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Fractional Calculus I

You can differentiate and integrate n times but what if n is not a whole number? This generalisation of calculus was introduced and discussed on askNRICH by some school students.

Curvy Catalogue

Stage: 5 Challenge Level: Challenge Level:1

Sketch as many different types of examples of the following curves that you can think of:

1. Continuous curves with exactly one point with zero gradient and exactly two zeros.
2. Continuous curves with exactly two points with zero gradient and exactly two zeros.
3. Continuous curves with exactly one point with zero gradient and exactly one zero.

4. Continuous curves with exactly two points with zero gradient and exactly one zero.

Once you have a feel for the problem, in each case make a catalogue of the different types of curves satisfying the different criteria and give them clear mathematical descriptions. Try to make your catalogue as complete as possible.

Prove that in a much larger catalogue you could construct examples of continuous curves with exactly $N$ points of zero gradient and exactly $M$ zeros for any non-negative whole numbers $N$ and $M$.
1. Give an example algebraic equation for various curve types in your catalogue. 
2. Create a clear argument that your catalogue is complete relative to you criteria.