• 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

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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。