In Problems 1-8, determine the first three nonzero termsin the Taylor polynomial approximationsinitial value problem.for the given5. x" + tx = 0;x(0) = 1,x' (0) = 0

Answered: In Problems 1-8, determine the first…

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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The goal of Taylor approximation is to approximate a function f(x) with a polynomial of degree n centered at c: Tn(x) = ao+a₁(x-c) + a₂(x-c)²+...+an(x-c)". f(k) (c) k! In this problem, you will show that a = assuming that f(k) (c) = T() (c), where k = 0,1,2, - - -. (a) If T₁ (x) = a + a₁(x − c) + a₂(x − c)² + + an(rc)", what is T₁(c)? (b) What are T (c), T (c) and T³) (c)? (c) Do you see a pattern? What is T) (c)? f(k) (c) (d) Using the equality: f(k) (c) = T(k) (c) and the result in (c), show that a = k!
4. Show that the Taylor polynomials for f(x) = In(1 + x) can be used to approximate f(s) for s e (-1, 1) by carrying our the following steps. First, show that for t E (-1,1), we have (-t)"+1 1 = 1- t+t? – ... +(-t)" + 1+t 1+t Next, by integrating both sides of this equation, get a new integral estimate for Rn for this function, and finally, show that Rn tends to zero for any s E (-1,1).
1. Use the definition to find the first five nonzero terms of the Taylor polynomial, centered at c = 1, for f(x) = -1. Then express your answer in X sigma notation.
Help finishing problem
Can I also get the work for this problem.
5. In many applications the error function erf(r) = du is important. Compute and graph the fourth order Taylor polynomial of er f(x) about a = 0.
A calculator may be used for the following questions. Let T(x) represent the Taylor Polynomial of degree n about x = 0 for f(x) = e*. If T,, (2) is used to approximate the value of f(2)= e2, then which of the following expressions is NOT less than ? 7! Select one: a. f (2)-T6(2) O b. 16(2)-f(2) O c. f(2) T5(2) O d. Ig(2)-f(2)
Suppose f(x) is a function whose 4th order Taylor polynomial at the base point 0 is T₁(x) = x² + √²x³ + x² + x + 1 18 Determine the value of f(4) (1). Of(4) (0) = 24 ƒ(4) (0) = 2/1/2 1 48 O f(4) (0) = = f(4) (0) = = 1 12 None of the listed answers is correct.
Problem 38 Let T2(z) be the Taylor polynomial of f(r) = V centered at a = 4. Use the Error Bound to estimate the accuracy of the approximation (i.e.to find the maximum value of the error) of VI=T;(x) for 3.9 <1<4

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In Problems 1-8, determine the first three in the Taylor polynomial approximations for the given initial value problem.