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please help number 2 & 3

for all v, w ER" is an inner product on K. 4.1.6. Prove that for two vectors v, w in a real inner product || v + w || ² = || v || ² + || w || ² if and only if u and w are orthogonal. is hasis for n 5 space

1. Show that the linear minimax approximation to $1+x_{2} $ on $[0,1]$ is $q_{1}(x)=0.955+0.414x$ 2. The Legendre polynomial has a weight $w(x)≡1$ on $[−1,1]$ which is defined by $P_{n}(x)=2_{n}n!(−1)_{n} dx_{n}d_{n} (1−x_{2})_{n},n≥1$ with $P_{0}(x)=1$. (a) Show that ${P_{n}(x)}$ is an orthogonal family. (b) $∥P_{n}∥_{2}=(2n+1)2 $, dengan $n≥1$. 3. Define $S_{n}(x)=n+11 T_{n+1}(x),n≥0$, 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 ${S_{n}(x):n≥0}$ is an orthogonal family on $[−1,1]$ with weight function $w(x)=1−x_{2} $. (b) Show that the family ${S_{n}(x)}$ satisfies the triple recursive relation $S_{n+1}(x)=2xS_{n}(x)−S_{n−1}(x),n≥1.$ 4. For the integral $I=∫_{−1}1−x_{2} f(x)dx$ with weight $w(x)=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 $∫_{a}f(x)dx$ on an interval is defined by $∫_{a}f(x)dx=452h (7f(a)+32f(a+h)+12f(2a+b )+32f(b−h)+7f(b))−9458h_{7} f_{(6)}(ξ) $ 6. Use Gauss-Laguere with $n=2$ and $n=4$ nodes to calculate the improper integral following: $∫_{0}e_{−x_{2}}dx=2π $

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