1- Prove the following relations:ag(1 – 2xt + t?)= (x - t) g(x, t)(1)atdg(1– 2xt + t?)=t g(x,t)(2)axag(x - t)atagt(3)axWhere g(x,t) is the generating function of Legendre's polynomials

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Answered: 1- Prove the following relations: ag (1…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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2 1. Consider a polynomial function f(x) = co + C₁ x + c₂x² +...+ cd-1xd-1 + caxd. A. Applying the addition rule, the constant multiple rule, and the power rule for derivatives, we can compute that ƒ'(x) = c₁ + 2c2x + … + (d − 1)cd-1xd-2 + (d)caxd-¹. Explain. B. Following the general outline from Part A, compute the derivatives of each of the following functions: g(x) = 5x³ 10x² + 7x - 11. p(x) = −7x² – 3x² + πx √17. O O 2. Consider the function f(x) = (2x³ – 5x + 1) (3x² − 4). A. Distribute out the product of the two polynomials multiplied in the expression of f and simplify your result to find an equivalent expression for f(x).
Find the linear approximation of f(x) = Vx +1 at a = 1. 1 V2 + (x – 1) 2 V2 V2 + 4 (= - 1) (ӕ — 1) 2 + 4 V2 V2 + 4 (x – 1) 4
Find the linearization L(x) of f(x) at x = a. flx) = a =0 7x +4 Lox) = -x+ 7 16 16 L(x) = 16 LW) = - *+ 7· 16 4 Lw - ** 7 L(x) 16 16
Prove that f(x) = ³ + x/2 is C¹-structurally stable.
3. Interpolate the function f(x) = 2ª over the interval [−1, 1] by a polynomial (a) Construct the interpolating polynomial p2(x) of degree n = 2 in the Newton form using uniformly spaced nodes. 1 (b) Find an upper bound for the error |ƒ(x) — p2(x)| at x = 2 (c) Find an upper bound for the error |ƒ(x) — p2(x)| on [−1,1]. Note that max f(x) p2(x)| ≤ x= [a,b] h³ M3 9√3 9 max | f(x) — pn(x)| ≤ x= [a,b] where h = (b − a)/2 and |ƒ""'(x)| ≤ M3 for x € [a, b]. (d) How many equally-spaced nodes are required to interpolate the function f(x) to within € = 0.01? Note that for n > 2, 1 max | f(n+¹) (§) | (n + 1)! E[a,b] hn + 1n! 4 where h = (b − a)/n.

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1- Prove the following relations: ag (1 – 2xt + t?)= (x - t) g(x, t) (1) at ag (1- 2xt + t2)= t g(x, t) (2) ax ag t at ag (x - t) ax (3) Where g(x,t) is the generating function of Legendre's polynomials