Calculus generally

Third in our series of problems on population dynamics for advanced students.

Find the relationship between the locations of points of inflection, maxima and minima of functions.

How would you sort out these integrals?

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?

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

Can you find the maximum value of the curve defined by this expression?

What functions can you make using the function machines RECIPROCAL and PRODUCT and the operator machines DIFF and INT?

Can you hit the target functions using a set of input functions and a little calculus and algebra?

Can you sketch these difficult curves, which have uses in mathematical modelling?

Can you find a quadratic equation which passes close to these points?