Calc1231W11T4 - Maclaurin series

Sunday, 14 October 2018

4:52 PM

A Maclaurin series for a function f is a special case of the Taylor series at the point a = 0. That is 1! 2! This kind of (possibly infinite) polynomial is often called a power series Note: Recall that f (4) (a) denotes the 4th derivative of f evaluated at x = a not (f(a))4 Additionally, f (0) (a) The most direct way to obtain a Maclaurin series is by calculating the derivatives of f and evaluating them at (b For example if f@) = cosh@), then and so om Thus the first terms of the Maclaurin series for cosh@) up to and including the degree 6 term is Colin Maciaurin 1698 - 1746 = f(a)

 

Let P6 (x) denote the Maclaurin series up to and including terms of degree 6 for a function f(x). • If f(x) = cos@) then P6(c) — • If f(x) = sin@) then P6 (x) 1 • If f(x) then P6@)

 

Identify the functions cos@), sin(x), exp@) and In(l + x) from their power series in powers of x: 2 2 3 2! 3! 2n (2n)! 3 4 4 exp(x) sin (x) In(l+x) 2 3

 

Let's explore the sense in which f@) By Taylor's theorem e equals its series expansion about a 2 2! where Rn+l@) is the Lagrange form of the remainder, which can be written as f (n+l) (c) n+l Rn+l (x) where c is somewhere between 0 and x Suppose 0 < c < x; then Hence Since by the Pinching Theorem So for all values of x: Note: the Maple syntax for ex is exp (x) exp(c) x"' (11+1)' O < ec exp(x) Rn+l@) e lim lim Rn+l@) — + Rn+1@) 2 2!

 

The following question was adapted from the MATH1231 2013s2 final exam Question 1 part vi_ Question: Let f@) = In 4x2 + 2m + 1 . The following MAPLE session may assist you with this question > f +2*x+1) : > taylor(f (x) , x = (), 7); Write down the values of f" (0) and f" Answer: 4 2 2x+2x 4 32x5 64x6 — 16/3x3 + 4x +

 

 

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