• 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

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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。