### 2D-3D

Two circles of equal size intersect and the centre of each circle is on the circumference of the other. What is the area of the intersection? Now imagine that the diagram represents two spheres of equal volume with the centre of each sphere on the surface of the other. What is the volume of intersection?

### Power Up

Show without recourse to any calculating aid that 7^{1/2} + 7^{1/3} + 7^{1/4} < 7 and 4^{1/2} + 4^{1/3} + 4^{1/4} > 4 . Sketch the graph of f(x) = x^{1/2} + x^{1/3} + x^{1/4} -x

### 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:

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 $M$.

Extension:
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.