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# 数学代写|微积分代写Calculus代考|MATH1051 The Limit of a Function

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## 数学代写|微积分代写Calculus代考|The Limit of a Function

Before diving into differential calculus, it is essential to understand the concept of the limit of a function. The idea of a limit may be new to you, but it is at the heart of calculus, and it is essential to understand the material in this section before going on. Once you understand the concept of limits, you should be able to grasp the ideas of differential calculus quite readily.
Limits are so important in calculus that we will discuss them from two different points of view. First, we will discuss limits from an intuitive point of view. Then, we will give a precise mathematical definition.

Here is a little bit of mathematical shorthand, which will be useful in this section. Suppose a variable $x$ has values lying in an interval with the following properties:

1. The interval surrounds some number $a$.
2. The difference between $x$ and $a$ is less than another number $B$, where $B$ is any number that you choose.
3. $x$ does not take the particular value $a$. (We will see later why this point is excluded.)
The above three statements can be summarized by the following:
$|x-a|>0 \quad$ (This statement means $x$ cannot have the value $a$.)
$|x-a|<B \quad$ (The magnitude of the difference between $x$ and $a$ is less than the arbitrary number $B$.)
These relations can be combined in the single statement:
$$0<|x-a|<B .$$

## 数学代写|微积分代写Calculus代考|Velocity

Our discussion has become a little abstract, so before we go on to differential calculus, let’s talk about something down to earth: motion. As a matter of fact, Leibniz and Newton invented calculus because they were concerned with problems of motion, so it is a good place to start. Besides, you already know quite a bit about motion.

In this chapter, we will only consider motion along a straight line. Here is a warm-up problem.

A train travels away at a velocity $v \mathrm{mph}$ (miles per hour). At $t=0$, it is distance $S(0)=S_0$ from us. (The subscript on $S_0$ is to avoid confusion. $S_0$ is a particular distance and is a constant; $S(t)$ is a function that gives the distance the train is from us at time $t$.) Write the equation for $S(t)$ in terms of time $t$. (Take the unit of $t$ to be hours.)
$$S(t)=$$

If you wrote $S(t)=S_0+v t$, you are correct. Go on to frame 119.
If your answer was not equivalent to the above, try to convince yourself that this answer is correct. Note that it yields $S_0$ when $t=0$, as required. The equation is that of a straight line, and it might be worthwhile reviewing the section on linear functions, frames $\mathbf{2 3 – 3 9}$, before continuing. Whenever you are satisfied with this result,

Here is a plot of the positions at different times of a train going in a straight line. Obviously, this represents a linear equation. Write the equation for the position of this train (in miles) in terms of time (in hours).
$$S(t)=$$
Find the velocity of the train from your equation.
$$v=$$

## 数学代写|微积分代写Calculus代考|The Limit of a Function

$x$ 不取特定值 $a$. (我们梢后会看到为什么要排除这一点。)

$|x-a|>0 \quad$ (这句话的意思 $x$ 不能有值 $a$. $|x-a|<B \quad$ (两者之间的差异幅度 $x$ 和 $a$ 小于任意数 $B$. 这些关系可以组合在一个语句中:
$$0<|x-a|<B .$$

## 数学代写微积分代写Calculus代考|Velocity

$$S(t)=$$

$$S(t)=$$

$$v=$$

## 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。