Math and roller coasters are two different things. One is popular with children, whereas the other is not. However, roller coasters, like many other aspects of our life, contain a significant amount of math.

When building a roller coaster, there are two major factors to consider: the roller coaster rail should be correctly attached, and the connecting points should not have any extremely dynamic curves. These two conditions may appear to be unconnected, yet they are both based on the same concept: a multivariable function's continuity and differentiability.

 Roller coaster rails are another feature. Consider a 2D parabola with the equation y=x2. Now, invert the parabola and place it on a three-dimensional plane. The form of the parabola mirrors some areas of the roller coaster rails, as can be seen. A 2D parabola, on the other hand, is far too simplistic to be a roller coaster rail. If one more variable is introduced, and numerous sections of functions are combined, the shape of a roller coaster that we are familiar with is created.

coaster model made by multivariable functions

 The first and most critical step in this procedure is to connect the roller coaster sections. The roller coaster would tumble off the rail if the sections were not even linked, posing major safety concerns. "Making the piecewise functions continuously" is the mathematical name for this operation.

The second element to consider is the rail's smoothness. Even though the roller coaster rails are continuous, the roller coaster cart may bounce off the rail if there are too dynamic bends in the rail. This is where calculus comes into play; we term a slope of a function "differentiable" when it is continuous at all points of the function. If a roller coaster's rail is not differentiable, it means there is a place in the rail where the slope on one side of the rail differs from the slope on the other side of the rail.
Mathematicians and physicists design unique piecewise functions of roller coasters that can make the trip the most dynamic while remaining safe, keeping these two essential requirements in mind. If you're worried about the roller coaster's safety, make sure the rail is continuous and differentiable!

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