### 数学代写|微积分代写Calculus代写|Distance formula

statistics-lab™ 为您的留学生涯保驾护航 在代写微积分Calculus方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写微积分Calculus代写方面经验极为丰富，各种代写微积分Calculus相关的作业也就用不着说。

• 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|Distance formula

How do we find the distance between two points in the plane, $P_{1}$ and $P_{2}$ ? If their coordinates are given as $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$, respectively, then consider the third point $\left(x_{2}, y_{1}\right)$ that aligns vertically with $P_{2}$ and horizontally with $P_{1}$, as pictured in figure 8 .Drawing the vertical and horizontal segments forms a right triangle, so the Pythagorean theorem applies. The length of the horizontal

segment is the difference in the $x$-coordinates: $\left|x_{2}-x_{1}\right|$. The length of the vertical segment is the difference in the $y$-coordinates: $\left|y_{2}-y_{1}\right|$. See figure 9 . Using the Pythagorean theorem, the distance between $P_{1}$ and $P_{2}$, written $d\left(P_{1}, P_{2}\right)$, is
\begin{aligned} d\left(P_{1}, P_{2}\right) &=\sqrt{\left|x_{2}-x_{1}\right|^{2}+\left\lfloor y_{2}-\left.y_{1}\right|^{2}\right.} \ &=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}} \end{aligned}

This is called the distance formula and it serves as our definition of the distance between two points in the plane.

Definition 3 DISTANCE FORMULA The distance between two points $P_{1}=\left(x_{1}, y_{1}\right)$ and $P_{2}=\left(x_{2}, y_{2}\right)$ is given by
$$d\left(P_{1}, P_{2}\right)=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}} .$$
Example 3 Find the distance between the points $(2,6)$ and $(1,-4)$.
Solution Setting $\left(x_{1}, y_{1}\right)=(2,6)$ and $\left(x_{2}, y_{2}\right)=(1,-4)$, the distance formula gives
\begin{aligned} d((2,6),(1,-4)) &=\sqrt{(1-2)^{2}+(-4-6)^{2}} \ &=\sqrt{(-1)^{2}+(-10)^{2}}=\sqrt{1+100} \ &=\sqrt{101} \end{aligned}
The distance between the two points is $\sqrt{101}$. If desired, a decimal approximation can be given.

## 数学代写|微积分代写Calculus代写|Slopes of lines

Consider a line $\ell$ that is neither vertical nor horizontal (figure 10 ). Draw two horizontal lines that cross line $\ell$. Because horizontal lines are parallel to one another, the corresponding angles at which the horizontal lines meet $\ell$ must be congruent (by the corresponding angles theorem from geometry). The same is true for vertical lines as well. Horizontal and vertical lines meet at right angles. Therefore, the triangles in figure 11 are similar (they have the same angles).

One important fact from geometry about similar triangles is that the ratios of corresponding sides are equal in the two triangles. Labeling sides $a, b, c$, and $d$ as in figure 12 , a pair of equal ratios is
$$\frac{a}{b}=\frac{c}{d}$$

The “rise” of the line as it moves along the hypotenuse of one of the right triangles is $a$ or $c$, whereas the “run” of the line is $b$ or $d$. This quantity of $\frac{\text { rise }}{\text { run }}$ is therefore the same for any such triangle we draw; it is a property of the line. We call this property the slope of the line.
How can the value of the slope be calculated? If we know the coordinates of two points on the line, $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$, then the rise is $y_{2}-y_{1}$ and the run is $x_{2}-x_{1}$ (see figure 13 ), and the slope is
$$\text { slope of line }=\frac{\text { rise }}{\text { run }}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \text {. }$$

## 数学代写|微积分代写Calculus代写|Point-slope form of the equation of a line

Suppose we wish to know the equation of a line with slope $m$ through the point $\left(x_{1}, y_{1}\right)$ (figure 19). The equation of the line gives the relationship between the $x$-and $y$-coordinates of points on the line. If the point $(x, y)$ is on the line, then the slope formula applied to the points $(x, y)$ and $\left(x_{1}, y_{1}\right)$ must yield the number $m$, which is given as the slope of the line:
$$\text { slope }=m=\frac{y-y_{1}}{x-x_{1}} .$$
Clearing the fraction by multiplying both sides of the equation by the denominator results in
$$y-y_{1}=m\left(x-x_{1}\right)$$
We call this the point-slope form of the equation of a line.

In the formula, $x$ and $y$ are variables whereas $x_{1}, y_{1}$, and $m$ represent specific numbers.

Example 5 Find the equation of the line with slope $-2$ through the point $(1,4)$.

Solution We are given the slope of the line, $m=-2$, and a point on the line, $\left(x_{1}, y_{1}\right)=(1,4)$. Using $m=-2, x_{1}=1$, and $y_{1}=4$ in the point-slope form of the equation of a line yields
$$\begin{array}{r} y-y_{1}=m\left(x-x_{1}\right) \ y-4=-2(x-1) . \end{array}$$

The equation of the line is $y-4=-2(x-1)$. However, the answer is traditionally expressed in a different form, the form $y=m x+b$. To place the equation in the traditional form, we first distribute the slope $-2$ through the parentheses:
$$y-4=-2 x+2$$
We finish by adding 4 to both sides of the equation:
$$y=-2 x+6$$
The equation of the line is $y=-2 x+6$.
The reason for expressing the equation in the form $y=m x+b$ is that the form lends itself readily to graphing and interpretation, as we shall see shortly.

In geometry you learned that two points determine a line. Therefore, given two points we ought to be able to determine the equation of the line through those points.

## 数学代写|微积分代写Calculus代写|Distance formula

d(磷1,磷2)=|X2−X1|2+⌊是2−是1|2 =(X2−X1)2+(是2−是1)2

d(磷1,磷2)=(X2−X1)2+(是2−是1)2.

d((2,6),(1,−4))=(1−2)2+(−4−6)2 =(−1)2+(−10)2=1+100 =101

## 数学代写|微积分代写Calculus代写|Slopes of lines

线的斜率 = 上升  跑 =是2−是1X2−X1.

坡 =米=是−是1X−X1.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。