# 数学代写|微积分代写Calculus代写|MTH125

#### Doug I. Jones

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• 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
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## 数学代写|微积分代写Calculus代写|The Mean Value Theorem

The two theorems which are at the heart of this section draw connections between the instantaneous rate of change and the average rate of change of a function. The Mean Value Theorem, of which Rolle’s theorem is a special case, says that if $f$ is differentiable on an interval, then there is some point in that interval at which the instantaneous rate of change of the function is equal to the average rate of change of the function over the entire interval. For example, if $f$ gives the position of an object moving in a straight line, the Mean Value Theorem says that if the average velocity over some interval of time is 60 miles per hour, then at some time during that interval the object was moving at exactly 60 miles per hour. This is not a surprising fact, but it does turn out to be the key to understanding many useful applications.

Before we turn to a consideration of Rolle’s theorem, we need to establish another fundamental result. Suppose an object is thrown vertically into the air so that its position at time $t$ is given by $f(t)$ and its velocity by $v(t)=f^{\prime}(t)$. Moreover, suppose it reaches its maximum height at time $t_0$. On its way up, the object is moving in the positive direction, and so $v(t)>0$ for $tt_0$. It follows, by the Intermediate Value Theorem and the fact that $v$ is a continuous function, that we must have $v\left(t_0\right)=0$. That is, at time $t_0$, when $f(t)$ reaches its maximum value, we have $f^{\prime}\left(t_0\right)=0$. This is an extremely useful fact which holds in general for differentiable functions, not only at maximum values but at minimum values as well. Before providing a general demonstration, we first need a few definitions.

Definition 3.7.1. A function $f$ is said to have a local maximum at a point $c$ if there exists an open interval $I$ containing $c$ such that $f(c) \geq f(x)$ for all $x$ in $I$. A function $f$ is said to have a local minimum at a point $c$ if there exists an open interval $I$ containing $c$ such that $f(c) \leq f(x)$ for all $x$ in $I$. If $f$ has either a local maximum or a local minimum at $c$, then we say $f$ has a local extremum at $c$.

In short, $f$ has a local maximum at a point $c$ if the value of $f$ at $c$ is at least as large as the value of $f$ at any nearby point, and $f$ has a local minimum at a point $c$ if the value of $f$ at $c$ is at least as small as the value of $f$ at any nearby point. The next example provides an illustration.

## 数学代写|微积分代写Calculus代写|Finding maximum and minimum values

Problems involving finding the maximum or minimum value of a quantity occur frequently in mathematics and in the applications of mathematics. A company may want to maximize its profit or minimize its costs; a farmer may want to maximize the yield from his crop or minimize the amount of irrigation equipment needed to water his fields; an airline may want to maximize its fuel efficiency or minimize the length of its routes. Methods for solving some optimization problems are so computationally intense that they challenge, and sometimes even go beyond, the fastest computers currently available. An example of such a problem is the famous traveling salesman problem, in which a salesman wishes to visit a certain set of cities using the shortest possible route. In this section we will not consider problems of this type, but rather we will confine ourselves to problems involving continuous functions of a single independent variable.
3.8.1 Closed intervals. We will start with the simplest case. Suppose $f$ is a continuous function on a closed interval $[a, b]$. From the Extreme Value Theorem we know that $f$ attains both a maximum value and a minimum value on the interval. We now look for candidates at which these values might occur. To start, an extreme value could occur at one of the endpoints. For example, the maximum value of $f(x)=x^2$ on $[0,1]$ occurs at $x=1$. If an extreme value occurs in the open interval $(a, b)$ at a point $c$ where $f$ is differentiable, then $f$ has a local extremum at $c$ and so, from our work in Section 3.7, we know that $f^{\prime}(c)=0$. For example, the minimum value of $f(x)=x^2$ on $[-1,1]$ occurs at $x=0$ and $f^{\prime}(0)=0$. Finally, the only other candidates for the locations of extreme values would be points where $f^{\prime}$ is undefined. For example, the minimum value of $f(x)=|x|$ on $[-1,1]$ occurs at $x=0$, where $f^{\prime}$ is not defined. Hence we are led to the following conclusion: The extreme values of a continuous function $f$ on a closed interval are located either at the endpoints of the interval, at points where $f^{\prime}$ is 0 , or at points where $f^{\prime}$ is undefined. The following terminology will help us state this more easily.

# 微积分代考

## 数学代写|微积分代写Calculus代写|Finding maximum and minimum values

3.8.1 封闭区间。我们将从最简单的情况开始。认为F是闭区间上的连续函数[一种,b]. 从极值定理我们知道F在区间上同时获得最大值和最小值。我们现在寻找可能出现这些值的候选者。首先，极值可能出现在端点之一。例如，最大值F(X)=X2在[0,1]发生在X=1. 如果极值出现在开区间(一种,b)在某一点C在哪里F是可微的，那么F有一个局部肢体C因此，根据我们在第 3.7 节中的工作，我们知道F′(C)=0. 例如，最小值F(X)=X2在[−1,1]发生在X=0和F′(0)=0. 最后，极值位置的唯一其他候选点是F′未定义。例如，最小值F(X)=|X|在[−1,1]发生在X=0， 在哪里F′没有定义。因此我们得出以下结论：连续函数的极值F在闭区间上位于区间的端点，在点F′是 0 ，或者在点F′未定义。以下术语将帮助我们更轻松地说明这一点。

## 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。

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