Wolfram|Alpha now has the ability to find and analyze the discontinuities of most functions of real numbers.

In technical language, a discontinuity of a function from reals to reals is a point where either the left- or right-hand limit does not exist, or where these limits exist but aren’t both equal to the value of the function at this point. Thought of another way, a continuous function is one you can graph without having to lift your pen up from the paper.Ĭonversely, a discontinuity of a function is a point where the value of the function experiences a sudden change. Most functions that we see every day, from the parabolic arc of a thrown ball to the exponential growth of money in a bank account, are “continuous.” That is, they don’t change their value suddenly. This first exposure comes through studying limits and discontinuities. It’s usually in precalculus class that students are first exposed to the more exotic and subtle aspects of functions on the real line. However, the acceleration you feel depends on how fast you are going along the track (the faster you go, the greater the acceleration), while curvature is a property intrinsic to the track itself.

The higher the curvature, the stronger the acceleration, all other things being equal.

The reason, of course, is that when you travel along a strongly curved section of track or road, you feel an acceleration. You want a curve that is fun to travel along, which means you want a lot of sharp curves-but not too sharp, unless you want your amusement park guests to get sick or pass out.Ī similar consideration is faced by engineers building a railroad or a highway: you want the path of the road to have curves that are not too sharp-in this case, to prevent the cars or trains from having to slow down (reducing efficiency) or even to wreck. You need to create a curved shape for the track, which will be designed on a computer before being built out of metal. Imagine you are building a roller coaster.

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