## 数学代写|微积分代写Calculus代写|MATH171

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## 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

$x=$到$y$轴的有向距离
$y=f(x)=$到$x$轴的有向距离

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$[-3,3]$。

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

$x$的函数$f$的零点是$x$ -值，$f(x)=0$。

A. $f(x)=3 x^2+x-10$
B. $g(x)=\sqrt{10-x^2}$
C. $h(t)=\frac{2 t-3}{t+5}$

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. $\sqrt{10-x^2}=0$

\begin{aligned} 10-x^2 & =0 \ 10 & =x^2 \ \pm \sqrt{10} & =x \end{aligned}

$g$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。

## 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|MTH191

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。

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$1. Each element in$A$must be matched with an element in$B$. 2. Some elements in$B$may not be matched with any element in$A$. 3. Two or more elements in$A$may be matched with the same element in$B$. 4. An element in$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$f(-1), f(0), f(2), and so on. To find these values, substitute the specified input values into the given equation. \begin{aligned} \text { For } x & =-1, & f(-1) & =3-2(-1)=3+2=5 . \ & \text { For } x=0, & f(0) & =3-2(0)=3-0=3 . \ \text { For } x & =2, & f(2) & =3-2(2)=3-4=-1 . \end{aligned} Althoughf$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。

$$\left{\left(1,9^{\circ}\right),\left(2,13^{\circ}\right),\left(3,15^{\circ}\right),\left(4,15^{\circ}\right),\left(5,12^{\circ}\right),\left(6,10^{\circ}\right)\right}$$

$A$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

$\begin{array}{ccc}\text { Input } & \text { Output } & \text { Equation } \ x & f(x) & f(x)=1-x^2\end{array}$

$$f(x)=3-2 x$$

\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}

$$f(x)=x^2-4 x+7, \quad f(t)=t^2-4 t+7, \quad \text { and } \quad g(s)=s^2-4 s+7$$

$$f(\square)=(\square)^2-4(\square)+7 .$$

## 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|MATH1141

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|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

$$y=m x+b$$

$$y=m x+b$$

## 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。