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 +

 

 

Created with Microsoft OneNote 2016.