## 数学代写|微积分代写Calculus代写|MAST10006

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。

• 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|微积分代写Calculus代写|Optimization example: maximum profit

Recall that for the revenue, cost, and profit functions from economics, marginal means “derivative.” Recall also that profit is revenue minus cost: $$P(x)=R(x)-C(x) .$$
If we wish to maximize profit, then we need to find the critical numbers of the profit function-that is, where $P^{\prime}(x)=0$ :
\begin{aligned} P^{\prime}(x) & =R^{\prime}(x)-C^{\prime}(x) \ R^{\prime}(x)-C^{\prime}(x) & =0 \ R^{\prime}(x) & =C^{\prime}(x) \end{aligned}
In other words, the critical numbers of the profit function occur where marginal revenue equals marginal cost. The traditional solution method for profit maximization problems is to equate marginal revenue and marginal cost. Because we also wish to check to ensure that profit is maximized rather than minimized, we still form the profit function and determine its maximum.

Example 4 Each month we can sell as many widgets as we can make for $\$ 12$each. The cost, in dollars, of making$x$widgets is given by $$C(x)=10000+7 x-0.002 x^2+\frac{1}{3} \cdot 10^{-6} x^3 .$$ How many widgets should we make to maximize profit? Solution Because we wish to maximize profit, this is an optimization problem. 1)-2 Notice that there is nothing geometric about this problem. No picture seems applicable, so we don’t draw one. Furthermore, the relevant variable has already been introduced;$x$is the number of widgets made in 1 month. ## 数学代写|微积分代写Calculus代写|Optimization example: minimum material Solution First we recognize this as an optimization problem because we are asked to minimize the cost. (1) We draw a picture of a utility on the bank of a straight river, with a manufacturer on the opposite side of the river but downstream. We also label the width of the river$(900 \mathrm{~m})$and the downstream distance to the manufacturer$(3000 \mathrm{~m})$. See figure 9. Visualizing different possibilities, we see the pipeline could go straight to the opposite shore to have the least amount of pipe under water (figure 10 , top). The pipeline could also go directly to the manufacturer, remaining under water the entire route, to have the least total amount of pipe (figure 10 , bottom). But we are not asked to minimize the amount of pipe under water or minimize the total length of pipe; instead, we are asked to minimize cost. It seems as if the least cost prompts us to follow a route like that in figure 9. (2) The variable amount in figure 9 appears to be the spot at which the pipe emerges from the river, which is in fact what we are asked for. Let’s let$x$represent the distance downstream from the utility at which the pipe emerges, and label this distance in the diagram; see figure 11 . We can then determine and label other lengths as well. The length of the pipe along the shore is$(3000-x) \mathrm{m}$, whereas the “vertical leg” of the right triangle is the width of the river,$900 \mathrm{~m}$. The length of the hypotenuse can be found using the Pythagorean theorem:$c^2=900^2+x^2$, or$c=\sqrt{900^2+x^2}$. These are labeled in figure 12 . (3) The quantity we are asked to optimize (minimize in this case) is the cost of the pipeline. The cost of the pipeline includes the cost of running pipe under the water and the cost of running pipe along the shoreline. Under water, the pipeline cost is$\$200 / \mathrm{m}$, and from the diagram we see that the length of pipe under the water is $\sqrt{900^2+x^2} \mathrm{~m}$. Therefore, the cost of the pipe under the water is
$$200 \sqrt{900^2+x^2}$$

Example 5 A cylindrical can must have volume $100 \mathrm{~cm}^3$. What dimensions should be used to minimize the amount of material used?

Solution We notice the phrase “minimize the amount of material used” and conclude that this is an optimization problem.
(1)-2) We are told the can is cylindrical, so we draw a cylindrical can (figure 13). We are asked for the dimensions to use, which include the can’s height and radius, so we visualize various possible shapes, such as tall and thin or short and wide (figure 14).
(3) We wish to minimize the amount of material used to make the can. The material of the can includes the top, bottom, and side of the can. Assuming a uniform thickness of the material, the material used is proportional to the surface area of the can. The formula for the surface area (SA) of a cylinder is
$$S A=2 \pi r h+2 \pi r^2 .$$

# 微积分代考

## 数学代写|微积分代写Calculus代写|Optimization example: maximum profit

$$P(x)=R(x)-C(x) .$$

$$P^{\prime}(x)=R^{\prime}(x)-C^{\prime}(x) R^{\prime}(x)-C^{\prime}(x) \quad=0 R^{\prime}(x)=C^{\prime}(x)$$

$$200 \sqrt{900^2+x^2}$$

(1)-2) 我们被告知罐子是圆柱形的，所以我们画一个圆柱形罐头 (图13)。我们被要求提供要使用的尺 寸，其中包括罐头的高度和半径，因此我们想象出各种可能的形状，例如又高又薄或又短又宽（图
14)。
（3）我们莃望尽量减少制造罐头所用的材料量。罐的材料包括罐的顶部、底部和侧面。假设材料厚度均 匀，则所用材料与罐的表面积成正比。圆柱表面积 (SA) 的公式为
$$S A=2 \pi r h+2 \pi r^2 .$$

## 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。

## 数学代写|微积分代写Calculus代写|MATH141

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。

• 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|微积分代写Calculus代写|Optimization example: maximum volume

Our next classical example involves making a box.
Example 2 An open-top box is to be made from a sheet of cardboard measuring 11 inches by 17 inches (legal size) by cutting squares from the corners and folding up the sides. What size squares should be cut from the corners to maximize the volume of the box?

Solution First we recognize this as an optimization problem because we are asked to “maximize the volume.”
(1) We draw a picture of a rectangular sheet of cardboard with its corners cut out (figure 5.) Visualizing different possibilities, we realize that cutting out small squares results in a wide, long, short box, whereas cutting out large squares results in a tall box that is less wide and long. Although we do not need to draw these diagrams if we can picture them mentally, they are given in figure 6.

(2) The fundamental quantity that seems to vary is the size of square that we cut from the corners. A square has the same length as width, so let’s call that length and width $x$. We then label the (original) diagram using the variable $x$; see figure 7 .
(3) The quantity to be optimized (in this case, maximized) is the volume of the box. The volume of a rectangular box is length times width times height:
$$V=\ell \cdot w \cdot h .$$
Consulting the diagram, we see that after the sides are folded up, the base of the box is the rectangle inside the creases. Thus, the length and width of the box are the lengths of these creases, 17 inches minus $2 x$ inches and 11 inches minus $2 x$ inches. It may be helpful to write these dimensions in the diagram as well; see figure 8 .

## 数学代写|微积分代写Calculus代写|Optimization example: best path

Solution First we recognize this as an optimization problem because we are asked to minimize the cost.
(1) We draw a picture of a utility on the bank of a straight river, with a manufacturer on the opposite side of the river but downstream. We also label the width of the river $(900 \mathrm{~m})$ and the downstream distance to the manufacturer $(3000 \mathrm{~m})$. See figure 9. Visualizing different possibilities, we see the pipeline could go straight to the opposite shore to have the least amount of pipe under water (figure 10 , top). The pipeline could also go directly to the manufacturer, remaining under water the entire route, to have the least total amount of pipe (figure 10 , bottom). But we are not asked to minimize the amount of pipe under water or minimize the total length of pipe; instead, we are asked to minimize cost. It seems as if the least cost prompts us to follow a route like that in figure 9.

(2) The variable amount in figure 9 appears to be the spot at which the pipe emerges from the river, which is in fact what we are asked for. Let’s let $x$ represent the distance downstream from the utility at which the pipe emerges, and label this distance in the diagram; see figure 11 . We can then determine and label other lengths as well. The length of the pipe along the shore is $(3000-x) \mathrm{m}$, whereas the “vertical leg” of the right triangle is the width of the river, $900 \mathrm{~m}$. The length of the hypotenuse can be found using the Pythagorean theorem: $c^2=$ $900^2+x^2$, or $c=\sqrt{900^2+x^2}$. These are labeled in figure 12 .
(3) The quantity we are asked to optimize (minimize in this case) is the cost of the pipeline. The cost of the pipeline includes the cost of running pipe under the water and the cost of running pipe along the shoreline. Under water, the pipeline cost is $\$ 200 / \mathrm{m}$, and from the diagram we see that the length of pipe under the water is$\sqrt{900^2+x^2} \mathrm{~m}$. Therefore, the cost of the pipe under the water is $$200 \sqrt{900^2+x^2}$$ # 微积分代考 ## 数学代写|微积分代写Calculus代写|Optimization example: maximum volume 我们的下一个经典示例涉及制作一个盒子。 示例 2 一个开顶盒将由一张 11 英寸$x 17$英寸（法定尺寸) 的纸板制成，方法是从角上切下正方形并将 边折炟起来。应该从角上切出多大的正方形才能使盒子的体积最大化? 解决方案 首先，我们将此视为优化问题，因为我们被要求“最大化音量”。 (1) 我们画了一张切掉角的长方形纸板（图 5)。可视化不同的可能性，我们意识到切出小方块会产生 宽、长、短的盒子，而切出大方块会产生在一个不太宽和不太长的高盒子里。虽然我们不需要画这些图， 如果我们可以在脑海中描绘它们，但它们在图 6 中给出。 (2) 似乎变化的基本量是我们从角上切出的正方形的大小。正方形的长度和宽度相同，所以我们称其为长 度和宽度$x$. 然后我们使用变量标记 (原始) 图表$x$；见图 7。 (3) 要优化的数量（在本例中为最大化）是盒子的体积。长方体的体积是长乘以宽乘以高： $$V=\ell \cdot w \cdot h .$$ 看图，边折起来后，盒子的底部就是折痕里面的长方形。因此，盒子的长度和宽度就是这些折痕的长度， 减去 17 英寸$2 x$英寸和 11 英寸负$2 x$英寸。将这些维度写在图表中也可能会有所帮助；见图 8。 ## 数学代写|微积分代写Calculus代写|Optimization example: best path 解决方案 首先我们认识到这是一个优化问题，因为我们被要求最小化成本。 (1) 我们在一条笔直的河岸上画了一个公用事业公司的图片，在河的对面下游有一个制造商。我们还标注 了河流的宽度$(900 \mathrm{~m})$以及到制造商的下游距离$(3000 \mathrm{~m})$. 参见图 9。可视化不同的可能性，我们看到 管道可以直接通向对岸，从而使水下管道数量最少（图 10，顶部) 。管道也可以直接通向制造商，在整 个路线中保持在水下，以获得最少的管道总量（图 10，底部）。但我们并没有要求我们尽量减少水下管 道的数量或尽量减少管道的总长度；相反，我们被要求最小化成本。似乎最低成本促使我们遵循图 9 中 的路线。 (2) 图 9 中的变量似乎是管道从河流中露出的位置，这实际上是我们所要求的。让我们让$x$代表公用设施 下游管道出现的距离，并在图中标出该距离；见图 11。然后我们也可以确定和标记其他长度。沿岸管道 的长度为$(3000-x) \mathrm{m}$，而直角三角形的“垂直边”是河流的宽度，$900 \mathrm{~m}$. 可以使用毕达哥拉斯定理找 到斜边的长度:$c^2=900^2+x^2$，要么$c=\sqrt{900^2+x^2}$. 这些在图 12 中进行了标记。 (3) 我们被要求优化的数量 (在这种情况下最小化) 是管道的成本。管道成本包括在水下铺设管道的成本 和沿海岸线铺设管道的成本。在水下，管道成本为$\$200 / \mathrm{m}$ ，从图中我们可以看出水下管道的长度是 $\sqrt{900^2+x^2} \mathrm{~m}$. 因此，水下管道的造价为
$$200 \sqrt{900^2+x^2}$$

## 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。

## 数学代写|微积分代写Calculus代写|MATH1051

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。

• 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|微积分代写Calculus代写|Complete curve-sketching examples

Example 1 provided the information needed to sketch the curve. What if we need to gather this information? The complete process is lengthy but highly informative. The next example illustrates the entire process for a relatively simple function.
Example 2 Sketch the graph of $f(x)=\sqrt[3]{x^2}$.
Solution For any function, a good place to start is to determine its domain. Because $f$ contains only an odd root and no denominator, it is defined on all real numbers.

Our earlier list of information that calculus can provide gives us a list of what to explore. We start with discontinuities. Because root functions are continuous where defined, the function is continuous everywhere; there are no discontinuities.

Next on the list is corners or vertical tangents, which may be identified as part of the process of finding intervals of increase/decrease and extreme points. We defer this item momentarily.

For intervals of increase/decrease, we need to find the derivative and identify critical numbers. Differentiating, we have
$$f^{\prime}(x)=\frac{2}{3} x^{-1 / 3}=\frac{2}{3 \sqrt[3]{x}} .$$
To find critical numbers, we set both the numerator and the denominator equal to zero and solve; $2=0$ has no solutions and $3 \sqrt[3]{x}=0$ has solution $x=0$. The only critical number is $x=0$, and our chart is completed easily:
\begin{tabular}{c|c|c|c}
interval & sign of $f^{\prime}$ & inc/dec & local extrema \
\hline$(-\infty, 0)$ & $-$ & decreasing & $\leftarrow$ local min at $x=0$ \
$(0, \infty)$ & $+$ & increasing &
\end{tabular}
For the purpose of graphing, we need the extreme points and not just their locations, so we find the $y$-coordinate as well:
$$f(0)=\sqrt[3]{0^2}=0 .$$
The local minimum point is $(0,0)$. Notice that because $f^{\prime}$ is undefined at $x=0$, there is no horizontal tangent line at this local min. In other words, the local min might be at a corner in the graph.

We continue by finding intervals of concavity and inflection points. The second derivative is
$$f^{\prime \prime}(x)=-\frac{2}{9} x^{-4 / 3}=-\frac{2}{9 x^{4 / 3}}=-\frac{2}{9 \sqrt[3]{x^4}}$$

## 数学代写|微积分代写Calculus代写|Optimization example: maximum enclosed area

The general solution strategy for an optimization problem is to determine the quantity to be optimized, make that quantity the value of a function, and then find the extreme values of that function. Let’s examine the details of this strategy using an example.

Example 1 A farmer’s child has purchased a piglet. The farmer has given the child 60 feet of fencing left over from another project. Using the side of the barn as one side of a rectangular pig pen, the child wishes to enclose the largest area possible. What dimensions should be used?

Solution Perhaps the first step in solving an optimization problem is to recognize that it is an optimization problem. This is accomplished by noticing that the stated task involves the largest, the smallest, the greatest, the best, the maximum, the minimum, or [insert optimum word here]. In this case, largest is the word used to indicate an optimum value.

As when working any word problem, we draw a picture, if possible. The pig pen is described as a rectangle, so we draw a rectangle. We are lonking at the ground from above (a top view, or aerial view). We depict a barn along one side of the rectangle. See figure 1.

Although nothing in this example is changing (this is not a related rates exercise), so our diagram is static (diagrams for related rates exercises are dynamic), it is still helpful to visualize various possibilities. We are told that there is 60 feet of fencing to make the rectangular pig pen.

We could make the pig pen very wide but not very long (figure 2 , left), very long but not very wide (figure 2, right), or something in between.

We cannot draw all possible configurations and check their areas, for there are infinitely many possibilities. For this reason, we introduce one or more variables to help. Let’s use $\ell$ for length and $w$ for width, as in figure 3 . Then area, which is the quantity we wish to maximize, is given by
$$A=\ell \cdot w .$$

# 微积分代考

## 数学代写|微积分代写Calculus代写|Complete curve-sketching examples

$$f^{\prime}(x)=\frac{2}{3} x^{-1 / 3}=\frac{2}{3 \sqrt[3]{x}} .$$

$$f(0)=\sqrt[3]{0^2}=0$$

$$f^{\prime \prime}(x)=-\frac{2}{9} x^{-4 / 3}=-\frac{2}{9 x^{4 / 3}}=-\frac{2}{9 \sqrt[3]{x^4}}$$

## 数学代写|微积分代写Calculus代写|Optimization example: maximum enclosed area

$$A=\ell \cdot w$$

## 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。

## 数学代写|微积分代写Calculus代写|MTH2010

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。

• 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|微积分代写Calculus代写|Stokes’ Theorem

Equation (1.25) defines the average of curl over a finite surface. That equation can be rewritten as
$$\int_S \mathbf{d A} \cdot(\boldsymbol{\nabla} \times \mathbf{F})=\oint_C \mathrm{dr} \cdot \mathbf{F}$$
for a vector $\mathbf{F}$. In this form, it is called Stokes’ theorem relating the integral of the curl of a vector over a surface $S$ to the line integral of the vector around the closed curve $C$ bounding the surface. From Stokes’ theorem it follows that, if $\nabla \times \mathbf{F}=0$ everywhere in a region, then
$$\oint_C \mathbf{d r} \cdot \mathbf{F}=0$$
around any closed path in the region.
We now have seen three general properties of the electric field vector $\mathbf{E}$. It is

• irrotational,
• conservative,
• derivable from a potential.

Although we have used the electric field as a simple example, these three properties also hold for a large class of vector fields. Some examples are the velocity vector of streamline fluid flow, the heat flow vector (with the temperature being the corresponding scalar field), the magnetic field $\mathbf{B}$ in a region with no current, and the gravitational field. The three properties have different physical manifestations, but they are mathematically equivalent.

## 数学代写|微积分代写Calculus代写|Algebraic Identities

Two useful algebraic identities (to be given as problems) that we will use in expanding vector derivatives are:

1. The triple scalar product of three vectors $\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})$ has the symmetry properties
$$\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})=\mathbf{b} \cdot(\mathbf{c} \times \mathbf{a})=\mathbf{c} \cdot(\mathbf{a} \times \mathbf{b})=(\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} .$$
That is, it is invariant under cyclic permutation (in either direction) or the interchange of the dot and the cross.
2. The triple vector product $\mathbf{a} \times(\mathbf{b} \times \mathbf{c})$ can be expanded as
$$\mathbf{a} \times(\mathbf{b} \times \mathbf{c})=\mathbf{b}(\mathbf{a} \cdot \mathbf{c})-\mathbf{c}(\mathbf{a} \cdot \mathbf{b}) .$$
This identity should be kept firmly in memory as the bac minus cab rule, using the mnemonic “a cross b cross c equals bac minus cab.”

Operations on combinations of functions of position can be simplified by using the two distinct properties of $\nabla$ :

1. $\nabla$ is a differential operator.
2. $\nabla$ is a vector.
Because $\boldsymbol{\nabla}$ is a differential operator, it acts on functions one at a time, just as in $\mathrm{d}(u v)=u d v+v d u$. We also follow the convention that the differential operator

acts only on functions to its right, so the order in which $\nabla$ appears must be to the left of the functions it acts on and to the right of the other functions. As a vector, $\boldsymbol{\nabla}$ must behave in any expansion like any other vector.

In every case, strict adherence to these two properties of $\boldsymbol{\nabla}$ will lead to the correct evaluation of vector derivatives. We give several examples below of the use of the two properties of $\nabla$ and the algebraic identities.

# 微积分代考

## 数学代写|微积分代写Calculus代写|Stokes’ Theorem

$$\int_S \mathbf{d A} \cdot(\boldsymbol{\nabla} \times \mathbf{F})=\oint_C \mathrm{dr} \cdot \mathbf{F}$$

$$\oint_C \mathbf{d r} \cdot \mathbf{F}=0$$

• 无旋的，
• 保守的，
• 可从势能推导出来。
虽然我们使用电场作为一个简单的例子，但这三个属性也适用于一大类矢量场。一些例子是流线型流体 流动的速度矢量、热流矢量 (温度是相应的标量场) 、磁场 $\mathbf{B}$ 在没有电流和引力场的区域。这三个性质 在物理上有不同的表现，但在数学上是等价的。

## 数学代写|微积分代写Calculus代写|Algebraic Identities

1. 三个向量的三重标量积 $\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})$ 具有对称性
$$\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})=\mathbf{b} \cdot(\mathbf{c} \times \mathbf{a})=\mathbf{c} \cdot(\mathbf{a} \times \mathbf{b})=(\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c}$$
也就是说，它在循环置换（在任一方向上）或点和叉的交换下是不变的。
2. 三向量积 $\mathbf{a} \times(\mathbf{b} \times \mathbf{c})$ 可以扩展为
$$\mathbf{a} \times(\mathbf{b} \times \mathbf{c})=\mathbf{b}(\mathbf{a} \cdot \mathbf{c})-\mathbf{c}(\mathbf{a} \cdot \mathbf{b})$$
这个身份应该作为 bac minus cab 规则牢牢记住，使用助记符”a cross b cross c equals bac minus cab”。
可以通过使用的两个不同属性来简化对位置函数组合的操作 $\nabla$ :
3. $\nabla$ 是溦分算子。
4. $\nabla$ 是一个向量。
因为 $\nabla$ 是一个微分算子，它一次作用于一个函数，就像在 $\mathrm{d}(u v)=u d v+v d u$. 我们还遵循微分 算子的约定
只作用于它右边的函数，所以顺序 $\nabla$ 出现必须在它作用的函数的左边和其他函数的右边。作为向量， $\nabla$ 必须像任何其他向量一样在任何扩展中表现。
在每种情况下，严格遵守这两个属性 $\boldsymbol{\nabla}$ 将导致对向量导数的正确评估。我们在下面给出几个使用的两个 属性的例子 $\nabla$ 和代数恒等式。

## 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。

## 数学代写|微积分代写Calculus代写|MATH141

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。

• 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|微积分代写Calculus代写|Divergence Theorem

Combining Eq. (1.16) for the average value of the divergence over a finite volume $V$ with the definition of a volume average that
$$\langle\nabla \cdot \mathbf{E}\rangle_V=\frac{1}{V} \int_V \nabla \cdot \mathbf{E} d^3 r,$$
we get
$$\int_V^r \nabla \cdot \mathbf{E} d^3 r=\oint_{J_S} \mathbf{d} \mathbf{A} \cdot \mathbf{E} .$$
In this form, it is called the divergence theorem.
The definition of the divergence given by Eq. (1.17) can be used to evaluate the divergence of the position vector. We apply the definition to a sphere of radius $R$, getting ${ }^2$
\begin{aligned} \nabla \cdot \mathbf{r} & =\lim {V \rightarrow 0} \frac{1}{V} \oint_S \mathbf{d A}^{\prime} \cdot \mathbf{r}^{\prime}=\lim {R \rightarrow 0} \frac{1}{V} \oint R^3 d \Omega^{\prime} \ & =\lim {R \rightarrow 0} \frac{4 \pi R^3}{(4 / 3) \pi R^3}=3 . \end{aligned} As we did with the gradient, we now show what the divergence would look like in Cartesian coordinates. Figure $1.3$ shows an infinitesimal volume (a parallelepipid in Cartesian coordinates) of dimensions $\Delta x \times \Delta y \times \Delta z$, that will shrink to zero at the point $x, y, z$. The surface integral in the definition of $\operatorname{div} \mathbf{E}$ is over the six faces of the parallelepipid, I-VI, so the integral can be written as $$\lim {V \rightarrow 0} \frac{1}{V} \oint_S \mathbf{d} \mathbf{A} \cdot \mathbf{r}=I+I I+I I I+I V+V+V I,$$
where $I$ indicates the integral over face $\mathrm{I}$, and similarly for the other faces.

## 数学代写|微积分代写Calculus代写|Curl

Next, we look at how a vector $\mathbf{E}$ can vary across its direction, and we give a physical definition of the curl of a vector field. Figure $1.4$ shows a vector field having such a variation, with the density of lines being proportional to the strength of the field. If this were a velocity field, such as the current of water in a stream, this variation could be measured experimentally by placing a paddle wheel in the stream as shown in the figure. Then the rotation of the paddle wheel would be a measure of the variation of the vector field. This can be done without getting wet by calculating a line integral around a typical closed curve $C$, as shown on the figure.

The line integral can be used to define an average value of the variation (called curl) over a surface $S$ bounded by the curve $C$. The average curl is defined by
$$\langle\hat{\mathbf{n}} \cdot \operatorname{curl} \mathbf{E}\rangle_S=\frac{1}{S} \oint_C \mathrm{dr}^{\prime} \cdot \mathbf{E}\left(\mathbf{r}^{\prime}\right),$$
where $\hat{\mathbf{n}}$ is the unit vector normal to the surface $S$ at any point. Note that, by this detinition, the vector average $\langle\hat{\mathbf{n}} \cdot \mathbf{c u r l} \mathbf{E}\rangle_S$ does not depend on the shape of the surface $S$, but only on the bounding path $C$ and the area of the surface. Since the variation will be different in different directions, it is the average value of the normal component of curl that is defined by Eq. (1.25).

The positive sign for the direction of $\hat{\mathbf{n}}$ is taken by convention to be the boreal direction. That is, if the integral around the contour $C$ is taken in the direction of the rotation of the earth, then the north pole is in the positive direction as shown on Fig. 1.5a.

This is also stated as the right hand rule: If the integral around the contour $C$ is taken in the direction that the four fingers of the right hand curl as they tend to close, then the right thumb points in the positive direction for $\hat{\mathbf{n}}$, as shown in Fig. 1.5b. This will be our general sign convention relating the direction of integration around a closed curve and the positive direction of the normal vector to any surface bounded by the curve.

I’he value of curl $\mathbf{E}$ at a point $\mathbf{r}$ can be defined by starting with a smooth surface through the point and taking the limit as the curve bounding the surface shrinks about the point, and the enclosed surface shrinks to zero area. This gives the definition of curl at a point:
$$[\operatorname{curl} \mathbf{E}(\mathbf{r})]n=\lim {S \rightarrow 0} \frac{1}{S} \oint_C \mathbf{d r}^{\prime} \cdot \mathbf{E}\left(\mathbf{r}^{\prime}\right) .$$

# 微积分代考

## 数学代写|微积分代写Calculus代写|Divergence Theorem

$$\langle\nabla \cdot \mathbf{E}\rangle_V=\frac{1}{V} \int_V \nabla \cdot \mathbf{E} d^3 r$$

$$\int_V^r \nabla \cdot \mathbf{E} d^3 r=\oint_{J_S} \mathbf{d} \mathbf{A} \cdot \mathbf{E} .$$

$$\nabla \cdot \mathbf{r}=\lim V \rightarrow 0 \frac{1}{V} \oint_S \mathbf{d} \mathbf{A}^{\prime} \cdot \mathbf{r}^{\prime}=\lim R \rightarrow 0 \frac{1}{V} \oint R^3 d \Omega^{\prime} \quad=\lim R \rightarrow 0 \frac{4 \pi R^3}{(4 / 3) \pi R^3}=3$$

$$\lim V \rightarrow 0 \frac{1}{V} \oint_S \mathbf{d A} \cdot \mathbf{r}=I+I I+I I I+I V+V+V I$$

## 数学代写|微积分代写Calculus代写|Curl

$$\langle\hat{\mathbf{n}} \cdot \operatorname{curl} \mathbf{E}\rangle_S=\frac{1}{S} \oint_C \mathrm{dr}^{\prime} \cdot \mathbf{E}\left(\mathbf{r}^{\prime}\right),$$

（1） 定义的是卷曲法向分量的平均值。(1.25)。

$$[\operatorname{curl} \mathbf{E}(\mathbf{r})] n=\lim S \rightarrow 0 \frac{1}{S} \oint_C \mathbf{d r}^{\prime} \cdot \mathbf{E}\left(\mathbf{r}^{\prime}\right)$$

## 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。

## 数学代写|微积分代写Calculus代写|MATH1051

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。

• 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|微积分代写Calculus代写|Vector Differential Operators

A scalar field (that is, a scalar function of the position vector $\mathbf{r}$ ) is conveniently pictured by means of surfaces (in three dimensions) or lines (in two dimensions) along which its magnitude is constant. Depending on the physical application, these constant magnitude surfaces or lines could be called equipotentials, isobars, isotherms, or whatever applies in the given situation.

A common example, shown in Fig. 1.1, is a topographic map of a hillside.

The lines of equal altitude shown on the map are equipotentials of the gravitational field. No work is done in moving along an equipotential, and the direction of steepest slope is everywhere perpendicular to the equipotential. That perpendicular direction is defined as the direction of the gradient of the potential. In our topographic example, this direction is the steepest direction up the hill.

Experimentally, this would be opposite to the direction a ball would roll if placed at rest on the hillside.

The magnitude of the gradient is defined to be the rate of change of the potential with respect to distance in the direction of maximum increase. This provides a mathematical definition of the gradient as
$$\operatorname{grad} \phi=\hat{\mathbf{n}} \frac{d \phi}{|\mathbf{d r}|} .$$
In Eq. (1.1), the unit vector $\hat{\mathbf{n}}(=\mathbf{n} /|\mathbf{n}|)$ is in the direction of maximum increase of $\phi$, and $\mathbf{d r}$ is taken in that direction of maximum increase.

For infinitesimal displacements, an equipotential surface can be approximated by its tangent plane (or tangent line in two dimensions), so the change in a scalar field in an infinitesimal displacement $\mathbf{d r}$ will vary as the cosine of the angle between the direction of maximum gradient and dr. Then the differential change of $\phi$ in any direction is given by
$$d \phi(\mathbf{r})=\mathbf{d r} \cdot \mathbf{g r a d} \phi .$$

## 数学代写|微积分代写Calculus代写|Divergence

A vector field can have two different types of variation. It can vary along its direction, for instance like the velocity field, v, of a stream as the slope gets steeper. The vector field can also vary across its direction, as when the velocity is faster in the middle of the stream than near the edges. How can these two variations be measured?

The rate of increase of a vector field along its direction is called the divergence of the vector field. A simple example for a vector field $\mathbf{E}$ is shown in Fig. 1.2. A measure of the strength of the field is the density of lines of force in the figure, with the increase in the field indicated by increasing lines of force.
We construct a mathematical volume $V$ enclosed by a surface $S$, as shown in the figure. The increase in $\mathbf{E}$ can be seen in the figure as more lines of $\mathbf{E}$ leaving the volume than entering it (‘diverging’ from the volume).

A quantitative measure of the excess of lines leaving the volume is given by the integral $\oint_S \mathbf{d A} \cdot \mathbf{E} .{ }^1$ This integral can be used to define an average divergence (written as ‘div’) of the lines of the vector field. That is
$$\langle\operatorname{div} \mathbf{E}\rangle_V=\frac{1}{V} \oint_S \mathbf{d A}^{\prime} \cdot \mathbf{E}\left(\mathbf{r}^{\prime}\right),$$
where the notation $\langle\operatorname{div} \mathbf{E}\rangle_V$ denotes the average of $\operatorname{div} \mathbf{E}$ over the volume $V$.
The value of $\operatorname{div} \mathbf{E}$ at a point $\mathbf{r}$ can be defined by shrinking the integral about the point, so
$$\operatorname{div} \mathbf{E}(\mathbf{r})=\lim _{V \rightarrow 0} \frac{1}{V} \oint_S \mathbf{d} \mathbf{A}^{\prime} \cdot \mathbf{E}\left(\mathbf{r}^{\prime}\right)$$ gives the divergence of the vector field at the point $\mathbf{r}$ (if the limit exists), and is a measure of its rate of increase along the direction of the vector field. We take Eq. (1.17) as the definition of the divergence operator.

We will show on the next page that div $\mathbf{E}$ can be written as $\boldsymbol{\nabla} \cdot \mathbf{E}$, corresponding to the dot product of the vector differential operator $\boldsymbol{\nabla}$ with a vector E. We start using that notation now, so that the following equations will be in the usual notation for the divergence.

# 微积分代考

## 数学代写|微积分代写Calculus代写|Vector Differential Operators

$$\operatorname{grad} \phi=\hat{\mathbf{n}} \frac{d \phi}{|\mathbf{d r}|} .$$

$$d \phi(\mathbf{r})=\mathbf{d r} \cdot \operatorname{grad} \phi$$

## 数学代写|微积分代写Calculus代写|Divergence

$$\langle\operatorname{div} \mathbf{E}\rangle_V=\frac{1}{V} \oint_S \mathbf{d} \mathbf{A}^{\prime} \cdot \mathbf{E}\left(\mathbf{r}^{\prime}\right)$$

$$\operatorname{div} \mathbf{E}(\mathbf{r})=\lim _{V \rightarrow 0} \frac{1}{V} \oint_S \mathbf{d} \mathbf{A}^{\prime} \cdot \mathbf{E}\left(\mathbf{r}^{\prime}\right)$$

## 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。

## 数学代写|微积分代写Calculus代写|MATH1111

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。

• 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|微积分代写Calculus代写|Nonlinear difference equations

In Section 1.4 we discussed the difference equation
$$x_{n+1}=\alpha x_n,$$
$n=0,1,2, \ldots$, as a model for either growth or decay and we saw that its solution is given by
$$x_n=\alpha^n x_0,$$
$n=0,1,2, \ldots$. Now
$$\lim {n \rightarrow \infty} \alpha^n= \begin{cases}0, & \text { if } 0<\alpha<1, \ 1, & \text { if } \alpha=1, \ \infty, & \text { if } \alpha>1,\end{cases}$$ from which it follows that if $\left{x_n\right}$ is a solution of (1.5.1) with $x_0>0$, then $$\lim {n \rightarrow \infty} x_n=x_0 \lim {n \rightarrow \infty} \alpha^n= \begin{cases}0, & \text { if } 0<\alpha<1, \ x_0, & \text { if } \alpha=1, \ \infty, & \text { if } \alpha>1 .\end{cases}$$ These limiting values are consistent with our radioactive decay cxample since, in that case, $0<\alpha<1$ and we would expect the amount of a radioactive element to decline toward 0 over time. The case $0<\alpha<1$ also may make sense for a population model if the population is declining and heading toward extinction. However, the unbounded growth indefinitely into the future implied by the case $\alpha>1$ is very unlikely for a population model: eventually ecological or even sociological problems come to the forefront, such as when the population begins to overreach the resources available to it, and the rate of growth of the population changes. Even for bacteria growing in a Petri dish, diminishing food and space eventually cause a change in the rate of growth. Hence the equation $$x{n+1}=\alpha x_n$$
for $n=0,1,2, \ldots$ and $\alpha>1$, called the uninhibited, or natural, growth model, although often accurate as a model of population growth over short periods of time, is usually too simplistic for predictions over long time spans.

## 数学代写|微积分代写Calculus代写|The inhibited growth model

Suppose we wish to model the growth of a certain population which, without ecological constraints, would grow at a rate of $100 \beta \%$ per unit of time. That is, if $x_n$ represents the size of the population after $n$ units of time and there are no constraints on the size of the population, then
$$x_{n+1}-x_n=\beta x_n$$
for $n=0,1,2, \ldots$. However, suppose that, because of the limitation of resources, the population will begin to decline if it ever has more than $M$ individuals. We call $M$ the carrying capacity of the available resources, the maximum population which is sustainable over time. Then it would be reasonable to modify our model by forcing the amount of increase over a unit of time to decrease as the size of the population approaches $M$ and to become negative if the size of the population ever exceeds $M$. One way to accomplish this is to multiply the term $\beta x_n$ in (1.5.5) by
$$\frac{M-x_n}{M},$$
a ratio which is close to 1 when $x_n$ is small, close to 0 when $x_n$ is close to $M$, and negative when $x_n$ exceeds $M$. This leads us to the difference equation
$$x_{n+1}-x_n=\beta x_n\left(\frac{M-x_n}{M}\right),$$
$n=0,1,2, \ldots$, or, equivalently,
$$x_{n+1}=x_n+\frac{\beta}{M} x_n\left(M-x_n\right),$$
$n=0,1,2, \ldots$, which we call the inhibited growth model, also known as the discrete logistic equation. This is an example of a nonlinear difference equation because if we multiply out the right-hand side of the equation we have a quadratic term, namely, $\frac{\beta}{M} x_n^2$. Such equations are, in general, far more difficult to solve than the linear difference equations we considered in Section 1.4; in fact, many nonlinear difference equations are not solvable in terms of the elementary functions of calculus. Hence we will not consider any methods for solving such equations, relying instead on computing specific solutions by iterating the equation using a calculator or, preferably, a computer.

## 数学代写|微积分代写Calculus代写|Nonlinear difference equations

$$x_{n+1}=\alpha x_n,$$
$n=0,1,2, \ldots$ ，作为增长或衰退的模型，我们看到它的解决方案由下式给出
$$x_n=\alpha^n x_0,$$
$n=0,1,2, \ldots$ 现在
$$\lim n \rightarrow \infty \alpha^n={0, \quad \text { if } 0<\alpha<1,1, \quad \text { if } \alpha=1, \infty, \quad \text { if } \alpha>1,$$

$\lim n \rightarrow \infty x_n=x_0 \lim n \rightarrow \infty \alpha^n=\left{0, \quad\right.$ if $0<\alpha<1, x_0, \quad$ if $\alpha=1, \infty, \quad$ if $\alpha>1$.

$$x n+1=\alpha x_n$$

## 数学代写|微积分代写Calculus代写|The inhibited growth model

$$x_{n+1}-x_n=\beta x_n$$

$$\frac{M-x_n}{M},$$

$$x_{n+1}-x_n=\beta x_n\left(\frac{M-x_n}{M}\right),$$
$n=0,1,2, \ldots$ ，或者，等价地，
$$x_{n+1}=x_n+\frac{\beta}{M} x_n\left(M-x_n\right),$$
$n=0,1,2, \ldots$ ，我们称之为抑制增长模型，也称为离散逻辑方程。这是一个非线性差分方程的例子， 因为如果我们将方程的右边相乘，就会得到一个二次项，即 $\frac{\beta}{M} x_n^2$.一般来说，这些方程比我们在 $1.4$ 节 中考虑的线性差分方程更难求解；事实上，许多非线性差分方程无法用微积分的初等函数求解。因此，我 们不会考虑任何求解此类方程的方法，而是依赖于使用计算器或最好是计算机迭代方程来计算特定解。

## 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。

## 数学代写|微积分代写Calculus代写|MATH141

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。

• 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|微积分代写Calculus代写|Monotone sequences

Definition 1.2.5. We say a sequence $\left{a_n\right}$ is monotone increasing if $a_n \leq a_{n+1}$ for all $n$. We say a sequence $\left{a_n\right}$ is monotone decreasing if $a_n \geq a_{n+1}$ for all $n$. We say a sequence is monotone if it is either monotone increasing or monotone decreasing.

Now suppose $\left{a_n\right}$ is a monotone increasing sequence. For such a sequence there either exists a number $P$ such that $a_n \leq P$ for all $n$ or there does not exist such a $P$. In the latter case, given any real number $M$, it is then possible to find an integer $N$ such that $a_N>M$. Since the sequence is monotone, it follows that $a_n>M$ for all $n>N$, and so the sequence diverges to infinity. On the other hand, if there does exist a number $P$ such that $a_n \leq P$ for all $n$, then there in fact exists a number $B$ such that $u_n \leq B$ for all $n$ and $B \leq P$ for any number $P$ with the property that $a_n \leq P$ for all $n$. The existence of $B$, known as the least upper bound of the sequence $\left{a_n\right}$, is not at all obvious; indeed, the subtle properties of the real numbers that imply the existence of $B$ were not fully understood until the middle part of the 19th century. However, given the existence of $B$, it is easy to see that given any $\epsilon>0$, there exists an integer $N$ for which $a_N>B-\epsilon$

(if not, then $B-\epsilon$ would be an upper bound for the sequence smaller than $B$ ). Since the sequence is monotone increasing and $a_nN$. That is, we have shown that the sequence converges and
$$\lim {n \rightarrow \infty} a_n=B \text {. }$$ Similar results hold for sequences which are monotone decreasing. Theurem 1.2.11 (Munutune sequence theurem). Suppose the sequence $\left{a_n\right}$ is monutone. If the sequence is monotone increasing and there exists a number $P$ such that $a_n \leq P$ for all $n$, then the sequence converges. If the sequence is monotone increasing and no such number $P$ exists, then $$\lim {n \rightarrow \infty} a_n=\infty .$$
If the sequence is monotone decreasing and there exists a number $Q$ such that $a_n \geq Q$ for all $n$, then the sequence converges. If the sequence is monotone decreasing and no such number $Q$ exists, then
$$\lim _{n \rightarrow \infty} a_n=-\infty$$

## 数学代写|微积分代写Calculus代写|The sum of a sequence

This section considers the problem of adding together the terms of a sequence. Of course, this is a problem only if more than a finite number of terms of the sequence are nonzero. In this case, we must decide what it means to add together an infinite number of nonzero numbers. The first example shows how a relatively simple question may lead to such infinite summations.

Example 1.3.1. Suppose a game is played in which a fair coin is tossed until the first time a head appears. What is the probability that a head appears for the first time on an even-numbered toss? To solve this problem, we first need to determine the probability of obtaining a head for the first time on any given even-numbered toss, and then we need to add all these probabilities together. Let $P_n$ denote the probability that the first head appears on the $n$th toss, $n=1,2,3, \ldots$. Then, since the coin is assumed to be fair,
$$P_1=\frac{1}{2} .$$
Now in order to get a head for the first time on the second toss, we must toss a tail on the first toss and then follow that with a head on the second toss. Since one-half of all first tosses will be tails and then one-half of those tosses will be followed by a second toss of heads, we should have
$$P_2=\left(\frac{1}{2}\right)\left(\frac{1}{2}\right)=\frac{1}{4} .$$
Similarly, since one-fourth of all sequences of coin tosses will begin with two tails and then half of these sequences will have a head for the third toss, we have
$$P_3=\left(\frac{1}{4}\right)\left(\frac{1}{2}\right)=\frac{1}{8} .$$
Continuing in this fashion, it should seem reasonable that, for any $n=1,2,3, \ldots$,
$$P_n=\frac{1}{2^n} .$$
Hence we have a sequence of probabilities $\left{P_n\right}$ for $n=1,2,3, \ldots$, and, in order to find the desired probability, we need to add up the even-numbered terms in this sequence. Namely, the probability that a head appears for the first time on an even toss is given by
$$P_2+P_4+P_6+\cdots=\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\cdots .$$

## 数学代写|微积分代写Calculus代写|Monotone sequences

$\backslash$ 左{a_n\右 $}$ 是单调递减的如果 $a_n \geq a_{n+1}$ 对所有人 $n$. 如果一个序列是单调递增或单调递减，我们就说它 是单调的。

(如果没有，那么 $B-\epsilon$ 将是小于的序列的上限 $B$ ). 由于序列是单调递增的，并且 $a_n N$. 也就是说，我们 已经证明序列收敛并且
$$\lim n \rightarrow \infty a_n=B .$$

$$\lim n \rightarrow \infty a_n=\infty .$$

$$\lim _{n \rightarrow \infty} a_n=-\infty$$

## 数学代写|微积分代写Calculus代写|The sum of a sequence

$$P_1=\frac{1}{2} .$$

$$P_2=\left(\frac{1}{2}\right)\left(\frac{1}{2}\right)=\frac{1}{4} .$$

$$P_3=\left(\frac{1}{4}\right)\left(\frac{1}{2}\right)=\frac{1}{8} .$$

$$P_n=\frac{1}{2^n} .$$

$$P_2+P_4+P_6+\cdots=\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\cdots$$

## 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。

## 数学代写|微积分代写Calculus代写|MATH1051

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。

• 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|微积分代写Calculus代写|Areas and tangents

The study of calculus begins with questions about change. What happens to the velocity of a swinging pendulum as its position changes? What happens to the position of a planet as time changes? What happens to a population of owls as its rate of reproduction changes? Mathematically, one is interested in learning to what extent changes in one quantity affect the value of another related quantity. Through the study of the way in which quantities change we are able to understand more deeply the relationships between the quantities themselves. For example, changing the angle of elevation of a projectile affects the distance it will travel; by considering the effect of a change in angle on distance, we are able to determine, for example, the angle which will maximize the distance.

Related to questions of change are problems of approximation. If we desire to approximate a quantity which cannot be computed directly (for example, the area of some planar region), we may develop a technique for approximating its value. The accuracy of our technique will depend on how many computations we are willing to make; calculus may then be used to answer questions about the relationship between the accuracy of the approximation and the number of calculations used. If we double the number of computations, how much do we gain in accuracy? As we increase the number of computations, do the approximations approach some limiting value? And if so, can we use our approximating method to arrive at an exact answer? Note that once again we are asking questions about the effects of change.

Two fundamental concepts for studying change are sequences and limits of sequences. For our purposes, a sequence is nothing more than a list of numbers. For example,
$$1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots$$
might represent the beginning of a sequence, where the ellipsis indicates that the list is to continue on indefinitely in some pattern. For example, the 5 th term in this sequence might be
the 8th term
$$\frac{1}{16}=\frac{1}{2^4},$$
$$\frac{1}{128}=\frac{1}{2^7} \text {, }$$
and, in general, the $n$th term
$$\frac{1}{2^{n-1}},$$
where $n=1,2,3, \ldots$. Notice that the sequence is completely specified only when we have given the general form of a term in the sequence. Also note that this list of numbers is approaching 0 , which we would call the limit of the sequence. In the next section of this chapter we will consider in some detail the basic question of determining the limit of a sequence.

## 数学代写|微积分代写Calculus代写|Sequences

Ás we noted in Section 1.1, listing the first few terms of a sequence does not uniquely specify the remaining terms of the sequence. To fully specify a sequence, we need a formula that describes an arbitrary term in the sequence. For example, the first example above lists the first four terms of the sequence $\left{a_n\right}$ with
$$a_n=n$$
for $n=1,2,3, \ldots ;$ the second example lists the first four terms of $\left{b_n\right}$ with
$$b_n=2 n$$
for $n=1,2,3, \ldots ;$ the third example lists the first four terms of $\left{c_n\right}$ with
$$c_n=1-\frac{1}{n}$$
for $n=1,2,3, \ldots ;$ the fourth lists the first four terms of $\left{d_n\right}$ with
$$d_n=\frac{(-1)^n}{2^n}$$
for $n=0,1,2,3, \ldots ;$ and the fifth lists the first four terms of $\left{e_n\right}$ with
$$e_n=(-1)^n$$
for $n=0,1,2, \ldots$. Note, however, that although these are in some sense the natural formulas for these sequences, they are not the only possibilities.

As indicated in Section 1.1, we are often interested in the value, if one exists, which a sequence approaches. For example, the sequences $\left{a_n\right}$ and $\left{b_n\right}$ increase beyond any possible bound as $n$ increases, and hence they have no limiting value. To visualize what is happening here, you might plot the points of the sequence on the real line. For both of these sequences, the plotted points will march off to the right without any upper limit. Although a limit does not exist in these cases, we usually write
$$\lim {n \rightarrow \infty} a_n=\infty$$ and $$\lim {n \rightarrow \infty} b_n=\infty$$
to express the fact that the limits do not exist because the terms in the sequence are eventually always larger than any specified positive bound. On the other hand, if we plot the points of the sequence $\left{c_n\right}$, as in Figure 1.2.1, we see that although they are always increasing (that is, moving toward the right), nevertheless they never increase beyond 1. Moreover, even though no term in the sequence is ever equal to 1 , we can see that the points become arbitrarily close to 1 . Hence we say that the limit of the sequence is 1 and we write
$$\lim _{n \rightarrow \infty} c_n=1$$

## 数学代写|微积分代写Calculus代写|Areas and tangents

$$1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots$$

$$\begin{gathered} \frac{1}{16}=\frac{1}{2^4}, \ \frac{1}{128}=\frac{1}{2^7}, \end{gathered}$$

$$\frac{1}{2^{n-1}}$$

## 数学代写|微积分代写Calculus代写|Sequences

$$a_n=n$$

$$b_n=2 n$$

$$c_n=1-\frac{1}{n}$$

$$d_n=\frac{(-1)^n}{2^n}$$

$$e_n=(-1)^n$$

$$\lim n \rightarrow \infty a_n=\infty$$

$$\lim n \rightarrow \infty b_n=\infty$$ 和
$$\lim n \rightarrow \infty b_n=\infty$$

$$\lim _{n \rightarrow \infty} c_n=1$$

## 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。

## 数学代写|微积分代写Calculus代写|MATH1111

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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|微积分代写Calculus代写|The Tangent Problem

Consider the problem of trying to find an equation of the tangent line $\ell$ to a curve with equation $y=f(x)$ at a given point $P$. (We will give a precise definition of a tangent line in Chapter 2 ; for now you can think of it as the line that touches the curve at $P$ and follows the direction of the curve at $P$, as in Figure 5.) Because the point $P$ lies on the tangent line, we can find the equation of $\ell$ if we know its slope $m$. The problem is that we need two points to compute the slope and we know only one point, $P$, on $\ell$. To get around the problem we first find an approximation to $m$ by taking a nearby point $Q$ on the curve and computing the slope $m_{P Q}$ of the secant line $P Q$.

Now imagine that $Q$ moves along the curve toward $P$ as in Figure 6 . You can see that the secant line $P Q$ rotates and approaches the tangent line $\ell$ as its limiting position. This means that the slope $m_{P Q}$ of the secant line becomes closer and closer to the slope $m$ of the tangent line. We write
$$m=\lim {Q \rightarrow P} m{P Q}$$
and say that $m$ is the limit of $m_{P Q}$ as $Q$ approaches $P$ along the curve.
Notice from Figure 7 that if $P$ is the point $(a, f(a))$ and $Q$ is the point $(x, f(x))$, then
$$m_{P Q}=\frac{f(x)-f(a)}{x-a}$$
Because $x$ approaches $a$ as $Q$ approaches $P$, an equivalent expression for the slope of the tangent line is
$$m=\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}$$
In Chapter 3 we will learn rules for calculating such limits.
The tangent problem has given rise to the branch of calculus called differential calculus; it is important because the slope of a tangent to the graph of a function can have different interpretations depending on the context. For instance, solving the tangent problem allows us to find the instantaneous speed of a falling stone, the rate of change of a chemical reaction, or the direction of the forces on a hanging chain.

## 数学代写|微积分代写Calculus代写|A Relationship between the Area and Tangent Problems

The area and tangent problems seem to be very different problems but, surprisingly, the problems are closely related-in fact, they are so closely related that solving one of them leads to a solution of the other. The relationship between these two problems is introduced in Chapter 5; it is the central discovery in calculus and is appropriately named the Fundamental Theorem of Calculus. Perhaps most importantly, the Fundamental Theorem vastly simplifies the solution of the area problem, making it possible to find areas without having to approximate by rectangles and evaluate the associated limits.

Isaac Newton (1642-1727) and Gottfried Leibniz (1646-1716) are credited with the invention of calculus because they were the first to recognize the importance of the Fundamental Theorem of Calculus and to utilize it as a tool for solving real-world problems. In studying calculus you will discover these powerful results for yourself.

We have seen that the concept of a limit arises in finding the area of a region and in finding the slope of a tangent line to a curve. It is this basic idea of a limit that sets calculus apart from other areas of mathematics. In fact, we could define calculus as the part of mathematics that deals with limits. We have mentioned that areas under curves and slopes of tangent lines to curves have many different interpretations in a variety of contexts. Finally, we have discussed that the area and tangent problems are closely related. After Isaac Newton invented his version of calculus, he used it to explain the motion of the planets around the sun, giving a definitive answer to a centuries-long quest for a description of our solar system. Today calculus is applied in a great variety of contexts, such as determining the orbits of satellites and spacecraft, predicting population sizes,forecasting weather, measuring cardiac output, and gauging the efficiency of an economic market.

## 数学代写|微积分代写Calculus代写|The Tangent Problem

$$m=\lim Q \rightarrow P m P Q$$

$$m_{P Q}=\frac{f(x)-f(a)}{x-a}$$

$$m=\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}$$

## 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。