Section: 平均变化率 | 1.4 平均变化率

Section outline

Suppose you and your family are planning to drive to Disney World. You know that the trip includes the following approximate sections:
::假设你和你的家人正计划开车去迪斯尼世界

200 Miles East

100 Miles South

350 Miles East

50 Miles North

150 Miles East

450 Miles South

How could you generalize the description of the journey?
::你怎么能概括地描述旅程?

Average Rate of Change
::平均变化率

Consider the following situation: you are on a week long road trip with your friend. When you begin to drive on the second day, you have already driven a total of 200 miles. After 6 hours of driving on the second day, you have driven a total of 500 miles. On average, how many miles did you drive per hour on the second day of the trip?
::考虑一下以下情形:您正与您的朋友一起在一周长的路上旅行。当您在第二天开始开车时,您已经驾驶了总共200英里的车程。在第二天开车6小时后,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多少英里?

The graph below shows this situation, with the x axis representing the number of hours driving (on the second day), and the y axis representing the number of miles driven. The first point on the graph, (0, 200), says that at the beginning of the second day you have already driven 200 miles. The second point on the graph, (6, 500), says that after 6 hours of driving on the second 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(第二天),Y轴代表驾驶的英里数。图上的第一个点(0,200)表示,在第二天初,你已经驾驶了200英里。图上的第二个点(6,500)表示,在第二天驾驶6小时之后,你总共驾驶了500英里。

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 where you drove 70 miles and another hour where 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。我们将此速率称为平均变化率,因为它是6小时的平均数。也就是说,你不一定每小时驾驶50英里。你开车的车速可能是一个小时70英里,另一个小时只开30英里。

We can represent the average rate of change on the graph by indicating how much each quantity has changed: The y values increased by 300, and the x 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:
::我们可以通过显示每个数量变化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 :

\begin{aligned} Average rate of change = \frac{change in y}{change in x} \end{aligned}

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.
::我们可以检查一个函数的平均变化率,无论它是作为数据(如上例所示)还是以等式表示的。

Notice that the average rate of change of the function $\begin{aligned} f (x) = 4 x \end{aligned}$ is the slope of the line, 4. While a linear function has a constant slope, other functions, such as $\begin{aligned} f (x) = x^{2} \end{aligned}$ , will not. You will explore this idea in greater detail in your study of calculus.
::注意函数 f( x) = 4x 的平均变化速率是线性斜度, 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。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about your trip to Disney World.
::早些时候,你对去迪士尼世界有问题

One way to generalize the information about your trip is to identify your average rate of change in position from home to Disney World.
::概括你旅行信息的一个方法就是确定你从家到迪士尼世界平均职位变动率

Average Rate of Change = (Change in y ) / (Change in x )
::平均变化率=(变化y) /(变化x)

The trip segments were:
::旅行部分有:

200 Miles East

100 Miles South

350 Miles East

50 Miles North

150 Miles East

450 Miles South

If we assume that travelling South is - y , and traveling East is + x :
::如果我们假设旅行的南方是 -y,旅行的东方是+x:

The total change in y = -100 +50 -450 = -500
::y=-100+50-450=-500的总变化

The total change in x = +200 +350 +150 = 700
::x=+200+350+150=700的总变化

Your trip includes a total of approximately 1200 miles of travel, and you will average 5 miles South for every 7 miles East. Your average rate of change is -5/7.
::您的行程包括总共大约1200英里的行程, 并且您平均每东经7英里, 以南5英里。您的平均变化速度是 - 5/ 7 。

Example 2
::例2

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

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

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。

Example 3
::例3

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

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

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 7 - 1 = 6. 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。

Example 4
::例4

What is the average rate of change of the function $\begin{aligned} f (x) = 3 x^{2} \end{aligned}$ 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?

The two points are (2, 12) and (5, 75). The average rate of change is 63/3 = 21.
::这两个点是(2、12)和(5、75),平均变动率为63/3=21。

Example 5
::例5

If Kelli spends $5 at 1pm, $7 at 2:30pm, $12 at 4pm, and $2 at 4:30pm, what is her average rate of spending?
::如果Kelli在下午1点支出5美元,下午2点30分支出7美元,下午4点支出12美元,下午4点30分支出2美元,她的平均支出率是多少?

Kelli spent a total of $26 over a total of 3 1/2 hours:
::Kelli在总共3个半小时中共花费了26美元:

$\begin{aligned} \frac{$ 26}{7 / 2 h r} \end{aligned}$
::267/2hr美元

$\begin{aligned} \frac{$ 52}{7} \end{aligned}$ Multiply top and bottom by 2 to remove the fraction in denominator.
::527美元乘以上下乘以2 来去除分母中的分数。

$\begin{aligned} \approx \frac{$ 7.43}{h r} \end{aligned}$
::+7.43美元小时

Example 6
::例6

What is the average rate of change shown in the image below?
::下面图像显示的平均变化率是多少?

The two points in the image are: $\begin{aligned} (- 6, - 1) : (7, 6) \end{aligned}$ .
::图像中的两点是-6)-17,6)

The average rate of change is the same as the slope of the line.
::平均变化率与线的斜坡相同。

Recall that $\begin{aligned} m = \frac{Δ y}{Δ x} ∴ \frac{y_{2} - y_{1}}{x_{2} - x_{1}} \end{aligned}$
::回想到 m xy2- y1x2- x1

$\begin{aligned} \frac{(6) - (- 1)}{(7) - (- 6)} \end{aligned}$

$\begin{aligned} \frac{7}{13} \end{aligned}$

Review
::回顾

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

x increases from 1 to 3
::x 从1增加为3

x increases from -4 to -1
::x x 从 -4 增加至 -1

Suppose $\begin{aligned} f (1) = 2 \end{aligned}$ and the average rate of change of $\begin{aligned} f \end{aligned}$ between 1 and 5 is 3. Find $\begin{aligned} f (5) \end{aligned}$
::假设f(1)=2,1至5之间平均变化率为F1至5。

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 first 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,他记录了离开家后的全部距离。行程前半段,他的平均速度是多少公里/小时?最后半段?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

On Monday, the price of a gallon of gas was $3.74, and on Saturday, the price had risen to $4.09. What is the average rate of change of the price of a gallon of gas?
::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。一加仑天然气价格的平均变动率是多少?

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?

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。

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?

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?

Let $\begin{aligned} y = f (x) = x^{2} + x + 2 \end{aligned}$
a. Find the average rate of change of y with respect to x between x = −1 and x = 2
b. Draw the graph of f and the graph of the secant line through the points from part a.
::Let y=f(x)=x2+x+2 a. 查找 x = = = 1 和 x = 2 b. 之间 x 之间的y 平均变化速率。通过部分a 的点绘制 f 的图和 seant 线的图。

Amy takes a trip from Chicago to Milwaukee. Due to road construction, she drives the first 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 McDonald’s for 10 minutes for a snack. She drives the next 45 miles at a constant speed of 45 mph.
a. What was Amy's average driving speed for the trip?
b. What is her average speed for the entire trip (including the stop at McDonald’s)?
::Amy从芝加哥到密尔瓦基旅行。由于道路建设,她开车前10英里,车速不变,以20英里的速度。接下来30英里,她保持60英里的固定速度,然后在麦当劳停留10分钟做点心。她开车45英里,车速不变,为45英里。 a。 Amy的平均开车速度是多少? b. 她整个行程的平均速度(包括麦当劳的停靠)是多少?

The weight w ( t ) (in grams) of a tumor t weeks after it forms is given by $\begin{aligned} w (t) = \frac{t^{2}}{15} \end{aligned}$ . Find the average rate at which the tumor is growing during the fifth week after it was formed.
::W(t)=t215给出肿瘤形成后几周的重量 w(t) (克) (单位:克)。找出肿瘤形成后第五个星期内肿瘤生长的平均速率。

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。

Section outline

Average Rate of Change ::平均变化率

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Example 6 ::例6

Review ::回顾

Review (Answers) ::回顾(答复)