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What are some applications of smoothing a piecewise polynomial?
For example, I am interested in learning from you: 1) In what future areas of my math studies will this be useful? and 2) Are there common real world applications of this technique?

This topic is presented as a worked example following the lecture on the product rule in Calculus I (see note below), at the following link: MIT OCW 18.01 SC Worked Example of Smoothing a Piecewise Polynomial

Hopefully, awareness of future applicability will help me learn this idea well enough to remember it permanently.

I look forward to your thoughts on the topic.

Note: The formal name of the MIT OCW class is presented here to facilitate searching by other participants interested in the same material: MIT OCW Math 18.01 SC Single Variable Calculus // MIT Open Courseware Math 18.01 OCW Scholar Single Variable Calculus

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Any time you have data at discrete locations that can be approximated by piecewise polynomials and you would like derived quantities, smoothing is important because you want all derivatives to exist.

Just about any experimental or computational work in almost any field will find this useful.

For example, imagine counting how many flowers are of a particular species in a field. It might make sense to split the field into discrete areas and count the number in each discrete area. Then you have a grid of piecewise constant values. You can of course construct higher order piecewise polynomials from this.

Then you want to know how rapidly the species of flowers changes as you move north in the field. Having a continuous and differentiable function is useful to develop models of how the species of flowers are distributed based on whatever conditions you are measuring.

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Insisting on the continuity of a piecewise polynomial and its derivatives is especially important in computer-aided manufacture (CAM). In the milling/machining of molds, parts and other such objects, splines (piecewise polynomials, or rational functions, that are continuous at least up to the second derivative) are important mathematical tools, in that one usually ensures that the path being taken by the cutting tool when a substrate like steel or a polymer is being cut is a spline whenever curved profiles are needed. This is due to the fact that the cutters are usually moving at high speed, and any abrupt changes in the acceleration/curvature (i.e., abrupt changes in the second derivative) of the path the cutter is taking can result in unwanted vibrations that could damage the cutter, the material being cut, or both.

Here is one book that talks about the use of splines in machining.

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