Project: Explain the mathematics of roller coaster construction. Imagine you are a roller coaster designer, and are asked to mathematically represent the curve of the first drop of a new ride. By looking at some of your favorite coasters, you determine that the optimal slope for the ascent, when the car is being pulled up the slope by a chain, is 0.7. The slope of the first drop that will be the most thrilling (without being dangerous) is –1.5. You decide to connect these two straight stretches, y = L1 and y = L2 , with a parabola of the form y = f(x) = ax 2 + bx + c, where x and f(x) are measured in meters. In order for the curve of the track to remain smooth, you must make sure that the linear segments, L 1 and L2 , have the same slope at each endpoint of the parabola, points P and Q. This ensures that the parabola is differentiable, as well as continuous at the endpoints. Choose point P, the left endpoint of your parabola, as your origin, to simplify calculations.