这个作业是完成二次逼近、线性逼近等数学问题

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

任务1.1
找到满足条件的函数()= cos的二次逼近()= + +>
(1),(2)和(3)= 0。
Graph和在同一坐标平面上的线性近似()= 1。
评论函数和近似函数的良好程度。

任务1.2
确定在任务1.1中二次逼近()=()精确到的值
0.1。
[提示:Graph =(),= cos − 0.1,并且= cos + 0.1在同一坐标平面上]

任务1.3
要用一个接近平方的二次函数逼近一个函数,最好以以下形式编写:
()= +(−)+(−)>
证明满足条件(1),(2)和(3)的泛型二次函数为:
()=()+’()(−)+
1个
2’(()(-)>

任务1.4
找出()=√+ 3 near = 1的线性近似和二次近似。
图,在同一坐标平面上的二次逼近和线性逼近。你是什​​么
得出结论?
圣彼得学院第二阶段数学方法
研究:泰勒多项式和泰勒/麦克劳林级数

我们希望找到更好的方法,而不是对()附近的线性或二次近似值感到满意,
高阶多项式的近似。我们寻找一个ℎ度多项式:
?()= @ + A(−)+>(−)> + B(−)B +⋯+?(−)?
这样吗?并且其一阶导数在=处的值与及其一阶导数相同。
任务2.1
通过反复微分并设置=,表明以上第1部分中所述的条件(1),(2)和(3)
如果@ =(),A =’(),> = A
>’’()
通常:
? =(?)()

哪里:
! = 1×2×3×⋯×(!被称为k阶乘
结果多项式:
?()=()+′()(−)+′′()
(-)2
2!
+′′′()
(-)3
3!
+(4)
()
(-)4
4!
+⋯+()
()
(-)

称为的中心为的度度泰勒多项式。
Task 2.2
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.
Task 3.1
By finding the th-degree Taylor polynomial about = 0, determine the Maclaurin series for the following
functions:
() = J
ℎ() = ln(1 + )
Task 3.2
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
Task 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


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