• 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

Line Graphs are most often displayed on a Cartesian coordinate system, which was presented in the previous section. As previously described, this coordinate system consists of two intersecting perpendicular lines called axes, as was shown in Figure $2.5$ and reproduced below as Figure $2.14$ for convenience.

Recall from Section $2.1$ that he intersection of the two axes is the Origin, which is the reference point for the system. The horizontal line is typically called the $x$-axis and the vertical line is typically called the $y$-axis. Although the letters $x$ and $y$ are the most widely used symbols for these axes, other letters are used when they are more meaningful in a particular problem.

Tick marks are then used to divide each axis into fixed units of length. Units to the right of the origin on the $x$-axis and units above the origin on the $y$-axis are assigned positive values. Units to the left of the origin on the $x$-axis and units below the origin on the $y$-axis are assigned negative values. Arrowheads are affixed to the positive portions of the $x$-and $y$-axes to indicate the directions of increasing values of $x$ and $y$. Successive tick marks must be equally spaced, which makes the units between successive tick marks the same, although the units or scale on the horizontal axis can, and frequently does, differ from the scale used on the vertical axis.

Graphing an equation can be considerably shortened if we already know the shape of the curve that the equation describes. One such case is provided by linear equations. These equations are singled out because they are one of the simplest and most important equations in both business and mathematics. They also have the geometric property that their graphs are always straight lines, among other useful features that are presented in the next section.
Formally, an equation is linear in two variables, $x$ and $y$, if it satisfies the following definition:

Definition 2.1: A linear equation in two variables $x$ and $y$ is an equation of the form:
$$A x+B y=C$$
where $A, B$, and $C$ are known real numbers and $A$ and $B$ are not both zero (this avoids the equations of the form $0=C$. The variables $x$ and $y$ can be replaced by any other convenient letters. Thus, if $x$ and $y$ are replaced by $p$ and $q$, respectively, then Equation $2.1$ becomes the linear equation $\mathrm{A} p+\mathrm{B} q=C$ in the variables $p$ and $q$.

Definitions are very precise mathematical statements. Unfortunately, this precision often makes a definition seem very complicated when, in fact, it is not. Usually, a few moments of thought is all that is needed to convert the given statement to an understandable concept.

For example, Definition $2.1$ simply states that any equation having the form of (i.e., looks like) the equation $A x+B y=\mathrm{C}$ where the letters $A, B$, and $C$ are replaced by numbers (e.g., $3 x+7 y=10$ ), is called a linear equation. Another important point is that because an exponent of 1 is understood but not written, a necessary feature of a linear equation is that the exponent of both the $x$ and $y$ terms must only be 1 .

The definition does not give any clues as to what a linear equation means geometrically-that will come later. What it does say, however, is that any equation that can be written in a form that looks like Equation 2.1, is a linear equation.

Linear equations and their corresponding straight lines have a number of extremely useful characteristics. These include a concept called the line’s slope, the line’s $y$-intercept, and the ease with which these two quantities can be found from the line’s equation and graph. Conversely, the line’s equation can easily be determined if two solutions, that is, two points on the line are known. Each of these topics is presented in this section. We begin with the concept of a lines slope.

Graphically, a line’s slope is the direction and steepness of the line. Mathematically, it provides the rate of change, that is, how fast or slow the $y$ variable changes with respect to a change in the $x$ variable.

Graphically, a line with a positive slope rises as you move from left to right along the $x$-axis, as shown in Figure $2.33$. Also, as shown in the figure, a positively sloped line will have an angle between 0 and 90 degrees between the line and the positively directed $x$-axis. Finally, a line with a large positive slope, such as 100 , is steeper upward (that is, rises more quickly) than a line with a less positive slope, such as 5 .

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。