7. Consider the Taylor polynomial T,(x) for f(x) = vT at a = 4. Use Taylor's inequality toestimate the accuracy of the approximation f(x) 2 T2(x) when x lies in the interval2

This is a problem of Taylor's polynomial and approximation.

Answered: 7. Consider the Taylor polynomial T;(x)…

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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Suppose that g(x) is a function that is continuous for all real values of x. Let's say that you want to compute the third-degree Taylor polynomial T3(x) centered at 8 for g(x), and you know that g(8) = 2, g' (8) = -5, g "(8) = 18, and g "(8) = 48. (In case you are having trouble reading the notation, I just gave you the value of g at 8, followed by the values of the first, second, and third derivatives of g at 8, in that order.) What is T3(x)?
How large should the value of (n) such that the Taylor polynomial of order (n) about 100 gives an approximate value for sqrt(101) with an error less than 10-10 in absolute value.
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(3) (a) Let p be a polynomial of degree n with n distinct root c1, c2, ..., Cn. Show that 1. 1 p(x) 7(c)(x – c;)' i=1 Hint. First show that p'(c;) # 0 for all i = 1, 2,...,n. (b) Use (a) to find dr. r(x +2)(x – 3)(r – 1)
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1. Let f(x) = xe*. (a) Find p4(x), the 4th Taylor polynomial for f(x), about x = 0. (b) Bound the error in the approximation f(x)~p4(x) when x € [-2,2], and when x = [1,1].
Consider the function f(x) = Vx + I. Compute f'(x) and f"(x). • (a) • (b) ) Write down the Taylor polynomials T1 and T2 of the function f at the point a = 0. • (c) DWhich value of b makes f(b) equal to V1.4? • (d) ) What are the estimates of V1.4 given by Tj and T2 from part (b)? (The actual value of V1.4 is about 1. 1832.)
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could you pleaae do all parts and could the answer be written out please , not typed up. Thank you

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7. Consider the Taylor polynomial T;(x) for f(x)= VTat a = 4. Use Taylor's inequality to estimate the accuracy of the approximation f(z) × T2(x) when z lies in the interval 2