1.15 平均变化率

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  平均变化率

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  Introduction
  ::导言

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  Consider the following situation: You are on a week-long road trip with your friend. When you begin to drive on the 2nd day, you have already driven a total of 200 miles. After 6 hours of driving on the 2nd day, you have driven a total of 500 miles. On average, how many miles did you drive per hour on the 2nd day of the trip?
  ::考虑一下以下情形: 您正与您的朋友一起旅行一周。 当您在第二天开始驾驶时, 您已经驾驶了总共200英里的车程。 在第二天驾驶了6小时之后, 您已经驾驶了总共500英里的车程。 平均而言, 您在旅行的第二天每小时驾驶了多少英里的车程 ?

  

   

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  Average Rate of Change
  ::平均变化率

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  The graph below shows this situation, with the x- axis representing the number of hours driving (on the 2nd day), and the y- axis representing the number of miles driven. The 1st point on the graph, (0, 200), says that at the beginning of the 2nd day you have already driven 200 miles. The 2nd point on the graph, (6, 500), says that after 6 hours of driving on the 2nd day, you have driven 500 miles total.
  ::下图显示了这种情况,以x轴代表驾驶时数(第2天),Y轴代表驾驶的英里数。图上的第一个点(0,200)表示,在第2天初,你已经驾驶了200英里。图上的第二个点(6,500)表示,在第2天驾驶了6小时之后,你总共驾驶了500英里。

  

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  Notice that, in total, during your 6 hours of driving, you have driven 300 miles. The rate at which you drove is 300 miles in 6 hours, or 50 miles per hour. We refer to this rate as the average rate of change because it is an average across the 6 hours. That is, you did not necessarily drive 50 miles every hour. There could have been one hour in which you drove 70 miles, and another in which you drove only 30 miles.
  ::请注意,在总共6小时的驾驶期间,你驾驶了300英里。你驾驶的车速是6小时300英里,或每小时50英里。我们把这个速度称为平均变化速度,因为这是6小时的平均速度。也就是说,你不一定每小时驾驶50英里。你可能开了一个小时70英里,另一个只开30英里。

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  We can represent the average rate of change on the graph by indicating how much each quantity has changed: The $\begin{align*}y\end{align*}$ - values increased by 300, and the $\begin{align*}x\end{align*}$ - values increased by 6. The average rate of change is the ratio of these changes in each variable. This is how we can define average rate of change in general:
  ::我们可以通过说明每个数量变化了多少来代表图中的平均变化率:Y值增加了300,x值增加了6。平均变化率是每个变量中这些变化的比率。这就是我们如何定义总体的平均变化率:

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     Average Rate of Change
  ::平均变化率

  $$\begin{align*} \text{Average rate of change }= \frac{\text{change in y}}{\text{change in x}} = \frac{f(x_2)-f(x_1)}{x_2-x_1} \end{align*}$$

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  We can examine the average rate of change of a function, whether it is represented as data, as in the previous example, or by an equation, or graphically.
  ::我们可以检查一个函数的平均变化率,无论它是作为数据(如前一例),还是以方程式或图形表示的。

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  Notice that the average rate of change of the function $\begin{align*}f(x)=4x\end{align*}$ over any interval is the slope of the line, 4. While a linear function has a constant slope, other functions, such as $\begin{align*}f(x)=x^{2}\end{align*}$ , will not. You will explore this idea in greater detail in your study of calculus. 
  ::注意函数 f( x) = 4x 在任何间隔的平均变化率是线的斜度, 4. 虽然线性函数有一个恒定的斜度, 其他函数, 如 f( x) =x2 则不会。 您将在对微积分的研究中更详细地探讨这个想法 。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Find the average rate of change on the given interval:
  ::查找给定间隔的平均变化率 :

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  $\begin{align*}f(x)=x^{2}\end{align*}$ on [0, 2]
  ::[ 0, 2] 上的 f(x) =x2

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  Solution:
  ::解决方案 :

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  $\begin{align*}f(x)=x^{2}\end{align*}$
  :xx)=x2

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  The endpoints of the interval are (0, 0) and (2, 4). Therefore, the change in y is 4 and the change in x is 2. The average rate of change is 4/2 = 2.
  ::间隔的终点是(0,0)和(2,4),因此,y的变动是4,x的变动是2。 平均变化率为4/2=2。

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  Example 2
  ::例2

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  Now let's check if our claim regarding the average rate of change of $\begin{align*}f(x)=4x\end{align*}$ is valid for a specific interval. Find the average rate of change on the given interval
  ::现在让我们检查一下我们关于f(x)=4x平均变化率的索赔是否对特定间隔有效。 查找给定间隔的平均变化率

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  $\begin{align*}f(x)=4x\end{align*}$ on [1, 7]
  ::[1, 7] 上的 f(x)=4x

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  Solution:
  ::解决方案 :

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  $\begin{align*}f(x)=4x\end{align*}$
  :xx)=4x

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  The endpoints of the interval are (1, 4) and (7, 28). Therefore, the change in y is 28 - 4 = 24 and the change in x is $\begin{align*}7-1=6\end{align*}$  The average rate of change is 24/6 = 4. 
  ::间隔的终点是(1,4)和(7,28),因此,y的变动为28-4=24,x的变动为7-1=6,平均变动率为24/6=4。

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  Example 3
  ::例3

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  What is the average rate of change of the function $\begin{align*}f(x)=3x^{2}\end{align*}$ on the interval [2, 5]?
  ::间隔 [2, 5] 的函数 f(x)=3x2 的平均变化速率是多少?

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  Solution:
  ::解决方案 :

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  The two endpoints are (2, 12) and (5, 75). The average rate of change is 63/3 = 21.
  ::两个终点是(2、12)和(5、75),平均变动率为63/3=21。

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  Example 4
  ::例4

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  a) Find the average rate of change of the function $\begin{align*}f(x) = x^2\end{align*}$ as $\begin{align*}x\end{align*}$ varies from 1 to 3.
  ::a) 查找函数f(x)=x2的平均变化率,因为x的变动率从1到3不等。

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  Solution:
  ::解决方案 :

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  The average rate of change is the slope of the line that passes through the two points $\begin{align*}(3, 9)\end{align*}$  and $\begin{align*} (1, 1)\end{align*}$  on the graph.
  ::平均变化率是穿过图中两个点(3,9)和(1,1)的线的斜坡。

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  $$\begin{align*}\frac{f(x_2)-f(x_1)}{x_2 - x_1}&=\frac{f(3)-f(1)}{3 - 1}\\ &=\frac{9-1}{3-1}\\ &=\frac{8}{2}\\ &=4 \end{align*}$$  b)  If the temperature at 1 p.m. was 82°F, and at 9 p.m. was 50°F, what was the average rate of change of temperature over the given time interval?
  ::f(x2)-f(x1)x2-x1=f(3)-f(1)3-1=9-13-1=1=82=4b)如果下午1时的温度为82°F,晚上9时的温度为50°F,在给定时间段的平均温度变化速度是多少?

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  Solution:
  ::解决方案 :

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  The temperature dropped by 32° over a period of 8 hours:  $\begin{align*}\frac{-32}{8} = -4\end{align*}$ , or 4 degrees per hour.
  ::温度在8小时(每小时328°C4或4°C)内下降32°C。

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  c) Brian drove to town to get some milk. He left at 9 a.m., drove 8 miles north, 5 miles west, 3 miles north, and 2 miles west. He arrived at the store at 9:45 a.m. What was his average rate of change in miles per minute ?
  ::Brian开车到镇上取牛奶,上午9时离开,向北8英里,向西5英里,向北3英里,向西2英里,上午9时45分抵达商店。

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  Solution: 
  ::解决方案 :

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  Brian drove a total of 18 miles in a total of 45 minutes: 

  $$\begin{align*}\frac{18 \text{ mi}}{45 \text{ min}} = \frac{2}{5} \text{ miles per minute}\end{align*}$$  d)  If Kelli spends $5 at 1 p.m., $7 at 2:30 p.m, $12 at 4 p.m., and $2 at 4:30 p.m., what is her average rate of spending in dollars per hour?
  ::Brian总共驾驶了18英里,总共45分钟:18 mi45分钟=每分钟25英里d) 如果Kelli在下午1时花费5美元,下午2时30分花费7美元,下午4时花费12美元,下午4时30分花费2美元,她平均每小时花费美元的比例是多少?

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  Solution: 
  ::解决方案 :

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  Kelli spent a total of $26 over a total of 3.5 or $\begin{align*}\tfrac{7}{2}\end{align*}$  hours. Set up the fraction and then multiply the top and bottom by 2 to remove the fraction from the denominator:
  ::Kelli在总共3.5或72小时的时间里总共花费了26美元。 设置分数, 然后将顶部和底部乘以2, 从分母中去除分数 :

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  $$\begin{align*}\frac{\$26}{7/2 \ \mathrm{hrs}} \ \frac{\times \ 2}{\times \ 2} = \frac{$52}{7 \ \mathrm{hrs}} \approx 7.43 \text{ dollars per hour}\end{align*}$$
  ::2x2=527美元 每小时7.43美元

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  e) What is the average rate of change shown in the graph  below?
  :e) 下图显示的平均变动率是多少?

  

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  Solution:
  ::解决方案 :

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  The two points in the graph are  $\begin{align*}(-6, -1) \ \mathrm{and} \ (7, 6)\end{align*}$ .
  ::图表中的两点是(-6,-1)和(7,6)。

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  The average rate of change is the same as the slope of the line.  Recall:  $\begin{align*}m= \frac{\Delta y}{\Delta x}= \frac{y_2 - y_1}{x_2 - x_1}\end{align*}$ .
  ::平均变化率与线的斜坡相同。 提醒: myx=y2-y1x2- x1 。

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  $$\begin{align*}m=\frac{(6) - (-1)}{(7) - (-6)} = \frac{7}{13}\end{align*}$$
  ::m=(6)-(-1)(7)-(-6)=713

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  Summary
  ::摘要

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  • Average rate of change formula:  $\begin{align*} \text{Average rate of change }= \frac{\text{change in y}}{\text{change in x}}\end{align*}$ .
    ::平均变化率公式:平均变化率=变化变化x。
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  • We can examine the average rate of change of a function, whether it is represented as data, as in the previous example, or by an equation, or in a graph.
    ::我们可以检查一个函数的平均变化率,无论它是作为数据(如前一个例子),还是以方程(或以图表)表示的。

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  Review
  ::回顾

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  For $\begin{align*}y = x^3\end{align*}$ , find the average rate of change as:
  ::对于y=x3, 发现平均变化率为:

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  1. x increases from 1 to 3
     ::x 从1增加为3
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  2. x increases from -4 to -1
     ::x x 从 -4 增加至 -1
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  3. Suppose $\begin{align*}f(1) = 2\end{align*}$ and the average rate of change of $\begin{align*}f\end{align*}$ between 1 and 5 is 3. Find $\begin{align*}f(5)\end{align*}$
     ::假设f(1)=2,1至5之间平均变化率为F1至5。
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  4. Jamie went on a bicycle trip and stopped regularly at half-hour intervals. At each break, he recorded his total distance since leaving home. What was his average speed in km/h during the 1st half of the trip? During the last half? Jamie hoped to average at least 11.5km/h over the course of the trip. Did he? Explain.
     ::Jamie骑自行车出行,每隔半小时定期停留一次。每次休息时,他记录了离开家的全程距离。行程第1半段的平均速度是每小时多少公里?最后半段是多少?Jamie希望行程中的平均速度至少为11.5公里/小时。他有没有解释?

  Stops	Time	Distance (km)
  1	0.5	7
  2	1.0	15
  3	1.5	21
  4	2.0	24
  5	2.5	28
  6	3.0	36

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  5. On Monday, the price of a gallon of gas was $3.74. On Saturday, the price had risen to $4.09. What is the average rate of change of the price of a gallon of gas per day?
     ::星期一,一加仑天然气的价格为3.74美元。星期六,价格升至4.09美元。 每天一加仑天然气价格的平均变动率是多少?
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  6. According to census figures, the population of Clovis was 31,194 in 1980 and 32,511 in 2001. What was the average rate of change of the population over that time interval?
     ::根据人口普查数字,1980年克洛维斯的人口为31,194人,2001年为32,511人,在此期间人口的平均变化率是多少?
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  7. Griego and Sons will deliver 12 yd ³ of gravel for $240 and 30 yd ³ for $575. What is the average rate of change of the cost as the number of cubic yards varies from 12 to 30?
     ::Griego和Sonss将交付12 yd3块砂砾,价值240美元,30 yd3美元,价值575美元。
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  8. When a load of 5 pounds is placed on a spring, its length is 6 inches, and when a load of 9 pounds is placed on the spring, its length is 8 inches. What is the average rate of change of the length of the spring as the load varies from 5 pounds to 9 pounds?
     ::当弹簧的长度为5磅时,弹簧的长度为6英寸,弹簧的长度为9磅,弹簧的长度为8英寸。 当弹簧的长度从5磅到9磅不等时,弹簧的平均变化速度是多少?
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  9. Driving at a speed of 75 mph, a Mini Cooper’s fuel efficiency is 29 miles per gallon. If the driver slows to a speed of 60 mph, he will have a fuel efficiency of 34 miles per gallon. What is the average rate of change of the fuel efficiency as the speed drops from 75 mph to 60 mph?
     ::迷你库珀的燃料效率以75英里的速度行驶,每加仑29英里。 如果司机慢到60英里的速度,他的燃料效率将达到每加仑34英里。 当速度从75英里下降到60英里时,燃料效率的平均变化速率是多少?
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  10. Let $\begin{align*}y = f(x) = x^2 + x + 2\end{align*}$ .
      a) Find the average rate of change of y with respect to x between x = −1 and x = 2.
      b) Draw the graph of $\begin{align*}f(x)\end{align*}$  and the graph of the line through (−1,−2) and (2, 4) on the function. c) Find the slope of the line through  (−1,−2) and (2, 4). d) What do you notice about the answers in parts b) and c)?
      ::Lets y=f(x) =x2+x+2. a) 查找 x = = -1 和 x = 2. b) 之间 y 的平均变化率; 在函数上绘制 f(x) 的图和线的图 (-1,-2) 和 2(, 4) 。 c) 查找线的斜度 (-1,-2) 和(2, 4) 。 (d) 关于b部分和c部分的答案,您注意到什么?
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  11. Amy takes a trip from Chicago to Milwaukee. Due to road construction, she drives the 1st 10 miles at a constant speed of 20 mph. For the next 30 miles, she maintains a constant speed of 60 mph and then stops at a diner for 10 minutes. She drives the next 45 miles at a constant speed of 45 mph. a) What is Amy's average driving speed for the trip? b) What is her average speed for the entire trip, including the stop at the diner?
      ::Amy从芝加哥到Milwaukee的行程。由于道路建设,她以20英里的恒定速度驾驶第10英里。接下来的30英里,她保持60英里的恒定速度,然后在餐厅停留10分钟。她以45英里的恒定速度驾驶后45英里。a)Amy的平均驾驶速度是多少?b)她整个行程的平均速度是多少,包括餐厅的停车点?
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  12. The weight w ( t ) (in grams) of a tumor t weeks after it forms is expressed by $\begin{align*}w(t) = \frac{t^2}{15}\end{align*}$ . Find the average rate at which the tumor is growing per day  during the 5th week after it formed.
      ::w(t)=t215表示肿瘤形成后数周的重量w(t) (克)。 找出肿瘤形成后第5周每天生长的平均速率。
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  13. The price of a house increases from $120,000 in 1995 to $250,000 in 2007. What was the average rate of change to the nearest dollar per year?
      ::房屋价格从1995年的120,000美元增加到2007年的250,000美元,年平均变化率是多少?
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  14. Find the rate of change for each set of ordered pairs below. What is the average rate of change of all the sets?
      $$\begin{align*}(2, 8) \ &\mathrm{and} \ (3, 12) \\ (\text{-}1, 14) \ &\mathrm{and} \ (3, 9) \\ (3, 2) \ &\mathrm{and} \ (\text{-}1, 8) \\\end{align*}$$  Write your answer to the average rate of change as a fraction.
      ::查找下方每组定购的一对的变动率。 所有各组的平均变动率是多少? (2,8) 和 (3,12) (1,14) 和 (3,9,3,2) 和 (1,8) 将您对平均变动率的回答写成一个分数。
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  15. The balance in Sam's savings account changed from $1,200 in June to $1,600 in August. What was the average rate of change per month to the nearest dollar? 
      ::Sam储蓄账户的余额从6月的1 200美元改为8月的1 600美元,月平均兑换率是多少?
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  16. Every hour a runner is able to run the same number of miles. If on his 3rd hour of running he has gone 15 miles, and on his 5th hour of running he has gone 25 miles, what is the rate at which he runs?
      ::如果跑步者在跑步的第三个小时跑了15英里,在跑步的第五个小时跑了25英里,他跑的速率是多少?

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  Review (Answers)
  ::回顾(答复)

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  Please see the Appendix.
  ::请参看附录。