Roller Coaster Design:
You will develop a piecewise curve with 5 pieces that is continuous and differentiable at all points along the curve and showing a possible roller coaster. You need to show the graph and prove that the function you generate is differentiable (and thus continuous) at the points where the piecewise curves "meet". You may use any type of function we have discussed SO far including polynomial, rational, exponential, trigonometric, and others. Try to incorporate at least four different types of functions in your design. You may hand-draw your roller coaster on a sheet of graph paper or use a computer program (e.g. Desmos.com works well for this) to generate your roller coaster.

Make sure you show the function and it's domain for each of your five curves. Use a bold dot to mark the transition points. Oh and name your roller coaster too!

Roller Coaster Analysis (50 points): Once you have your model roller coaster graphed along with piecewise functions for each curve, it's time to do some analysis using calculus!

(1) Show the necessary steps that your roller coaster is differentiable at each transition point. Remember that differentiobility implies continuity so you don't need to show any work related to continuity at each transition point. This part is challenging!
(2) For each "piece" of your roller coaster, calculate the average rate of change using the beginning and ending points of each section.
(3) What is the absolute maximum and minimum heights of your roller coaster? Label this on your roller coaster.
(4) List all the points (x,y) on your roller coaster where there's a horizontal tangent line.
(5) List all the points (x,y) on your roller coaster where there's a vertical tangent line.
(6) Which portion of your roller coaster did it achieve the most positive average rate of change? Which portion had the most negative average rate of change?
(7) Identify any changes in concavity on your roller coaster and label them on your coaster. You do not need to calculate the second derivative here.

Things to Remember:
Your roller coaster should end at the same height it starts. You may want to have a horizontal line for your loading/unloading area. But don't use the x-axis (how many "good" roller coasters start at ground level? Exactly.) This does not count as one of your five curves. Mark on the graph the starting/ending height of your roller-coaster. The next part of your roller coaster can go higher or lower than the previous part. This is necessary to maintain differentiability at the transition points. Are you good enough to build in a loop? That would be awesome (and worth five bonus points). Turn in your roller coaster graph with functions and domains clearly displayed and labeled, your completed analysis and responses to the questions along with supporting work.

Roller Coaster Design: You will develop a piecewise curve with 5...