Question

1. Taylor series are special power series that are defined from a function f(z) atz = a by fitting higher and higher degree polynomials T, a(x) to the curve at the point (a, f(a)), with the goal of getting a better and better fit as we not only let the degree grow larger, but take a series whose partial sums are these so-called Taylor polynomials Tm,a(x) We will explore how this is done by determine the Taylor series of f(z) cos(z) at x = 0 by hand through a series of steps, as follows. (a) Comput You can get a formula for this derivative; do obtain one result when n 2m is even, and another when n = 2m + 1 is odd. (b) Compute (0). You can get a formula for this. The formula will depend on n, and whether n = 2m or n 2m + 1. (c) Let T,no(x)-Pa(z) = Σ ak (z-0). The a's are just unknown constants at this point. This is a polynomial of degree m for now.1 i. Write out Po(x), P(x), P2(), Ps(x), P4(x), Ps(), and Po(x) explicitly in terms of the unknown constants ak, per the given formula; here, P()i given for you: ii. Now, for each of n 0,12,, 6, compute each derivative of each Pm() here, the result for Ps() (m 3) is given for you: P(r) 2.1 a2+3-2 a(a F33)(z) 3.2.1-аз 0)

Answer 1

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