(Solved): please help number 2 & 3for all v, w ER" is an inner product on K. 4.1.6. Prove that for two vec ...

1. Show that the linear minimax approximation to $\sqrt{1+x^2}$ on $[0,1]$ is $q_1^{\ast }(x)=0.955+0.414x$ 2. The Legendre polynomial has a weight $w(x)\equiv 1$ on $[-1,1]$ which is defined by $P_n(x)=\frac{(-1{)}^n}{2^{n}n!}\frac{d^n}{d{x}^n}{\left(1-x^2\right)}^n,n\ge 1$ with $P_0(x)=1$. (a) Show that $\left\{P_n(x)\right\}$ is an orthogonal family. (b) ${\Vert P_n\Vert }_2=\sqrt{\frac{2}{(2n+1)}}$, dengan $n\ge 1$. 3. Define $S_n(x)=\frac{1}{n+1}{T}_{n+1}(x),n\ge 0$, where $T_{n+1}(x)$ is a Chebyshev polynomial of degree $n+1$. The $n+1$ polynomial $S_n(x)$ is called the second type of Chebyshev polynomial. (a) Show that $\left\{S_n(x):n\ge 0\right\}$ is an orthogonal family on $[-1,1]$ with weight function $w(x)=\sqrt{1-x^2}$. (b) Show that the family $\left\{S_n(x)\right\}$ satisfies the triple recursive relation $S_{n+1}(x)=2x{S}_n(x)-S_{n-1}(x),n\ge 1.$ 4. For the integral $I={\int }_{-1}^1\sqrt{1-x^2}f(x)dx$ with weight $w(x)=\sqrt{1-x^2}$, find an explicit formula for nodes and weights from Gauss's quadrature formula and also give a formula for the error. 5. Boole's rule for calculating ${\int }_a^{b}f(x)dx$ on an interval is defined by $\begin{array}{rl}{\int }_a^{b}f(x)dx=\frac{2h}{45}& \left(7f(a)+32f(a+h)+12f\left(\frac{a+b}{2}\right)+32f(b-h)+7f(b)\right)\\ & -\frac{8h^7}{945}{f}^{(6)}(\xi )\end{array}$ 6. Use Gauss-Laguere with $n=2$ and $n=4$ nodes to calculate the improper integral following: ${\int }_0^{\infty }{e}^{-x^2}dx=\frac{\sqrt{\pi }}{2}$