## 数学代写|微积分代写Calculus代写|MATH1111

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
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|The Set R

We are ultimately concerned with one and only one set, the set $\mathbf{R}$ of real numbers. The properties of $\mathbf{R}$ that we use are

• the arithmetic properties,
• the ordering properties, and
• the completeness property.
Throughout, we use ‘real’ to mean ‘real number’, i.e., an element of $\mathbf{R}$.
The arithmetic properties start with the fact that reals $a, b$ can be added to produce a real $a+b$, the $s u m$ of $a$ and $b$. The rules for addition are $a+b=b+a$ and $a+(b+c)=(a+b)+c$, valid for all reals $a, b$, and $c$. There is also a real 0 , called zero, satisfying $a+0=0+a=a$ for all reals $a$, and each real $a$ has a negative $-a$ satisfying $a+(-a)=0$. As usual, we write subtraction $a+(-b)$ as $a-b$.

Reals $a, b$ can also be multiplied to produce a real $a \cdot b$, the product of $a$ and $b$, also written $a b$. The rules for multiplication are $a b=b a, a(b c)=(a b) c$, valid for all reals $a, b$, and $c$. There is also a real 1 , called one, satisfying $a 1=1 a=a$ for all reals $a$, and each real $a \neq 0$ has a reciprocal $1 / a$ satisfying $a(1 / a)=1$. As usual, we write division $a(1 / b)$ as $a / b$.

Addition and multiplication are related by the property $a(b+c)=a b+a c$ for all reals $a, b$, and $c$ and the assumption $0 \neq 1$. Let us show how the above properties imply there is a unique real number 0 satisfying $0+a=a+0=a$ for all $a$. If $0^{\prime}$ were another real satisfying $0^{\prime}+a=a+0^{\prime}=a$ for all $a$, then, we would have $0^{\prime}=0+0^{\prime}=0^{\prime}+0=0$, hence, $0=0^{\prime}$. Also it follows that there is a unique real playing the role of one and $0 a=0$ for all $a$. These are the arithmetic properties of the reals.

## 数学代写|微积分代写Calculus代写|The Subset N and the Principle of Induction

A subset $S \subset \mathbf{R}$ is inductive if
A. $1 \in S$ and
B. $S$ is closed under addition by $1: x \in S$ implies $x+1 \in S$.
For example, $\mathbf{R}^{+}$is inductive. The subset $\mathbf{N} \subset \mathbf{R}$ of natural numbers or naturals is the intersection of all inductive subsets of $\mathbf{R}$,
$$\mathbf{N}=\bigcap{S: S \subset \mathbf{R} \text { inductive }}$$

Then, $\mathbf{N}$ itself is inductive. Indeed, since $1 \in S$ for every inductive set $S$, we conclude that $1 \in \bigcap{S: S \subset \mathbf{R}$ inductive $}=\mathbf{N}$. Similarly, $n \in \mathbf{N}$ implies $n \in S$ for every inductive set $S$. Hence, $n+1 \in S$ for every inductive set $S$. hence, $n+1 \in \bigcap{S: S \subset \mathbf{R}$ inductive $}=\mathbf{N}$. This shows that $\mathbf{N}$ is inductive.
From the definition, we conclude that $\mathbf{N} \subset S$ for any inductive $S \subset \mathbf{R}$. For example, since $\mathbf{R}^{+}$is inductive, we conclude that $\mathbf{N} \subset \mathbf{R}^{+}$, i.e., every natural is positive.

From the definition, we also conclude that $\mathbf{N}$ is the only inductive subset of $\mathbf{N}$. For example, $S={1} \cup(\mathbf{N}+1)$ is a subset of $\mathbf{N}$, since $\mathbf{N}$ is inductive. Clearly, $1 \in S$. Moreover, $x \in S$ implies $x \in \mathbf{N}$ implies $x+1 \in \mathbf{N}+1$ implies $x+1 \in S$, so, $S$ is inductive. Hence, $S=\mathbf{N}$ or ${1} \cup(\mathbf{N}+1)=\mathbf{N}$, i.e., $n-1$ is a natural for every natural $n$ other than 1 .

The conclusions above are often paraphrased by saying $\mathbf{N}$ is the smallest inductive subset of $\mathbf{R}$, and they are so important they deserve a name.

Theorem 1.3.1 (Principle of Induction). If $S \subset \mathbf{R}$ is inductive, then, $S \supset \mathbf{N}$. If $S \subset \mathbf{N}$ is inductive, then, $S=\mathbf{N}$.

Let $2=1+1>1$; we show that there are no naturals between 1 and 2 . For this, let $S={1} \cup{n \in \mathbf{N}: n \geq 2}$. Then, $1 \in S$. If $n \in S$, there are two possibilities. Either $n=1$ or $n \neq 1$. If $n=1$, then, $n+1=2 \in S$. If $n \neq 1$, then, $n \geq 2$, so, $n+1>n \geq 2$ and $n+1 \in \mathbf{N}$, so, $n+1 \in S$. Hence, $S$ is inductive. Since $S \subset \mathbf{N}$, we conclude that $S=\mathbf{N}$. Thus, $n \geq 1$ for all $n \in \mathbf{N}$, and there are no naturals between 1 and 2. Similarly (Exercise 1.3.1), for any $n \in \mathbf{N}$, there are no naturals between $n$ and $n+1$.

## 数学代写|微积分代写微积分代写|集合R

• 表示算术属性，
• 表示排序属性，
• 表示完整性属性。在整个过程中，我们用’real’表示’实数’，即 $\mathbf{R}$.
算术属性从实数开始 $a, b$ 能不能加个真 $a+b$， $s u m$ 的 $a$ 和 $b$。加法的规则是 $a+b=b+a$ 和 $a+(b+c)=(a+b)+c$，对所有实数都有效 $a, b$，以及 $c$。还有一个真正的0，叫0，令人满意 $a+0=0+a=a$ 对于所有实数 $a$，每一个真实的 $a$ 它是负的 $-a$ 令人满意的 $a+(-a)=0$。像往常一样，我们写减法 $a+(-b)$ 作为 $a-b$.

## 数学代写|微积分代写微积分代写|子集N和归纳法原理

A。 $1 \in S$ 和
B。 $S$ 在加法下封闭 $1: x \in S$ 暗示 $x+1 \in S$.

$$\mathbf{N}=\bigcap{S: S \subset \mathbf{R} \text { inductive }}$$

## 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。

## 数学代写|微积分代写Calculus代写|MATH141

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
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|A Note to the Reader

This text consists of many assertions, some big, some small, some almost insignificant. These assertions are obtained from the properties of the real numbers by logical reasoning. Assertions that are especially important are called theorems. An assertion’s importance is gauged by many factors, including its depth, how many other assertions it depends on, its breadth, how many other assertions are explained by it, and its level of symmetry. The later portions of the text depend on every single assertion, no matter how small, made in Chapter 1.

The text is self-contained, and the exercises are arranged linearly: Every exercise can be done using only previous material from this text. No outside material is necessary.

Doing the exercises is essential for understanding the material in the text. Sections are numbered linearly within each chapter; for example, $\S 4.3$ means the third section in Chapter 4 . Equation numbers are written within parentheses and exercise numbers in bold. Theorems, equations, and exercises are numbered linearly within each section; for example, Theorem 4.3.2 denotes the second theorem in $\$ 4.3$, (4.3.1) denotes the first numbered equation in$\S 4.3$, and 4.3.3 denotes the third exercise at the end of$\S 4.3$. Throughout, we use the abbreviation ‘iff’ to mean ‘if and only if’ and to signal the end of a derivation. ## 数学代写|微积分代写Calculus代写|Sets and Mappings We assume the reader is familiar with the usual notions of sets and mappings, but we review them to fix the notation. Strictly speaking, some of the material in this section should logically come after we discuss natural numbers ($\$1.3)$. However we include this material here for convenience.

A set is a collection $A$ of objects, called elements. If $x$ is an element of $A$ we write $x \in A$. If $x$ is not an element of $A$, we write $x \notin A$. Let $A, B$ be sets. If every element of $A$ is an element of $B$, we say $A$ is a subset of $B$, and we write $A \subset B$. Equivalently, we say $B$ is a superset of $A$ and we write $B \supset A$. When we write $A \subset B$ or $A \supset B$, we allow for the possibility $A=B$, i.e., $A \subset A$ and $A \supset A$.

The union of sets $A$ and $B$ is the set $C$ whose elements lie in $A$ or lie in $B$; we write $C=A \cup B$, and we say $C$ equals $A$ union $B$. The intersection of sets $A$ and $B$ is the set $C$ whose elements lie in $A$ and lie in $B$; we write $C=A \cap B$ and we say $C$ equals $A$ inter $B$. Similarly, one defines the union $A_1 \cup \ldots \cup A_n$ and the intersection $A_1 \cap \ldots \cap A_n$ of finitely many sets $A_1, \ldots, A_n$.

More generally, given any infinite collection of sets $A_1, A_2, \ldots$, their union is the set $\bigcup_{n=1}^{\infty} A_n$ whose elements lie in at least one of the given sets. Similarly, their intersection $\bigcap_{n=1}^{\infty} A_n$ is the set whose elements lie in all the given sets.
Let $A$ and $B$ be sets. If they have no elements in common, we say they are disjoint, $A \cap B$ is empty, or $A \cap B=\emptyset$, where $\emptyset$ is the empty set, i.e., the set with no elements. By convention, we consider $\emptyset$ a subset of every set.
The set of all elements in $A$, but not in $B$, is denoted $A \backslash B={x \in A$ : $x \notin B}$ and is called the complement of $B$ in $A$. For example, when $A \subset B$, the set $A \backslash B$ is empty. Often the set $A$ is understood from the context; in these cases, $A \backslash B$ is denoted $B^c$ and called the complement of $B$.
We will have occasion to use De Morgan’s law,
\begin{aligned} &\left(\bigcup_{n=1}^{\infty} A_n\right)^c=\bigcap_{n=1}^{\infty} A_n^c \ &\left(\bigcap_{n=1}^{\infty} A_n\right)^c=\bigcup_{n=1}^{\infty} A_n^c \end{aligned}

## 数学代写|微积分代写Calculus代写|读者注

$A$中除$B$外的所有元素的集合记为$A \backslash B={x \in A$: $x \notin B}$，称为$A$中$B$的补集。例如，当$A \subset B$时，设置$A \backslash B$为空。通常可以从上下文理解集合$A$;在这些情况下，$A \backslash B$记为$B^c$，称为$B$的补集。

\begin{aligned} &\left(\bigcup_{n=1}^{\infty} A_n\right)^c=\bigcap_{n=1}^{\infty} A_n^c \ &\left(\bigcap_{n=1}^{\infty} A_n\right)^c=\bigcup_{n=1}^{\infty} A_n^c \end{aligned}

## 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。

## 数学代写|离散数学作业代写discrete mathematics代考|Math1030Q

statistics-lab™ 为您的留学生涯保驾护航 在代写离散数学discrete mathematics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。我们的专家在代写离散数学discrete mathematics代写方面经验极为丰富，各种代写离散数学discrete mathematics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|离散数学作业代写discrete mathematics代考|Permutations and Combinations

A permutation is an arrangement of a given number of objects, by taking some or all of them at a time. A combination is a selection of a number of objects where the order of the selection is unimportant. Permutations and combinations are defined in terms of the factorial function, which was defined in Chap. 4.
Principles of Counting
(a) Suppose one operation has $m$ possible outcomes and a second operation has $n$ possible outcomes, then the total number of possible outcomes when performing the first operation followed by the second operation is $m \times n$ (Product Kule).
(b) Suppose one operation has $m$ possible outcomes and a second operation has $n$ possible outcomes, then the total number of possible outcomes of the first operation or the second operation is given by $m+n$ (Sum Rule).
Example (Counting Principle $(a)$ )
Suppose a dice is thrown and a coin is then tossed. How many different outcomes are there and what are they?
Solution
There are six possible outcomes from a throw of the dice, $1,2,3,4,5$ or 6 , and there are two possible outcomes from the toss of a coin, $\mathrm{H}$ or $\mathrm{T}$. Therefore, the total number of outcomes is determined from the product rule as $6 \times 2=12$. The outcomes are given by
$$(1, \mathrm{H}),(2, \mathrm{H}),(3, \mathrm{H}),(4, \mathrm{H}),(5, \mathrm{H}),(6, \mathrm{H}),(1, \mathrm{~T}),(2, \mathrm{~T}),(3, \mathrm{~T}),(4, \mathrm{~T}),(5, \mathrm{~T}),(6, \mathrm{~T}) .$$
Example (Counting Principle $(b)$ )
Suppose a dice is thrown and if the number is even a coin is tossed and if it is odd then there is a second throw of the dice. How many different outcomes are there?

## 数学代写|离散数学作业代写discrete mathematics代考|Algebra

Algebra is the branch of mathematics that uses letters in the place of numbers, where the letters stand for variables or constants that are used in mathematical expressions. Algebra is the study of such mathematical symbols and the rules for manipulating them, and it is a powerful tool for problem-solving in science and engineering.

The origins of algebra are in the work done by Islamic mathematicians during the Golden age in Islamic civilization, and the word ‘algebra’ comes from the Arabic term ‘al-jabr’, which appears as part of the title of a book by the Islamic mathematician, Al-Khwarizmi, in the ninth century A.D. The third century A.D. Hellenistic mathematician, Diophantus, also did early work on algebra, and we mentioned in Chap. 1 that the Babylonians employed an early form of algebra.
Algebra covers many areas such as elementary algebra, linear algebra and abstract algebra. Elementary algebra includes the study of symbols and rules for manipulating them to form valid mathematical expressions, simultaneous equations, quadratic equations, polynomials, indices and logarithms. Linear algebra is concerned with the solution of a set of linear equations, and includes the study of matrices (see Chap. 8) and vectors. Abstract algebra is concerned with the study of abstract algebraic structures such as monoids, groups, rings, integral domains, fields and vector spaces.

## 数学代写|离散数学作业代写discrete mathematics代考|排列与组合

.

(a)假设一个操作有 $m$ 可能的结果和第二次手术 $n$ 可能的结果，那么执行第一个操作和第二个操作时可能的结果总数为 $m \times n$ (Product Kule).
(b)假设一个操作有 $m$ 可能的结果和第二次手术 $n$ 可能的结果，那么第一个操作或第二个操作可能的结果的总数由 $m+n$ (求和规则)。

## 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。

## 数学代写|离散数学作业代写discrete mathematics代考|MATH200

statistics-lab™ 为您的留学生涯保驾护航 在代写离散数学discrete mathematics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。我们的专家在代写离散数学discrete mathematics代写方面经验极为丰富，各种代写离散数学discrete mathematics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|离散数学作业代写discrete mathematics代考|Simple and Compound Interest

Savers receive interest on placing deposits at the bank for a period of time, whereas lenders pay interest on their loans to the bank. We distinguish between simple and compound interest, where simple interest is always calculated on the original principal, whereas for compound interest, the interest is added to the principal sum, so that interest is also earned on the added interest for the next compounding period.
For example, if Euro 1000 is placed on deposit at a bank with an interest rate of $10 \%$ per annum for 2 years, it would earn a total of Euro 200 in simple interest. The interest amount is calculated by
$$\frac{1000 * 10 * 2}{100}=\text { Euro } 200 .$$
The general formula for calculating simple interest on principal $P$, at a rate of interest $I$, and for time $T$ (in years:), is
$$A=\frac{P \times I \times T}{100} .$$
The calculation of compound interest is more complicated as may be seen from the following example.
Example (Compound Interest)
Calculate the interest earned and what the new principal will be on Euro 1000 , which is placed on deposit at a bank, with an interest rate of $10 \%$ per annum (compound) for 3 years.
Solution
At the end of year 1, Euro 100 of interest is earned, and this is capitalized making the new principal at the start of year 2 Euro 1100. At the end of year 2, Euro 110 is earned in interest, making the new principal at the start of year 3 Euro 1210. Finally, at the end of year 3, a further Euro 121 is earned in interest, and so the new principal is Euro 1331 and the total interest earned for the 3 years is the sum of the interest earned for each year (i.e. Euro 331). This may be seen from Table 5.1.

## 数学代写|离散数学作业代写discrete mathematics代考|Time Value of Money and Annuities

The time value of money discusses the concept that the earlier that cash is received the greater value it has to the recipient. Similarly, the later that a cash payment is made, the lower its value to the recipient, and the lower its cost to the payer.
This is clear if we consider the example of a person who receives $\$ 1000$now and a person who receives$\$10005$ years from now. The person who receives $\$ 1000$now is able to invest it and to receive annual interest on the principal, whereas the other person who receives$\$1000$ in 5 years earns no interest during the period. Further, the inflation during the period means that the purchasing power of $\$ 1000$is less in 5 years time is less than it is today. We presented the general formula for what the future value of a principal$P$invested for$n$years at a compound rate$r$of interest is$A=P(1+r)^n$. We can determine the present value of an amount$A$received in$n$years time at a discount rate$r$by $$P=\frac{A}{(1+r)^n}$$ An annuity is a series of equal cash payments made at regular intervals over a period of time, and so there is a need to calculate the present value of the series of payments made over the period. The actual method of calculation is clear from Table 5.2. ## 离散数学代写 ## 数学代写|离散数学作业代写离散数学代考|单利和复利 储蓄者在银行存入一段时间就能获得利息，而出借人则要向银行支付贷款的利息。我们区分单利和复利，其中单利总是按原始本金计算，而复利则是将利息加到本金总额中，这样就可以在下一个复利期间从增加的利息中获得利息。例如，如果1000欧元存入一家银行，利率是$10 \%$两年，它将获得200欧元的单利收益。利息金额由 计算$$\frac{1000 * 10 * 2}{100}=\text { Euro } 200 .$$ 计算本金单利的一般公式$P$，以利率计算$I$，为了时间$T$(以年为单位:)，是 $$A=\frac{P \times I \times T}{100} .$$复利的计算更复杂，从下面的例子可以看出。 示例(复利) 计算获得的利息和1000欧元的新本金是多少，这是存在银行的存款，利率为$10 \%$每年(复利)，为期三年。在第1年年底，100欧元的利息赚了，这是资本化的，使第二年年初的新本金1100欧元。第二年年底，110欧元的利息收入，第三年年初的新本金为1210欧元。最后，在第三年年底，又获得了121欧元的利息，所以新的本金是1331欧元，3年的总利息是每年的利息的总和(即331欧元)。这可以从表5.1中看出 ## 数学代写|离散数学作业代写离散数学代考|货币和年金的时间价值 货币的时间价值讨论的概念是，现金越早收到，它对接受者的价值就越大。同样，现金支付的时间越晚，它对接受者的价值就越低，对支付人的成本就越低。如果我们考虑这样一个例子:一个人现在收到$\$1000$，而另一个人多年后收到$\$ 10005$。现在收到$\$1000$的人可以投资并获得本金的年利息，而另一个在5年后收到$\$ 1000$的人在此期间没有利息。此外，这一时期的通货膨胀意味着5年后$\$1000$的购买力比今天要低。

$$P=\frac{A}{(1+r)^n}$$来确定$A$在$n$年的时间内以贴现率$r$收到的金额的现值。年金是在一段时间内定期支付的一系列等额现金，因此有必要计算在这段时间内支付的一系列现金的现值。实际的计算方法见表5.2

## 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。

## 数学代写|代数学代写Algebra代考|Math4120

statistics-lab™ 为您的留学生涯保驾护航 在代写代数学Algebra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。我们的专家在代写代数学Algebra代写方面经验极为丰富，各种代写代数学Algebra相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|代数学代写Algebra代考|Scalar Multiplication

The other basic operation on vectors that we introduce at this point is one that changes a vector’s length and/or reverses its direction, but does not otherwise change the direction in which it points.

Suppose $\mathbf{v}=\left(v_1, v_2, \ldots, v_n\right) \in \mathbb{R}^n$ is a vector and $c \in \mathbb{R}$ is a scalar. Then their scalar multiplication, denoted by $c \mathbf{v}$, is the vector
$$c \mathbf{v} \stackrel{\text { dff }}{=}\left(c v_1, c v_2, \ldots, c v_n\right) .$$
We remark that, once again, algebraically this is exactly the definition that someone would likely expect the quantity $c \mathbf{v}$ to have. Multiplying each entry of $\mathbf{v}$ by $c$ seems like a rather natural operation, and it has the simple geometric interpretation of stretching $\mathbf{v}$ by a factor of $c$, as in Figure 1.4. In particular, if $|c|>1$ then scalar multiplication stretches $\mathbf{v}$, but if $|c|<1$ then it shrinks $\mathbf{v}$. When $c<0$ then this operation also reverses the direction of $\mathbf{v}$, in addition to any stretching or shrinking that it does if $|c| \neq 1$.

Two special cases of scalar multiplication are worth pointing out:

• If $c=0$ then $c v$ is the zero vector, all of whose entries are 0 , which we denote by 0 .
• If $c=-1$ then $c \mathbf{v}$ is the vector whose entries are the negatives of $\mathbf{v}$ ‘s entries, which we denote by $-\mathbf{v}$.
We also define vector subtraction via $\mathbf{v}-\mathbf{w} \stackrel{\text { dif }}{=} \mathbf{v}+(-\mathbf{w})$, and we note that it has the geometric interpretation that $\mathbf{v}-\mathbf{w}$ is the vector pointing from the head of $\mathbf{w}$ to the head of $\mathbf{v}$ when $\mathbf{v}$ and $\mathbf{w}$ are in standard position. It is perhaps easiest to keep this geometric picture straight (“it points from the head of which vector to the head of the other one?”) if we just think of $\mathbf{v}-\mathbf{w}$ as the vector that must be added to $\mathbf{w}$ to get $\mathbf{v}$ (so it points from $\mathbf{w}$ to $\mathbf{v}$ ). Alternatively, $\mathbf{v}-\mathbf{w}$ is the other diagonal (besides $\mathbf{v}+\mathbf{w}$ ) in the parallelogram with sides $\mathbf{v}$ and $\mathbf{w}$, as in Figure 1.5.
• It is straightforward to verify some simple properties of the zero vector, such as the facts that $\mathbf{v}-\mathbf{v}=\mathbf{0}$ and $\mathbf{v}+\mathbf{0}=\mathbf{v}$ for every vector $\mathbf{v} \in \mathbb{R}^n$, by working entry-by-entry with the vector operations. There are also quite a few other simple ways in which scalar multiplication interacts with vector addition, some of which we now list explicitly for easy reference.

## 数学代写|代数学代写Algebra代考|Linear Combinations

One common task in linear algebra is to start out with some given collection of vectors $\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_k$ and then use vector addition and scalar multiplication to construct new vectors out of them. The following definition gives a name to this concept.

For example, $(1,2,3)$ is a linear combination of the vectors $(1,1,1)$ and $(-1,0,1)$ since $(1,2,3)=2(1,1,1)+(-1,0,1)$. On the other hand, $(1,2,3)$ is not a linear combination of the vectors $(1,1,0)$ and $(2,1,0)$ since every vector of the form $c_1(1,1,0)+c_2(2,1,0)$ has a 0 in its third entry, and thus cannot possibly equal $(1,2,3)$.

When working with linear combinations, some particularly important vectors are those with all entries equal to 0 , except for a single entry that equals 1 . Specifically, for each $j=1,2, \ldots, n$, we define the vector $\mathbf{e}_j \in \mathbb{R}^n$ by
$$\mathbf{e}_j \stackrel{\text { df }}{=}(0,0, \ldots, 0,1,0, \ldots, 0) .$$
For example, in $\mathbb{R}^2$ there are two such vectors: $\mathbf{e}_1=(1,0)$ and $\mathbf{e}_2=(0,1)$. Similarly, in $\mathbb{R}^3$ there are three such vectors: $\mathbf{e}_1=(1,0,0), \mathbf{e}_2=(0,1,0)$, and $\mathbf{e}_3=(0,0,1)$. In general, in $\mathbb{R}^n$ there are $n$ of these vectors, $\mathbf{e}_1, \mathbf{e}_2, \ldots, \mathbf{e}_n$, and we call them the standard basis vectors (for reasons that we discuss in the next chapter). Notice that in $\mathbb{R}^2$ and $\mathbb{R}^3$, these are the vectors that point a distance of 1 in the direction of the $x-, y$-, and $z$-axes, as in Figure 1.6.

For now, the reason for our interest in these standard basis vectors is that every vector $\mathbf{v} \in \mathbb{R}^n$ can be written as a linear combination of them. In particular, if $\mathbf{v}=\left(v_1, v_2, \ldots, v_n\right)$ then
$$\mathbf{v}=v_1 \mathbf{e}_1+v_2 \mathbf{e}_2+\cdots+v_n \mathbf{e}_n,$$
which can be verified just by computing each of the entries of the linear combination on the right. This idea of writing vectors in terms of the standard basis vectors (or other distinguished sets of vectors that we introduce later) is one of the most useful techniques that we make use of in linear algebra: in many situations, if we can prove that some property holds for the standard basis vectors, then we can use linear combinations to show that it must hold for all vectors.

## 数学代写|代数学代写Algebra代考|标量乘法

$$c \mathbf{v} \stackrel{\text { dff }}{=}\left(c v_1, c v_2, \ldots, c v_n\right) .$$我们注意到，再一次，从代数上讲，这正是某人可能期望的量的定义 $c \mathbf{v}$ 拥有。乘以的每一项 $\mathbf{v}$ 通过 $c$ 看起来是一个很自然的操作，它对拉伸有简单的几何解释 $\mathbf{v}$ 乘以 $c$，如图1.4所示。特别是，如果 $|c|>1$ 那么标量乘法就会延伸 $\mathbf{v}$，但如果 $|c|<1$ 然后收缩 $\mathbf{v}$。什么时候 $c<0$ 那么这个操作的方向也就颠倒了 $\mathbf{v}$除了它所做的任何拉伸或收缩 $|c| \neq 1$.

• $c=0$ 然后 $c v$ 是零向量，它的所有元素都是0，我们用0表示。
• If $c=-1$ 然后 $c \mathbf{v}$ 这个向量的分量是负数吗 $\mathbf{v}$ 的条目，我们用 $-\mathbf{v}$.
我们还通过定义向量减法 $\mathbf{v}-\mathbf{w} \stackrel{\text { dif }}{=} \mathbf{v}+(-\mathbf{w})$，我们注意到它的几何解释是 $\mathbf{v}-\mathbf{w}$ 向量是否指向的头部 $\mathbf{w}$ 到 $\mathbf{v}$ 何时 $\mathbf{v}$ 和 $\mathbf{w}$ 处于标准位置。也许最容易保持这个几何图形的直线(“它从哪个向量的头部指向另一个向量的头部?”)，如果我们只是想 $\mathbf{v}-\mathbf{w}$ 作为必须加到的向量 $\mathbf{w}$ 得到 $\mathbf{v}$ (所以它指向 $\mathbf{w}$ 到 $\mathbf{v}$ )。或者， $\mathbf{v}-\mathbf{w}$ 另一条对角线(除了? $\mathbf{v}+\mathbf{w}$ )在有边的平行四边形中 $\mathbf{v}$ 和 $\mathbf{w}$，如图1.5所示。
• 验证零向量的一些简单性质是很直接的，比如 $\mathbf{v}-\mathbf{v}=\mathbf{0}$ 和 $\mathbf{v}+\mathbf{0}=\mathbf{v}$ 对于每一个向量 $\mathbf{v} \in \mathbb{R}^n$，通过用向量运算进行逐入口运算。还有许多其他简单的方法可以使标量乘法与向量加法相互作用，我们现在显式列出其中一些方法，以方便参考
数学代写|代数学代写Algebra代考|线性组合
线性代数中的一个常见任务是，从某个给定的向量集合$\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_k$开始，然后使用向量加法和标量乘法从它们中构造出新的向量。下面的定义给出了这个概念的名称例如，$(1,2,3)$是由$(1,2,3)=2(1,1,1)+(-1,0,1)$开始的向量$(1,1,1)$和$(-1,0,1)$的线性组合。另一方面，$(1,2,3)$不是向量$(1,1,0)$和$(2,1,0)$的线性组合，因为$c_1(1,1,0)+c_2(2,1,0)$形式的每个向量在第三个条目中都有一个0，因此不可能等于$(1,2,3)$当处理线性组合时，一些特别重要的向量是那些所有项都等于0的向量，只有一个项等于1。具体来说，对于每个$j=1,2, \ldots, n$，我们通过
$$\mathbf{e}_j \stackrel{\text { df }}{=}(0,0, \ldots, 0,1,0, \ldots, 0) .$$
来定义向量$\mathbf{e}_j \in \mathbb{R}^n$。例如，在$\mathbb{R}^2$中有两个这样的向量:$\mathbf{e}_1=(1,0)$和$\mathbf{e}_2=(0,1)$。类似地，在$\mathbb{R}^3$中有三个这样的向量:$\mathbf{e}_1=(1,0,0), \mathbf{e}_2=(0,1,0)$和$\mathbf{e}_3=(0,0,1)$。一般来说，在$\mathbb{R}^n$中有$n$这些向量，$\mathbf{e}_1, \mathbf{e}_2, \ldots, \mathbf{e}_n$，我们称它们为标准基向量(原因我们将在下一章讨论)。注意，在$\mathbb{R}^2$和$\mathbb{R}^3$中，这些是指向$x-, y$ -和$z$ -轴方向上距离为1的向量，如图1.6所示现在，我们对这些标准基向量感兴趣的原因是，每个向量$\mathbf{v} \in \mathbb{R}^n$都可以写成它们的线性组合。特别是，如果$\mathbf{v}=\left(v_1, v_2, \ldots, v_n\right)$那么
$$\mathbf{v}=v_1 \mathbf{e}_1+v_2 \mathbf{e}_2+\cdots+v_n \mathbf{e}_n,$$
这可以通过计算右边线性组合的每一项来验证。这种用标准基向量(或我们稍后介绍的其他不同的向量集)来表示向量的想法是我们在线性代数中使用的最有用的技巧之一:在许多情况下，如果我们能证明某些性质适用于标准基向量，那么我们就可以使用线性组合来证明它一定适用于所有向量

## 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。

## 数学代写|代数学代写Algebra代考|MATH355

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|代数学代写Algebra代考|Vectors and Vector Operations

In earlier math courses, focus was on how to manipulate expressions involving a single variable. For example, we learned how to solve equations like $4 x-3=7$ and we learned about properties of functions like $f(x)=3 x+8$, where in each case the one variable was called ” $x$ “. One way of looking at linear algebra is the natural extension of these ideas to the situation where we have two or more variables. For example, we might try solving an equation like $3 x+2 y=1$, or we might want to investigate the properties of a function that takes in two independent variables and outputs two dependent variables.

To make expressions involving several variables easier to deal with, we use vectors, which are ordered lists of numbers or variables. We say that the number of entries in the vector is its dimension, and if a vector has $n$ entries, we say that it “lives in” or “is an element of” $\mathbb{R}^n$. We denote vectors themselves by lowercase bold letters like $\mathbf{v}$ and $\mathbf{w}$, and we write their entries within parentheses. For example, $\mathbf{v}=(2,3) \in \mathbb{R}^2$ is a 2 -dimensional vector and $\mathbf{w}=(1,3,2) \in \mathbb{R}^3$ is a 3-dimensional vector (just like $4 \in \mathbb{R}$ is a real number).
In the 2 – and 3-dimensional cases, we can visualize vectors as arrows that indicate displacement in different directions by the amount specified in their entries. The vector’s first entry represents displacement in the $x$-direction, its second entry represents displacement in the $y$-direction, and in the 3-dimensional case its third entry represents displacement in the $z$-direction, as in Figure 1.1.
The front of a vector, where the tip of the arrow is located, is called its head, and the opposite end is called its tail. One way to compute the entries of a vector is to subtract the coordinates of its tail from the corresponding coordinates of its head. For example, the vector that goes from the point $(-1,1)$ to the point $(2,2)$ is $(2,2)-(-1,1)=(3,1)$. However, this is also the same as the vector that points from $(1,0)$ to $(4,1)$, since $(4,1)-(1,0)=(3,1)$ as well.

It is thus important to keep in mind that the coordinates of a vector specify its length and direction, but not its location in space; we can move vectors around in space without actually changing the vector itself, as in Figure 1.2. To remove this ambiguity when discussing vectors, we often choose to display them with their tail located at the origin – this is called the standard position of the vector.

Even though we can represent vectors in 2 and 3 dimensions via arrows, we emphasize that one of our goals is to keep vectors (and all of our linear algebra tools) as dimension-independent as possible. Our visualizations involving arrows can thus help us build intuition for how vectors behave, but our definitions and theorems themselves should work just as well in $\mathbb{R}^7$ (even though we cannot really visualize this space) as they do in $\mathbb{R}^3$. For this reason, we typically introduce new concepts by first giving the algebraic, dimension-independent definition, followed by some examples to illustrate the geometric significance of the new concept. We start with vector addition, the simplest vector operation that there is.

Vector addition can be motivated in at least two different ways. On the one hand, it is algebraically the simplest operation that could reasonably be considered a way of adding up two vectors: most students, if asked to add up two vectors, would add them up entry-by-entry even if they had not seen Definition 1.1.1. On the other hand, vector addition also has a simple geometric picture in terms of arrows: If $\mathbf{v}$ and $\mathbf{w}$ are positioned so that the tail of $\mathbf{w}$ is located at the same point as the head of $\mathbf{v}$ (in which case we say that $\mathbf{v}$ and $\mathbf{w}$ are positioned head-to-tail), then $\mathbf{v}+\mathbf{w}$ is the vector pointing from the tail of $\mathbf{v}$ to the head of $\mathbf{w}$, as in Figure 1.3(a). In other words, $\mathbf{v}+\mathbf{w}$ represents the total displacement accrued by following $\mathbf{v}$ and then following $\mathbf{w}$.

If we instead work entirely with vectors in standard position, then $\mathbf{v}+$ $\mathbf{w}$ is the vector that points along the diagonal between sides $\mathbf{v}$ and $\mathbf{w}$ of a parallelogram, as in Figure 1.3(b).

## 数学代写|代数学代写代数代考|向量与向量运算

. . 数学代写|代数学代写代数代考|

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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。