91Ó°ÊÓ

Determine the trigonometric interpolating polynomial \(S_{2}(x)\) of degree 2 on \([-\pi, \pi]\) for the following functions, and graph \(f(x)-S_{2}(x)\) : a. \(\quad f(x)=\pi(x-\pi)\) b. \(\quad f(x)=x(\pi-x)\) c. \(f(x)=|x|\) d. \(f(x)= \begin{cases}-1, & -\pi \leq x \leq 0 \\ 1, & 0

Find the linear least squares polynomial approximation to \(f(x)\) on the indicated interval if a. \(\quad f(x)=x^{2}+3 x+2, \quad[0,1]\); h. \(f(x)=x^{3}, \quad[0,2]\) c. \(\quad f(x)=\frac{1}{x}, \quad[1,3]\); d. \(\quad f(x)=e^{x}, \quad[0,2]\); e. \(\quad f(x)=\frac{1}{2} \cos x+\frac{1}{3} \sin 2 x, \quad[0,1]\) f. \(f(x)=x \ln x, \quad[1,3]\)

Find the linear least squares polynomial approximation on the interval \([-1,1]\) for the following functions. a. \(\quad f(x)=x^{2}-2 x+3\) b. \(\quad f(x)=x^{3}\) c. \(\quad f(x)=\frac{1}{x+2}\) d. \(\quad f(x)=e^{x}\) e. \(\quad f(x)=\frac{1}{2} \cos x+\frac{1}{3} \sin 2 x\) f. \(f(x)=\ln (x+2)\)

Find the continuous least squares trigonometric polynomial \(S_{3}(x)\) for \(f(x)=e^{x}\) on \([-\pi, \pi]\).

Find the least squares polynomials of degrees 1,2, and 3 for the data in the following table. Compute the error \(E\) in each case. Graph the data and the polynomials. $$ \begin{array}{lllllll} \hline x_{i} & 0 & 0.15 & 0.31 & 0.5 & 0.6 & 0.75 \\ y_{i} & 1.0 & 1.004 & 1.031 & 1.117 & 1.223 & 1.422 \\ \hline \end{array} $$

Given the data: $$\begin{array}{llllllllll} \hline x_{i} & 4.0 & 4.2 & 4.5 & 4.7 & 5.1 & 5.5 & 5.9 & 6.3 & 6.8 & 7.1 \\ y_{i} & 102.56 & 113.18 & 130.11 & 142.05 & 167.53 & 195.14 & 224.87 & 256.73 & 299.50 & 326.72 \\ \hline \end{array}$$ a. Construct the least squares polynomial of degree 1, and compute the error. b. Construct the least squares polynomial of degree 2, and compute the error. c. Construct the least squares polynomial of degree 3, and compute the error. d. Construct the least squares approximation of the form \(b e^{a x}\), and compute the error. e. Construct the least squares approximation of the form \(b x^{a}\), and compute the error.

Use the zeros of \(\tilde{T}_{3}\) and transformations of the given interval to construct an interpolating polynomial of degree 2 for the following functions. a. \(\quad f(x)=\frac{1}{x}, \quad[1,3]\) b. \(\quad f(x)=e^{-x}, \quad[0,2]\) c. \(\quad f(x)=\frac{1}{2} \cos x+\frac{1}{3} \sin 2 x, \quad[0,1]\) d. \(\quad f(x)=x \ln x, \quad[1,3]\)

Find the general continuous least squares trigonometric polynomial \(S_{n}(x)\) in for $$ f(x)= \begin{cases}-1, & \text { if }-\pi

Find the sixth Maclaurin polynomial for \(\sin x\), and use Chebyshev economization to obtain a lesserdegree polynomial approximation while keeping the error less than \(0.01\) on \([-1,1]\).

Show that for each \(n\), the Chebyshev polynomial \(T_{n}(x)\) has \(n\) distinct zeros in \((-1,1)\).