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# 数学代写|微积分代写Calculus代考|Even Functions and Odd Functions: Symmetry

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## 数学代写|微积分代写Calculus代考|Even Functions and Odd Functions: Symmetry

The graphs of even and odd functions have special symmetry properties.
DEFINITIONS A function $y=f(x)$ is an
\begin{aligned} & \text { even function of } x \text { if } f(-x)=f(x), \ & \text { odd function of } x \quad \text { if } f(-x)=-f(x), \end{aligned}
for every $x$ in the function’s domain.
The names even and odd come from powers of $x$. If $y$ is an even power of $x$, as in $y=x^2$ or $y=x^4$, it is an even function of $x$ because $(-x)^2=x^2$ and $(-x)^4=x^4$. If $y$ is an odd power of $x$, as in $y=x$ or $y=x^3$, it is an odd function of $x$ because $(-x)^1=-x$ and $(-x)^3=-x^3$

The graph of an even function is symmetric about the $\boldsymbol{y}$-axis. Since $f(-x)=f(x)$, a point $(x, y)$ lies on the graph if and only if the point $(-x, y)$ lies on the graph (Figure 1.12a). A reflection across the $y$-axis leaves the graph unchanged.

The graph of an odd function is symmetric about the origin. Since $f(-x)=-f(x)$, a point $(x, y)$ lies on the graph if and only if the point $(-x,-y)$ lies on the graph (Figure 1.12b). Equivalently, a graph is symmetric about the origin if a rotation of $180^{\circ}$ about the origin leaves the graph unchanged. Notice that the definitions imply that both $x$ and $-x$ must be in the domain of $f$.

## 数学代写|微积分代写Calculus代考|Common Functions

A variety of important types of functions are frequently encountered in calculus.
Linear Functions A function of the form $f(x)=m x+b$, where $m$ and $b$ are fixed constants, is called a linear function. Figure 1.14a shows an array of lines $f(x)=m x$. Each of these has $b=0$, so these lines pass through the origin. The function $f(x)=x$ where $m=1$ and $b=0$ is called the identity function. Constant functions result when the slope is $m=0$ (Figure 1.14b).

DEFINITION Two variables $y$ and $x$ are proportional (to one another) if one is always a constant multiple of the other-that is, if $y=k x$ for some nonzero constant $k$.
If the variable $y$ is proportional to the reciprocal $1 / x$, then sometimes it is said that $y$ is inversely proportional to $x$ (because $1 / x$ is the multiplicative inverse of $x$ ).

Power Functions A function $f(x)=x^a$, where $a$ is a constant, is called a power function. There are several important cases to consider.

(a) $f(x)=x^a$ with $a=n$, a positive integer.
The graphs of $f(x)=x^n$, for $n=1,2,3,4,5$, are displayed in Figure 1.15. These functions are defined for all real values of $x$. Notice that as the power $n$ gets larger, the curves tend to flatten toward the $x$-axis on the interval $(-1,1)$ and to rise more steeply for $|x|>1$. Each curve passes through the point $(1,1)$ and through the origin. The graphs of functions with even powers are symmetric about the $y$-axis; those with odd powers are symmetric about the origin. The even-powered functions are decreasing on the interval $(-\infty, 0]$ and increasing on $[0, \infty)$; the odd-powered functions are increasing over the entire real line $(-\infty, \infty)$.

(b) $f(x)=x^a$ with $a=-1$ or $a=-2$.
The graphs of the functions $f(x)=x^{-1}=1 / x$ and $g(x)=x^{-2}=1 / x^2$ are shown in Figure 1.16. Both functions are defined for all $x \neq 0$ (you can never divide by zero). The graph of $y=1 / x$ is the hyperbola $x y=1$, which approaches the coordinate axes far from the origin. The graph of $y=1 / x^2$ also approaches the coordinate axes. The graph of the function $f$ is symmetric about the origin; $f$ is decreasing on the intervals $(-\infty, 0)$ and $(0, \infty)$. The graph of the function $g$ is symmetric about the $y$-axis; $g$ is increasing on $(-\infty, 0)$ and decreasing on $(0, \infty)$.

(c) $a=\frac{1}{2}, \frac{1}{3}, \frac{3}{2}$, and $\frac{2}{3}$.
The functions $f(x)=x^{1 / 2}=\sqrt{x}$ and $g(x)=x^{1 / 3}=\sqrt[3]{x}$ are the square root and cube root functions, respectively. The domain of the square root function is $[0, \infty)$, but the cube root function is defined for all real $x$. Their graphs are displayed in Figure 1.17, along with the graphs of $y=x^{3 / 2}$ and $y=x^{2 / 3}$. (Recall that $x^{3 / 2}=\left(x^{1 / 2}\right)^3$ and $x^{2 / 3}=\left(x^{1 / 3}\right)^2$.)
Polynomials A function $p$ is a polynomial if
$$p(x)=a_n x^n+a_{n-1} x^{n-1}+\cdots+a_1 x+a_0$$
where $n$ is a nonnegative integer and the numbers $a_0, a_1, a_2, \ldots, a_n$ are real constants (called the coefficients of the polynomial). All polynomials have domain $(-\infty, \infty)$. If the leading coefficient $a_n \neq 0$, then $n$ is called the degree of the polynomial. Linear functions with $m \neq 0$ are polynomials of degree 1 . Polynomials of degree 2 , usually written as $p(x)=a x^2+b x+c$, are called quadratic functions. Likewise, cubic functions are polynomials $p(x)=a x^3+b x^2+c x+d$ of degree 3. Figure 1.18 shows the graphs of three polynomials. Techniques to graph polynomials are studied in Chapter 4 .

## 数学代写|微积分代写Calculus代考|Even Functions and Odd Functions: Symmetry

\begin{aligned} & \text { even function of } x \text { if } f(-x)=f(x), \ & \text { odd function of } x \quad \text { if } f(-x)=-f(x), \end{aligned}

## 数学代写|微积分代写Calculus代考|Common Functions

(a) $f(x)=x^a$与$a=n$为正整数。

(b) $f(x)=x^a$与$a=-1$或$a=-2$。

(c) $a=\frac{1}{2}, \frac{1}{3}, \frac{3}{2}$和$\frac{2}{3}$。

$$p(x)=a_n x^n+a_{n-1} x^{n-1}+\cdots+a_1 x+a_0$$

## 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。