Section: 摘要:权力、一夫多妻制和合理功能 | 3.12 摘要:权力、一夫多妻制和合理职能

Section outline

Chapter Summary
::章次摘要

In this chapter, we learned about:
::在本章中,我们了解到:

Power Functions
::权力功能

A power function is a function of the form $\begin{aligned} f (x) = a x^{n}, \end{aligned}$ where $\begin{aligned} a \neq 0 \end{aligned}$ and $\begin{aligned} n \end{aligned}$ is a real number.
::功率函数是窗体f(x)=axn的函数,其中 a0 和 n 是实际数字。

If $\begin{aligned} n \end{aligned}$ is even, then the power function is even.
::如果 n 是偶数, 那么权力功能是偶数。

If $\begin{aligned} n \end{aligned}$ is odd, then the power function is odd.
::如果 n 是奇数, 那么功率函数是奇数。

Polynomial Functions
::多元多边函数

A polynomial function has the form $\begin{aligned} P (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + a_{n - 2} x^{n - 2} + \dots + a_{1} x + a_{0} . \end{aligned}$
::多式函数的表单为 P(x) = anxn+an- 1xn-1+an- 2xn-2a1x+a0。

A quadratic function is a special type of polynomial function. Its graph is a parabola that opens up if $\begin{aligned} a > 0, \end{aligned}$ and down if $\begin{aligned} a < 0 \end{aligned}$ . Quadratic functions can be written in the following forms:

Standard form: $\begin{aligned} f (x) = a x^{2} + b x + c \end{aligned}$
::标准窗体: f( x) = 轴2+bx+c

Vertex form: $\begin{aligned} f (x) = a (x - h)^{2} + k \end{aligned}$
::vertex 窗体: f(x)=a(x-h)2+k

Factored form: $\begin{aligned} f (x) = a (x - r_{1}) (x - r_{2}) \end{aligned}$
::乘数表: f(x)=a(x-r1)(x-r2)

::二次函数是多元函数的一种特殊类型。其图形是一个抛物体, 如果 a>0, 则打开。二次曲线函数可以以下列形式写成: 标准窗体: f( x) = ax2+bx+c Vertex 窗体: f( x) = a( x- h)2+k 系数窗体: f( x)= a( x-r1) (x-r2)

Rational Functions
::理性函数

A rationa l function has the form $\begin{aligned} r (x) = \frac{P (x)}{Q (x)}, \end{aligned}$ where $\begin{aligned} P (x) \end{aligned}$ and $\begin{aligned} Q (x) \neq 0 \end{aligned}$ are polynomials.
::合理函数的窗体为 r( x) = P( x)Q( x) , P( x) 和 Q( x) 0 是多数值。

The vertical asymptote is found by setting $\begin{aligned} Q (x) = 0 \end{aligned}$ and solving for $\begin{aligned} x \end{aligned}$ , assuming $\begin{aligned} P (x) \end{aligned}$ and $\begin{aligned} Q (x) \end{aligned}$ have no common factors.
::假设P(x)和Q(x)没有共同因素,则通过设定 Q(x)=0 和解析 x 来发现垂直静态。

The horizontal asymptote is found by this method:

If the degree of the numerator is smaller than the degree of the denominator, then the horizontal asymptote is $\begin{aligned} y = 0 \end{aligned}$ .
::如果分子的大小小于分母的分母的分量,则水平的分数为y=0。

If the degree of the denominator and the numerator are the same, then the horizontal asymptote equals the ratio of the leading coefficients.
::如果分母和分子的大小相同,则水平零点等于主要系数的比率。

If the degree of the numerator is larger than the degree of the denominator, then there is no horizontal asymptote. If the degree is one larger than the denominator, you can use long division to find the oblique asymptote.
::如果分子的大小大于分母的分母的分量,那么就不会有水平的同位数。如果该分量大于分母的大小之一,那么您可以使用长的分隔线来找到倾斜的同位数。

::通过这种方法发现了水平值:如果分子的量小于分母的量小于分母的量小,则水平值的量小为y=0。如果分母和分子的量相同,则水平值等于主要系数的比重。如果分子的量大于分母的量大,则没有水平值的量小。如果分母比分母大,则您可以使用长的分界线来找到负分母。

Holes occur in the graph when the same factor exists in the numerator and denominator.
::当分子和分母中存在相同系数时,在图形中出现空洞。

Find $\begin{aligned} x \end{aligned}$ -intercepts by setting the numerator equal to zero. Solve for any value that doesn't also make the denominator zero.
::通过将分子设置为 0 来查找 x 界面。解决任何不会同时使分母为零的值。

Find $\begin{aligned} y \end{aligned}$ -intercepts by plugging in 0 for $\begin{aligned} x \end{aligned}$ .
::x 以 0 插头查找 Y 界面。

Polynomial and Rational Inequalities
::多元和理性不平等

Isolate the zero and solve for zeros of the function.
::将函数的零分解为零和零分解。

Divide the domain into intervals and use test points to help you build the solution set.
::将域除以间隔, 并使用测试点来帮助您构建解决方案集。

Chapter Application Problem
::应用章节问题

The chief operating officer of a young media company prepared some reports for a presentation to investors. The graph of the production cost data in thousands is below:
::一家年轻媒体公司的首席业务主管编写了一些报告,向投资者介绍。

The portion of the costs displayed would be modeled by a polynomial function. From the appearance of the graph, the polynomial would:
::显示的成本部分将用多面函数模拟。从图形的外观看,多面函数将:

Have domain [0,5] and range [0,20].
::拥有域[0,5]和范围[0,20]。

Be continuous on its domain.
::在其范围内保持连续性。

Decrease on [0, 2] and on [4,5].
::[0,2]和[4,5]的减少。

Increase on [2,4].
::增加[2,4]。

Have zeros when 2,000 and 5,000 units are produced.
::生产2 000个和5 000个单位时为零。

Have maximum costs when 4,000 units are produced.
::生产4 000个单位时成本最高。

In this chapter, we showed that polynomials are appropriate to model this type of graph because they:
::在本章中,我们表明多面体适合于模拟这种类型的图表,因为它们:

Are defined and continuous everywhere.
::到处都有定义和连续性。

Have intervals where they are strictly increasing or decreasing.
::其间隔期严格增加或减少。

Have zeros that can all be found, though some zeros may not be real numbers.
::虽然有些零数可能不是真实数字,

Have relative maximum and minimum values.
::具有相对最高值和最低值。

For the graph of production costs, w hat would be the minimum degree of this polynomial? Does it have a maximum degree?
::对于生产成本图来说,这一多面性的最低程度是多少?它是否具有最高程度?

The function that was found to best fit the production cost data is C(x) = -x ³ + 9x ² -24x + 20 on [0,5].
::发现最符合生产成本数据的函数是 [0,5] C(x) = -x3 + 9x2 - 24x + 20。

The minimum degree for this polynomial is 3, because the end behavior shows the degree is odd. There is no maximum degree, but it must be odd.
::此多元度的最低度为 3, 因为最终行为显示该度为奇特。没有最高度, 但肯定很奇怪。

Th e production costs were maximum when the production level was at 4,000 units. If the costs kept increasing, the company would have an increasingly difficult time ensuring profitability of the company.
::如果生产水平为4 000个单位,生产成本最高,如果成本继续增加,公司将越来越难以确保公司盈利。

Some situations do exhibit continual strong growth rather than the ebb and flow of polynomials. Usually this type of growth in not sustainable in the real world. Hypothetical models for strong consistent growth are helpful to study in theory, just as we studied polynomials in this chapter.
::有些情况确实表现出持续强劲的增长,而不是多边货币的起伏和流动。通常,这种增长在现实世界中是不可持续的。强势持续增长的假想模型有助于理论研究,正如我们在本章中研究多元货币一样。

Review
::回顾

Try the following cumulative review problems to practice the concepts in this chapter:
::尝试下列累积审查问题来实践本章中的概念:

Section outline

Chapter Summary ::章次摘要

Chapter Application Problem ::应用章节问题

Review ::回顾

Chapter Summary
::章次摘要

Chapter Application Problem
::应用章节问题

Review
::回顾