Question: Integral and derivative of polynomial. Suppose the \(n\) -vector \(c\) gives the coefficients of a polynomial \(p(x)=c_{1}+c_{2} x+\cdots+c_{n} x^{n-1}\).

1. Let \(\alpha\) and \(\beta\) be numbers with \(\alpha<\beta\). Find an \(n\) -vector \(a\) for which
   \[a^{T} c=\int_{\alpha}^{\beta} p(x) d x\]
   always holds. This means that the integral of a polynomial over an interval is a linear function of its coefficients.
2. Let \(\alpha\) be a number. Find an \(n\) -vector \(b\) for which

This means that the derivative of the polynomial at a given point is a linear function of its coefficients.

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