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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|微积分代写Calculus代写|Approximation

You are familiar with the idea of approximating numbers using a calculator:
Example 1 Approximate $\sqrt{2}$ to four decimal places.
Solution Using a calculator, $\sqrt{2} \approx 1.4142$.
You may be used to writing $\sqrt{2}=1.4142$. However, technically speaking, this statement is false. The numbers $\sqrt{2}$ and $1.4142$ are not exactly the same. But, for most practical purposes, they are the same, which is one reason why we often calculate with $1.4142$ instead of $\sqrt{2}$.
We use decimal approximations because of their convenience. As long as we keep enough digits during intermediate calculations, our final answers are close enough. If we need a rod of length $10 \sqrt{2} \mathrm{~cm}$, we write the answer as $14.142 \mathrm{~cm}$ because we know how to measure the decimal expression much more easily than the square root expression, and if we are off in the fourth decimal place, it’s just thousandths of a millimeter and, for most purposes, is totally inconsequential.

Remember approximations of $\pi$ ? You may have used $\pi=\frac{22}{7}$ or $\pi=3.14$. Again, these statements as written are actually false, so we often write $\pi \approx \frac{22}{7}$ and $\pi \approx 3.14$ instead; but, for many applications, these approximations do just fine.

The difference between an approximation and the actual value it is meant to approximate is called the error in the approximation. For instance, the error in the approximation $\sqrt{2} \approx 1.4142$ can be calculated as
\begin{aligned} 1.4142 &=1.4142 \ -\quad \sqrt{2} &=1.414213562 \ldots \ \hline &-0.000013562 \ldots, \end{aligned}
whereas the error in $\pi \approx \frac{22}{7}$ is
\begin{aligned} \frac{22}{7} &=3.14285714 \ldots \ -\quad \pi &=3.14159265 \ldots \ \hline & \end{aligned}

Now consider approximating an expression such as $7+\omega$. The infinitesimal $\omega$ is so much smaller than the real number 7 that $7+\omega$ is, for all practical purposes, the same as 7 . We therefore say that they are approximately equal.

## 数学代写|微积分代写Calculus代写|Hyperreal approximations: a few details

Now that we have studied approximation, it is time to look at a few of the technical details before returning to more examples. Two hyperreal numbers are approximately equal if their relative difference is infinitesimal.

Definition 3 APPROXIMATION Let $x$ and $y$ be hyperreal numbers. If $x \neq 0, x \approx y$ if $\frac{y-x}{x}$ is zero or infinitesimal. Also, $0 \approx 0$.

By this definition, numbers that are equal are also approximately equal. Also, the only number that is approximately equal to zero is zero itself. This definition does not give approximation as an operation, but rather as an “equivalence relation” (think “=” rather than “+”). But theorem 3 shows how we can often use approximation as if it is an operation, by throwing away lower-level terms.

Proof. Notice $\frac{y+x-y}{y}=\frac{x}{y}$ is infinitesimal by definition of lower level. Therefore, $y+x \approx y$ by definition of approximation.

In section $1.1$ we looked at lower and higher levels, which were defined formally, but not the same level. Approximation allows us to assign this definition now.

Definition 4 SAME LEVEL Two hyperreal numbers $x$ and $y$ are on the same level if $\frac{x}{y} \approx r$ for some nonzero real number $r$.

Therefore, $3 \omega+\omega^2$ and $12 \omega$ are on the same level because $\frac{3 \omega+\omega^2}{12 \omega} \approx$ $\frac{3 \omega}{12 \omega}=\frac{1}{4}$ is a nonzero real number.

# 微积分代考

## 有限元方法代写

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## 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。

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