分类：微积分代写

数学代写|微积分代写Calculus代写|MATH171

Posted on 2023年8月31日2023年10月10日 by statistics-lab

如果你也在怎样代写微积分Calculus 这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。微积分Calculus 基本上就是非常高级的代数和几何。从某种意义上说，它甚至不是一门新学科——它采用代数和几何的普通规则，并对它们进行调整，以便它们可以用于更复杂的问题。(当然，问题在于，从另一种意义上说，这是一门新的、更困难的学科。)

微积分Calculus数学之所以有效，是因为曲线在局部是直的;换句话说，它们在微观层面上是直的。地球是圆的，但对我们来说，它看起来是平的，因为与地球的大小相比，我们在微观层面上。微积分之所以有用，是因为当你放大曲线，曲线变直时，你可以用正则代数和几何来处理它们。这种放大过程是通过极限数学来实现的。

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。

数学代写|微积分代写Calculus代写|Analyzing Graphs of Functions

In Section 1.4, you studied functions from an algebraic point of view. In this section, you will study functions from a graphical perspective.

The graph of a function $f$ is the collection of ordered pairs $(x, f(x))$ such that $x$ is in the domain of $f$. As you study this section, remember that
$x=$ the directed distance from the $y$-axis
$y=f(x)=$ the directed distance from the $x$-axis
as shown in Figure 1.52.

Use the graph of the function $f$, shown in Figure 1.53, to find (a) the domain of $f$, (b) the function values $f(-1)$ and $f(2)$, and (c) the range of $f$.
Solution
a. The closed dot at $(-1,1)$ indicates that $x=-1$ is in the domain of $f$, whereas the open dot at $(5,2)$ indicates that $x=5$ is not in the domain. So, the domain of $f$ is all $x$ in the interval $[-1,5)$.
b. Because $(-1,1)$ is a point on the graph of $f$, it follows that $f(-1)=1$. Similarly, because $(2,-3)$ is a point on the graph of $f$, it follows that $f(2)=-3$.
c. Because the graph does not extend below $f(2)=-3$ or above $f(0)=3$, the range of $f$ is the interval $[-3,3]$.
VCHECKPOINT Now try Exercise 1.
The use of dots (open or closed) at the extreme left and right points of a graph indicates that the graph does not extend beyond these points. If no such dots are shown, assume that the graph extends beyond these points.

By the definition of a function, at most one $y$-value corresponds to a given $x$-value. This means that the graph of a function cannot have two or more different points with the same $x$-coordinate, and no two points on the graph of a function can be vertically above or below each other. It follows, then, that a vertical line can intersect the graph of a function at most once. This observation provides a convenient visual test called the Vertical Line Test for functions.
Vertical Line Test for Functions
A set of points in a coordinate plane is the graph of $y$ as a function of $x$ if and only if no vertical line intersects the graph at more than one point.

数学代写|微积分代写Calculus代写|Zeros of a Function

If the graph of a function of $x$ has an $x$-intercept at $(a, 0)$, then $a$ is a zero of the function.
Zeros of a Function
The zeros of a function $f$ of $x$ are the $x$-values for which $f(x)=0$.
Example 3 Finding the Zeros of a Function
Find the zeros of each function.
a. $f(x)=3 x^2+x-10$
b. $g(x)=\sqrt{10-x^2}$
c. $h(t)=\frac{2 t-3}{t+5}$
Solution
To find the zeros of a function, set the function equal to zero and solve for the independent variable.
a.
$$
\begin{aligned}
& 3 x^2+x-10=0 \quad \text { Set } f(x) \text { equal to } 0 . \
& (3 x-5)(x+2)=0 \
& \text { Factor. } \
& 3 x-5=0 \
& x=\frac{5}{3} \
& \text { Set } 1 \text { st factor equal to } 0 \text {. } \
& x+2=0 \
& x=-2 \
& \text { Set } 2 \text { nd factor equal to } 0 \text {. } \
&
\end{aligned}
$$
The zeros of $f$ are $x=\frac{5}{3}$ and $x=-2$. In Figure 1.55, note that the graph of $f$ has $\left(\frac{5}{3}, 0\right)$ and $(-2,0)$ as its $x$-intercepts.
b. $\sqrt{10-x^2}=0$
Set $g(x)$ equal to 0 .
$$
\begin{aligned}
10-x^2 & =0 \
10 & =x^2 \
\pm \sqrt{10} & =x
\end{aligned}
$$
Square each side.
Add $x^2$ to each side.
Extract square roots.
The zeros of $g$ are $x=-\sqrt{10}$ and $x=\sqrt{10}$. In Figure 1.56, note that the graph of $g$ has $(-\sqrt{10}, 0)$ and $(\sqrt{10}, 0)$ as its $x$-intercepts.

微积分代考

数学代写|微积分代写Calculus代写|Analyzing Graphs of Functions

在第1.4节中，您从代数的角度学习了函数。在本节中，您将从图形的角度学习函数。

函数$f$的图是有序对$(x, f(x))$的集合，使得$x$在$f$的域中。在学习本节时，请记住这一点
$x=$到$y$轴的有向距离
$y=f(x)=$到$x$轴的有向距离
如图1.52所示。

使用如图1.53所示的函数$f$的图形来找到(a) $f$的定义域，(b) $f(-1)$和$f(2)$的函数值，以及(c) $f$的范围。
解决方案
a.“$(-1,1)$”表示“$x=-1$”在$f$的域中，“$(5,2)$”表示“$x=5$”不在该域中。所以，$f$的定义域都是$x$在$[-1,5)$区间内。
b.因为$(-1,1)$是$f$图上的一个点，所以可知$f(-1)=1$。同样，因为$(2,-3)$是$f$图上的一个点，所以$f(2)=-3$。
c.由于图在$f(2)=-3$以下或$f(0)=3$以上不扩展，所以$f$的范围为区间$[-3,3]$。
现在试试练习1。
在图形的最左和最右点使用点(开点或闭点)表示图形不会超出这些点。如果没有显示这样的点，则假定图形超出了这些点。

根据函数的定义，最多有一个$y$ -值对应于给定的$x$ -值。这意味着一个函数的图形不能有两个或多个具有相同$x$ -坐标的不同点，并且函数图形上的两个点不能垂直地位于彼此的上方或下方。因此，一条垂直线最多只能与一个函数的图形相交一次。这种观察提供了一种方便的视觉测试，称为函数的垂直线测试。
函数的垂直线测试
坐标平面上的一组点是$y$作为$x$的函数的图形，当且仅当没有垂直线与图形相交于一个以上的点。

数学代写|微积分代写Calculus代写|Zeros of a Function

如果$x$函数的图形在$(a, 0)$处的截距为$x$，则$a$是该函数的零点。
函数的零点
$x$的函数$f$的零点是$x$ -值，$f(x)=0$。
例3求函数的零点
找出每个函数的零点。
A. $f(x)=3 x^2+x-10$
B. $g(x)=\sqrt{10-x^2}$
C. $h(t)=\frac{2 t-3}{t+5}$
解决方案
要找到一个函数的零点，先将函数设为零，然后解出自变量。
a。
$$
\begin{aligned}
& 3 x^2+x-10=0 \quad \text { Set } f(x) \text { equal to } 0 . \
& (3 x-5)(x+2)=0 \
& \text { Factor. } \
& 3 x-5=0 \
& x=\frac{5}{3} \
& \text { Set } 1 \text { st factor equal to } 0 \text {. } \
& x+2=0 \
& x=-2 \
& \text { Set } 2 \text { nd factor equal to } 0 \text {. } \
&
\end{aligned}
$$
$f$的零点分别是$x=\frac{5}{3}$和$x=-2$。在图1.55中，请注意$f$的图形有$\left(\frac{5}{3}, 0\right)$和$(-2,0)$作为其$x$ -截点。
B. $\sqrt{10-x^2}=0$
设置$g(x)$ = 0。
$$
\begin{aligned}
10-x^2 & =0 \
10 & =x^2 \
\pm \sqrt{10} & =x
\end{aligned}
$$
每条边平方。
两边各加$x^2$。
取平方根。
$g$的零点分别是$x=-\sqrt{10}$和$x=\sqrt{10}$。在图1.56中，请注意$g$的图形有$(-\sqrt{10}, 0)$和$(\sqrt{10}, 0)$作为其$x$ -截点。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

微观经济学代写

微观经济学是主流经济学的一个分支，研究个人和企业在做出有关稀缺资源分配的决策时的行为以及这些个人和企业之间的相互作用。my-assignmentexpert™ 为您的留学生涯保驾护航在数学Mathematics作业代写方面已经树立了自己的口碑, 保证靠谱, 高质且原创的数学Mathematics代写服务。我们的专家在图论代写Graph Theory代写方面经验极为丰富，各种图论代写Graph Theory相关的作业也就用不着说。

线性代数代写

线性代数是数学的一个分支，涉及线性方程，如：线性图，如：以及它们在向量空间和通过矩阵的表示。线性代数是几乎所有数学领域的核心。

博弈论代写

现代博弈论始于约翰-冯-诺伊曼（John von Neumann）提出的两人零和博弈中的混合策略均衡的观点及其证明。冯-诺依曼的原始证明使用了关于连续映射到紧凑凸集的布劳威尔定点定理，这成为博弈论和数学经济学的标准方法。在他的论文之后，1944年，他与奥斯卡-莫根斯特恩（Oskar Morgenstern）共同撰写了《游戏和经济行为理论》一书，该书考虑了几个参与者的合作游戏。这本书的第二版提供了预期效用的公理理论，使数理统计学家和经济学家能够处理不确定性下的决策。

微积分代写

微积分，最初被称为无穷小微积分或 “无穷小的微积分”，是对连续变化的数学研究，就像几何学是对形状的研究，而代数是对算术运算的概括研究一样。

它有两个主要分支，微分和积分；微分涉及瞬时变化率和曲线的斜率，而积分涉及数量的累积，以及曲线下或曲线之间的面积。这两个分支通过微积分的基本定理相互联系，它们利用了无限序列和无限级数收敛到一个明确定义的极限的基本概念。

计量经济学代写

什么是计量经济学？
计量经济学是统计学和数学模型的定量应用，使用数据来发展理论或测试经济学中的现有假设，并根据历史数据预测未来趋势。它对现实世界的数据进行统计试验，然后将结果与被测试的理论进行比较和对比。

根据你是对测试现有理论感兴趣，还是对利用现有数据在这些观察的基础上提出新的假设感兴趣，计量经济学可以细分为两大类：理论和应用。那些经常从事这种实践的人通常被称为计量经济学家。

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，工具箱允许您学习和应用专业技术。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。

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|微积分代写Calculus代写|MTH191

Posted on 2023年8月31日2023年8月31日 by statistics-lab

数学代写|微积分代写Calculus代写|Introduction to Functions

Many everyday phenomena involve two quantities that are related to each other by some rule of correspondence. The mathematical term for such a rule of correspondence is a relation. In mathematics, relations are often represented by mathematical equations and formulas. For instance, the simple interest $I$ earned on $\$ 1000$ for 1 year is related to the annual interest rate $r$ by the formula $I=1000 \mathrm{r}$.
The formula $I=1000 \mathrm{r}$ represents a special kind of relation that matches each item from one set with exactly one item from a different set. Such a relation is called a function.
Definition of Function
A function $f$ from a set $A$ to a set $B$ is a relation that assigns to each element $x$ in the set $A$ exactly one element $y$ in the set $B$. The set $A$ is the domain (or set of inputs) of the function $f$, and the set $B$ contains the range (or set of outputs).
To help understand this definition, look at the function that relates the time of day to the temperature in Figure 1.47.

This function can be represented by the following ordered pairs, in which the first coordinate ( $x$-value) is the input and the second coordinate ( $y$-value) is the output.
$$
\left{\left(1,9^{\circ}\right),\left(2,13^{\circ}\right),\left(3,15^{\circ}\right),\left(4,15^{\circ}\right),\left(5,12^{\circ}\right),\left(6,10^{\circ}\right)\right}
$$
Characteristics of a Function from Set $A$ to Set $B$

Each element in $A$ must be matched with an element in $B$.
Some elements in $B$ may not be matched with any element in $A$.
Two or more elements in $A$ may be matched with the same element in $B$.
An element in $A$ (the domain) cannot be matched with two different elements in $B$.

数学代写|微积分代写Calculus代写|Function Notation

When an equation is used to represent a function, it is convenient to name the function so that it can be referenced easily. For example, you know that the equation $y=1-x^2$ describes $y$ as a function of $x$. Suppose you give this function the name ” $f$.” Then you can use the following function notation.
$\begin{array}{ccc}\text { Input } & \text { Output } & \text { Equation } \ x & f(x) & f(x)=1-x^2\end{array}$
The symbol $f(x)$ is read as the value of $f$ at $x$ or simply $f$ of $x$. The symbol $f(x)$ corresponds to the $y$-value for a given $x$. So, you can write $y=f(x)$. Keep in mind that $f$ is the name of the function, whereas $f(x)$ is the value of the function at $x$. For instance, the function given by
$$
f(x)=3-2 x
$$
has function values denoted by $f(-1), f(0), f(2)$, and so on. To find these values, substitute the specified input values into the given equation.
$$
\begin{aligned}
\text { For } x & =-1, & f(-1) & =3-2(-1)=3+2=5 . \
& \text { For } x=0, & f(0) & =3-2(0)=3-0=3 . \
\text { For } x & =2, & f(2) & =3-2(2)=3-4=-1 .
\end{aligned}
$$

Although $f$ is often used as a convenient function name and $x$ is often used as the independent variable, you can use other letters. For instance,
$$
f(x)=x^2-4 x+7, \quad f(t)=t^2-4 t+7, \quad \text { and } \quad g(s)=s^2-4 s+7
$$
all define the same function. In fact, the role of the independent variable is that of a “placeholder.” Consequently, the function could be described by
$$
f(\square)=(\square)^2-4(\square)+7 .
$$

微积分代考

数学代写|微积分代写Calculus代写|Introduction to Functions

许多日常现象涉及两个量，它们通过某种对应规则相互关联。这种对应规则的数学术语是关系。在数学中，关系通常用数学方程和公式来表示。例如，在$\$ 1000$上赚取的1年单利$I$与年利率$r$通过公式$I=1000 \mathrm{r}$相关联。
公式$I=1000 \mathrm{r}$表示一种特殊的关系，它将一个集合中的每个项目与另一个集合中的一个项目进行匹配。这样的关系称为函数。
函数的定义
从集合$A$到集合$B$的函数$f$是一个关系，它将集合$A$中的每个元素$x$精确地分配给集合$B$中的一个元素$y$。集合$A$是函数$f$的域(或一组输入)，集合$B$包含范围(或一组输出)。
为了帮助理解这个定义，请看图1.47中将一天中的时间与温度联系起来的函数。

该函数可以用以下有序对表示，其中第一个坐标($x$ -value)是输入，第二个坐标($y$ -value)是输出。
$$
\left{\left(1,9^{\circ}\right),\left(2,13^{\circ}\right),\left(3,15^{\circ}\right),\left(4,15^{\circ}\right),\left(5,12^{\circ}\right),\left(6,10^{\circ}\right)\right}
$$
从集合$A$到集合的函数特征 $B$

$A$中的每个元素必须与$B$中的一个元素匹配。

$B$中的某些元素可能与$A$中的任何元素不匹配。

$A$中的两个或多个元素可以与$B$中的相同元素匹配。

$A$(域)中的一个元素不能与$B$中的两个不同元素匹配。

数学代写|微积分代写Calculus代写|Function Notation

当用一个方程来表示一个函数时，给函数命名是很方便的，这样就可以很容易地引用它。例如，您知道方程$y=1-x^2$将$y$描述为$x$的函数。假设您将此函数命名为“$f$”，那么您可以使用以下函数表示法。
$\begin{array}{ccc}\text { Input } & \text { Output } & \text { Equation } \ x & f(x) & f(x)=1-x^2\end{array}$
符号$f(x)$读取为$f$在$x$处的值，或者直接读取为$x$的$f$。符号$f(x)$对应于给定$x$的$y$ -值。你可以写$y=f(x)$。请记住，$f$是函数的名称，而$f(x)$是函数在$x$处的值。例如，函数由
$$
f(x)=3-2 x
$$
具有用$f(-1), f(0), f(2)$表示的函数值，等等。要找到这些值，将指定的输入值代入给定的方程。
$$
\begin{aligned}
\text { For } x & =-1, & f(-1) & =3-2(-1)=3+2=5 . \
& \text { For } x=0, & f(0) & =3-2(0)=3-0=3 . \
\text { For } x & =2, & f(2) & =3-2(2)=3-4=-1 .
\end{aligned}
$$

虽然$f$常被用作方便的函数名，$x$常被用作自变量，但您也可以使用其他字母。例如，
$$
f(x)=x^2-4 x+7, \quad f(t)=t^2-4 t+7, \quad \text { and } \quad g(s)=s^2-4 s+7
$$
都定义了相同的函数。事实上，自变量的作用是“占位符”。因此，函数可以用
$$
f(\square)=(\square)^2-4(\square)+7 .
$$

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

微观经济学代写

线性代数代写

线性代数是数学的一个分支，涉及线性方程，如：线性图，如：以及它们在向量空间和通过矩阵的表示。线性代数是几乎所有数学领域的核心。

博弈论代写

微积分代写

计量经济学代写

Matlab代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|微积分代写Calculus代写|MATH1141

Posted on 2023年8月31日2023年8月31日 by statistics-lab

数学代写|微积分代写Calculus代写|Linear Equations in Two Variables

The simplest mathematical model for relating two variables is the linear equation in two variables $y=m x+b$. The equation is called linear because its graph is a line. (In mathematics, the term line means straight line.) By letting $x=0$, you can see that the line crosses the $y$-axis at $y=b$, as shown in Figure 1.28. In other words, the $y$-intercept is $(0, b)$. The steepness or slope of the line is $m$.
$$
y=m x+b
$$
The slope of a nonvertical line is the number of units the line rises (or falls) vertically for each unit of horizontal change from left to right, as shown in Figure 1.28 and Figure 1.29.

A linear equation that is written in the form $y=m x+b$ is said to be written in slope-intercept form.
The Slope-Intercept Form of the Equation of a Line
The graph of the equation
$$
y=m x+b
$$
is a line whose slope is $m$ and whose $y$-intercept is $(0, b)$.
Exploration
Use a graphing utility to compare the slopes of the lines $y=m x$, where $m=0.5,1,2$, and 4 . Which line rises most quickly? Now, let $m=-0.5$, $-1,-2$, and -4 . Which line falls most quickly? Use a square setting to obtain a true geometric perspective. What can you conclude about the slope and the “rate” at which the line rises or falls?

Once you have determined the slope and the $y$-intercept of a line, it is a relatively simple matter to sketch its graph. In the next example, note that none of the lines is vertical. A vertical line has an equation of the form
$$
x=a .
$$
Vertical line
The equation of a vertical line cannot be written in the form $y=m x+b$ because the slope of a vertical line is undefined, as indicated in Figure 1.30.

数学代写|微积分代写Calculus代写|Writing Linear Equations in Two Variables

If $\left(x_1, y_1\right)$ is a point on a line of slope $m$ and $(x, y)$ is any other point on the line, then
$$
\frac{y-y_1}{x-x_1}=m .
$$
This equation, involving the variables $x$ and $y$, can be rewritten in the form
$$
y-y_1=m\left(x-x_1\right)
$$
which is the point-slope form of the equation of a line.
Point-Slope Form of the Equation of a Line
The equation of the line with slope $m$ passing through the point $\left(x_1, y_1\right)$ is
$$
y-y_1=m\left(x-x_1\right) .
$$
The point-slope form is most useful for finding the equation of a line. You should remember this form.
Example 3 Using the Point-Slope Form
Find the slope-intercept form of the equation of the line that has a slope of 3 and passes through the point $(1,-2)$.
Solution
Use the point-slope form with $m=3$ and $\left(x_1, y_1\right)=(1,-2)$.
$$
\begin{aligned}
y-y_1 & =m\left(x-x_1\right) & & \text { Point-slope form } \
y-(-2) & =3(x-1) & & \text { Substitute for } m, x_1, \text { and } y_1 . \
y+2 & =3 x-3 & & \text { Simplify. } \
y & =3 x-5 & & \text { Write in slope-intercept form. }
\end{aligned}
$$
The slope-intercept form of the equation of the line is $y=3 x-5$. The graph of this line is shown in Figure 1.39.

微积分代考

数学代写|微积分代写Calculus代写|Linear Equations in Two Variables

关系两个变量的最简单的数学模型是两个变量的线性方程$y=m x+b$。这个方程被称为线性方程，因为它的图形是一条直线。(在数学中，“线”一词的意思是直线。)通过设置$x=0$，您可以看到这条线在$y=b$处穿过$y$ -轴，如图1.28所示。也就是说，$y$截距等于$(0, b)$。直线的陡度或斜率为$m$。
$$
y=m x+b
$$
非垂直线的斜率是直线从左向右每发生一个单位的水平变化，直线垂直上升(或下降)的单位数，如图1.28和图1.29所示。

表示为$y=m x+b$的线性方程被称为斜率-截距式。
直线方程的斜截式
方程的图形
$$
y=m x+b
$$
是一条斜率为$m$, $y$截距为$(0, b)$的直线。
探索
使用绘图工具来比较直线$y=m x$的斜率，其中$m=0.5,1,2$和4。哪条线上升最快?现在，让$m=-0.5$$-1,-2$和-4。哪条线掉得最快?使用正方形设置来获得真正的几何透视。关于斜率和直线上升或下降的“速率”，你能得出什么结论?

一旦确定了直线的斜率和$y$ -截距，绘制其图形就相对简单了。在下一个示例中，请注意没有一条线是垂直的。一条垂直线的方程是这样的
$$
x=a .
$$
垂直线
垂直线的方程不能写成$y=m x+b$的形式，因为垂直线的斜率没有定义，如图1.30所示。

数学代写|微积分代写Calculus代写|Writing Linear Equations in Two Variables

如果$\left(x_1, y_1\right)$是斜率为$m$的直线上的一点$(x, y)$是直线上的任意一点，则
$$
\frac{y-y_1}{x-x_1}=m .
$$
这个方程，包含变量$x$和$y$，可以重写为
$$
y-y_1=m\left(x-x_1\right)
$$
也就是直线方程的点斜式。
直线方程的点斜式
斜率为$m$的直线经过点$\left(x_1, y_1\right)$的方程为
$$
y-y_1=m\left(x-x_1\right) .
$$
点斜式在求直线方程时是最有用的。你应该记住这个表格。
例3点斜式的使用
求斜率为3且经过$(1,-2)$点的直线的斜截式方程。
解决方案
使用点斜形式与$m=3$和$\left(x_1, y_1\right)=(1,-2)$。
$$
\begin{aligned}
y-y_1 & =m\left(x-x_1\right) & & \text { Point-slope form } \
y-(-2) & =3(x-1) & & \text { Substitute for } m, x_1, \text { and } y_1 . \
y+2 & =3 x-3 & & \text { Simplify. } \
y & =3 x-5 & & \text { Write in slope-intercept form. }
\end{aligned}
$$
直线方程的斜截式是$y=3 x-5$。该线的曲线图如图1.39所示。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

微观经济学代写

线性代数代写

线性代数是数学的一个分支，涉及线性方程，如：线性图，如：以及它们在向量空间和通过矩阵的表示。线性代数是几乎所有数学领域的核心。

博弈论代写

微积分代写

计量经济学代写

Matlab代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|微积分代写Calculus代写|MTH-211

Posted on 2023年8月14日2023年8月28日 by statistics-lab

数学代写|微积分代写Calculus代写|JOHANN KEPLBR

Among the first of the seventeenth-century mathematicians to abandon the formal proof structure of Archimedes and to introduce the free use of infinitesimals in the determination of areas and volumes was Johann Kepler. As an astronomer Kepler was primarily concerned to interpret the universe in terms of ordered mathematical harmony and throughout his entire life he pursued this goal with tireless energy and unremitting determination. In so doing he was willing to go to any lengths to develop from a mathematical hypothesis sufficient data to test a theory but he was nevertheless prepared at any time to abandon the theory and start afresh should it fail to fit the observed facts. ${ }^{\dagger}$

As a mathematician Kepler sought to demonstrate in the simplest possible way the existence of mathematical form and structure in the external world and, upheld by his Platonic-Pythagorean philosophy, he often allowed faith, analogy and intuition to guide him when traditional methods failed. Many of the mathematical problems which Kepler attempted to resolve in the interests of astronomy were extremely difficult by any standards and well beyond the scope of existing seventeenth-century techniques. Although Kepler became conversant with the whole range of Greek mathematics he was more interested in results than in proofs. If an Archimedean demonstration supplied him with a required property he was glad to make use of it, but when formal methods failed to provide the answers to his problems the practical necessities of the situation drove him onwards to devise new techniques and processes.

Many of the astronomical problems with which he was faced required the evaluation of trigonometric integrals and in some of these cases, finding no theoretical method available, Kepler was obliged to resort to numerical summation. One such problem arose in connection with the theory of planetary magnetism $^{\dagger}$ in which the entire sphere of the planet was regarded as a magnet, one pole being attracted and the other repelled by the sun. After many attempts to determine the relation between the attractive and repulsive forces he finally concluded that it was necessary to divide off the arc into infinitely many sman and equal parts and to find the sum of the sines (or chords) of all the arcs. This was duly accomplished and he arrived (presumably by incomplete induction) at a conclusion formally equivalent to the definite integral
$$
\int_0^\alpha \sin \theta d \theta=1-\cos \alpha .
$$

数学代写|微积分代写Calculus代写|INDIVISIBLES IN ITALY

Although there is no record of any direct exchange of views between Kepler and Galileo on the subject of infinitesimals the publication of the Stereometria may well have acted as a spur, both to Galileo and to his pupil Cavalieri, in extending and developing their own work in this field. Although Galileo never actually published a work on indivisible methods as such the whole trend of the discussion in the Two New Sciences” throws light not only on his own approach to such matters but also on the background of ideas which he transmitted to Cavalieri.

Galileo had a profound respect for the Greeks and in consequence greatly admired the achievements of those mathematicians such as Valerio who had expanded and developed the work of Archimedes.” Nevertheless it is clear that for the most part his own ideas led him along entirely different paths. The influence of Scholastic thought on Galileo is, of course, most pronounced in his early works but even in Two New Sciences it is very much to the fore when he discusses the nature of indivisibles and the infinite. In this work Galileo reviews most of the mediaeval paradoxes and at the same time provides some new ones of his own. Most of these paradoxes are based on the concept of indivisible quantities and Galileo makes no distinction between physical atoms and line and surface elements: in fact he finds support for his belief in the efficacy of mathematical indivisibles from his desire to demonstrate the possibility of the existence of a vacuum.

In discussing the Wheel of Aristotle, $\ddagger$ still a popular subject in the seventeenth century, Galileo first allows the hexagon, $A B C D E F$, to roll upon the line $A S$ (see Fig. 4.12) carrying with it a concentric hexagon, $H I K L M N$, inscribed within it. As the sides $A B, B C, C D, D E, E F, F A$ take up successive positions on the line $A B$, the sides of the concentric hexagon fall into positions $H I, I^{\prime} K, K^{\prime} L, \ldots, N^{\prime} H$ on the line $H T$, parallel to $A S$. Hence, in one revolution of the wheel, the total distance $H H^{\prime}$ moved by the point $H$ along the line $H T$ equals the distance $A A^{\prime}$ moved by the point $A$ along the line $A S$. The distance $H H^{\prime}$, however, is greater than the perimeter of the lesser polygon by the length of the spaces $I I^{\prime}, K K^{\prime}, L L^{\prime}, \ldots, H H^{\prime}$. Now, if the number of sides of the polygon be increased to 1000 , then in one complete revolution the lesser polygon will lay off 1000 sides and the same number of spaces along the line $H T$. Finally, if the rolling polygons be replaced by concentric circles, rigidly connected, the point $C$ on the lesser circle will move through the same distance, $C C^{\prime}$, as the point $B$ on the greater circle (see Fig. 4.13).

微积分代考

数学代写|微积分代写Calculus代写|JOHANN KEPLBR

约翰·开普勒是17世纪最早抛弃阿基米德的形式证明结构，并在确定面积和体积时引入无限小的自由使用的数学家之一。作为一名天文学家，开普勒主要关注的是用有序的数学和谐来解释宇宙，在他的一生中，他以不懈的精力和不懈的决心追求这一目标。在这样做的时候，他愿意尽一切努力从数学假设中发展出足够的数据来检验一种理论，但他也随时准备放弃这种理论，如果它与观察到的事实不相符，他又重新开始。 ${ }^{\dagger}$

作为一名数学家，开普勒试图以最简单的方式证明数学形式和结构在外部世界的存在。在他的柏拉图-毕达哥拉斯哲学的支持下，当传统方法失败时，他经常让信仰、类比和直觉来指导他。开普勒为了天文学的兴趣而试图解决的许多数学问题，以任何标准来看都是极其困难的，而且远远超出了17世纪现有技术的范围。尽管开普勒对整个希腊数学都很熟悉，但他对结果比对证明更感兴趣。如果阿基米德的论证为他提供了所需的性质，他很乐意利用它，但是当正式的方法不能为他的问题提供答案时，实际情况的需要驱使他继续设计新的技术和过程。

他所面临的许多天文问题都需要计算三角积分，在某些情况下，由于找不到可用的理论方法，开普勒不得不求助于数值求和。其中一个这样的问题出现在行星磁学理论$^{\dagger}$中，在这个理论中，行星的整个球体被视为一个磁体，一极被太阳吸引，另一极被太阳排斥。在多次尝试确定引力和斥力之间的关系之后，他最终得出结论，有必要将弧线分成无限多等分，并找出所有弧线的正弦(或弦)之和。这是如期完成的，他(大概是通过不完全归纳法)得出了一个形式上等同于定积分的结论
$$
\int_0^\alpha \sin \theta d \theta=1-\cos \alpha .
$$

数学代写|微积分代写Calculus代写|INDIVISIBLES IN ITALY

尽管没有记录表明开普勒和伽利略在无穷小问题上有过任何直接的意见交换，立体测量的出版对伽利略和他的学生卡瓦列里来说都是一种刺激，在这一领域扩展和发展他们自己的工作。虽然伽利略从未真正发表过关于不可分割方法的著作，但《两门新科学》中讨论的整个趋势不仅揭示了他自己对这些问题的研究方法，也揭示了他传递给卡瓦列里的思想背景。

伽利略对希腊人有很深的敬意，因此非常钦佩那些数学家的成就，比如瓦莱里奥，他们扩展和发展了阿基米德的工作。”然而，很明显，在很大程度上，他自己的思想引导他走上了完全不同的道路。经院思想对伽利略的影响，当然，在他早期的作品中是最明显的，但即使在《两门新科学》中，当他讨论不可分割和无限的本质时，这种影响也非常突出。在这部著作中，伽利略回顾了大部分中世纪的悖论，同时也提出了一些他自己的新悖论。这些悖论大多是基于不可分量的概念，伽利略对物理原子、线元素和面元素没有区别:事实上，他从证明真空存在的可能性的愿望中找到了对数学不可分有效性的信念的支持。

在讨论亚里士多德的轮子时(在17世纪仍然是一个流行的话题)，伽利略首先允许六边形(a B C D E F)在线上滚动(见图4.12)，它带着一个同心六边形(H I K L M N)，上面刻着一个同心六边形(H I K L M N)。由于边$A B, B C, C D, D E, E F, F A$在直线$A B$上占据连续位置，同心六边形的边落在直线$H T$上的位置$H I， $ I^{\素数}K, K^{\素数}L， $ ldots, N^{\素数}H$，平行于$A S$。因此，在车轮转一圈时，H H^{\ ‘}$被点H$沿直线H T$移动的总距离等于A A^{\ ‘}$被点A$沿直线A S$移动的距离。然而，距离$H H^{\prime}$大于较小多边形的周长$I I^{\prime}， K K^{\prime}， L L^{\prime}， \ldots, H H^{\prime}$。现在，如果多边形的边数增加到1000条，那么在一次完整的旋转中，较小的多边形将沿直线H T$释放1000条边和相同数量的空间。最后，如果将滚动多边形替换为刚性连接的同心圆，则较小圆上的点$C$将与较大圆上的点$B$移动相同的距离$C C^{\ ‘}$(见图4.13)。

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微观经济学代写

线性代数代写

线性代数是数学的一个分支，涉及线性方程，如：线性图，如：以及它们在向量空间和通过矩阵的表示。线性代数是几乎所有数学领域的核心。

博弈论代写

微积分代写

计量经济学代写

Matlab代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|微积分代写Calculus代写|MTH-211

Posted on 2023年8月12日2023年8月28日 by statistics-lab

数学代写|微积分代写Calculus代写|ThE Growth OF KinEMatics IN thE WeST

Whereas in dynamics movement is studied in relation to the forces associated with it, in kinematics only its spatial and temporal aspects are considered. It follows that in kinematics we are concerned with the description of movement and not with its causes and it can therefore properly be regarded as the geometry of movement.

If a particle be known to trace a given curve the geometric properties of that curve can be used to predict the subsequent positions of the particle; conversely, if a curve be defined as the path of a point moving under specified conditions, then the laws of kinematics can be utilised to provide information as to certain geometric properties of the curve. For example, knowledge of the instantaneous motion at a given point on the curve enables us to draw the tangent to the curve at that point. The development, in the fourteenth century, of certain important concepts of motion including instantaneous velocity can therefore be seen to have direct and immediate bearing on the study of the tangent properties of curves. Furthermore, the introduction of graphical methods of representation led to the establishment of a link between the velocity-time graph, the total distance covered and the area under the curve, and this in turn is closely connected with the integral calculus (see Figs. 2.8 and 2.9).

The imaginative insights gained by the use of kinematic concepts in geometry were responsible for some of the more powerful methods developed during the seventeenth century for the study of curves. The work of Isaac Barrow, for instance, which certainly influenced Newton, is dominated by the idea of curves generated by moving points and lines.

数学代写|微积分代写Calculus代写|The Latitude OF Forms

At Merton College, Oxford, between the years 1328 and 1350, the distinction between kinematics and dynamics was made explicit. In the work of Thomas Bradwardine, William Heytesbury, Richard Swineshead and John Dumbleton the foundations for further study in this field were laid through the clarification and formalisation of a number of important concepts including the notion of instantaneous velocity (velocitas instantanea). ${ }^{\dagger}$

The study of space and motion at Merton College arose from the mediaeval discussion of the intension and remission of forms, i.e. the increase and decrease of the intensity of qualities. The distinction between intension and extension is exemplified in the case of heat by the difference between temperature, or degree of heat, and quantity of heat; in the case of weight between density, or weight per unit volume and total weight. For local motion the distinction is between velocity (or motion) at a given instant (instantaneous velocity) and total motion over a period of time, i.e. distance covered.

William Heytesbury distinguishes between uniform and difform (nonuniform) motion. In the case of difform motion he says: $\ddagger if, in a period of time, it were moved uniformly at the same degree of velocity (uniformiter illo gradu velocitatis) with which it is moved in that given instant, whatever [instant] be assigned.

The most notable single achievement of the Merton School was the establishment of the mean speed theorem for uniformly accelerated (uniformly difform) motion, i.e. $s=\left(v_1+v_2\right) t / 2$, where $s$ is the distance covered in time $t$ and $v_1$ and $v_2$ are the initial and final velocities. Various proofs of this theorem were presented, some purely arithmetical and others depending on some skilful manipulation of infinite series. Although in some cases velocities (or intensions) were represented by single straight lines and Richard Swineshead actually uses a geometric analogy ${ }^{\dagger}$ to explain intension and remission of qualities it was not until knowledge of the Merton College work reached France and Italy that it received the full benefits of geometric representation and so linked kinematics with the geometry of straight and curved lines.

微积分代考

数学代写|微积分代写Calculus代写|ThE Growth OF KinEMatics IN thE WeST

在动力学中，运动是根据与之相关的力来研究的，而在运动学中，只考虑其空间和时间方面。由此可见，在运动学中，我们所关心的是运动的描述，而不是运动的原因，因此，运动学可以恰当地看作是运动的几何学。

如果已知粒子沿给定曲线运动，则该曲线的几何特性可用于预测粒子的后续位置;相反，如果一条曲线被定义为一个点在特定条件下运动的路径，那么运动学定律可以用来提供关于曲线的某些几何性质的信息。例如，知道曲线上某一点的瞬时运动，我们就能画出该点与曲线的切线。因此，在14世纪，包括瞬时速度在内的某些重要运动概念的发展，可以看作对曲线的切线性质的研究有直接和直接的影响。此外，图形表示方法的引入使速度-时间图、覆盖的总距离和曲线下的面积之间建立了联系，而这又与积分学密切相关(见图2.8和图2.9)。

利用几何学中的运动学概念所获得的富有想象力的洞见，促成了17世纪研究曲线的一些更有力的方法的发展。例如，艾萨克·巴罗(Isaac Barrow)的工作，当然影响了牛顿，主要是由移动的点和线产生曲线的想法。

数学代写|微积分代写Calculus代写|The Latitude OF Forms

1328年至1350年间，牛津大学默顿学院明确区分了运动学和动力学。在Thomas Bradwardine, William Heytesbury, Richard Swineshead和John Dumbleton的工作中，通过澄清和形式化一些重要的概念，包括瞬时速度(velocitas instantanea)的概念，为该领域的进一步研究奠定了基础。${} ^{\匕首}$

默顿学院对空间和运动的研究源于中世纪对形式的强度和缓解的讨论，即质量强度的增加和减少。在热的情况下，强度和延伸之间的区别可以通过温度或热度与热量之间的差异来举例说明;在密度或单位体积重量与总重量之间的情况下。对于局部运动，区别在于给定瞬间(瞬时速度)的速度(或运动)和一段时间内的总运动，即所走过的距离。

威廉·海茨伯里区分了均匀运动和非均匀运动。在不均匀运动的情况下，他说:“如果在一段时间内，物体以与它在给定瞬间的运动速度相同的速度均匀运动，无论给定的是什么。”

默顿学派最著名的成就是建立了均匀加速(均匀均匀)运动的平均速度定理，即$s=\左(v_1+v_2\右)t / 2$，其中$s$是时间所覆盖的距离$t$， $v_1$和$v_2$是初始和最终速度。给出了这个定理的各种证明，有些是纯粹的算术证明，有些则依赖于对无穷级数的一些巧妙的处理。虽然在某些情况下，速度(或强度)是用一条直线来表示的，Richard Swineshead实际上使用了一个几何类比来解释强度和质量的缓解，但直到默顿学院的研究成果传播到法国和意大利，它才充分受益于几何表示，并将运动学与直线和曲线的几何联系起来。

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线性代数代写

线性代数是数学的一个分支，涉及线性方程，如：线性图，如：以及它们在向量空间和通过矩阵的表示。线性代数是几乎所有数学领域的核心。

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PYTHON代写	回归分析与线性模型代写
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数学代写|微积分代写Calculus代写|MATH1111

Posted on 2023年8月12日2023年8月28日 by statistics-lab

数学代写|微积分代写Calculus代写|ON Hindu MathemÁtics

The history of early Hindu mathematics has always presented considerable problems for the West. Nineteenth century studies, although sufficiently startling in some cases” as to arouse interest, were nevertheless often undertaken by Sanskrit scholars from the West whose knowledge of mathematics was not sufficiently profound to enable them to do much more than note results. Although more substantial studies in recent years by Hindu mathematicians have done a great deal towards filling some of the gaps in our knowledge it is still not possible to form a clear picture of either method or motivation in Hindu mathematics. Notwithstanding, even a straightforward ordering of results becomes meaningful when enough material has been collected to form a coherent pattern.

The Hindus seem to have been attracted by the computational aspect of mathematics, partly for its own sake and partly as a tool in astrological prediction. No trace has been found of any proof structure such as that established by Euclid for Greek mathematics nor is there any evidence which suggests Greek influence. Nevertheless, some of the results achieved in connection with numerical integration by means of infinite series anticipate developments in Western Europe by several centuries.

The invention of the place-value system is assigned by some authorities ${ }^{\dagger}$ to the first century B.C. and its use appears to have become fairly widespread by A.D. 700. The Pythagorean problem of incommensurable numbers was either unknown or not taken seriously and it was therefore possible to look forward with increasing confidence to the perfection of computational methods and the calculation of numerical magnitudes with ever-increasing accuracy.

In arithmetic zero ranked as a number and operations with zero were defined by Brahmagupta $\ddagger(628)$ as follows:
$$
a-a=0, \quad a+0=a, \quad a-0=a, \quad 0 . a=0, \quad a \cdot b=0 .
$$

数学代写|微积分代写Calculus代写|ThE ARABS

The original Arab contribution was more important in optics and perspective than in pure mathematics. Advances made in trigonometry arose in connection with observational astronomy and had no immediate impact on math

ematics as such. Nevertheless, although the Arabs had no special feeling for the rigorous methods of the Greek geometers they understood the Exhaustion proofs in Euclid’s Elements and knew how to use an argument by reductio ad absurdum.

Ibn-al-Haitham ${ }^{\dagger}$ (Alhazen) in particular made striking advances in applying such methods in the calculation of the volumes of solids of revolution. Whereas Archimedes had concerned himself in the case of the parabola with rotation about the axis only, Ibn-al-Haitham extended and developed this work by considering the volumes of solids formed by the rotation of parabolic segments about lines other than the axis.

If $O$ be any point on a parabola, vertex $V$ and focus $F$, then $O X$, drawn parallel to $V F$, is a diameter and $X P$, drawn parallel to the tangent $D Y$ to meet the parabola at $P$, is an ordinate (see Fig. 2.3). By a standard theorem in geometrical conics, $X P^2=4 O F . O X$, so that the equation of the parabola referred to oblique axes $O X$ and $O Y$ can in general be written in the form $y^2=k x$. In the special case where $O$ is the vertex the axes are rectangular. Ibn-al-Haitham makes use of this relation to determine the volumes of solids formed by rotating parabolic segments about any diameter or ordinate.

微积分代考

数学代写|微积分代写Calculus代写|ON Hindu MathemÁtics

早期印度数学的历史总是给西方带来相当大的问题。19世纪的研究，虽然在某些情况下足以令人吃惊，引起兴趣，但往往是由西方的梵文学者进行的，他们的数学知识不够渊博，除了记录结果之外，还不能做更多的事情。尽管近年来印度数学家进行了大量的研究，填补了我们知识上的一些空白，但仍然不可能对印度数学的方法或动机形成一个清晰的图景。尽管如此，当收集到足够的材料形成一个连贯的模式时，即使是简单的结果排序也会变得有意义。

印度人似乎被数学的计算方面所吸引，一部分是为了它本身，另一部分是作为占星预测的工具。没有发现欧几里得为希腊数学建立的那种证明结构的痕迹，也没有任何证据表明希腊的影响。然而，用无穷级数方法进行数值积分所取得的一些成果，却比西欧的发展早了几个世纪。

位置价值系统的发明被一些权威人士认为是在公元前1世纪${ }^{\dagger}$，到公元700年，它的使用似乎已经相当广泛。毕达哥拉斯关于不可通约数的问题要么是未知的，要么是没有被认真对待的，因此，人们有可能满怀信心地期待计算方法的完善，以及对数值大小的计算越来越精确。

在算术中，0被列为一个数字，Brahmagupta $\ddagger(628)$定义了与零相关的操作如下:
$$
a-a=0, \quad a+0=a, \quad a-0=a, \quad 0 . a=0, \quad a \cdot b=0 .
$$

数学代写|微积分代写Calculus代写|ThE ARABS

阿拉伯人最初的贡献在光学和透视学方面比在纯数学方面更重要。三角学的进步与观测天文学有关，对数学没有直接影响

Ematics就是这样的。然而，尽管阿拉伯人对希腊几何学家的严谨方法没有特别的感情，他们却理解欧几里得《几何原》中的竭竭证明，知道如何使用还原法和反证法进行论证。

特别是Ibn-al-Haitham ${ }^{\dagger}$ (Alhazen)在应用这种方法计算旋转固体体积方面取得了惊人的进展。阿基米德只关注抛物线绕轴旋转的情况，而海瑟姆则扩展并发展了这一工作，他考虑了抛物线绕轴以外的直线旋转所形成的固体体积。

如果$O$是抛物线上的任何一点，顶点$V$和焦点$F$，则平行于$V F$的$O X$是直径，平行于切线$D Y$与抛物线相交$P$的$X P$是纵坐标(见图2.3)。利用几何二次曲线中的一个标准定理$X P^2=4 O F . O X$，使得关于斜轴$O X$和$O Y$的抛物线方程一般可以写成$y^2=k x$的形式。在特殊情况下$O$是顶点坐标轴是矩形的。Ibn-al-Haitham利用这种关系来确定旋转任何直径或纵坐标的抛物线段形成的固体的体积。

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线性代数代写

线性代数是数学的一个分支，涉及线性方程，如：线性图，如：以及它们在向量空间和通过矩阵的表示。线性代数是几乎所有数学领域的核心。

博弈论代写

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计量经济学代写

Matlab代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|微积分代写Calculus代写|MATH1051

Posted on 2023年8月12日2023年8月28日 by statistics-lab

数学代写|微积分代写Calculus代写|Early Greex Mathematics and the Physical World

Early Egyptian mathematics consisted for the most part of empirical rules for performing operations on concrete objects by means of numerical processes, and generalisations which emerged seem to have resulted from the simple extension of a rule found to be valid in a number of special cases to all others. Babylonian texts, on the other hand, exhibit a much greater degree of generalisation but unfortunately we know little of the manner in which these generalisations were arrived at. With the Greeks, however, we find for the first time the idea of a general proof, valid in all cases. The developing concept of mathematics as a deductive system linked, at least in its initial assumptions and in its final conclusions with the physical world, led to increasing speculation as to the relation between the abstract elements of such a system and the concrete objects, shapes and forms belonging to the externai world. Much thought was given as to the fundamental basis of knowledge and to the processes of abstraction, idealisation, generalisation and deduction through which, in one way or another, such knowledge appeared to be derived.

The only physical science with any substantial basis of observation at this time was astronomy and, since in this case the objects observed were remote and their constitution unknown, it was natural that abstract and concrete elements should be regarded as interchangeable. ${ }^{\dagger}$

By a process of abstraction the early Pythagoreans developed the fruitfulconcept of figurate numbers, i.e. numbers made up of discrete elements arranged in geometric forms. $\ddagger$ This concept was then projected back upon the physical universe and its construction explained in terms of number. The discovery of the relation between musical harmony and the theory of proportion, attributed to Pythagoras himself, reinforced this view of the structure of the universe. The physical line segments and triangles of Pythagorean geometry could similarly be considered to be made up of discrete numerical elements until it was discovered that the diagonal of the square was incommensurable with the side. In this case, if the side be constructed of a finite number of discrete elements, how can the hypotenuse be constructed $?^{\dagger}$

Through geometry the idea of finite indivisible elements found a link with the materialist doctrine of physical atomism $\ddagger$ which arose at Abdera (ca. 430 B.C.). According to the Atomists, mind and soul as well as physical objects were considered to be made up of indivisible particles moving in the void. The belief that Democritus of Abdera also made use of the concept of indivisibles in volumetric determinations is confirmed by certain comments made by Archimedes in his treatise The Method. In particular, he attributes to Democritus, some fifty years before the formal proofs of Eudoxus, the discovery of the relationships between the volumes of a cone and a pyramid and those of a cylinder and a prism respectively.’ Of interest in this context is Plutarch’s account” of the queries said to have been raised by Democritus in connection with the sections of a cone cut by a plane parallel to the base.

数学代写|微积分代写Calculus代写|The Axiomatisation of Greek Mathematics

The final axiomatic structure of Greek mathematics was laid down in the third century B.C. and has been available for detailed scrutiny for over 2000 years in the Elements of Euclid, the treatises of Archimedes and the Conics of Apollonius.

It is never easy to establish with any degree of certitude the relationship between logic and philosophy on the one hand, and the structure of mathematical and scientific systems on the other. For this reason alone any attempt to assess the influence of the philosophy of Plato and the logic of Aristotle on the work of systematisation undertaken by Euclid and culminating in the thirteen books of the Elements is fraught with difficulty. Between the logic of Aristotle and the geometry of Euclid, at any rate, the interaction seems to have been mutual, for the impact of mathematics on Aristotle’s thought is only paralleled by the respect for formal logic displayed by Euclid in the Elements. This monumental work rapidly established itself as one of the supreme models of rigorous reasoning and has influenced all views on the nature of mathematical form and structure to the present day. The axiomatic or postulational approach laid down by Euclid was extended and developed in the field of geometry by Archimedes who not only showed himself a supreme master of the Euclidean method but made substantial and original advances by establishing the study of the physical sciences of statics and hydrostatics on a similar axiomatic basis.

In the Elements $\ddagger$ Euclid unified a collection of isolated discoveries and theorems into a single deductive system based upon a set of initial postulates, definitions and axioms. The definitions, although in some respects obscure and unhelpful, nevertheless suffice to identify the points, lines, planes and circles of physico-geometric experience and the postulates describe the technical requirements for their construction: the axioms (or common notions) represent rules of logic by means of which consequences may be deduced from the postulates. Although both Euclid and Archimedes make use of a great many logical processes not listed among Euclid’s axioms, those which are given constitute a satisfactory definition of equivalence of measure, a notion of fundamental significance in the construction of any formal deductive system.

微积分代考

数学代写|微积分代写Calculus代写|Early Greex Mathematics and the Physical World

早期的埃及数学大部分由经验法则组成，这些法则是通过数值过程对具体物体进行运算的，而普遍化似乎是由在一些特殊情况下有效的规则的简单推广而产生的。另一方面，巴比伦文本表现出更大程度的概括，但不幸的是，我们对这些概括的方式知之甚少。然而，在希腊人那里，我们第一次发现一般证明的观念，它在一切情况下都是有效的。数学作为一个演绎系统，至少在其最初的假设和最终的结论中，与物理世界联系在一起，这一概念的发展，导致人们越来越多地思考这种系统的抽象元素与属于外部世界的具体对象、形状和形式之间的关系。对于知识的基本基础，对于抽象、理想化、概括、演绎等过程，人们都作了大量的思考。

当时唯一有观测基础的自然科学是天文学，由于观测到的物体很遥远，它们的构成也不为人所知，所以很自然地，抽象元素和具体元素应该被认为是可以互换的。${} ^{\匕首}$

通过一个抽象的过程，早期的毕达哥拉斯学派发展了富有成果的形数概念，即由以几何形式排列的离散元素组成的数。这个概念随后被投射到物质宇宙上，它的结构用数字来解释。毕达哥拉斯发现了音乐和谐与比例理论之间的关系，这一发现强化了他对宇宙结构的看法。毕达哥拉斯几何中的物理线段和三角形同样可以被认为是由离散的数值元素组成的，直到人们发现正方形的对角线与边是不可通约的。在这种情况下，如果边是由有限个离散单元构成的，那么斜边是如何构成的呢?^{\匕首}$

通过几何学，有限的不可分割的元素的观念与在Abdera(约公元前430年)产生的物理原子论的唯物主义学说联系在一起。根据原子论者的观点，思想和灵魂以及物质对象都被认为是由在虚空中运动的不可分割的粒子组成的。阿基米德在他的论文《方法》中所作的某些评论证实了阿比德拉的德谟克利特在体积测定中也使用了不可分的概念。他特别指出，在欧多克索斯正式证明之前大约50年，德谟克利特发现了锥体和金字塔的体积与圆柱体和棱镜的体积之间的关系。在这种背景下，普鲁塔克对德谟克利特提出的问题的描述令人感兴趣，这些问题与平行于底座的平面切割的锥体部分有关。

数学代写|微积分代写Calculus代写|The Axiomatisation of Greek Mathematics

希腊数学的最终公理结构是在公元前3世纪确立的，并在2000多年前的欧几里得的《数学原理》、阿基米德的论文和阿波罗尼乌斯的《圆锥论》中得到了详细的研究。

一方面是逻辑和哲学，另一方面是数学和科学体系的结构，要确定两者之间的关系，从来都不是一件容易的事。仅仅因为这个原因，任何评估柏拉图的哲学和亚里士多德的逻辑对欧几里得的系统化工作的影响的尝试都充满了困难，欧几里得的系统化工作最终以13本《自然要素》告终。在亚里士多德的逻辑学和欧几里得的几何学之间，无论如何，似乎是相互作用的，因为数学对亚里士多德思想的影响，只有欧几里得在《几何原理》中对形式逻辑的尊重才能与之相提并论。这部巨著迅速确立了它作为严格推理的最高模型之一的地位，并影响了所有关于数学形式和结构本质的观点，直到今天。欧几里得提出的公理化或公设方法在几何领域得到了阿基米德的扩展和发展，他不仅表明自己是欧几里得方法的最高大师，而且通过在类似的公理化基础上建立静力学和流体静力学的物理科学研究，取得了实质性的原创进展。

在《数学基本原理》中，欧几里得将一系列孤立的发现和定理统一成一个基于一组初始公设、定义和公理的单一演绎系统。这些定义，虽然在某些方面模糊不清，毫无帮助，但足以识别物理几何经验的点、线、面和圆，以及公设描述了它们的构造的技术要求:公理(或共同概念)代表了逻辑规则，通过这些规则可以从公设中推导出结果。虽然欧几里得和阿基米德都使用了许多欧几里得公理中没有列出的逻辑过程，但这些过程构成了一个令人满意的度量等价的定义，这是一个在任何形式演绎系统的构造中具有根本意义的概念。

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微观经济学代写

线性代数代写

线性代数是数学的一个分支，涉及线性方程，如：线性图，如：以及它们在向量空间和通过矩阵的表示。线性代数是几乎所有数学领域的核心。

博弈论代写

微积分代写

计量经济学代写

Matlab代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|微积分代写Calculus代写|MTH-211

Posted on 2023年7月28日2023年8月25日 by statistics-lab

数学代写|微积分代写Calculus代写|Trigonometric Substitutions

Trigonometric substitutions occur when we replace the variable of integration by a trigonometric function. The most common substitutions are $x=a \tan \theta, x=a \sin \theta$, and $x=a \sec \theta$. These substitutions are effective in transforming integrals involving $\sqrt{a^2+x^2}, \sqrt{a^2-x^2}$, and $\sqrt{x^2-a^2}$ into integrals we can evaluate directly since they come from the reference right triangles in Figure 8.2.

With $x=a \tan \theta$,
$$
a^2+x^2=a^2+a^2 \tan ^2 \theta=a^2\left(1+\tan ^2 \theta\right)=a^2 \sec ^2 \theta .
$$
With $x=a \sin \theta$,
$$
a^2-x^2=a^2-a^2 \sin ^2 \theta=a^2\left(1-\sin ^2 \theta\right)=a^2 \cos ^2 \theta .
$$
With $x=a \sec \theta$,
$$
x^2-a^2=a^2 \sec ^2 \theta-a^2=a^2\left(\sec ^2 \theta-1\right)=a^2 \tan ^2 \theta .
$$
We want any substitution we use in an integration to be reversible so that we can change back to the original variable afterward. For example, if $x=a \tan \theta$, we want to be able to set $\theta=\tan ^{-1}(x / a)$ after the integration takes place. If $x=a \sin \theta$, we want to be able to set $\theta=\sin ^{-1}(x / a)$ when we’re done, and similarly for $x=a \sec \theta$.

As we know from Section 7.6, the functions in these substitutions have inverses only for selected values of $\theta$ (Figure 8.3). For reversibility,
$$
\begin{aligned}
& x=a \tan \theta \text { requires } \theta=\tan ^{-1}\left(\frac{x}{a}\right) \text { with }-\frac{\pi}{2}<\theta<\frac{\pi}{2}, \
& x=a \sin \theta \text { requires } \theta=\sin ^{-1}\left(\frac{x}{a}\right) \text { with }-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}, \
& x=a \sec \theta \text { requires } \theta=\sec ^{-1}\left(\frac{x}{a}\right) \text { with }\left{\begin{array}{l}
0 \leq \theta<\frac{\pi}{2} \text { if } \frac{x}{a} \geq 1, \
\frac{\pi}{2}<\theta \leq \pi \text { if } \frac{x}{a} \leq-1 .
\end{array}\right.
\end{aligned}
$$

数学代写|微积分代写Calculus代写|Integration of Rational Functions by Partial Fractions

This section shows how to express a rational function (a quotient of polynomials) as a sum of simpler fractions, called partial fractions, which are easily integrated. For instance, the rational function $(5 x-3) /\left(x^2-2 x-3\right)$ can be rewritten as
$$
\frac{5 x-3}{x^2-2 x-3}=\frac{2}{x+1}+\frac{3}{x-3} \text {. }
$$
You can verify this equation algebraically by placing the fractions on the right side over a common denominator $(x+1)(x-3)$. The skill acquired in writing rational functions as such a sum is useful in other settings as well (for instance, when using certain transform methods to solve differential equations). To integrate the rational function $(5 x-3) /\left(x^2-2 x-3\right)$ on the left side of our previous expression, we simply sum the integrals of the fractions on the right side:
$$
\begin{aligned}
\int \frac{5 x-3}{(x+1)(x-3)} d x & =\int \frac{2}{x+1} d x+\int \frac{3}{x-3} d x \
& =2 \ln |x+1|+3 \ln |x-3|+C .
\end{aligned}
$$
The method for rewriting rational functions as a sum of simpler fractions is called the method of partial fractions. In the case of the preceding example, it consists of finding constants $A$ and $B$ such that
$$
\frac{5 x-3}{x^2-2 x-3}=\frac{A}{x+1}+\frac{B}{x-3} .
$$
(Pretend for a moment that we do not know that $A=2$ and $B=3$ will work.) We call the fractions $A /(x+1)$ and $B /(x-3)$ partial fractions because their denominators are only part of the original denominator $x^2-2 x-3$. We call $A$ and $B$ undetermined coefficients until suitable values for them have been found.

微积分代考

数学代写|微积分代写Calculus代写|Trigonometric Substitutions

三角函数替换是指用三角函数替换积分变量。最常见的替换是$x=a \tan \theta, x=a \sin \theta$和$x=a \sec \theta$。这些替换在将涉及$\sqrt{a^2+x^2}, \sqrt{a^2-x^2}$和$\sqrt{x^2-a^2}$的积分转换为我们可以直接计算的积分时是有效的，因为它们来自图8.2中的参考直角三角形。

通过$x=a \tan \theta$，
$$
a^2+x^2=a^2+a^2 \tan ^2 \theta=a^2\left(1+\tan ^2 \theta\right)=a^2 \sec ^2 \theta .
$$
通过$x=a \sin \theta$，
$$
a^2-x^2=a^2-a^2 \sin ^2 \theta=a^2\left(1-\sin ^2 \theta\right)=a^2 \cos ^2 \theta .
$$
通过$x=a \sec \theta$，
$$
x^2-a^2=a^2 \sec ^2 \theta-a^2=a^2\left(\sec ^2 \theta-1\right)=a^2 \tan ^2 \theta .
$$
我们希望我们在积分中使用的任何替换都是可逆的，这样我们就可以在之后把它换回原来的变量。例如，如果是$x=a \tan \theta$，我们希望能够在集成发生后设置$\theta=\tan ^{-1}(x / a)$。对于$x=a \sin \theta$，我们希望在完成后能够设置$\theta=\sin ^{-1}(x / a)$，对于$x=a \sec \theta$也是如此。

正如我们在7.6节中所知道的，这些替换中的函数只对$\theta$的选定值有逆(图8.3)。对于可逆性，
$$
\begin{aligned}
& x=a \tan \theta \text { requires } \theta=\tan ^{-1}\left(\frac{x}{a}\right) \text { with }-\frac{\pi}{2}<\theta<\frac{\pi}{2}, \
& x=a \sin \theta \text { requires } \theta=\sin ^{-1}\left(\frac{x}{a}\right) \text { with }-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}, \
& x=a \sec \theta \text { requires } \theta=\sec ^{-1}\left(\frac{x}{a}\right) \text { with }\left{\begin{array}{l}
0 \leq \theta<\frac{\pi}{2} \text { if } \frac{x}{a} \geq 1, \
\frac{\pi}{2}<\theta \leq \pi \text { if } \frac{x}{a} \leq-1 .
\end{array}\right.
\end{aligned}
$$

数学代写|微积分代写Calculus代写|Integration of Rational Functions by Partial Fractions

本节展示如何将有理函数(多项式的商)表示为易于积分的简单分数(称为部分分数)的和。例如，有理函数$(5 x-3) /\left(x^2-2 x-3\right)$可以重写为
$$
\frac{5 x-3}{x^2-2 x-3}=\frac{2}{x+1}+\frac{3}{x-3} \text {. }
$$
你可以用代数方法验证这个等式，把分数放在右边的公分母$(x+1)(x-3)$上。将有理函数写成这样的和的技巧在其他情况下也很有用(例如，当使用某些变换方法来解微分方程时)。要对前面表达式左边的有理函数$(5 x-3) /\left(x^2-2 x-3\right)$积分，我们只需对右边分数的积分求和:
$$
\begin{aligned}
\int \frac{5 x-3}{(x+1)(x-3)} d x & =\int \frac{2}{x+1} d x+\int \frac{3}{x-3} d x \
& =2 \ln |x+1|+3 \ln |x-3|+C .
\end{aligned}
$$
把有理函数写成简单分数和的方法叫做部分分数法。在前面的例子中，它包括查找常量$A$和$B$，以便
$$
\frac{5 x-3}{x^2-2 x-3}=\frac{A}{x+1}+\frac{B}{x-3} .
$$
(暂时假设我们不知道$A=2$和$B=3$会正常工作。)我们称分数为$A /(x+1)$和$B /(x-3)$部分分数因为它们的分母只是原分母$x^2-2 x-3$的一部分。我们称$A$和$B$为待定系数，直到找到适合它们的值。

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数学代写|微积分代写Calculus代写|MTH251

Posted on 2023年7月28日2023年8月25日 by statistics-lab

数学代写|微积分代写Calculus代写|Inverse Hyperbolic Functions

The inverses of the six basic hyperbolic functions are very useful in integration . Since $d(\sinh x) / d x=\cosh x>0$, the hyperbolic sine is an increasing function of $x$. We denote its inverse by
$$
y=\sinh ^{-1} x .
$$
For every value of $x$ in the interval $-\infty<x<\infty$, the value of $y=\sinh ^{-1} x$ is the number whose hyperbolic sine is $x$. The graphs of $y=\sinh x$ and $y=\sinh ^{-1} x$ are shown in Figure 7.32a.

The function $y=\cosh x$ is not one-to-one because its graph in Table 7.5 does not pass the horizontal line test. The restricted function $y=\cosh x, x \geq 0$, however, is oneto-one and therefore has an inverse, denoted by
$$
y=\cosh ^{-1} x .
$$

For every value of $x \geq 1, y=\cosh ^{-1} x$ is the number in the interval $0 \leq y<\infty$ whose hyperbolic cosine is $x$. The graphs of $y=\cosh x, x \geq 0$, and $y=\cosh ^{-1} x$ are shown in Figure 7.32b.

Like $y=\cosh x$, the function $y=\operatorname{sech} x=1 / \cosh x$ fails to be one-to-one, but its restriction to nonnegative values of $x$ does have an inverse, denoted by
$$
y=\operatorname{sech}^{-1} x .
$$
For every value of $x$ in the interval $(0,1], y=\operatorname{sech}^{-1} x$ is the nonnegative number whose hyperbolic secant is $x$. The graphs of $y=\operatorname{sech} x, x \geq 0$, and $y=\operatorname{sech}^{-1} x$ are shown in Figure 7.32c.

The hyperbolic tangent, cotangent, and cosecant are one-to-one on their domains and therefore have inverses, denoted by
$$
y=\tanh ^{-1} x, \quad y=\operatorname{coth}^{-1} x, \quad y=\operatorname{csch}^{-1} x .
$$
These functions are graphed in Figure 7.33.

数学代写|微积分代写Calculus代写|Useful Identities

We use the identities in Table 7.9 to calculate the values of $\operatorname{sech}^{-1} x, \operatorname{csch}^{-1} x$, and $\operatorname{coth}^{-1} x$ on calculators that give only $\cosh ^{-1} x, \sinh ^{-1} x$, and $\tanh ^{-1} x$. These identities are direct consequences of the definitions. For example, if $0<x \leq 1$, then
$$
\operatorname{sech}\left(\cosh ^{-1}\left(\frac{1}{x}\right)\right)=\frac{1}{\cosh \left(\cosh ^{-1}\left(\frac{1}{x}\right)\right)}=\frac{1}{\left(\frac{1}{x}\right)}=x .
$$
We also know that $\operatorname{sech}\left(\operatorname{sech}^{-1} x\right)=x$, so because the hyperbolic secant is one-to-one on $(0,1]$, we have
$$
\cosh ^{-1}\left(\frac{1}{x}\right)=\operatorname{sech}^{-1} x
$$

微积分代考

数学代写|微积分代写Calculus代写|Inverse Hyperbolic Functions

六个基本双曲函数的逆在积分中非常有用。由于$d(\sinh x) / d x=\cosh x>0$，双曲正弦函数是$x$的递增函数。我们用
$$
y=\sinh ^{-1} x .
$$
对于区间$-\infty<x<\infty$中每个$x$的值，$y=\sinh ^{-1} x$的值是双曲正弦值为$x$的数。$y=\sinh x$和$y=\sinh ^{-1} x$的曲线图如图7.32a所示。

函数$y=\cosh x$不是一对一的，因为它在表7.5中的图形没有通过水平线检验。然而，受限函数$y=\cosh x, x \geq 0$是一对一的，因此有一个逆，表示为
$$
y=\cosh ^{-1} x .
$$

对于$x \geq 1, y=\cosh ^{-1} x$的每个值，表示在$0 \leq y<\infty$区间内的双曲余弦值为$x$的数。$y=\cosh x, x \geq 0$和$y=\cosh ^{-1} x$的曲线图如图7.32b所示。

像$y=\cosh x$一样，函数$y=\operatorname{sech} x=1 / \cosh x$不是一对一的，但是它对$x$的非负值的限制确实有一个逆，表示为
$$
y=\operatorname{sech}^{-1} x .
$$
对于区间内$x$的每一个值$(0,1], y=\operatorname{sech}^{-1} x$都是其双曲正割为$x$的非负数。$y=\operatorname{sech} x, x \geq 0$和$y=\operatorname{sech}^{-1} x$的图形如图7.32c所示。

双曲正切，余切和余割在它们的定义域上是一对一的，因此有逆，表示为
$$
y=\tanh ^{-1} x, \quad y=\operatorname{coth}^{-1} x, \quad y=\operatorname{csch}^{-1} x .
$$
这些函数如图7.33所示。

数学代写|微积分代写Calculus代写|Useful Identities

我们使用表7.9中的恒等式在只给出$\cosh ^{-1} x, \sinh ^{-1} x$和$\tanh ^{-1} x$的计算器上计算$\operatorname{sech}^{-1} x, \operatorname{csch}^{-1} x$和$\operatorname{coth}^{-1} x$的值。这些恒等式是定义的直接结果。例如，如果$0<x \leq 1$，则
$$
\operatorname{sech}\left(\cosh ^{-1}\left(\frac{1}{x}\right)\right)=\frac{1}{\cosh \left(\cosh ^{-1}\left(\frac{1}{x}\right)\right)}=\frac{1}{\left(\frac{1}{x}\right)}=x .
$$
我们还知道$\operatorname{sech}\left(\operatorname{sech}^{-1} x\right)=x$，因为双曲正割在$(0,1]$上是一对一的，我们有
$$
\cosh ^{-1}\left(\frac{1}{x}\right)=\operatorname{sech}^{-1} x
$$

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数学代写|微积分代写Calculus代写|MATH2310

Posted on 2023年7月28日2023年8月25日 by statistics-lab

数学代写|微积分代写Calculus代写|Tricky Trig Integrals

In this section, you integrate powers of the six trigonometric functions, like $\int \sin ^3(x) d x$ and $\int \sec ^4(x) d x$, and products or quotients of trig functions, like $\int \sin ^2(x) \cos ^3(x) d x$ and $\int \frac{\csc ^2(x)}{\cot (x)} d x$. This is a bit tedious – time for some caffeine. To use the following techniques, you must have an integrand that contains just one of the six trig functions like $\int \csc ^3(x) d x$ or a certain pairing of trig functions, like $\int \sin ^2(x) \cos (x) d x$. If the integrand has two trig functions, the two must be one of these three pairs: sine with cosine, secant with tangent, or cosecant with cotangent. For an integrand containing something other than one of these pairs, you can convert the problem into one of these pairs by using trig identities like $\sin (x)=1 / \csc (x)$ and $\tan (x)=\sin (x) / \cos (x)$. For instance,
$$
\begin{aligned}
& \int \sin ^2(x) \sec (x) \tan (x) d x \
= & \int \sin ^2(x) \frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} d x \
= & \int \frac{\sin ^3(x)}{\cos ^2(x)} d x
\end{aligned}
$$
After any needed conversions, you get one of three cases:
$$
\begin{aligned}
& \int \sin ^m(x) \cos ^n(x) d x \
& \int \sec ^m(x) \tan ^n(x) d x \
& \int \csc ^m(x) \cot ^n(x) d x
\end{aligned}
$$
where either $m$ or $n$ is a positive integer.

数学代写|微积分代写Calculus代写|Sines and cosines

Case 1: The power of sine is odd and positive
If the power of sine is odd and positive, lop off one sine factor and put it to the right of the rest of the expression, convert the remaining sine factors to cosines with the Pythagorean identity, and then integrate with the substitution method where $u=\cos (x)$.
The Pythagorean identity tells you that, for any angle $x, \sin ^2(x)+$ $\cos ^2(x)=1$. And thus $\sin ^2(x)=1-\cos ^2(x)$ and $\cos ^2(x)=1-$ $\sin ^2(x)$.
Now integrate $\int \sin ^3(x) \cos ^4(x) d x$.

Lop off one sine factor and move it to the right.
$$
\int \sin ^3(x) \cos ^4(x) d x=\int \sin ^2(x) \cos ^4(x) \sin (x) d x
$$
Convert the remaining sines to cosines using the Pythagorean identity and simplify.
$$
\begin{aligned}
& \int \sin ^2(x) \cos ^4(x) \sin (x) d x \
= & \int\left(1-\cos ^2(x)\right) \cos ^4(x) \sin (x) d x \
= & \int\left(\cos ^4(x)-\cos ^6(x)\right) \sin (x) d x
\end{aligned}
$$
Integrate with substitution, where $u=\cos (x)$.
$$
\begin{gathered}
u=\cos (x) \
\frac{d u}{d x}=-\sin (x) \
d u=-\sin (x) d x
\end{gathered}
$$

微积分代考

数学代写|微积分代写Calculus代写|Tricky Trig Integrals

在本节中，将对六个三角函数(如$\int \sin ^3(x) d x$和$\int \sec ^4(x) d x$)的幂和三角函数(如$\int \sin ^2(x) \cos ^3(x) d x$和$\int \frac{\csc ^2(x)}{\cot (x)} d x$)的积或商进行积分。这有点乏味——是时候喝点咖啡因了。要使用下面的技巧，必须有一个只包含六个三角函数中的一个的被积函数，如$\int \csc ^3(x) d x$或某些三角函数对，如$\int \sin ^2(x) \cos (x) d x$。如果被积函数有两个三角函数，这两个三角函数必须是以下三对中的一个:正弦与余弦，正割与正切，或余割与余切。对于包含这些对之外的东西的被积函数，您可以通过使用像$\sin (x)=1 / \csc (x)$和$\tan (x)=\sin (x) / \cos (x)$这样的三角恒等式将问题转换为这些对中的一个。例如，
$$
\begin{aligned}
& \int \sin ^2(x) \sec (x) \tan (x) d x \
= & \int \sin ^2(x) \frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} d x \
= & \int \frac{\sin ^3(x)}{\cos ^2(x)} d x
\end{aligned}
$$
在任何必要的转换之后，您将得到以下三种情况之一:
$$
\begin{aligned}
& \int \sin ^m(x) \cos ^n(x) d x \
& \int \sec ^m(x) \tan ^n(x) d x \
& \int \csc ^m(x) \cot ^n(x) d x
\end{aligned}
$$
其中$m$或$n$为正整数。

数学代写|微积分代写Calculus代写|Sines and cosines

情形1:sin的幂是奇数且正的
如果sin的幂是奇数且正的，则去掉一个sin因子并将其放在表达式的其余部分的右侧，用勾股定理将剩余的sin因子转换为余弦，然后用代换法积分$u=\cos (x)$。
勾股定理告诉你，对于任意角度$x, \sin ^2(x)+$$\cos ^2(x)=1$。因此是$\sin ^2(x)=1-\cos ^2(x)$和$\cos ^2(x)=1-$$\sin ^2(x)$。
现在积分$\int \sin ^3(x) \cos ^4(x) d x$。

去掉一个正弦因子，向右移动。
$$
\int \sin ^3(x) \cos ^4(x) d x=\int \sin ^2(x) \cos ^4(x) \sin (x) d x
$$

用勾股定理把剩下的正弦转换成余弦化简。
$$
\begin{aligned}
& \int \sin ^2(x) \cos ^4(x) \sin (x) d x \
= & \int\left(1-\cos ^2(x)\right) \cos ^4(x) \sin (x) d x \
= & \int\left(\cos ^4(x)-\cos ^6(x)\right) \sin (x) d x
\end{aligned}
$$

用代换法积分，其中$u=\cos (x)$。
$$
\begin{gathered}
u=\cos (x) \
\frac{d u}{d x}=-\sin (x) \
d u=-\sin (x) d x
\end{gathered}
$$

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