# 数学代写|微积分代写Calculus代考|MTH/280

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## 数学代写|微积分代写Calculus代考|The Cartesian Plane

Just as you can represent real numbers by points on a real number line, you can represent ordered pairs of real numbers by points in a plane called the rectangular coordinate system, or the Cartesian plane, named after the French mathematician René Descartes (1596-1650).

The Cartesian plane is formed by using two real number lines intersecting at right angles, as shown in Figure 1.1. The horizontal real number line is usually called the $\boldsymbol{x}$-axis, and the vertical real number line is usually called the $\boldsymbol{y}$-axis. The point of intersection of these two axes is the origin, and the two axes divide the plane into four parts called quadrants.

Each point in the plane corresponds to an ordered pair $(x, y)$ of real numbers $x$ and $y$, called coordinates of the point. The $x$-coordinate represents the directed distance from the $y$-axis to the point, and the $y$-coordinate represents the directed distance from the $x$-axis to the point, as shown in Figure 1.2.

The notation $(x, y)$ denotes both a point in the plane and an open interval on the real number line. The context will tell you which meaning is intended.

## 数学代写|微积分代写Calculus代考|The Pythagorean Theorem and the Distance Formula

The following famous theorem is used extensively throughout this course.
Pythagorean Theorem
For a right triangle with hypotenuse of length $c$ and sides of lengths $a$ and $b$, you have $a^2+b^2=c^2$, as shown in Figure 1.5. (The converse is also true. That is, if $a^2+b^2=c^2$, then the triangle is a right triangle.)
Suppose you want to determine the distance $d$ between two points $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$ in the plane. With these two points, a right triangle can be formed, as shown in Figure 1.6. The length of the vertical side of the triangle is $\left|y_2-y_1\right|$, and the length of the horizontal side is $\left|x_2-x_1\right|$. By the Pythagorean Theorem, you can write
\begin{aligned} d^2 & =\left|x_2-x_1\right|^2+\left|y_2-y_1\right|^2 \ d & =\sqrt{\left|x_2-x_1\right|^2+\left|y_2-y_1\right|^2}=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2} . \end{aligned}
This result is the Distance Formula.
The Distance Formula
The distance $d$ between the points $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$ in the plane is
$$d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2} .$$

## 数学代写|微积分代写Calculus代考|The Pythagorean Theorem and the Distance Formula

\begin{aligned} d^2 & =\left|x_2-x_1\right|^2+\left|y_2-y_1\right|^2 \ d & =\sqrt{\left|x_2-x_1\right|^2+\left|y_2-y_1\right|^2}=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2} . \end{aligned}

$$d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2} .$$

## 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。