Legendre polynomial least square approximation

Using Legendre polynomials of degree 1, 2, and 3, find the least-squares approximation for the function e^(−x) on [2, 4]. I am confused as to how to do this on the interval [2,4] and not [-1,1] I have included the notes on this section, including an example.

Convergence and Error Analysis

Word Doc also attached, easier to view. Please give the complete solution, include reasoning and calculations used to arrive at answer.

Lagrange Polynomial Interpolation

Please see the attached problem. (Word and pdf attached). Please give the complete solution, include reasoning and calculations used to arrive at answer. If you use a particular theorem please identify it. We cannot use any high level command from Maple, Mathematica or MatLab.

Doolittle LU decomposition

Please see the attached problem: Please give the complete solution, include reasoning and calculations used to arrive at answer.

Third Order Taylor Method

Please see problem attached.

Differential Equations & Lagrange Polynomials

Please see attached problem. Please give the complete solution, include reasoning and calculations used to arrive at answer. If you use a particular theorem please identify it. We cannot use any high level command from Maple, Mathematica or MatLab. Thank you kindly.

Explanation of Trapezoid Rule & Extrapolation

A step by step solution is provided to improve the approximate value of definite integral obtained by using composite trapezoidal rule. Richardson’s Extrapolation to improve these approximations in this particular solution. This detailed explantion will help the students to understand the concept of extrapolation.

Gaussian Elimination with Scaled Partial Pivoting

Please see attached problem (both Word and pdf version attached). Please give the complete solution, include reasoning and calculations used to arrive at answer. If you use a particular theorem please identify it. We cannot use any high level command from Maple, Mathematica or MatLab.

Lagrange Polynomials

Please see attached problem Please give the complete solution, include reasoning and calculations used to arrive at answer. If you use a particular theorem please identify it. We cannot use any high level command from Maple, Mathematica or MatLab. Thank you kindly and please contact me with any questions. Kind regards.

Sets

See attachment 1) Let > 0 and >0, and . Show that and are - neighborhoods of for appropriate values of . 2)Let and be nonempty sets and let : have bounded range in . Let : and be defined by , Prove that We sometimes express this by writing ...continues