### 数学代写|数学物理方法代写Mathematical Methods代考|MATH1012

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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|数学物理方法代写Mathematical Methods代考|DEFINITION OF A DERIVATIVE

The derivative of a function is the slope (or gradient) of the tangent to the graph of the function at any point; i.e. it is the slope approached by the line between two points on the graph as they get closer (Figure 1.1). By the “slope” we mean (as usual) the change in height divided by the horizontal displacement. To use this definition to calculate the slopes of graphs we put it into more formal language.

More formally, the clerivative of a function $y=f(x)$ is the function $f^{\prime}(x)$ or $d y / d x$ defined by
$$f^{\prime}(x)=\lim {h \rightarrow 0} \frac{f(x+h)-f(x)}{h} .$$ Equivalently, in a commonly used alternative notation, we write $\delta x$ in place of $h$ to emphasize that we are making a small increment in $x$. We also sometimes write $y=y(x)$ instead of $y=f(x)$ to indicate that $y$ is a function of $x$. The definition is then written $$\frac{d y}{d x}=\lim {\delta x \rightarrow 0} \frac{y(x+\delta x)-y(x)}{\delta x} .$$
For a function of the form $y=f(x)$ we use the notation $f^{\prime}(x)$, $y^{\prime}$ and $d y / d x$ interchangeably. The definition expresses in an exact manner the fact that the derivative at $x$ is the slope of the tangent at $x$. In order to use it we need a way of taking the limit. The following example shows how this is done for a simple function such as a power of $x$.

## 数学代写|数学物理方法代写Mathematical Methods代考|SOME BASIC DERIVATIVES

You should already know (or be willing to take on trust for the present) the derivatives in Table 1.1.

Note: the trigonometric functions are introduced in section A.4 of Appendix A and the logarithm in Section A.3. Throughout this course $\ln (x)$ will be written for $\log {e}(x)$, the “natural logarithm.” It is common to use $\log (x)=\log {10}(x)$, and the use of any other base is indicated explicitly, as in $\log _{2}(x)$. The derivatives in the table may be taken as given.

The derivative of the sum of two functions is the sum of the derivatives. This means that if $f(x)$ and $g(x)$ are both functions of $x$ then
$$\frac{d}{d x}(f+g)=\frac{d f}{d x}+\frac{d g}{d x}$$
Exercise 1.2 What is $d y / d x$ if
(i) $y=x^{n}+c$ where $c$ is a constant and $n$ an integer
(ii) $y=x^{-3}+x^{4}$
(iii) $y=2 \sin (x)=\sin (x)+\sin (x)$
(iv) $y=\sin (x)+\cos (x)$.
We also learn rules for the derivative of a “function of a function” and of a product, which enable us to construct the derivatives of quite complicated expressions from the table without having to return to the basic definition.

## 数学代写|数学物理方法代写Mathematical Methods代考|DEFINITION OF A DERIVATIVE

$$f^{\prime}(x)=\lim h \rightarrow 0 \frac{f(x+h)-f(x)}{h} .$$

$$\frac{d y}{d x}=\lim \delta x \rightarrow 0 \frac{y(x+\delta x)-y(x)}{\delta x} .$$

## 数学代写|数学物理方法代写Mathematical Methods代考|SOME BASIC DERIVATIVES

$$\frac{d}{d x}(f+g)=\frac{d f}{d x}+\frac{d g}{d x}$$

(一) $y=x^{n}+c$ 在哪里 $c$ 是一个常数并且 $n$ 一个整数
\begin{aligned} &\text { (ii) } y=x^{-3}+x^{4} \ &\text { ( }) y=2 \sin (x)=\sin (x)+\sin (x) \ &\text { (四) } y=\sin (x)+\cos (x) \text {. } \end{aligned}

## 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。