In this exercise, we extend the analysis of the Legendre polynomials presented in Example 2.1.

(a) With only the first three Legendre polynomials, \(P_{0}(x), P_{1}(x)\), and \(P_{2}(x)\), this is only a complete, orthonormal basis for general quadratic polynomials on \(x \in[-1,1]\). For a polynomial expressed as:

\[\begin{equation*}p(x)=a x^{2}+b x+c, \tag{2.91}\end{equation*}\]for some constants \(a, b, c\), re-write it as a linear combination of the Legendre polynomials. Express the coefficients of this linear combination as a threedimensional vector.

(b) Act on this three-dimensional vector with the derivative matrix that you constructed in Example 2.1. Remember, the result represents another linear combination of Legendre polynomials. Does the result agree with what you would find from just differentiating the polynomial in part (a)?

(c) Now, construct the matrix that corresponds to the second derivative operator on the Legendre polynomials through explicit calculation of matrix elements, as in Example 2.1. Compare your result to simply squaring the (first) derivative matrix.

(d) Now, determine the matrix that corresponds to exponentiation of the derivative matrix; that is, determine the \(3 \times 3\) matrix

\[\begin{equation*}\mathbb{M}=e^{\Delta x \frac{d}{d x}} \tag{2.92}\end{equation*}\]

where \(d / d x\) is shorthand for the matrix you constructed in Example 2.1. Note that the Taylor series of the exponential should terminate at a finite order in \(\Delta x\). Act this on a general quadratic function \(p(x)\) as defined in part (a) and show that the argument \(x\) just translates as \(p(x+\Delta x)\).