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# 计算机代写|计算方法代写ALGORITHMIC METHODS代写|CSC4800 Interpretations of the Derivative

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## 计算机代写|计算方法代写ALGORITHMIC METHODS代写|Interpretations of the Derivative

We introduced the derivative geometrically as the slope of the tangent, and we saw that the tangent to a graph of a differentiable function $f$ at the point $\left(x_{0}, f\left(x_{0}\right)\right)$ is given by
$$y=f^{\prime}\left(x_{0}\right)\left(x-x_{0}\right)+f\left(x_{0}\right) .$$
Example 7.14 Let $f(x)=x^{4}+1$ with derivative $f^{\prime}(x)=4 x^{3}$.
(i) The tangent to the graph of $f$ at the point $(0,1)$ is
$$y=f^{\prime}(0) \cdot(x-0)+f(0)=1$$
and thus horizontal.

(ii) The tangent to the graph of $f$ at the point $(1,2)$ is
$$y=f^{\prime}(1)(x-1)+2=4(x-1)+2=4 x-2 .$$
The derivative allows further interpretations.
Interpretation as linear approximation. We start off by emphasising that every differentiable function $f$ can be written in the form
$$f(x)=f\left(x_{0}\right)+f^{\prime}\left(x_{0}\right)\left(x-x_{0}\right)+R\left(x, x_{0}\right),$$
where the remainder $R\left(x, x_{0}\right)$ has the property
$$\lim {x \rightarrow x{0}} \frac{R\left(x, x_{0}\right)}{x-x_{0}}=0 .$$
This follows immediately from
$$R\left(x, x_{0}\right)=f(x)-f\left(x_{0}\right)-f^{\prime}\left(x_{0}\right)\left(x-x_{0}\right)$$
by dividing by $x-x_{0}$, since
$$\frac{f(x)-f\left(x_{0}\right)}{x-x_{0}} \rightarrow f^{\prime}\left(x_{0}\right) \quad \text { as } x \rightarrow x_{0}$$

## 计算机代写|计算方法代写ALGORITHMIC METHODS代写|Differentiation Rules

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In this section $I \subset \mathbb{R}$ denotes an open interval. We first note that differentiation is a linear process.

Proposition $7.18$ (Linearity of the derivative) Let $f, g: I \rightarrow \mathbb{R}$ be two functions which are differentiable at $x \in I$ and take $c \in \mathbb{R}$. Then the functions $f+g$ and $c \cdot f$ are differentiable at $x$ as well and
\begin{aligned} (f(x)+g(x))^{\prime} &=f^{\prime}(x)+g^{\prime}(x), \ (c f(x))^{\prime} &=c f^{\prime}(x) \end{aligned}
Proof The result follows from the corresponding rules for limits. The first statement is true because
$$\frac{f(x+h)+g(x+h)-(f(x)+g(x))}{h}=\underbrace{\frac{f(x+h)-f(x)}{h}}{\rightarrow f^{\prime}(x)}+\underbrace{\frac{g(x+h)-g(x)}{h}}{\rightarrow g^{\prime}(x)}$$
as $h \rightarrow 0$. The second statement follows similarly.
Linearity together with the differentiation rule $\left(x^{m}\right)^{\prime}=m x^{m-1}$ for powers implies that every polynomial is differentiable. Let
$$p(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0} .$$
Then its derivative has the form
$$p^{\prime}(x)=n a_{n} x^{n-1}+(n-1) a_{n-1} x^{n-2}+\cdots+a_{1} .$$
For example, $\left(3 x^{7}-4 x^{2}+5 x-1\right)^{\prime}=21 x^{6}-8 x+5$.
The following two rules allow one to determine the derivative of products and quotients of functions from their factors.

## 计算机代写|计算方法代写ALGORITHMIC METHODS代写|Interpretations of the Derivative

$$y=f^{\prime}\left(x_{0}\right)\left(x-x_{0}\right)+f\left(x_{0}\right) .$$

(i) 图形的切线 $f$ 在这一点上 $(0,1)$ 是
$$y=f^{\prime}(0) \cdot(x-0)+f(0)=1$$

(ii) 图形的切线 $f$ 在这一点上 $(1,2)$ 是
$$y=f^{\prime}(1)(x-1)+2=4(x-1)+2=4 x-2 .$$

$$f(x)=f\left(x_{0}\right)+f^{\prime}\left(x_{0}\right)\left(x-x_{0}\right)+R\left(x, x_{0}\right),$$

$$\lim x \rightarrow x 0 \frac{R\left(x, x_{0}\right)}{x-x_{0}}=0$$

$$R\left(x, x_{0}\right)=f(x)-f\left(x_{0}\right)-f^{\prime}\left(x_{0}\right)\left(x-x_{0}\right)$$

$$\frac{f(x)-f\left(x_{0}\right)}{x-x_{0}} \rightarrow f^{\prime}\left(x_{0}\right) \quad \text { as } x \rightarrow x_{0}$$

## 计算机代写|计算方法代写ALGORITHMIC METHODS代写|Differentiation Rules

$$(f(x)+g(x))^{\prime}=f^{\prime}(x)+g^{\prime}(x),(c f(x))^{\prime} \quad=c f^{\prime}(x)$$

$$\frac{f(x+h)+g(x+h)-(f(x)+g(x))}{h}=\underbrace{\frac{f(x+h)-f(x)}{h}} \rightarrow f^{\prime}(x)+\underbrace{\frac{g(x+h)-g(x)}{h}} \rightarrow g^{\prime}(x)$$

$$p(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0} .$$

$$p^{\prime}(x)=n a_{n} x^{n-1}+(n-1) a_{n-1} x^{n-2}+\cdots+a_{1} .$$

## 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。