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# 数学代写|微积分代写Calculus代考|Differential Equations

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## 数学代写|微积分代写Calculus代考|Differential Equations

Here is an example of the motion of a particle in which a differential equation arises from Newton’s second law and some methods for solving it, to find the velocity and position of the particle.

Consider the motion of a particle moving in a straight line with a constant force $F$ acting on the particle along that line. Then the differential equation describing its motion, i.e. the rate of change of its velocity, can be written as
$$F=m \frac{d v}{d t} \quad \text { or } \quad \frac{d v}{d t}=\frac{F}{m}$$
At time $t=0$, the particle is located at distance $x_0$ from the origin and is traveling with velocity $v_0$. The problem is to find the velocity and position of the particle at time $t$.

One often sees Newton’s second law written as $d v=(F / m) d t$, and then both sides of this equation are integrated to determine the change in velocity as a function of time. However, this is actually an application of the fundamental theorem of calculus (frame 349) that
$$v(t)-v_0=\int_{t^{\prime}=0}^{t^{\prime}=t} \frac{d v}{d t^{\prime}} d t^{\prime}$$
where $v_0=v(0)$. We can use Newton’s second law that $d v / d t=(F / m) d t$, and then perform the integration
$$v(t)-v(0)=\int_{t^{\prime}=0}^{t^{\prime}=t} \frac{d v}{d t^{\prime}} d t^{\prime}=\int_{t^{\prime}=0}^{t^{\prime}=t} \frac{F}{m} d t^{\prime} .$$

## 数学代写|微积分代写Calculus代考|Taylor’s Theorem

Taylor’s theorem states that if a function $f(x)$ is finite at $x=0$, and has finite derivatives of every order in an interval about $x=0$, then the function can be written as an infinite polynomial-a power series called the Taylor series about $x=0$ – with the form
$$f(x)=\sum_{n=0}^{n=\infty} a_n x^n=\sum_{n=0}^{n=\infty} \frac{f^{(n)}(0)}{n !} x^n,$$
where the $n$th derivative is written as $f^{(n)}=\frac{d^{n \prime} f}{d x^n}$. In expressions such as $f^{(n)}(0)$, the function is first differentiated $n$ times and the result is then evaluated at the argument-here 0 . Examples of Taylor series about $x=0$ for some well-known functions are:
\begin{aligned} \sin x & =x-\frac{x^3}{3 !}+\frac{x^5}{5 !}-\frac{x^7}{7 !}+\cdots+(-1)^{n-1} \frac{x^{2 n-1}}{(2 n-1) !}+\cdots, \ \cos x & =1-\frac{x^2}{2 !}+\frac{x^4}{4 !}-\frac{x^6}{6 !}+\cdots+(-1)^n \frac{x^{2 n}}{(2 n) !}+\cdots \ \ln (1+x) & =x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots+(-1)^{n-1} \frac{x^n}{n}+\cdots \ \frac{1}{(1+x)^{1 / 2}} & =1-\frac{1}{2} x+\frac{3}{8} x^2-\frac{5}{16} x^3+\cdots \end{aligned}

## 数学代写|微积分代写Calculus代考|Differential Equations

$$F=m \frac{d v}{d t} \quad \text { or } \quad \frac{d v}{d t}=\frac{F}{m}$$

$$v(t)-v_0=\int_{t^{\prime}=0}^{t^{\prime}=t} \frac{d v}{d t^{\prime}} d t^{\prime}$$

$$v(t)-v(0)=\int_{t^{\prime}=0}^{t^{\prime}=t} \frac{d v}{d t^{\prime}} d t^{\prime}=\int_{t^{\prime}=0}^{t^{\prime}=t} \frac{F}{m} d t^{\prime}$$

## 数学代写|微积分代写Calculus代考|Taylor’s Theorem

$$f(x)=\sum_{n=0}^{n=\infty} a_n x^n=\sum_{n=0}^{n=\infty} \frac{f^{(n)}(0)}{n !} x^n$$

\begin{aligned} \sin x & =x-\frac{x^3}{3 !}+\frac{x^5}{5 !}-\frac{x^7}{7 !}+\cdots+(-1)^{n-1} \frac{x^{2 n-1}}{(2 n-1) !}+\cdots, \ \cos x & =1-\frac{x^2}{2 !}+\frac{x^4}{4 !}-\frac{x^6}{6 !}+\cdots+(-1)^n \frac{x^{2 n}}{(2 n) !}+\cdots \ \ln (1+x) & =x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots+(-1)^{n-1} \frac{x^n}{n}+\cdots \ \frac{1}{(1+x)^{1 / 2}} & =1-\frac{1}{2} x+\frac{3}{8} x^2-\frac{5}{16} x^3+\cdots \end{aligned}

## 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。