### 数学代写|线性代数代写linear algebra代考|MAST10007

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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|线性代数代写linear algebra代考|Euler’s Number and Natural Logarithms

There is a special number that shows up quite a bit in math called Euler’s number $e$. It is a special number much like Pi $\pi$ and is approximately 2.71828. $e$ is used a lot because it mathematically simplifies a lot of problems. We will cover $e$ in the context of exponents and logarithms.

Back in high school, my calculus teacher demonstrated Euler’s number in several exponential problems. Finally I asked, “Mr. Nowe, what is $e$ anyway? Where does it come from?” I remember never being fully satisfied with the explanations involving rabbit populations and other natural phenomena. I hope to give a more satisfying explanation here.
Why Euler’s Number Is Used So Much
A property of Euler’s number is its exponential function is a derivative to itself, which is convenient for exponential and logarithmic functions. We will learn about derivatives later in this chapter. In many applications where the base does not really matter, we pick the one that results in the simplest derivative, and that is Euler’s number. That is also why it is the default base in many data science functions.

Here is how I like to discover Euler’s number. Let’s say you loan $\$ 100$to somebody with$20 \%$interest annually. Typically, interest will be compounded monthly, so the interest each month would be$.20 / 12=.01666$. How much will the loan balance be after two years? To keep it simple, let’s assume the loan does not require payments (and no payments are made) until the end of those two years. Putting together the exponent concepts we learned so far (or perhaps pulling out a finance textbook), we can come up with a formula to calculate interest. It consists of a balance$A$for a starting investment$P$, interest rate$r$, time span$t$(number of years), and periods$n$(number of months in each year). Here is the formula: $$A=P \times\left(1+\frac{r}{n}\right)^{n t}$$ So if we were to compound interest every month, the loan would grow to$\$148.69$ as calculated here:
$$A=P \times\left(1+\frac{r}{n}\right)^{n t}$$

$$100 \times\left(1+\frac{.20}{12}\right)^{12 \times 2}=148.6914618$$
If you want to do this in Python, try it out with the code in Example 1-13.

## 数学代写|线性代数代写linear algebra代考|Natural Logarithms

When we use $e$ as our base for a logarithm, we call it a natural logarithm. Depending on the platform, we may use $\ln ()$ instead of $\log ()$ to specify a natural logarithm. So rather than express a natural logarithm expressed as $\log {e} 10$ to find the power raised on $e$ to get 10 , we would shorthand it as $\ln (10)$ : $$\log {e} 10=\ln (10)$$
However, in Python, a natural logarithm is specified by the log() function. As discussed earlier, the default base for the $\log ()$ function is $e$. Just leave the second argument for the base empty and it will default to using $e$ as the base shown in Example 1-15.
Example 1-15. Calculating the natural logarithm of 10 in Python
from nath import loge raised to what power gives us 10 ?

$x=\log (10)$ We will use $e$ in a number of places throughout this book. Feel free to experiment with exponents and logarithms using Excel, Python, Desmos.com, or any other calculation platform of your choice. Make graphs and get comfortable with what these functions look like.

## 数学代写|线性代数代写linear algebra代考|Euler’s Number and Natural Logarithms

$$\begin{gathered} A=P \times\left(1+\frac{r}{n}\right)^{n t} \ 100 \times\left(1+\frac{.20}{12}\right)^{12 \times 2}=148.6914618 \end{gathered}$$

## 数学代写|线性代数代写linear algebra代考|Natural Logarithms

$$\log e 10=\ln (10)$$

$x=\log (10)$ 我们将使用 $e$ 在本书的许多地方。随意使用 Excel、Python、Desmos.com 或您选择的任何其他计 算平台来试验指数和对数。制作图表并熟悉这些函数的外观。

## 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。