Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 68 / 2 / Solution

The degree-$n$ Bernstein basis isFor $0\le t\le 1$, every basis function is nonnegative, and the binomial theorem givesTherefore $P(t)={\sum }_i{P}_i{b}_i^n(t)$ is a convex combination of its control points. By the definition of their convex hull,This includes the endpoints because $P(0)=P_0$ and $P(1)=P_n$. It depends on the specified parameter interval: outside it the weights need not be nonnegative.

For $n\ge 1$, differentiating the basis and using the elementary binomial identities giveswhere out-of-range basis indices are zero. ThusHence the Bézier derivative control polygon has points $n(P_{i+1}-P_i)$, andIn particular the endpoint tangent vectors are $n(P_1-P_0)$ and $n(P_n-P_{n-1})$. For $n=0$ the curve is constant and its derivative is zero.