### 分类： 商业数学代写

• 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

## 商科代写|商业数学代写business mathematics代考|Probability and Expected Value

First, let us discuss games of chance. Games of chance include, but are not limited to, games involving flipping coin, rolling dice, drawing cards from a deck, spinning a wheel, and such. Suppose you are flipping a coin with a friend. If you both flip the same, both heads or both tails, you win $\$ 1$. If you flip a head and a tail each, you lose$\$1$. If you play this game with this bet 100 times over the course of the evening, how much do you expect to win? Or lose? To answer such questions, we need two concepts: the probability of an event and the expected value of a random variable.

In situations like this, it makes sense to count the number of times something occurs. An efficient way to do this is to use the frequency definition of the probability of an event. The probability of the event two heads or two tails is the number of ways we can achieve these results divided by the total number of possible outcomes. That is, we define both coins being flipped that land on both heads or both tails as a favorable event $A$. We can define event $A$ as the set of outcomes that include ${\mathrm{HH}, \mathrm{TT}}$.
Favorable outcomes are those that consist of ${\mathrm{HH}, \mathrm{TT}}$.
$$\begin{array}{r} \text { Probability of an event }=\frac{\text { Favorable outcomes }}{\text { Total outcomes }} \ \text { Probability of an event }{A}=\frac{\text { Number of outcomes of }{A}}{\text { Total outcomes }} \end{array}$$

Of course, the probability of an event (flip of a fair coin) must be equal or greater than zero, and equal to or less than 1. And the sum of the probabilities of all possible events must equal 1 . That is,
\begin{aligned} &0 \leq p_{i} \leq 1 \ &\sum_{i=1}^{n} p_{i}=1, i=1,2, \ldots, n \end{aligned}
We need to compute all the possible outcomes of flipping two coins, and then determine how many result in the same results defined by event $A$. A tree is useful for visualizing the outcomes. These outcomes constitute the sample space. On the first flip, the possible results are $\mathrm{H}$ or $\mathrm{T}$. And on the second flip, the same outcomes are still available. We assume that these events are equally likely to occur based on flipping and obtaining either a head or tail of each flip.

First, we define a random variable as a rule that assigns a number to every outcome of a sample.

We use $E[X]$, which is stated as the expected value of $X$. We define $E[X]$ as follows:
Expected value, $E[X]$, is the mean or average value.
Further, we provide the following two formulas: in the discrete case, $E[X]=\sum_{i=1}^{n} x_{i} p\left(x_{i}\right)$ and in the continuous case, $E[X]=\int_{-\infty}^{+\infty} x * f(x) d x$.

There are numerous ways to calculate the average value. We present a few common methods that you could use in decision theory.

If you had 2 quiz grades, an 82 and a 98 , almost intuitively you would add the two numbers and divide by 2 , giving an average of 90 .

Average scores: Two scores that were earned were 82 and 98 . Compute the average.
$$E[X]=\frac{(82+98)}{2}=90$$
If after 5 quizzes, you had three 82 s and two 98 s, you would add them and divide by $5 .$
$$\text { Average }=\frac{3(82)+2(98)}{5}=88.4$$
Rearranging the terms, we obtain
$$\text { Average }=\frac{3}{5}(82)+\frac{2}{5}(98)$$
In this form, we have two payoffs, 82 and 98 , each multiplied by the weights, $3 / 5$ and $2 / 5$. This is analogous to the definition of expected value.

Suppose a game has outcomes $a_{1}, a_{2}, \ldots, a_{n}$, each with a payoff $w_{1}, w_{2}, \ldots, w_{n}$ and a corresponding probability $p_{1}, p_{2}, \ldots, p_{n}$ where $p_{1}+p_{2}+\ldots+p_{n}=1$ and $0 \leq p_{i} \leq 1$, then the quantity
$$E=w_{1} p_{1}+w_{1} p_{2}+\ldots+w_{1} p_{n}$$
is the expected value of the game. Note that expected value is analogous to weighted average, but the weights must be probabilities $\left(0 \leq p_{i} \leq 1\right)$ and the weights must sum to 1 .

## 商科代写|商业数学代写business mathematics代考|Probability and Expected Value

Probability of an event $=\frac{\text { Favorable outcomes }}{\text { Total outcomes }}$ Probability of an event $A=\frac{\text { Number of outcomes of } A}{\text { Total outcomes }}$

$$0 \leq p_{i} \leq 1 \quad \sum_{i=1}^{n} p_{i}=1, i=1,2, \ldots, n$$

$$E[X]=\frac{(82+98)}{2}=90$$

$$\text { Average }=\frac{3(82)+2(98)}{5}=88.4$$

$$\text { Average }=\frac{3}{5}(82)+\frac{2}{5}(98)$$

$$E=w_{1} p_{1}+w_{1} p_{2}+\ldots+w_{1} p_{n}$$

## 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。

• 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

We will illustrate some mathematical models that describe change in the real world. We will solve some of these models and will analyze how good our resulting mathematical explanations and predictions are in context of the problem. The solution techniques that we employ take advantage of certain characteristics that the various models enjoy as realized through the formulation of the model.

When we observe change, we are often interested in understanding or explaining why or how a particular change occurs. Maybe we need or want to analyze the effects under different conditions or perhaps to predict what could happen in the future. Consider the firing of a weapon system or the shooting of a ball from a catapult as shown in Figure 1.3. Understanding how the system behaves in different environments under differing weather or operators, or predicting how well it hits the targets are all of interest. For the catapult, the critical elements of the ball, the tension, and angle of the firing arm are found as important elements (Fox, 2013b). For our purposes, we will consider a mathematical model to be a mathematical construct that is designed to study a particular real-world system or behavior (Giordano et al., 2014). The model allows us to use mathematical operations to reach mathematical conclusions about the model as illustrated in Figure 1.4. It is the arrow going from real-world system and observations to the mathematical model using the assumptions, variables, and formulations that are critical in the process.

We define a system as a set of objects joined by some regular interaction or interdependence in order for the complete system to work together. Think of a larger business with many companies that work independently and interact together to make the business prosper. Other examples might include a bass and trout population living in a lake, a communication, cable TV, or weather satellite orbiting the earth, delivering Amazon Prime packages, U.S. postal service mail or packages, locations of emergency services or computer terminals, or large companies’ online customer buying systems. The person modeling is interested in understanding how a system works, what causes change in a system, and the sensitivity of the system to change. Understanding all these elements will help in building an adequate model to replicate reality. The person modeling is also interested in predicting what changes might occur and when these changes might occur.

## 商科代写|商业数学代写business mathematics代考|Steps in Model Construction

An outline is presented as a procedure to help construct mathematical models. In the next section, we will illustrate this procedure with a few examples. We suggest a nine-step process.

These nine steps are summarized in Figure 1.6. These steps act as a guide for thinking about the problem and getting started in the modeling process. We choose these steps from the compilation of steps by other authors listed in additional readings and put them together in these nine steps.

We illustrate the process through an example. Consider building a model where we want to identify the spread of a contagious disease.
Step 1: Understand the decision to be made, the question to be asked, or the problem to be solved.

Understanding the decision is the same as identifying the problem to be solved. Identifying the problem to study is usually difficult.

In real life, no one walks up to you and hands you an equation to be solved. Usually, it is a comment like “we need to make more money” or “we need to improve our efficiency.” Perhaps, we need to make better decisions or we need all our units that are not $100 \%$ efficient to become more efficient. We need to be precise in our formulation of the mathematics to actually describe the situation that we need to solve. In our example, we want to identify the spread of a contagious disease to determine how fast it will spread within our region. Perhaps, we will want to use the model to answer the following questions:

1. How long will it take until one thousand people get the disease?
2. What actions may be taken to slow or eradicate the disease?
Step 2: Make simplifying assumptions.
Giordano et al. (2014, pp. 62-65) described this well. Again, we suggest starting by brain storming the situation. Make a list of as many factors, or variables, as you can. Now, we realize that we usually cannot capture all these factors influencing a problem in our initial model. The task now is simplified by reducing the number of factors under consideration. We do this by making simplifying assumptions about the factors, such as holding certain factors as constants or ignoring some in the initial modeling phase. We might then examine to see if relationships exist between the remaining factors (or variables). Assuming simple relationships might reduce the complexity of the problem. Once you have a shorter list of variables, classify them as independent variables, dependent variables, or neither.

In our example, we assume we know the type of disease, how it is spread, the number of susceptible people within our region, and what type of medicine is needed to combat the disease. Perhaps, we assume that we know the size of population and the approximate number susceptible to getting the disease.

## 商科代写|商业数学代写business mathematics代考|Steps in Model Construction

1. 一千人得这种病需要多长时间？
2. 可以采取哪些行动来减缓或根除这种疾病？
第 2 步：做出简化假设。
佐丹奴等人。(2014, pp. 62-65) 很好地描述了这一点。同样，我们建议从头脑风暴开始。尽可能多地列出因素或变量。现在，我们意识到我们通常无法在初始模型中捕获影响问题的所有这些因素。现在通过减少所考虑因素的数量来简化任务。我们通过简化对因素的假设来做到这一点，例如将某些因素保持为常数或在初始建模阶段忽略一些因素。然后我们可能会检查其余因素（或变量）之间是否存在关系。假设简单的关系可能会降低问题的复杂性。一旦你有一个较短的变量列表，将它们分类为自变量、因变量或两者都不是。

## 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。

• 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

## 商科代写|商业数学代写business mathematics代考|Overview and Process of Mathematical Modeling

Bender (2000, pp. 1-8) first introduced a process for modeling. He highlighted the following: formulate the model, outline the model, ask if it is useful, and test the model. Others have expanded this simple outlined process. Giordano et al. (2014, p. 64) presented a six-step process: identify the problem to be solved, make assumptions, solve the model, verify the model, implement the model, and maintain the model. Myer (2004, pp. 13-15) suggested some guidelines for modeling, including formulation, mathematical manipulation, and evaluation. Meerschaert (1999) developed a five-step process: ask the question, select the modeling approach, formulate the model, solve the model, and answer the question. Albright (2010) subscribed mostly to concepts and process described in previous editions of Giordano et al. (2014). Fox (2012, pp. 21-22) suggested an eight-step approach: understand the problem or question, make simplifying assumptions, define all variables, construct the model, solve and interpret the model, verify the model, consider the model’s strengths and weaknesses, and implement the model.
Most of these pioneers in modeling have suggested similar starts in understanding the problem or question to be answered and in making key assumptions to help enable the model to be built. We add the need for sensitivity analysis and model testing in this process to help ensure that we have a model that is performing correctly to answer the appropriate questions.

For example, student teams in the Mathematical Contest in Modeling were building models to determine the all-time best college sports coach. One team picked a coach who coached less than a year, went undefeated for the remaining part of the year, and won their bowl game. Thus, his season was a perfect season. Their algorithm picked this person as the all-time best coach. Sensitivity analysis and model testing could have shown the fallacy to their model.

Someplace between the defining of the variables and the assumptions, we begin to consider the model’s form and technique that might be used to solve the model. The list of techniques is boundless in mathematics, and we will not list them here. Suffice it to say that it might be good to initially decide among the forms: deterministic or stochastic for the model, linear or nonlinear for the relationship of the variables, and continuous or discrete.

We introduce the process of modeling and examine many different scenarios in which mathematical modeling can play a role.

The art of mathematical modcling is learned through expericnce of building and solving models. Modelers must be creative, innovative, inquisitive, and willing to try new techniques as well as being able to refine their models, if necessary. A major step in the process is passing the common sense test for use of the model.
In its basic form, modeling consists of three steps:

1. Make assumptions
2. Do some math
3. Derive and interpret conclusions
To that end, one cannot question the mathematics and its solution, but one can always question the assumptions used.

To gain insight, we will consider one framework that will enable the modeler to address the largest number of problems. The key is that there is something changing for which we want to know the effects and the results of the effects. The problem might involve any system under analysis. The realworld system can be very simplistic or very complicated. This requires both types of real-world systems to be modeled with the same logical stepwise process.

Consider modeling an investment. Our first inclination is to use the equations about compound interest rates that we used in high school or college algebra. The compound interest formula calculates the value of a compound interest investment after ” $n$ ” interest periods.
$$A=P(1-i)^{n}$$

where:
$A$ is the amount after $n$ interest periods
$P$ is the principal, the amount invested at the start $i$ is the interest rate applying to each period $n$ is the number of interest periods

## 商科代写|商业数学代写business mathematics代考|Overview and Process of Mathematical Modeling

Bender (2000, pp. 1-8) 首先介绍了一种建模过程。他强调了以下几点：制定模型、概述模型、询问它是否有用以及测试模型。其他人已经扩展了这个简单的概述过程。佐丹奴等人。(2014, p. 64) 提出了一个六步过程：识别要解决的问题、做出假设、解决模型、验证模型、实施模型和维护模型。Myer (2004, pp. 13-15) 提出了一些建模指南，包括公式化、数学操作和评估。Meerschaert (1999) 制定了一个五步流程：提出问题、选择建模方法、制定模型、解决模型和回答问题。Albright (2010) 主要赞同 Giordano 等人先前版本中描述的概念和过程。（2014）。福克斯（2012 年，第

1. 做出假设
2. 做一些数学
3. 推导和解释结论
为此，人们不能质疑数学及其解决方案，但人们总是可以质疑所使用的假设。

## 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。

• 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

## 商科代写|商业数学代写business mathematics代考|CASH FLOW NET PRESENT VALUES

In Sections $4.1$ through $4.3$, we concerned ourselves with single lump-sum payments. Thus, we either calculated the future value of a lump-sum invested now, or we calculated the present value of a lump sum payment to be made in the future. In this section we consider investments consisting of a set of payments due at different times, a situation known as a cash flow.

As an example of a cash flow, consider an investment that returns $\$ 500$in 1 year, another$\$300$ in 3 years, and a final $\$ 400$in 4 years, with interest rates of$5 \%$compounded annually. What is the present value of such an opportunity? That is, what is the cash equivalent now of the entire transaction? A simple approach is to compute the present value of each of the individual payments using Equation 4.10, repeated below as Equation$4.11$for convenience, and then sum the individual present values to obtain the present value of the entire cash flow. $$P V=F V(1+i)^{-\mathrm{n}}$$ Example 1 Compute the present value of the cash flow that returns$\$500$ in 1 year, another $\$ 300$in 3 years, and a final$\$400$ in 4 years, with interest rates of $5 \%$ compounded annually.

Solution The first payment of $\$ 500$is due in 1 year. The present value of this amount, computed using Equation$4.11$is $$P V_{1}=(\ 500)(1+0.05)^{-1}=\ 476.19$$ The second payment of$\$300$ is due in 3 years. Again using Equation 4.11, we find its present value as:
$$P V_{2}=(\ 300)(1+0.05)^{-3}=\ 259.15 .$$
Similarly, the present value of the last payment is
$$P V_{3}=(\ 400)(1+0.05)=\ 329.08$$
Summing these three present values, we obtain the present value of the entire investment as:
$$P V=P V_{1}+P V_{2}+P V_{3}=\ 476.19+\ 259.15+\ 329.08=\ 1,064.42$$
In most present-value problems, a time diagram illustrating the contributions to the total present value from the individual payments is helpful. The time diagram for the cash flow given in Example 1 is shown as Figure 4.4.

The present and future values of a cash flow can always be determined by calculating the present or future values, respectively, of each individual payment using the appropriate equation – either Equation $4.9$ or 4.10, repeated below as Equations $4.13$ and $4.14$ for convenience – and then summing the results.
$$F V=P V(1+i)^{n}$$
or
$$P V=F V(1+i)^{-n}$$
For a specific type of investment, however, known as an annuity, the final sum can be calculated using a single formula.

Definition 4.1 An annuity is a set of equal payments made at equal intervals of time.

Car loans, mortgages, life insurance premiums, social security payments, and bond coupon payments are all examples of annuities. In each, one party, be it an individual, company, or government, pays to another party a set of equal payments, called periodic installments or payments, denoted as $P M T$, at equal periods of time, called the rent period, payment period, payment interval, or compounding period. Each of these terms can be used interchangeably.
Annuities are classified as either ordinary or due. With an ordinary annuity, payments are made at the end of each payment period, whereas with an annuity due, payments are made at the beginning of each period. Examples of ordinary annuities are car loan payments, mortgages, and bond coupon payments. Examples of annuities due are typically savings plans, pension plans, and lottery winnings that are paid over time.

An annuity is simple if the compounding period at which interest is paid coincides with the payment dates. In this section, we consider simple ordinary annuities; simple annuities due are presented in Section 4.7.

One of the most common types of ordinary annuities is a mortgage on a house or land. The mortgage is a loan used to pay for the property, with the property serving as collateral for the loan. This gives the lender, known as the mortgagor, a claim on the property should the borrower, known as the mortgagee, default on paying the mortgage. Full title to the property is only transferred to the mortgagee when the loan is fully paid.

In a traditional fixed-rate mortgage the monthly payment and interest rate are fixed for the life of the mortgage. Each payment is used to pay both the interest and principal for the loan. First, the monthly interest charge on loan is determined and paid, with the remaining portion of the monthly payment applied to paying off the loan.

Although the monthly payment is fixed, the interest due changes each month, decreasing with every payment. This occurs because the interest is computed each month anew on the unpaid loan balance. As the loan gets paid off, the unpaid balance decreases, which means that the interest due each month also decreases. Thus, each month more and more of the payment gets applied to paying off the loan. This method of payment is commonly referred to as the United States Rule.

The main consideration with mortgages is to determine the amount of the monthly payment, which depends on the original amount of the loan, the interest rate, and the length of the loan. For all mortgages that adhere to the United States Rule, the payment, PMT, is determined as
$$P M T=\frac{P V}{\left[\frac{1-(1+i)^{-n}}{i}\right]}$$
where:
$P M T=$ the monthly payment
$P V=$ the original amount of the loan
$i=$ the monthly interest rate $=$ (the annual interest rate) $/ 12$
$n=$ the length of the loan, in months, $=12$ * (the number of years of the loan)

Notice that Equation $4.18$ is the same as Equation 4.15, except that it is used to solve for the value of $P M T$ given $P V$, rather than solving for $P V$ given the PMT amount.

## 商科代写|商业数学代写business mathematics代考|CASH FLOW NET PRESENT VALUES

F在=磷在(1+一世)n

n=贷款期限，以月为单位，=12*（贷款年数）

## 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。

• 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

Most interest payments, from common savings accounts in banks to unpaid balances on credit cards, involve compound interest calculations. Additionally, compound interest forms the primary foundation of finance, investment analyses, and modern portfolio theory. As such, compound interest is an essential topic required for delving into these more advanced financial topics.

The defining property of compound interest is that once interest is paid on the initial principal amount, the interest is immediately added to the principal. This new principal amount, which now consists of the original principal amount plus the interest, earns interest during the next time period. Thus, the interest earned in one time period (referred to as a compounding period) earns interest in succeeding periods; this is known as interest being paid on interest and is the defining characteristic of compound interest calculations.
As an example, consider the deposit of $\$ 2,000$in an account paying an annual interest rate of$3 \%$, with compound interest computed and paid once a year. In the first year the principal earns$3 \%$of$\$2,000$ or $(0.03)(\$ 1,000)=\$60$. The new principal is now $\$ 2,060$(which is the original investment of$\$2,000$ plus the $\$ 60$interest payment). Thus, the second year’s interest payment is now based on this new amount, which becomes$3 \%$of$\$2,060$ or $\$ 61.80$. This makes the balance at the end of the second year$\$2,121,80$. Interest payments for the third year are now computed based on this new balance. The results of all interest computations through the fifth year have been collected in Table 4.1.

The yearly interest payments listed in Table $4.1$ are illustrated in Figure 4.2. Notice that the interest payments for each year is greater than that of the previous year. The reason for this is that each year’s interest is calculated on the sum of the initial principal and all prior interest payments (not just on the initial principal, as in simple interest calculations). Compare Figure $4.2$ with the analogous simple interest payments shown in Figure 4.1.

Interest rates are generally quoted on an annual basis but are typically compounded over shorter intervals of time. The annual rate, referred to as either the nominal interest rate or the stated interest rate is denoted by the symbol $\boldsymbol{r}$. The time between successive interval payments is called the compounding period, or the period, for short. The interest rate per period is denoted by the symbol $\boldsymbol{i}$; it is calculated by dividing the stated annual rate, $\boldsymbol{r}$ by the number of compounding periods in a year, which is denoted as $N$
$$i=r / N$$
If the interest is compounded quarterly, then $N$ is 4 (there are fourquarters in a year, and $i=r / 4$. For interest compounded semiannually, $N=2$ and $i=r / 2$; for interest compounded monthly $N=12$ and $i=r / 2$; and for interest compounded. Weekly $N=52$ and $i=r / 52$. If no compounding period is stated, compounding periods are assumed to be annual and $i=r$. This information is summarized in Table $4.2$, which lists the most commonly used compounding periods and the interest rates that apply to them, where $i$ is the stated annual interest rate. ${ }^{2}$

As seen in the last column of Table $4.2$, the interest rate per compounding period is the annual rate divided by the number of compounding periods in a year.

Equation $4.7$ remains valid for all the compound periods listed in Table $4.2$, as long as we realize that $i$ signifies the interest rate per compound period, and $P_{n}$ is the balance after $n$ compound periods. For example, if the interest is $2 \%$ compounded quarterly, $i=0.02 / 4=0.005$, which is the interest rate per quarter. Also, $P_{10}$, for example, denotes the principal after 10 compounding periods which, in this case, is 10 quarters and corresponds to $2^{1 / 2}$ years.

## 商科代写|商业数学代写business mathematics代考|LUMP-SUM FUTURE AND PRESENT VALUES

A lump-sum is a dollar amount made as a one-time single payment. Examples of lump-sum payments are an initial deposit, a single one-time dollar investment, or a final, single loan repayment. Equation $4.6$ relates a lump-sum principal amount at two points in time-the present, when the principal is first deposited, and its value in the future. The reason these values differ is due to the interest that is earned.

In this section, we rewrite and use Equation $4.6$ in two different ways to emphasize this time relationship. To do this, we will use standard financial notation that emphasizes the two unique usages. The first usage emphasizes determining $P_{n}$, the future value of an initial principal amount, given that we know $P_{0^{*}}$ The second usage emphasizes the equation’s use in determining the initial amount deposited, that is $P_{0}$, given that we know $P_{n}$, its future value. In financial applications, this second usage is typically much more important and the key to comparing investment alternatives.

For convenience, we first reproduce Equation $4.6$ as Equation $4.8$, so that we can rewrite it using standard financial notation. The advantage of this new notation is that it clearly relates the values of the principal amounts at two differing points in time, the present and the future.
$$P_{n}=P_{0}(1+i)^{n}$$
Financially, $P_{0}$, the initial principal, is referred to as the present value of the principal, or present value, for short. The notation used for this quantity is $\boldsymbol{P V}$. Similarly, $P_{n}$, which denotes the value of this principal sometime in the future, is referred to as the future value of the principal, or future value, for short. The notation used for this quantity is $\boldsymbol{F V}$. Note that this notation emphasizes what these quantities actually represent in time (now and in the future), as opposed to their strictly mathematical relationship.

## 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。

• 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

## 商科代写|商业数学代写business mathematics代考|SUMMARY OF KEY POINTS

Key Terms

• Degree of a polynomial
• Domain
• Euler’s number, e
• Exponential function
• First-degree polynomial function
• Function
• Linear function
• Polynomial function
• Power function
• Rational function
• Second-degree polynomial function
• Zero function
Key Concepts
Functions
• A function consists of two sets and an assignment rule between them, which assigns every value in the first set a unique, but not necessarily different, element of the second set.
• Functions can be specified by words, equation, graphs, or tables.
• When a graph depicts a function, the domain is always placed on the horizontal axis and the range on the vertical axis. The assignment rule assigns a number on the vertical axis to each value on the horizontal axis.
• A graph represents a function if and only if the graph passes the vertical line test. This test requires that any and all vertical lines that cross the horizontal axis at a value in the domain must intersect the graph at one and only one point.

The solutions to equations of the form $a x^{2}+b x+c=0$ are given by the quadratic formula ${ }^{4}$
$$x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}$$
To solve any quadratic equation, substitute the values of its coefficients $a$, $b$, and $c$ into the quadratic formula and simplify.
Example 1 Solve the equation $x^{2}+2 x-3=0$ for $x$.
Solution This is a quadratic equation with $a=1, b=2$, and $c=-3$. Substituting these values into the quadratic formula, we obtain
$$x=\frac{-2 \pm \sqrt{2^{2}-4(1)(-3)}}{2(1)}=\frac{-2 \pm \sqrt{4+12}}{2}=\frac{-2 \pm \sqrt{16}}{2}=\frac{-2 \pm 4}{2}$$
Using the plus sign, we obtain one solution as $x=(-2+4) / 2=1$. Using the minus sign, we find a second solution as $x=(-2-4) / 2=-3$.

Example 2 Solve the equation $4 y^{2}-2 y=3$ for $y$.
Solution We first rewrite this equation in the form as $4 y^{2}-2 y-3=0$, which is a quadratic equation with $a=4, b=-2$, and $c=-3$. Substituting these values into the quadratic formula, we have
$$y=\frac{-(-2) \pm \sqrt{(-2)^{2}-4(4)(-3)}}{2(4)}=\frac{2 \pm \sqrt{4+48}}{8}=\frac{-2 \pm \sqrt{52}}{8}=\frac{2 \pm 7.21}{8}$$
The solutions are then $y=(2+7.21) / 8=1.15$ and $y=(2-7.21) / 8=-0.65$, with all calculations rounded to two decimals.

The quadratic formula does not always yield two solutions. If $b^{2}-4 a c=0$, the formula reduces to
$$x=\frac{-b \pm \sqrt{0}}{2 a}=-\frac{b}{2 a}$$
In these cases, the quadratic equation has only one solution. If $b^{2}-4 a c$ is negative, the square root cannot be taken, and no real solutions exist. Readers familiar with complex numbers will note that this case has complex solutions. Because complex numbers have no use in commercial situations, we do not consider them here.

Individuals organizations, businesses, and countries exchange their goods and services for the products of others. Bartering was one of the earliest means of establishing trade – a farmer and a weaver might exchange one bushel of corn for one wool scarf – but bartering soon gave way to currency, first in silver and gold coins and more recently script (paper money), as the primary unit of trade. Script itself has little intrinsic worth; the real value of money is its acceptance as a recognized unit of trade, just as bitcoin is being similarly recognized. With money as a medium, a bushel of corn worth $\$ 10$and a wool scarf worth$\$13$ can be traded fairly, generally through a succession of wholesalers and distributors.

Money can either be saved, borrowed, and lent. Money is saved to buy consumer goods, such as television sets and iPhones; it is borrowed to finance purchases such as homes, cars, and college educations, and it is lent by banks and other financial institutions to make these purchases. Each dollar, pound, mark, shilling, yen, rubble, or peso that is lent or borrowed exact a charge or cost called interest.

The amount of money lent or borrowed is called the principal, usually denoted as $\boldsymbol{P}$, and the duration of the loan is its maturity, denoted as $\boldsymbol{t}$. In the simplest type of interest computation, the interest payment is directly proportional to the product of the principal and maturity. The constant of proportionality is the interest rate, denoted as $r$.

If we let $\boldsymbol{I}$ denoted the total interest, $\boldsymbol{t}$ the duration of the loan, and write $\boldsymbol{r}$ as a decimal value in terms of the same unit of time as $t$, then
$$I=\boldsymbol{P} t$$
Equation $4.1$ is the simple interest formula.

## 商科代写|商业数学代写business mathematics代考|SUMMARY OF KEY POINTS

• 多项式的次数
• 领域
• 欧拉数，e
• 指数函数
• 一阶多项式函数
• 功能
• 线性函数
• 多项式函数
• 电源功能
• 二次公式
• 二次函数
• 有理函数
• 二阶多项式函数
• 零功能
关键概念
功能
• 一个函数由两个集合和它们之间的分配规则组成，它将第一个集合中的每个值分配给第二个集合的唯一但不一定不同的元素。
• 函数可以用文字、方程式、图形或表格来指定。
• 当图形描述一个函数时，域总是放在水平轴上，而范围放在垂直轴上。分配规则将垂直轴上的数字分配给水平轴上的每个值。
• 当且仅当图形通过垂直线测试时，图形才表示函数。此测试要求在域中的某个值处与水平轴相交的任何和所有垂直线必须在一个且仅一个点与图形相交。

X=−b±b2−4一个C2一个

X=−2±22−4(1)(−3)2(1)=−2±4+122=−2±162=−2±42

X=−b±02一个=−b2一个

## 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。

• 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

An important class of functions that are more complex than first degree linear functions and their resulting straight-line graphs are second-degree polynomials. These functions are referred to as quadratic functions, and have the form:
$$f(x)=a_{2} x^{2}+a_{1} x+a_{0}$$
where $a_{2} \neq 0$. If we replace the constants $a_{2}, a_{1}$, and $a_{0}$, by $a, b$, and $c$, respectively, this second-degree polynomial function is written in its more conventional form as:
$$y=a x^{2} b x+c$$
In Equation 3.8, the variable that is squared is referred to as the quadratic variable, which in this case is $x$. Note that what determines if an equation is a function are not the symbols used in the equation, but whether the equation, domain, and range satisfy the definition of a function provided in Section 3.1.

Example 1 Determine which of the following functions are quadratic functions. For those that are, state their coefficients, a, b, and c.
a. $y=2 x^{2}-1 / 2$
b. $y=3 x-x^{2}$
c. $n^{2}=2 p+4$
Solution
a. This is a quadratic function in the variable $x$ with $a=2, b=0$, and $c=-1 / 2$.
b. Rewriting this equation as $y=-x^{2}+3 x$, we see that this is a quadratic function in the variable $x$ with $a=-1, b=3$, and $c=0$.
c. Rewriting this equation as $f(p)=1 / 2 n^{2}-2$, we see that it is a quadratic function in the variable $n$, with $a=1 / 2, b=0$, and $c=-2$.

As in the case of linear equations and in part (c) of this example, the letters $y$ and $x$ used in Equation $3.8$ are arbitrary; any other two letters are equally appropriate. The essential point is the form of the relationship between the variables. That is, a quadratic equation is one in which one variable can be written as the sum of a constant times the second variable squared, plus a constant times the second variable, plus a constant.

The graph of a quadratic function is a parabola, which is a shape similar to the cone of a rocket. Figures $3.11$ and $3.12$ are graphs of two different quadratic function.

In general, whenever we wish to solve a quadratic equation, it is easier to select values of the variable that is squared $(x$ in Equation $3.8)$ and $n$ in Example 2, and then use the given equation to find the value of the second variable, rather than the other way around, Sometimes, however, we have no choice. As an example of this, consider the following:

Example 3 Based on observations of prices, the demand D for oranges at a local fruit stand satisfies the equation $D=-0.25 P^{2}+6 P+900$, where $P$ is the price per orange (in cents). On a given Saturday morning, the store has 100 oranges in stock. Determine the price the store should charge for oranges if it wishes to deplete its inventory by the end of the day.

Solution Here, we seek the price that results in zero inventory. Mathematically, this means we are asked to find the value of the quadratic term $P$, for a given value of 100 for the linear term, $D$. Substituting $D=100$ into the demandprice equation, we find that $P$ must satisfy the quadratic equation
$$100=-0.25 P^{2}+6 P+900$$
which can be rewritten as
$$0.25 P^{2}-6 P-800=0$$
Solving this requires using the quadratic equation, ${ }^{3}$ with $a=0.25, b=-6$, and $c=-800$ (see the chapter appendix if you are not familiar with the quadratic formula). Using these values in the quadratic formula we obtain:
\begin{aligned} P_{1} &=\frac{-(-6)+\sqrt{(-6)^{2}-4(.25)(-800)}}{2(0.25)}=\frac{6+\sqrt{36+800}}{.5} \ &=\frac{6+\sqrt{836}}{0.5}=\frac{6+28.91}{0.5}=\frac{34.91}{0.5}=69.82 \end{aligned}
and
$$P_{2}=\frac{6-\sqrt{836}}{0.5}=\frac{6-28.91}{0.5}=\frac{-22.91}{0.5}=-45.82$$

Straight-line and quadratic functions are some of the simplest and yet valuable function in business and science. By themselves, however, they are not sufficient for modeling all real-world phenomena. Many such processes follow other functions. One of the most important of these remaining functions is the exponential function, which is a keystone of modern portfolio theory and environmental science.

In particular, most natural phenomena can be accurately modeled or represented by an exponential function. Examples of such situations are pollution levels, the use of natural resources, and the radioactive decay of certain materials. In practice, phenomena such as these can be misleading because their graphs stay relatively constant or flat for many years, very much like the graph of a linear equation. As the value of the exponent builds, however, the value of the $y$ variable suddenly “takes off” beyond any expectation based on what a linear or quadratic model would predict. Such a situation is shown in Figure 3.14, which illustrates the pollution level of nitrogen oxide versus time (in centuries).

$f(x)=a\left(b^{x}\right) x$ a real number
and is typically written using the form as
$$y=a\left(b^{x}\right)$$
where $a$ is a known non-zero real numbers and $b$ is a positive real number not equal to 1. The number $b$ is called the base. The distinguishing feature of an exponential function and the reason for its name is that the variable $x$ is the exponent.

## 商业数学代考

F(X)=一个2X2+一个1X+一个0

C。n2=2p+4

C。将此等式重写为F(p)=1/2n2−2，我们看到它是变量中的二次函数n， 和一个=1/2,b=0， 和C=−2.

100=−0.25磷2+6磷+900

0.25磷2−6磷−800=0

F(X)=一个(bX)X一个实数
，通常使用以下形式写成

## 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。

## 商科代写|商业数学代写business mathematics代考|CONCEPT OF A FUNCTION

• 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

## 商科代写|商业数学代写business mathematics代考|CONCEPT OF A FUNCTION

When mathematics is used to model real-world conditions, it must account for relationships between quantities. For example, the growth of bacteria over time, the amount of sales that follow from a given advertising budget, or the increase in the amount earned when a sum of money is invested in a bond.
As an example, Table $3.1$ illustrates a relationship between the number of cars sold over time, from 2016 to 2021 by Village Distributors, a small new car dealer. Each year’s sales are arranged under the corresponding year, which clearly shows the relationship between the two quantities, year and number of cars’ sold. Now consider Report 3.1, which is both wordier than Table $3.1$ and less useful. The reason is that a clear and direct assignment between individual years and the number of cars sold during each year is not immediately evident.

The notion of two distinct sets of quantities (like years and number of cars sold) and a rule of assignment between the sets, as presented in Table $3.1$ by arranging corresponding entries under each other, is central to the concept of a function. In fact, it describes a function.

Definition 3.1 A function is an assignment rule between two sets, which assigns to each element in the first set exactly one element (but not necessarily a different one) in the second set.

A function therefore has three components: (1) a first set (perhaps years),
(2) a second set (perhaps numbers), and (3) an assignment rule between the

two sets. This rule must be complete in that an assignment must be made to each and every element of the first set. As an example, take the first set to be all the people in the world, the second set as all positive numbers, and use the rule, “Assign to each person his or her exact weight.” This is a function. We have two sets and a rule which assigns to every element in the first set (people) exactly one element in the second set (his or her weight).

We know that a function consists of three components: a domain, a range, and a rule. The domain and range can be any two sets (people, cars, colors, numbers, etc.), while the rule can be given in a variety of ways (arrows, tables, words, etc.). In business situations, the primary concern is with sets of numbers (representing price, demand, advertising expenditures, cost, or profit, etc.) and rules defined by mathematical equations.

At first glance, it may seem strange to think of an equation as a rule, but it is. Consider two identical sets of real numbers and the equation $y=15 x+10$, where $x$ represents a number in the domain and $y$ represents a number in the range. The equation is nothing more than the rule “Multiply each element in

the domain by 15 and add 10 to the result.” Similarly, the equation $y=x^{2}-7$ is the rule “Square each element in the domain and then subtract 7 from the result.”

Whenever we have two sets of numbers and a rule given by an equation, where the variable $x$ denotes an element in the domain and the variable $y$ denotes an element in the range, we simply say that $y$ is a function of $x$ and write $y=f(x)$, although symbols other than $x$ and $y$ labels are frequently used when they are more appropriate to a particular problem.

Mathematical functions are the core of real-world applications and essential tools for decision makers. In this section, we present a class of one of the most useful business functions – polynomials
A function $f(x)$ is a polynomial function if it has the form
$$f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{2} x^{2}+a_{1} x+a_{0}$$
here $a_{n}, a_{n-1}, \cdots, a_{2}, a_{1}$, and $a_{0}$ are all known numbers.
The powers of $x$ (that is, $n, n-1$, and so on) are required to be nonnegative integers 1 , and the highest power with a corresponding non-zero coefficient a, is called the degree of the polynomial function. The lead coefficient, $a_{n}$, is the coefficient of the $x$ term with the highest power and cannot be zero, but any of the other following coefficients can be. The constant term $a_{0}$ is the coefficient of $x^{0}=1$.

Example 1 Determine which of the following functions are polynomial functions. For those that are, state their degree, and coefficients.
a. $f(x)=3 x^{2}-2 x$
b. $f(x)=0.8 x^{5}-2.25 x^{3}-\sqrt{7}$
c. $f(x)=\sqrt{x}$
d. $f(x)=\frac{1}{3}$
e. $f(x)=\frac{1}{x}$
Solution
a. This is a polynomial function of degree 2 , with $a_{2}=3, a_{1}=-2$, and $a_{0}=0$.
b. This is a polynomial function of degree 5 , with $a_{5}=0.8, a_{4}=0, a_{3}=2.25$. $a_{2}=a_{1}=0$, and $a_{0}=\sqrt{7}$.
c. This is not a polynomial function because $\sqrt{x}=x^{\frac{1}{2}}$. Here $x$ is raised to the $\frac{1}{2}$ power, which is not a non-zero integer.
d. This is a polynomial function of degree 0 , with $a_{0}=\frac{1}{3}$.
e. This is not a polynomial function because $\frac{1}{x}=x^{-1}$. Here $x$ is raised to the $-1$ power, which is a negative integer (this is a rational function, as presented at the end of this section).

## 商科代写|商业数学代写business mathematics代考|CONCEPT OF A FUNCTION

（2）第二组（可能是数字），以及（3）

F(X)=一个nXn+一个n−1Xn−1+⋯+一个2X2+一个1X+一个0

C。F(X)=X
d。F(X)=13

C。这不是多项式函数，因为X=X12. 这里X被提升到12幂，它不是一个非零整数。
d。这是一个 0 次多项式函数，其中一个0=13.
e. 这不是多项式函数，因为1X=X−1. 这里X被提升到−1幂，它是一个负整数（这是一个有理函数，如本节末尾所示）

## 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。

• 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

Linear equations are extremely useful in business applications for determining the relationship between short-term revenue and short-term costs. Conventionally, the term short-term refers to a time period in which both the price and the cost of an item remain constant. Over more extended time periods, economic conditions, such as inflation, supply and demand, and other economic factors typically act to change the cost and price structures. Over the short term, which is generally defined as a year or less, these other factors tend to have little direct influence.

The Break-Even point is the point at which the income from the sale of manufactured or purchased items exactly matches the cost of the items being sold. When this happens, the seller neither makes nor loses money but simply breaks even.

The reason the break-even point is so important is that it provides business information about the sales at which the company switches over from incurring a loss to making a profit. Should it be decided that sales can be higher than the break-even point, it means a profit can be made; otherwise, any sales less than the break-even point indicates that the venture will result in a loss. As such, it also forms as a lower bound for marketing, because if the break-even point cannot be reached, spending time and effort in marketing becomes a futile endeavor.

In determining the break-even point, both the revenue obtained by selling, and the cost involved in acquiring the items being sold must be taken into account.

The cost of items sold are commonly separated into two categories: fixed costs and variable costs.

Fixed costs include rent, insurance, property taxes, and other expenses that are present regardless of the number of items produced or purchased. Over the short run these costs are fixed because they exist and must be paid even if no items are purchased for resale or produced and sold. We will represent the fixed cost by the variable $F$.

Variable costs are those expenses that are directly attributable to the manufacture or purchase of the items themselves, such as labor and raw materials. Variable costs depend directly on the number of items manufactured or purchased – the more items manufactured or purchased, the higher the variable costs. If we restrict ourselves to short-run conditions, the cost-per-item is a fixed number, which makes the variable cost equal to this cost-per-item times the number of items purchased or manufactured. Designating the variable cost by $V$, the cost-per-item by $a$, and the number of items manufactured or purchased by $x$, we have
$$V=a x$$
Because the total cost is the sum of the variable cost plus fixed cost, the total cost equation becomes
$$C=V+F$$
Substituting Equation $2.13$ for $V$ into equation 2.14, the final cost equation becomes
$$C=a x+F$$
That is, the total cost is the sum of the variable cost and the fixed cost. The numbers $a$ and $F$ are assumed known and fixed; hence Equation2.15 is a linear equation in $C$ and $x$.

Example 2 A company manufacturing electronic calculators have recently signed contracts with its suppliers. For the duration of these contracts, the cost of manufacturing each calculator is $\$ 1.20$. The company estimates that the fixed costs for this period will be$\$8,000$. Determine the total cost function for this process and the actual cost incurred if only 500 calculators are actually manufactured.
Solution Using Equation $2.15$ with $a=1.20$ and $F=\$ 8,000$, we have $$C=\ 1.20 x+\ 8,000$$ ## 商科代写|商业数学代写business mathematics代考|The Break-Even Point From Examples 1 and 2, we note that a production run of 500 calculators will result in a total cost of$\$8,600$ and a sales revenue of only $\$ 2,500$. The company will experience a loss of$\$6,100$. Such embarrassing situations can be avoided with a break-even analysis. As the name suggests, this analysis involves finding the level of sales below which it will be unprofitable to produce items and above which sales revenue exceeds costs so that a profit is made. This level is the break-even point. The break-even point occurs when total cost exactly equals sales revenue.

If we restrict ourselves to the short run and assume that all items produced can be sold, the break-even point is obtained by setting the right side of Equation $2.12$ equal to the right side of Equation 2.15. That is, the break-even point occurs when $R=C$. Substituting for both the revenue, $R$, and cost, $C$, from Equations $2.12$ and $2.15$ yields
$$p x=a x+F \quad \text { (Eq. 2.16) }$$
Equation $2.16$ is one equation in the one unknown, $x$. Solving for $x$ using the algebraic methods presented in Section $1.2$ yields the break-even point, $B E P$, as
$$\mathbf{B E P}=x=\mathbf{F} /(p-a)$$
For the electronic calculator described in Examples 1 and 2, we found $C=\$ 1.20 x+\$8,000$ and $R=\$ 5.00 x$. The BEP occurs when$R=C$, or, from Equation$2.17$, when$x=8,000 /(5.00-1.20)=2,106$calculators. Any production and sales below 2,106 calculators results in a loss, while any production and sales above 2,106 units produces a profit. Example 3 A lamp component manufacturer determines that the manufacturing costs associated with each component are$\$5$ and that the fixed costs are $\$ 7,000$. Determine the BEP if each component sells for$\$7$. Assume that each unit made can beld.

Solution The total cost for this process, using Equation 2.15, is $C=\$ 5 x+\$7,000$. The sales revenue is $R=\$ 7 x$. The BEP is the value of$x$for which$R=C$. This point can be found by directly using Equation$2.17$, which yields,$x=7000 /(7-5)=3,500$components as the BEP. ## 商业数学代考 ## 商科代写|商业数学代写business mathematics代考|BREAK-EVEN ANALYSIS 线性方程在商业应用中对于确定短期收入和短期成本之间的关系非常有用。通常，术语短期是指项目的价格和成本都保持不变的时间段。在更长的时间段内，通货膨胀、供需和其他经济因素等经济条件通常会改变成本和价格结构。在通常定义为一年或更短时间的短期内，这些其他因素往往几乎没有直接影响。 盈亏平衡点是制造或购买物品的销售收入与所售物品的成本完全匹配的点。发生这种情况时，卖方既不赚钱也不亏钱，而只是收支平衡。 盈亏平衡点之所以如此重要，是因为它提供了有关公司从亏损转为盈利的销售业务信息。如果确定销售额可以高于盈亏平衡点，则意味着可以获利；否则，任何低于盈亏平衡点的销售额都表明该合资企业将导致亏损。因此，它也形成了营销的下限，因为如果无法达到收支平衡点，那么在营销上花费时间和精力将成为徒劳的努力。 在确定盈亏平衡点时，必须同时考虑通过销售获得的收入以及获取所售物品所涉及的成本。 ## 商科代写|商业数学代写business mathematics代考|The Cost Equation 出售物品的成本通常分为两类：固定成本和可变成本。 固定成本包括租金、保险、财产税和其他费用，无论生产或购买的物品数量如何。在短期内，这些成本是固定的，因为它们存在并且必须支付，即使没有购买用于转售或生产和销售的物品。我们将用变量表示固定成本F. 可变成本是直接归因于制造或购买物品本身的费用，例如劳动力和原材料。可变成本直接取决于制造或购买的物品数量——制造或购买的物品越多，可变成本就越高。如果我们将自己限制在短期条件下，则每件成本是一个固定数字，这使得可变成本等于每件成本乘以购买或制造的物品数量。指定可变成本在，每件商品的成本为一个，以及制造或购买的物品数量X， 我们有 在=一个X 因为总成本是可变成本加上固定成本的总和，所以总成本方程变为 C=在+F 代入方程2.13为了在进入方程 2.14，最终成本方程变为 C=一个X+F 也就是说，总成本是可变成本和固定成本的总和。号码一个和F假定已知且固定；因此方程 2.15 是一个线性方程C和X. 示例 2 一家制造电子计算器的公司最近与其供应商签订了合同。在这些合同期间，每个计算器的制造成本为$1.20. 公司预计本期固定成本为$8,000. 确定此过程的总成本函数以及实际仅制造 500 个计算器时产生的实际成本。 使用方程式的解决方案2.15和一个=1.20和F=$8,000， 我们有

C=$1.20X+$8,000

## 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。

## 商科代写|商业数学代写business mathematics代考|Graphs and Linear Equations

• 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

Line Graphs are most often displayed on a Cartesian coordinate system, which was presented in the previous section. As previously described, this coordinate system consists of two intersecting perpendicular lines called axes, as was shown in Figure $2.5$ and reproduced below as Figure $2.14$ for convenience.

Recall from Section $2.1$ that he intersection of the two axes is the Origin, which is the reference point for the system. The horizontal line is typically called the $x$-axis and the vertical line is typically called the $y$-axis. Although the letters $x$ and $y$ are the most widely used symbols for these axes, other letters are used when they are more meaningful in a particular problem.

Tick marks are then used to divide each axis into fixed units of length. Units to the right of the origin on the $x$-axis and units above the origin on the $y$-axis are assigned positive values. Units to the left of the origin on the $x$-axis and units below the origin on the $y$-axis are assigned negative values. Arrowheads are affixed to the positive portions of the $x$-and $y$-axes to indicate the directions of increasing values of $x$ and $y$. Successive tick marks must be equally spaced, which makes the units between successive tick marks the same, although the units or scale on the horizontal axis can, and frequently does, differ from the scale used on the vertical axis.

Graphing an equation can be considerably shortened if we already know the shape of the curve that the equation describes. One such case is provided by linear equations. These equations are singled out because they are one of the simplest and most important equations in both business and mathematics. They also have the geometric property that their graphs are always straight lines, among other useful features that are presented in the next section.
Formally, an equation is linear in two variables, $x$ and $y$, if it satisfies the following definition:

Definition 2.1: A linear equation in two variables $x$ and $y$ is an equation of the form:
$$A x+B y=C$$
where $A, B$, and $C$ are known real numbers and $A$ and $B$ are not both zero (this avoids the equations of the form $0=C$. The variables $x$ and $y$ can be replaced by any other convenient letters. Thus, if $x$ and $y$ are replaced by $p$ and $q$, respectively, then Equation $2.1$ becomes the linear equation $\mathrm{A} p+\mathrm{B} q=C$ in the variables $p$ and $q$.

Definitions are very precise mathematical statements. Unfortunately, this precision often makes a definition seem very complicated when, in fact, it is not. Usually, a few moments of thought is all that is needed to convert the given statement to an understandable concept.

For example, Definition $2.1$ simply states that any equation having the form of (i.e., looks like) the equation $A x+B y=\mathrm{C}$ where the letters $A, B$, and $C$ are replaced by numbers (e.g., $3 x+7 y=10$ ), is called a linear equation. Another important point is that because an exponent of 1 is understood but not written, a necessary feature of a linear equation is that the exponent of both the $x$ and $y$ terms must only be 1 .

The definition does not give any clues as to what a linear equation means geometrically-that will come later. What it does say, however, is that any equation that can be written in a form that looks like Equation 2.1, is a linear equation.

## 商科代写|商业数学代写business mathematics代考|PROPERTIES OF STRAIGHT LINES

Linear equations and their corresponding straight lines have a number of extremely useful characteristics. These include a concept called the line’s slope, the line’s $y$-intercept, and the ease with which these two quantities can be found from the line’s equation and graph. Conversely, the line’s equation can easily be determined if two solutions, that is, two points on the line are known. Each of these topics is presented in this section. We begin with the concept of a lines slope.

Graphically, a line’s slope is the direction and steepness of the line. Mathematically, it provides the rate of change, that is, how fast or slow the $y$ variable changes with respect to a change in the $x$ variable.

Graphically, a line with a positive slope rises as you move from left to right along the $x$-axis, as shown in Figure $2.33$. Also, as shown in the figure, a positively sloped line will have an angle between 0 and 90 degrees between the line and the positively directed $x$-axis. Finally, a line with a large positive slope, such as 100 , is steeper upward (that is, rises more quickly) than a line with a less positive slope, such as 5 .

## 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。