# Create Problems for Polynomial Interpolation

I want to create problems concerning polynomial interpolation for students. However, I need those problems to differ in difficulty for the different students.

What I could always do is solve some problem and classify them by difficulty. However, it would be nice if I could find a "system" with I could create such problems. This would also have the benefit that the problems for the students do not repeat themselves (most of the time).

The students are supposed to either use the classic approach, the newton approach, or the lagrange approach. Each one of them needs to be understood.

Is there a way to create such problems? and where in the process would I be able to increase / decrease difficulty?

The problems btw only ought to consist of the data points, from which the polynomial has to be interpolated.

I already tried building relatively easy polynomials and then calculating the data points for the problem, however calculation would still get complicated for the students. Nice, clean and easy numbers would be great. Numbers like 1/4, 1/2, 1/5 etc. would also be fine. I just want to avoid the students having to work with numbers such as 83.23461.

Any ideas?

Thank you in advance.

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• I'd suggest not asking students to find interpolating polynomials by hand... it's just mindless busywork if they know basic algebra; and if they don't know basic algebra then they should be taught that, and not interpolation. – Normal Human Aug 18 '15 at 16:15
• A litany of questions: What age/class are you teaching? What preceded this topic in the curriculum and what comes next? Is the task to carry out these computations by hand or to implement e.g. Neville's algorithm (wiki, mathematica)? What are the specific learning goals accomplished by teaching three methods of interpolation? Can you perhaps give one example (at any level of difficulty) in the body of your question and work through it the ways that you want to see students do it? – Benjamin Dickman Aug 19 '15 at 8:32
• Depending on how much creative license you have, it might be more interesting to have students do some graphics coding using interpolation or something. – Steven Gubkin Aug 26 '15 at 15:16

## 2 Answers

From my point of view as a student, you should not give many basic interpolation problems to students, since they are just, as Normal Human stated, mindless work. However, I suggest you use this plan:
First, you ask for a polynomial interpolating 3 points, without any condition on the said polynomial. Then, you increase the number of points to interpolate to 4 or 5 and you ask a polynomial of minimal degree. These are the very basic (and slightly boring) exercises, to ensure your students understood your lecture. I recommend you give at most 4 or 5 of those to your students.
After that, you may step up the difficulty by asking your students to find every interpolating polynomial whose degree is lesser or equal to a given constant. You may also create an exercise where you add constraints on the k$^{th}$ derivative of the interpolating polynomial at certain points, for a given k. These problems should be more challenging and interesting.

• Matching derivatives gives spline interpolation, doing that for e.g. 5 points with cubic polynomials might be enlightening. – vonbrand Aug 19 '15 at 22:54

You could grab some experimental data points, and use those. That way (specially if the data have experimental errors) show that the interpolating polynomial isn't always a good option, and one should go for some sort of curve fitting instead. If you use data that are in some way interesting to the students by themselves, much better. Can even show off e.g. spline interpolation. Having tools at hand to plot the results is a requirement in any case.

It isn't the case that much anymore, but reading data out of tables (and interpolating) used to be a critical skill for an engineering student like me.