the same height.The roller coaster must start and end with a zero-degree incline.
2. This is a gravity-driven coaster so it never gets as high as the previous high point. No ascent or descent can be steeper than 80 degrees from the horizontal.
3. Continuous means NO BREAKS in the track. Differentiable means no jumps, cusps or angled turns.
4. Are you good enough to actually put in a loop? That would be spectacular.
5. The thrill of the coaster is the sum of the thrills of the drops
6. Critically reflect on the safety as well.