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# 数学代写|微积分代写Calculus代考|Math323 Area Under a Curve and the Definite Integral

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## 数学代写|微积分代写Calculus代考|Area Under a Curve and the Definite Integral

The first section of this chapter focused on the techniques of integration, i.e. finding antiderivatives, all of which are embodied in the term “indefinite integral.” Just as differentiation is useful for many applications besides finding slopes of curves – for instance, calculating rates of growth or finding maxima and minima-so integral calculus has many applications such as finding the volumes of solids or finding the distance traveled by a body moving with a velocity $v(t)$. These applications all rely on the second type of integration, called the definite integral. This originated in the geometric problem of finding the area under a curve. We start by explaining what “area under a curve” means.

To illustrate what is meant by “the area under a curve,” here is a graph of the simplest of all curves-a straight line given by $f(x)=3$.

What is the area $A$ between the line $f(x)=3$, and the $x$-axis between the interval $x=a$ and $x=b$ ?
$$A=[3 a b|3(a+b)| 3(a-b) \mid 3(b-a)]$$

## 数学代写|微积分代写Calculus代考|Higher-Order Derivatives

Suppose a function $f$ depends on $x$, and we have differentiated it to obtain $d f / d x$. If we then differentiate $d f / d x$ with respect to $x$, the result is called the second derivative of $f$ with respect to $x$. This is written $\frac{d^2 f}{d x^2}$. Sometimes this is written as $f^{(2)}$, where the ” $(2)$ ” superscript indicates the second derivative of $f$, not the square of $f$. The variable $x$ is suppressed.
Answer: Frame 238: $\frac{-1}{(\ln x)^2 x}$

Try the following:
$$\text { If } f=2 x^3 \text {, then } f^{(2)}=\frac{d^2 f}{d x^2}=\left[6 x^2|12 x| 0\left|x^2\right| x\right]$$

Here’s how to do the problem in $\mathbf{2 4 2}$.
\begin{aligned} f &=2 x^3, \frac{d f}{d x}=6 x^2, \ f^{(2)} &=\frac{d^2 f}{d x^2}=\frac{d}{d x}\left(\frac{d f}{d x}\right) \frac{d}{d x}\left(6 x^2\right)=12 x . \end{aligned}
Try this:
\begin{aligned} f(x) &=x+\frac{1}{x} \ f^{(2)} &=\frac{d^2 f}{d x^2}=\left[-\frac{1}{x^2}\left|\frac{1}{x}\right|+\frac{2}{x^3} \mid \text { none of these }\right] \end{aligned}

## 数学代写微积分代写Calculus代考|Area Under a Curve and the Definite Integral

$$A=[3 a b|3(a+b)| 3(a-b) \mid 3(b-a)]$$

## 数学代写|微积分代写Calculus代考|Higher-Order Derivatives

If $f=2 x^3$, then $f^{(2)}=\frac{d^2 f}{d x^2}=\left[6 x^2|12 x| 0\left|x^2\right| x\right]$

$$f=2 x^3, \frac{d f}{d x}=6 x^2, f^{(2)} \quad=\frac{d^2 f}{d x^2}=\frac{d}{d x}\left(\frac{d f}{d x}\right) \frac{d}{d x}\left(6 x^2\right)=12 x .$$

$$f(x)=x+\frac{1}{x} f^{(2)} \quad=\frac{d^2 f}{d x^2}=\left[-\frac{1}{x^2}\left|\frac{1}{x}\right|+\frac{2}{x^3} \mid \text { none of these }\right]$$

## 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。