Section outline

  • General

    Collapse all Expand all

  • 分立和连续函数

    Is a bank account balance a continuous function? How about number of pets per household? Gallons of gas in your car? Number of days that students rode bikes to school in a given week?
    ::银行账户余额是否是一个连续的功能?每个家庭有多少宠物?你的汽车里加仑的气体?学生在一周内骑自行车上学的天数?

    These are all functions, but they are different types of functions. This Concept is all about learning the difference.
    ::所有这些功能都是功能,但它们是不同类型的功能。这个概念都是为了了解差异。

    Discrete and Continuous Functions
    ::分立和连续函数

    Imagine taking a poll to learn the most popular band in school. You interview a large and representative cross-section of students, asking each one how many CD's they have purchased supporting his/her favorite band. You decide that the band with the most CD's sold should be the 'winner'. Wouldn't you be surprised to see numbers like 3.2 or 5.7 on your graph? Who buys .2 or .7 CD's?
    ::想象一下民意测验来学习学校中最受欢迎的乐队。 你采访了一大批具有代表性的学生, 询问每个学生购买了多少CD支持他/她最喜欢的乐队。 你决定, 最有名的CD乐队应该“ 赢家 ” 。 您在图表上看到3.2 或 5.7 这样的数字难道不感到惊讶吗? 谁买了 2 或 7 CD ?

    Now imagine comparing the CD count to the age of the students to see if some bands are more popular with particular age groups. You add birthdate information to your graph. Would you be surprised to see an average student age of 15.4 or 16.7 years? Of course not, you would probably be a lot more surprised to see an average of exactly 15 or 16 years old.
    ::现在想象一下,将CD数与学生的年龄比较,看看某些乐队是否在特定年龄组中更受欢迎。你在图表中加入出生日期信息。你是否惊讶地看到平均15.4岁或16.7岁的学生年龄? 当然,你可能更惊讶地看到平均15或16岁的学生年龄。

    The difference between these functions is the topic of this lesson. The number of CD's in the first group is a discrete function, since it is very unlikely that someone would purchase a fraction of a CD. The ages in the second count are a continuous function, since people age constantly and particularly at younger ages keep track more accurately than just "years old".
    ::这些功能之间的差异是这一教训的主题。 裁谈会第一组中的裁谈会数量是一个互不关联的功能, 因为某人购买一张CD的可能性极小。 第二组中的年龄是一个连续的功能, 因为人们不断老化, 特别是年龄更小的人比“ 年老” 更准确地跟踪。

    Discrete Functions
    ::分立函数

    A discrete function is a function in which the are each a discrete set of values, rather than an interval in R . Recall from a prior lesson that an interval includes all values between the specified minimum and maximum. If a function is discrete, it does not include all of the values between two given numbers, but rather only specific values in a particular range.
    ::离散函数是一个函数,其中每个函数都是一组离散的值,而不是R.中的间隔。从先前的教训中回顾,一个间隔包括指定最小值和最大值之间的所有值。如果函数是离散的,它不包括两个给定数字之间的全部值,而只是特定范围内的特定值。

    Non-Discrete Functions
    ::非区分函数

    A non-discrete function is one that is continuous either on its entire domain, or on intervals within its domain. The term continuous refers to a function whose graph has no holes or breaks. (Note that this is not a formal definition. To formally define continuity requires that we use the , which we will examine in the next lesson. For now it is sufficient to focus on what the graph looks like.)
    ::非分立函数是在其整个域或在其域内的间隔上连续持续的函数。 连续术语是指图形无孔或断裂的函数。 (注意这不是一个正式定义。 要正式定义连续性, 需要我们使用该函数, 并在下一个课程中加以研究。 现在, 只需关注图表的外观即可 。)

    Examples
    ::实例

    Example 1
    ::例1

    Earlier, you were given some examples of different types of functions.
    ::早些时候,有人向你们举例说明了不同类型的职能。

    Identify the examples given as either discrete or continuous:
    ::确定作为离散或连续实例提供的例子:

    1. A bank account
      ::银行账户

    The balance in a bank account is counted in dollars and cents, any change is countable and quantifiable. This is an example of a discrete function.
    ::银行账户的余额以美元和美分计,任何变动都是可计算和可量化的,这是独立功能的一个例子。

    1. The number of pets in a household
      ::家庭宠物数量

    Discrete function, since one does not generally have a fraction of a pet.
    ::分立功能,因为一个人一般没有宠物的一小部分。

    1. Age of students taking this class
      ::参加这一班的学生年龄

    Continuous function, there is no limit to the level of accuracy you could apply to the age of each student since time is continuous.
    ::连续功能,由于时间是连续的,因此对每个学生的年龄可以适用的准确度没有限制。

    1. Number of days that students rode bikes to school
      ::学生骑自行车上学天数

    Discrete, the question suggests a specific count of number of days.
    ::分明地说,问题暗示了具体天数的计数。

    Example 2
    ::例2

    Identify the function as either continuous or discrete based on the graph:
    ::根据图表将函数指定为连续函数或离散函数:

    This graph shows a continuous function, as there are no holes identified on the line, and also no endpoints.
    ::此图显示一个连续函数, 因为线条上没有识别孔, 也没有端点 。

    Example 3
    ::例3

    Identify the function as either continuous or discrete based on the equation:
    ::根据等式确定该函数为连续函数或离散函数:

    y = x 3 − 3 x

    The function is continuous, as there is no restriction on the values which may be input for x .
    ::函数是连续的,因为对 x 输入的值没有限制。

    Note that this is not a straight line, or even a simple curve like y = x 2 . A graph need not be straight or simple to be continuous.
    ::请注意,这不是一条直线,甚至也不是象 y = x2 这样的简单曲线。 图表不必是直线或简单即可连续。

    For Examples 4 and 5, use the following information.
    ::关于例4和例5,请使用以下信息。

    Mark is working at the local fast food restaurant and earns $7.15 per hour. The following table shows the amount of money he earns by working a particular number of hours per week.
    ::Mark在当地快餐餐厅工作,每小时挣7.15美元,下表显示每周工作数小时挣得的钱。

    Hours Worked Money Earned
    1 $7.15
    3 $21.45
    7 $50.05
    12 $85.80
    15 $107.25

    Example 4
    ::例4

    Does the table above represent data that is "continuous" or "discrete"? Explain your answer. Write an equation that models the data.
    ::上表是否代表“ 连续” 或“ 分层” 的数据? 解释您的答案 。 写入一个模拟数据的方程式 。

    The data can be considered continuous because Mark might work any length of time, resulting in any amount of income.
    ::这些数据可被视为连续数据,因为Mark可能工作任何时间,从而产生任何收入。

    Mark earns $7.15 in each hour he works. His income can be represented by: income = hours X $7.15 or y = $ 7.15 x .
    ::马克每工作一个小时挣7.15美元,其收入可以表现为:收入=7.15美元或7.15美元。

    Example 5
    ::例5

    Use your equation to predict what Mark’s salary will be if he works 40 hours.
    ::使用您的方程式来预测Mark工作40小时后的工资。

    Mark’s income after 40 hours will be $ 7.15 ⋅ 40 = $ 286.00
    ::40小时后马克的收入为7.1540美元=286.00美元。

    Review
    ::回顾

    Identify each of the following variables as being either discrete or continuous.
    ::确定以下每个变量是互不关联的或连续的。

    1. The number of telephone calls received at school in a given week.
      ::一周内学校接到的电话数量。
    2. The weight of a bag of oranges.
      ::一袋橘子的重量
    3. The length of a piece of rope.
      ::一根绳子的长度
    4. Speed of a truck.
      ::一辆卡车的速度。
    5. The number of misdemeanor arrests in a town.
      ::一个城镇的轻罪逮捕数量。
    6. Number of flaws in a bolt of fabric.
      ::织物螺栓的缺陷数量。
    7. The population of the Bald Eagle.
      ::秃鹰岛的居民
    8. A person's age.
      ::一个人的年龄。
    9. Does the graph below represent a Continuous or a Discrete Domain?
      lesson content

      ::下图是否代表一个连续或分立域 ?
    10. The equation  f = 0.305 m can be used to convert meters into feet. Is the domain of this function discrete or continuous?
      ::F=0.305米的方程式可用于将米转换成脚。此函数的域是离散的还是连续的?
    11. Your local gardener tells you that your corn plant will grow 1.25” taller each month. It is now 6’ tall. Write a formula that will tell you how tall your plant is at any time in the future. Is there a continuous or a discrete domain?
      ::您的本地园丁告诉你,您的玉米种植将每月增长1.25” ” , 月增长率将更高。 现在,玉米已经高6 。 写一个公式来说明您的植物在未来任何时候的高度。 是否有连续的或独立的域名?
    12. You can buy T-shirts for $12.00, or hats for $15.00. Write an equation showing how much you will spend Yes for any combination of hats heart and t-shirts (t) that you purchase. Is the domain discrete or continuous?
      ::您可以以$12.00的价格购买T恤衫, 或以$15.00的价格购买帽子。 写一个方程式, 显示您购买的帽子( h) 和 t恤( t) 的组合将花费多少Yes 。 域名是离散的还是连续的 ?

    For questions 13-15, use the following information:
    ::对于问题13-15,请使用以下信息:

    A local neighborhood homeowners association is asking the community residents to participate in a recycling initiative. At the end of each week, each resident is asked submit the number of plastic containers they recycle to the HOA. The data collected was compiled into the following table:
    ::当地邻里住房所有者协会要求社区居民参加回收倡议,每星期结束时,要求每个居民向住房管理局提交回收塑料容器的数量。

    House # Plastic Containers
    1 28
    2 49
    3 35
    4 62
    5 41
    1. Does this table represent data that is "continuous" or "discrete"? Explain your answer
      ::此表格是代表“ 连续” 还是“ 分辨” 的数据吗 ? 解释您的答复
    2. Why can you not really write an equation to model the data?
      ::为什么你不能写一个方程式来模拟数据?
    3. Can you predict how many plastic containers the 6th house on the block will recycle the next week?
      ::你能预测下个星期 街区第6栋房子的塑料容器 将回收多少个塑料容器吗?

    Review (Answers)
    ::回顾(答复)

    Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
    ::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。