Do the homework from last class. Without further assumptions we can say nothing about the derivatives. This gives the correct answer if f really is constant, linear, or a parabola.
Numerical integration is similar.
If the function is on an interval, we have to use some sort of polynomial approximation. Similarly, we can lump N intervals together at a time and use polynomials of degree N. With a fixed equispaced sampling, this is basically all you can do. If you get to choose the sample points there is a whole science to it. Of particular use are Chebychev nodes, which smooth away edge effects, and Gaussian quadratures which use a trick to double the order of the integration. Other topics we could have done Fourier and wavelet transforms algebraic manipulations sampling and approximations searches and sorts Back to the Syllabus .