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# 数学代写|数值分析代写Numerical analysis代考|AMATH352 THE TAYLOR POLYNOMIAL

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## 数学代写|数值分析代写Numerical analysis代考|THE TAYLOR POLYNOMIAL

Most functions $f(x)$ that occur in mathematics cannot be evaluated exactly in any simple way. For example, consider evaluating $f(x)=\cos (x), e^x$, or $\sqrt{x}$, without using a calculator or computer. To evaluate such expressions, we use functions $\hat{f}(x)$ which are almost equal to $f(x)$ and are easier to evaluate. The most common class of approximating functions $\hat{f}(x)$ are the polynomials. They are easy to work with and they are usually an efficient means of approximating $f(x)$. Among polynomials, the most widely used is the Taylor polynomial. There are other more efficient approximating polynomials, and we study some of them in Chapter 6 . But the Taylor polynomial is comparatively easy to construct, and it is often a first step in obtaining more efficient approximations. The Taylor polynomial is also important in several other areas of mathematics.

Let $f(x)$ denote a given function, for example, $e^x$ or $\log (x)$. The Taylor polynomial is constructed to mimic the behavior of $f(x)$ at some point $x=a$. As a result, it will be nearly equal to $f(x)$ at points $x$ near to $a$.
To be more specific, find a linear polynomial $p_1(x)$ for which
\begin{aligned} &p_1(a)=f(a) \ &p_1^{\prime}(a)=f^{\prime}(a) \end{aligned}
Then, it is easy to verify that the polynomial is uniquely given by
$$p_1(x)=f(a)+(x-a) f^{\prime}(a)$$
The graph of $y=p_1(x)$ is tangent to that of $y=f(x)$ at $x=a$.

## 数学代写|数值分析代写Numerical analysis代考|THE ERROR IN TAYLOR’S POLYNOMIAL

To make practical use of the Taylor polynomial approximation to $f(x)$, we need to know its accuracy. The following theorem gives the main way of estimating this accuracy. We present it without proof, since it is given in most calculus texts.
(Taylor’s Remainder) Assume that $f(x)$ has $n+1$ continuous derivatives on an interval $\alpha \leq x \leq \beta$, and let the point $a$ belong to that interval. For the Taylor polynomial

$p_n(x)$ of (1.6), let $R_n(x) \equiv f(x)-p_n(x)$ denote the remainder in approximating $f(x)$ by $p_n(x)$. Then,
$$R_n(x)=\frac{(x-a)^{n+1}}{(n+1) !} f^{(n+1)}\left(c_x\right), \quad \alpha \leq x \leq \beta$$
with $c_x$ an unknown point between $a$ and $x$.

## 数学代写|数值分析代写Numerical analysis代考|THE TAYLOR POLYNOMIAL

$$p_1(a)=f(a) \quad p_1^{\prime}(a)=f^{\prime}(a)$$

$$p_1(x)=f(a)+(x-a) f^{\prime}(a)$$

## 数学代写数值分析代写Numerical analysis代考|THE ERROR IN TAYLOR’S POLYNOMIAL

(泰勒余数) 假设 $f(x)$ 有 $n+1$ 区间上的连续导数 $\alpha \leq x \leq \beta$, 并让点 $a$ 属于那个区间。对于泰勒多项式
$p_n(x)$ 的 (1.6)，让 $R_n(x) \equiv f(x)-p_n(x)$ 表示近似中的余数 $f(x)$ 经过 $p_n(x)$. 然后，
$$R_n(x)=\frac{(x-a)^{n+1}}{(n+1) !} f^{(n+1)}\left(c_x\right), \quad \alpha \leq x \leq \beta$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。