课程： 3.1 指数函数 | The Way To Learn

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指数函数

is one of the most powerful forces in nature. A famous legend states that the inventor of chess was asked to state his own reward from the king. The man asked for a single grain of rice for the first square of the chessboard, two grains of rice for the second square and four grains of rice for the third. He asked for the entire 64 squares to be filled in this way and that would be his reward. Did the man ask for too little, or too much?

Exponential Functions
::指数函数

take the form $\begin{aligned} f (x) = a \cdot b^{x} \end{aligned}$ where $\begin{aligned} a \end{aligned}$ and $\begin{aligned} b \end{aligned}$ are constants. $\begin{aligned} a \end{aligned}$ is the starting amount when $\begin{aligned} x = 0 \end{aligned}$ . $\begin{aligned} b \end{aligned}$ tells the story about the growth. If the growth is doubling then $\begin{aligned} b \end{aligned}$ is 2. If the growth is halving (which would be decay), then $\begin{aligned} b \end{aligned}$ is $\begin{aligned} \frac{1}{2} \end{aligned}$ . If the growth is increasing by 6% then $\begin{aligned} b \end{aligned}$ is 1.06.
::以 f(x) = a-bx 的形式显示,其中 a 和 b 是常数。 a 是 x=0. b 表示增长的起始值。如果增长翻了一番,b 则为 2. 如果增长减半(会衰减),b 等于 12. 如果增长增加6%,b 等于 1.06。

Exponential growth is everywhere. Money grows exponentially in banks. Populations of people, bacteria and animals grow exponentially when their food and space aren’t limited.
::市场增长无处不在。银行的货币成倍增长。当食物和空间不受限制时,人口、细菌和动物的数量就会成倍增长。

Radioactive isotopes like Carbon 14 have something called a half-life that indicates how long it takes for half of the molecules present to decay into other more stable molecules. It takes about 5,730 years for this process to occur which is how scientists can date artifacts of ancient humans.
::碳14等放射性同位素的半衰期表明一半的分子需要多长时间才能腐烂成其他更稳定的分子。这一过程需要大约5,730年的时间才能发生,科学家们可以将古代人类的文物的日期定下来。

Let's say a mummified animal is found preserved on the slopes of an ice covered mountain. After testing, you see that exactly one fourth of the carbon-14 has yet to decay and you want to find out how long ago was this animal alive. How would you do that?
::假设木乃伊动物被发现保存在冰盖山的斜坡上。经过测试,你可以看到,碳-14的四分之一还没有腐烂,你想知道这只动物存活了多久。你会怎么做呢?

Since this problem does not give specific amounts of carbon, it can be inferred that the time will not depend on the specific amounts. One technique that makes the problem easier to work with could be to create an example scenario that fits the one fourth ratio. Suppose 60 units were present when the animal was alive at time zero. This means that 15 units must be present today.
::由于这个问题没有给出具体数量的碳,因此可以推断时间并不取决于具体的数量。使问题更容易解决的一种技术是创造一个符合四分之一比例的范例。假设动物在零时存活时有60个单位存在,这意味着今天必须有15个单位存在。

\begin{aligned} 15 & = a \cdot {(\frac{1}{2})}^{x} \\ 60 & = a \cdot {(\frac{1}{2})}^{0} \end{aligned}

::15=a(12)x60=a(12)0

The second equation yields $\begin{aligned} a = 60 \end{aligned}$ and then the first equation becomes:
::第二个方程式产生a=60,然后第一个方程式产生:

$\begin{aligned} 15 = 60 \cdot {(\frac{1}{2})}^{x} \end{aligned}$
::15=60(12)x

Although you may not yet have the algebraic tools to solve for $\begin{aligned} x \end{aligned}$ , you should still be able to see that $\begin{aligned} x \end{aligned}$ is 2. This does not mean that two years ago the animal was alive, it means that two half life cycles ago the animal was alive. The half life cycle for carbon 14 is 5,730 years so this animal was alive over 11,000 years ago.
::虽然你可能还没有解答x的代数工具,但你还是应该能看到x为2,这并不意味着两年前该动物还活着,它意味着该动物在2个半生命周期之前是活的。碳14的半衰期是5,730年,因此该动物在11,000年前就活了下来。

Now, suppose you invested $100 the day you were born and it grew by 6% every year until you were 100 years old. How would you use an exponential function to determine how much would this investment be worth then?
::现在,假设你出生那天投资了100美元,每年增长6%,直到你100岁。你将如何使用指数函数来决定这个投资值多少呢?

The starting amount is 100 and the growth is 1.06 because it grows by 6% each year. This is enough information to write an exponential function. The $\begin{aligned} x \end{aligned}$ stands for time in years and the $\begin{aligned} f (x) \end{aligned}$ stands for the amount of money in the account. Plugging in to the formula for an exponential function, you will get the equation:
::起始金额为100, 增长为1.06, 因为它每年增长6%。这是足以写入指数函数的信息。 x 代表年数时间, f( x) 代表账户中的钱。插入一个指数函数的公式时, 您将得到公式 :

$\begin{aligned} f (x) = 100 \cdot {1.06}^{x} \end{aligned}$
::f(x) = 1001.06x

Then, you need to substitute in 100 years for $\begin{aligned} x \end{aligned}$ .
::然后,你需要在100年后代替x。

\begin{aligned} f (x) & = 100 \cdot {1.06}^{x} \\ f (x) & = 100 \cdot {1.06}^{100} \\ f (x) & \approx 33, 930.21 \end{aligned}

::f(x) = 1001.06xf(x) = 1001.06100f(x) = 33,930.21

After a century, there will be almost $34,000 in the account. Interest has greatly increased the $100 initial investment.
::在一个世纪之后,帐户中将有将近34 000美元,利息大大增加了100美元的初始投资。

Examples
::实例

Example 1
::例1

Earlier you were asked if the man asked for too little or too much rice if he gets one grain for the first square, two grains for the second, four grains for the third and so on. The number of grains of rice on the last square, the 64th square, would be almost ten quintillion (million million million). That is more rice than is produced in the world in an entire year.
::早些时候,有人问这个男人是否要求过少或过大米,如果他得到第一广场一粒粮食,第二广场两粒粮食,第三广场四粒粮食,等等。最后一个广场即第64广场的稻米数量几乎是十万五千万(百万),这比全世界全年的稻米产量还要多。

$\begin{aligned} 2^{63} = 9, 223, 372, 036, 854, 775, 808 \end{aligned}$

Example 2
::例2

Suppose forty rabbits are released on an island. The rabbits mate once every four months and produce up to 4 offspring who also produce more offspring four months later. Estimate the number of rabbits on the island in 3 years if their population grows exponentially. Assume half the population is female.
::假设40只兔子在一个岛上被释放。兔子队每四个月生一次,并产生4个后代,四个月后还生出更多后代。如果岛上的兔子人口成倍增长,估计3年内岛上的兔子数量。占人口的一半是女性。

Even though parts of this problem are unrealistic, it serves to illustrate how quickly exponential growth works. Forty is the initial amount so $\begin{aligned} a = 40 \end{aligned}$ . At the end of the first 4 month period 20 female rabbits could have their litters and up to 80 newborn rabbits could be born. The population has grown from 40 to 120 which means tripled. Thus, $\begin{aligned} b = 3 \end{aligned}$ . The last thing to remember is that the time period is in 4 month periods. Three years must be 9 periods.
::尽管部分问题不切实际,但它有助于说明指数增长的迅速程度。40是a=40的初始数量。在头4个月期结束时,20只母兔子可以生垃圾,80只新生兔子可以出生。人口从40个增加到120个,这意味着人口增长了三倍。b=3最后要记住的是,时间期限是4个月。3年必须是9年。

$\begin{aligned} f (x) = 40 \cdot 3^{9} = 787, 320 \end{aligned}$
::f(x)=4039=787,320

So after three years, there could be up to 787,320 rabbits!
::所以三年后可能有多达787,320只兔子!

Example 3
::例3

Completely analyze the following exponential function.
::完全分析以下指数函数。

$\begin{aligned} f (x) = e^{x} \end{aligned}$
:fx) =ex

Analyze in this context means to define all the characteristics of a function .
::在这方面分析意味着界定函数的所有特性。

Domain : $\begin{aligned} x ϵ (- \infty, \infty) \end{aligned}$
::域 : x (,)

Range : $\begin{aligned} y ϵ (0, \infty) \end{aligned}$
::范围: y (0, )

Increasing: $\begin{aligned} x ϵ (- \infty, \infty) \end{aligned}$
::增加: x (,)

Decreasing: NA
::下降: NA

Zeroes : None
::零:无

Intercepts : (0, 1)
::拦截: (0, 1)

Maximums: None
::最大值:无

Minimums: None
::最低限数:无

Asymptotes : $\begin{aligned} y = 0 \end{aligned}$ as $\begin{aligned} x \end{aligned}$ gets infinitely small (Horizontal asymptote)
::单位数 : y=0 x 变得无限小时, y=0

Holes: None
::洞洞:无

Example 4
::例4

Identify which of the following functions are exponential functions and which are not.
::标明下列哪些函数是指数函数,哪些不是指数函数。

$\begin{aligned} y = x^{6} \end{aligned}$
::y=x6 y=x6

$\begin{aligned} y = 5^{x} \end{aligned}$
::y=5x y=5x

$\begin{aligned} y = 1^{x} \end{aligned}$
::y=1x y=1x

$\begin{aligned} y = x^{x} \end{aligned}$
::y=xxx y=xx

$\begin{aligned} y = x^{\frac{1}{2}} \end{aligned}$
::y=x12 y=12

Exponential functions are of the form $\begin{aligned} y = a \cdot b^{x} \end{aligned}$
::指数函数为y=abx

a. $\begin{aligned} y = x^{6} \end{aligned}$ is not an exponential function because $\begin{aligned} x \end{aligned}$ is not in the exponent .
::a. y=x6 不是一个指数函数,因为 x 不在指数内。

b. $\begin{aligned} y = 5^{x} \end{aligned}$ Exponential function.
::b. y=5x指数函数。

c. $\begin{aligned} y = 1^{x} \end{aligned}$ Not a true exponential function because $\begin{aligned} y \end{aligned}$ is always 1 which is a constant function.
::c. y=1x 不是一个真正的指数函数,因为 y总是 1, 是一个不变函数。

d. $\begin{aligned} y = x^{x} \end{aligned}$ Not an exponential function because $\begin{aligned} x \end{aligned}$ is both the base and power of the exponent.
::d. y=xx 不是一个指数函数,因为x既是指数的基数,也是指数的功率。

e. $\begin{aligned} y = x^{\frac{1}{2}} \end{aligned}$ Not an exponential function.
::e. y=x12 不是一个指数函数。

Example 5
::例5

Write the exponential function that passes through the following points: $\begin{aligned} (0, 3), (1, \frac{3}{e}) \end{aligned}$ .
::写入通过下列点的指数函数: (0) 3,(1,3) (1,3e)。

The starting number is $\begin{aligned} a = 3 \end{aligned}$ . This number is changed by a factor of $\begin{aligned} \frac{1}{e} \end{aligned}$ which is $\begin{aligned} b \end{aligned}$ .
::起始数为 a=3。此数字被乘以 1 e 的乘数改为 b 。

$\begin{aligned} f (x) = 3 {(\frac{1}{e})}^{x} = 3 e^{- x} \end{aligned}$
:xx)=3(1e)x=3e-x

Summary

Exponential functions take the form $\begin{aligned} f (x) = a b^{x}, \end{aligned}$ where $\begin{aligned} a \end{aligned}$ represents the start and $\begin{aligned} b \end{aligned}$ represents how fast the growth is.
::指数函数为 f(x) =abx,其中表示起始数,b表示增长速度。

Review
::回顾

1. Explain what makes a function an exponential function. What does its equation look like?
::1. 解释一个函数是什么使指数函数成为指数函数。它的方程式长什么样?

2. Is the domain for all exponential functions all real numbers?
::2. 所有指数函数的域是否都是真实数字?

3. How can you tell from its equation whether or not the graph of an exponential function will be increasing?
::3. 你如何从其方程式中看出指数函数的图形是否会增加?

4. How can you tell from its equation whether or not the graph of an exponential function will be decreasing?
::4. 你如何从其方程式中看出指数函数的图形是否会下降?

5. What type of asymptotes do exponential functions have? Explain.
::5. 指数函数具有何种类型的微量函数?解释。

6. Suppose you invested $4,500 and it grew by 4% every year for 30 years. How much would this investment be worth after 30 years?
::6. 假设你投资了4,500美元,30年来每年增长4%。 30年后,这项投资值多少?

7. Suppose you invested $10,000 and it grew by 12% every year for 40 years. How much would this investment be worth after 40 years?
::7. 假设你投资了10 000美元,40年来每年增长12%。 40年后,这项投资值多少?

Write the exponential function that passes through the following points.
::写入通过下列点的指数函数。

8. (0, 5) and (1, 25)
::8. (0,5)和(1,25)

9. (0, 2) and (1, 8)
::9. (0,2)和(1,8)

10. (0, 16) and (2, 144)
::10. (0,16)和(2,144)

11. (1, 4) and (3, 36)
::11. (第1、4和第3、36段)

12. (0, 16) and (3, 2)
::12. (0,16)和(3,2)

13. (0, 81) and (2, 9)
::13. (0,81)和(2,9)

14. (1, 144) and (3, 12)
::14. (第1、144段)和(第3、12段)

15. Explain why for exponential functions of the form $\begin{aligned} y = a \cdot b^{x} \end{aligned}$ the $\begin{aligned} y \end{aligned}$ -intercept is always the value of $\begin{aligned} a \end{aligned}$ .
::15. 解释为什么y=abx表单的指数函数y-interview总是值a。

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。

章节大纲

常规

指数函数

Exponential Functions ::指数函数

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)