Stage 2 Mathematical Methods
Assessment Type 2: Mathematical Investigation
Topic 1: Further Differentiation and Applications

（1），（2）和（3）= 0。
Graph和在同一坐标平面上的线性近似（）= 1。

0.1。
[提示：Graph =（），= cos − 0.1，并且= cos + 0.1在同一坐标平面上]

（）= +（−）+（−）>

（）=（）+’（）（−）+
1个
2’（（）（-）>

？（）= @ + A（−）+>（−）> + B（−）B +⋯+？（−）？

>’’（）

？ =（？）（）

！ = 1×2×3×⋯×（！被称为k阶乘

？（）=（）+′（）（−）+′′（）
（-）2
2！
+′′′（）
（-）3
3！
+（4）
（）
（-）4
4！
+⋯+（）
（）
（-）

Find the 8th degree Taylor polynomial centred at = 0 for the function () = cos .
Graph together with the Taylor polynomials >, E, F, G on the coordinate plane bounded by −5 ≤ ≤ 5
and −1.4 ≤ () ≤ 1.4 and comment on how well they approximate .
St. Peter’s College Stage 2 Mathematical Methods
Investigation: Taylor Polynomials and Taylor/Maclaurin Series

In this part we introduce the concept of a Taylor polynomial with not just terms, but infinitely many terms.
With any polynomial in which we are adding and/or subtracting infinitely many terms, two possibilities
become apparent:
Case 1: the sum of terms never gets larger than a certain number (i.e. the infinite sum converges to a finite
limit).
Case 2: the sum of terms either becomes, positively or negatively, infinitely large or does not reach a finite
limit in some other way (i.e. the infinite sum diverges).
Such an infinite sum is called the Taylor series of the function centred at , represented below using Sigma
notation:
() = p (?)
()
?
!
H
?I@
NOTE: For the special case of = 0, the infinite sum is given the name Maclaurin series.
One implication of Case 2 above is that a Taylor or Maclaurin series may no longer approximate function values
accurately beyond a certain threshold.
This threshold is called the interval of convergence. When this interval is the entire set of real numbers, you can
use the series to find the value of () for every real value of .
However, when the interval of convergence for a series is bounded (for example −2 ≤ ≤ 6), that is, when it
diverges for some values of (in our example, for < −2 and > 6), you can use it to find the value of ()
only on its interval of convergence.
By finding the th-degree Taylor polynomial about = 0, determine the Maclaurin series for the following
functions:
() = J
ℎ() = ln(1 + )
Explain, giving evidence, whether the two Maclaurin series you found in Task 3.1 converge or diverge.
[Hint: consider the behaviour of the ℎ-term of the Maclaurin series]
Part 4
Determine the Maclaurin series of a function () of your choosing. This function must differ from those
investigated so far.
Investigate the relationship between the first few Taylor polynomials and ().
Investigate whether the function () converges for every real value of .
St. Peter’s College Stage 2 Mathematical Methods
Investigation: Taylor Polynomials and Taylor/Maclaurin Series

EasyDue™ 支持PayPal, AliPay, WechatPay, Taobao等各种付款方式!

E-mail: easydue@outlook.com  微信:easydue

EasyDue™是一个服务全球中国留学生的专业代写公司