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编号e

The width $\begin{aligned} w \end{aligned}$ in centimeters of a tortoise can be modeled by the function $\begin{aligned} w = 119 - 77 e^{- 0.178 t}, \end{aligned}$ where $\begin{aligned} t \end{aligned}$ is the tortoise's age in years ¹ . The $\begin{aligned} e \end{aligned}$ in the function is a number. We discuss this number below.
::乌龟的宽度(以厘米计)可以通过函数 w=119-77e-0.178t 来模拟。其中 t 是乌龟的岁数。函数中的 e 是一个数字。我们下面讨论这个数字。

lesson content

The Number e
::编号e

There are many special numbers in mathematics: $\begin{aligned} π \end{aligned}$ , 0, 1, $\begin{aligned} i \end{aligned}$ . Here, we will introduce another special number that is known only by a letter, $\begin{aligned} e \end{aligned}$ . It is called the Euler number , named after the Swiss mathematician Leonhard Euler, who popularized the use of the letter $\begin{aligned} e \end{aligned}$ for the constant . Credit for discovery of the constant itself goes to another important Swiss mathematician, Jacob Bernoulli, and his study of sequences in compound interest, a topic we will cover later in this chapter ² .
::数学中有许多特殊数字: , 0, 1, i。在这里, 我们将引入另一个特殊数字, 仅以字母为名, 即。它叫做 Euler 号, 以瑞士数学家Leonhard Euler 命名, 他为常数普及了字母 e 的使用。发现常数本身的功劳, 由另一位重要的瑞士数学家Jacob Bernoulli 提供, 以及他对复合兴趣序列的研究, 我们将在本章稍后讨论这个专题。

One way to calculate $\begin{aligned} e \end{aligned}$ is with the following formula :
::计算e的一个方法就是使用以下公式:

The Value of $\begin{aligned} e \end{aligned}$
::e 值的e值

\begin{aligned} e = {(1 + \frac{1}{n})}^{n} as n \to \infty \end{aligned}

As $\begin{aligned} n \to \infty \end{aligned}$ ( $\begin{aligned} n \end{aligned}$ approaches infinity) means that the values of $\begin{aligned} n \end{aligned}$ get increasingly large. Let's see how this works in an investigation.
::当NQQQ(无穷无尽)意味着n的值越来越大。让我们看看在调查中如何运作。

Investigation: Finding the values of $\begin{aligned} {(1 + \frac{1}{n})}^{n} \end{aligned}$ as $\begin{aligned} n \end{aligned}$ gets larger
::调查:1+1nn)n值随n值增加而查找

1. Copy the table below and fill in the blanks. Round each entry to the nearest 4 decimal places.
::1. 复制下表并填写空白。将每个条目四舍五入到最接近小数点后4位。

$\begin{aligned} n \end{aligned}$	1	2	3	4	5	6	7	8
$\begin{aligned} {(1 + \frac{1}{n})}^{n} \end{aligned}$	$\begin{aligned} {(1 + \frac{1}{1})}^{1} = 2 \end{aligned}$	$\begin{aligned} {(1 + \frac{1}{2})}^{2} = 2.25 \end{aligned}$

2. Does it seem like the numbers in the table are approaching a certain value? What do you think the number is?
::2. 表中的数字似乎接近某一数值吗?你认为数字是什么?

3. Find $\begin{aligned} {(1 + \frac{1}{100})}^{100} \end{aligned}$ and $\begin{aligned} {(1 + \frac{1}{1, 000})}^{1, 000} \end{aligned}$ . Does this change your answer from Number 2?
::3. 查找(1+1100100100)100和(1+11 000)1 000。这是否改变了您对2号的答复?

4. Fill in the blanks: As $\begin{aligned} n \end{aligned}$ approaches ___________, ___________ approaches $\begin{aligned} e \approx 2.718281828459 \dots \end{aligned}$
::4. 填空:随着n接近_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

We define $\begin{aligned} e \end{aligned}$ as the number that $\begin{aligned} {(1 + \frac{1}{n})}^{n} \end{aligned}$ approaches as $\begin{aligned} n \to \infty \end{aligned}$ . $\begin{aligned} e \end{aligned}$ is an irrational number with the first 12 decimal places above. We can approximate it as 2.72.
::我们将(1+1n)n接近的数值定义为n.e.e.是一个不合理的数字,前12位小数点位数以上。我们可以将其近似为2.72。

by Numberphile covers other ways to determine $\begin{aligned} e \end{aligned}$ .
::= 数字哲学涵盖确定e的其他方法。

Example 1
::例1

Simplify the following expressions with $\begin{aligned} e \end{aligned}$ :
::以 e 简化以下表达式:

a. $\begin{aligned} e^{2} \cdot e^{4} \end{aligned}$
::a. e2当e4

b. $\begin{aligned} 2 e^{- 3} \cdot e^{2} \end{aligned}$
::b. 2e-3e2

c. $\begin{aligned} \frac{4 e^{6}}{16 e^{2}} \end{aligned}$
::c. 4e616e2

Solution: Even though the base is a letter that represents a number, the rules for exponents still hold.
::解决方案:即使基数是代表数字的字母,但推论者的规则依然有效。

a. According to the product rule of exponents, if the bases are the same, you can just add the exponents: $\begin{aligned} e^{2} \cdot e^{4} = e^{2 + 4} = e^{6} \end{aligned}$ .
::a. 根据指数的产物规则,如果基准相同,只需加上指数:e2e4=e2+4=e6。

b. $\begin{aligned} 2 e^{- 3} \cdot e^{2} = 2 e^{- 3 + 2} = 2 e^{- 1} = \frac{2}{e} \end{aligned}$
::b. 2e-3e2=2e-3+2=2e-1=2e

c. The quotient rule of expo nents holds as well; if the bases are the same, you can subtract the exponents: $\begin{aligned} \frac{4 e^{6}}{16 e^{2}} = \frac{e^{6 - 2}}{4} = \frac{e^{4}}{4} \end{aligned}$ .
::c. 引文的商数规则也保持不变;如果基数相同,可以减去引文:4e616e2=e6-24=e44。

by CK-12 demonstrates how to apply the rules of exponents with $\begin{aligned} e \end{aligned}$ in an example.
::CK-12 显示如何在示例中应用参赛者的规则。

The Function $\begin{aligned} y = e^{x} \end{aligned}$
::函数 y=ex

$\begin{aligned} e \end{aligned}$ can be the base of an exponential function because it is positive and not equal to 1. As we will see in this chapter, there are many applications for this function.
::e 可以是指数函数的基数,因为它是正数,不等于1。正如我们在本章中看到的那样,这一函数有许多应用。

Example 2
::例2

Sketch the graph $\begin{aligned} y = e^{x} \end{aligned}$ . Identify the asymptote , $\begin{aligned} y \end{aligned}$ - intercept , domain and range .
::绘制图形 y = ex 。标明无线、 y 界面、域和范围。

Solution: As you would expect, because $\begin{aligned} e \approx 2.72 \end{aligned}$ , the graph of $\begin{aligned} e^{x} \end{aligned}$ will lie between the curves of $\begin{aligned} y = 2^{x} \end{aligned}$ and $\begin{aligned} y = 3^{x} . \end{aligned}$
::解答: 正如您所期望的, 因为 e2. 72, ex 的图将位于 y= 2x 和 y= 3x 的曲线之间。

lesson content

Like other that have not been transformed, the asymptote is $\begin{aligned} y = 0 \end{aligned}$ and the $\begin{aligned} y \end{aligned}$ -intercept is (0, 1). The domain is all real numbers and the range is all positive real numbers, $\begin{aligned} y > 0 \end{aligned}$ .
::和尚未变换的其他数据一样,最小值为y=0, y 拦截值为( 0, 1) 。域名是所有真实数字, 范围为正数, y > 0 。

by utexascnsquest shows how to graph $\begin{aligned} y = e^{x} \end{aligned}$ .
::以 utxascnsquest 显示如何图形 y=ex 。

Example 3
::例3

Sketch the graph of $\begin{aligned} y = e^{x} - 4 \end{aligned}$ .
::绘制 y=ex - 4 的图形。

Solution: This is the same as the graph above, except the y -values have all been reduced by 4, resulting in a vertical shift down 4 units.
::溶液:这与上图相同,只有y值减少了4,导致垂直向下移动4个单位。

lesson content

Example 4
::例4

Sketch the graph of $\begin{aligned} y = e^{(x + 2)} + 1 \end{aligned}$ .
::绘制 y=e( x+2) +1 的图形。

Solution: There are two transformations here. Like Example 3, there is a vertical shift of 1 unit up. The $\begin{aligned} x + 2 \end{aligned}$ in the exponent indicates a horizontal shift to the left by 2 units. A sketch appears below.
::解决方案 : 这里有两个转换。与例 3 一样, 垂直移动 1 个单位向上。引言中的 x+ 2 表示向左的 2 个单位向左水平移动。以下为草图。

lesson content

Example 5
::例5

The width w in centimeters of a tortoise can be modeled by the function $\begin{aligned} w = 119 - 77 e^{- 0.178 t}, \end{aligned}$ where t is the tortoise's age in years. Sketch a graph of the function and approximate the width of a tortoise after 3 years.
::乌龟的宽度(以厘米计)可以通过函数 w=119-77e-0.178t 来模拟,其中 t 是乌龟的岁数。绘制一个函数图,并大致显示三年后乌龟的宽度。

Solution: To sketch a graph of this function, it will be helpful to restate it more like a transformed version of an exponential function.
::解答: 要绘制此函数的图解, 重述它更像是指数函数的变换版本, 将会很有帮助。

\begin{aligned} w & = 119 - 77 e^{- 0.178 t} \\ w & = - 77 e^{- 0.178 t} + 119 & vertical shift up by 119 \\ w & = - 77 (e^{- 0.178})^{t} + 119 & power rule of exponents \\ w & = - 77 (0.84)^{t} + 119 & dilation by 77 and reflection across the t-axis \end{aligned}

::w=119-77e-0178tw=-77e-0.178tw=-77e-0.178t+119 以119w=-77(e-0.178t+119 power exponentsw=-77(0.84t+119clation)=-77(0.84t+119clation)=-77e-0.178tw=-77e-0.178t+119 以119w=-77w=-77(e-0.178t+119)+ 垂直移动上移以119w=-77(e-0.178t+119power 规则) exponentsw=-77(0.84t+119clationxxx)+77(77)xxxxxx和反射整个t-xasis

A graph of the function in Desmos yields
::Desmos 收益函数图

Notice that the width of the tortoise's shell after 3 years is approximately 74 centimeters.
::注意龟壳3年后的宽度大约是74厘米

by CK-12 reviews this example.
::由 CK-12 审查此示例。

Feature: The Bacteria Battle
::特色:细菌战斗

by Meredith Beaton
::梅雷迪思·贝顿(签名)

Many infectious diseases, including some strains of pneumonia, tuberculosis (which has infected approximately 1/3 of the world's population), gonorrhea, strep, and many others, are caused by bacteria. Bacteria are microscopic, single-celled organisms shaped as spheres, rods, or spirals. Not all bacteria are bad; in fact, only about 1% of bacteria cause infection in humans.
::许多传染性疾病,包括肺炎、肺结核(感染了世界人口的三分之一左右)、淋病、病菌病和其他许多传染病,都是细菌造成的。细菌是微小的、单细胞的有机体,形成如球体、棒子或螺旋状。并非所有细菌都是坏的;事实上,只有大约1%的细菌对人类造成感染。

Why It Matters
::为何重要

Antibiotics have a variety of mechanisms by which they kill bacteria and cure infectious diseases. However, killing the bacteria is not always as simple as it seems: bacteria are living organisms and have the ability to adapt to changes in the environment, including developing defenses and resistance against antibiotics. When bacteria mutate in such a way as to become resistant to antibiotics, they can no longer be killed by that antibiotic. These antibiotic-resistant bacteria can then reproduce and multiply freely. Not only can bacteria pass on antibiotic resistance by multiplying, they can also transfer resistance to other bacteria via special messaging systems.
::抗生素具有杀菌和治疗传染病的各种机制。然而,杀菌并非总能像看起来那样简单:细菌是活生物体,有能力适应环境的变化,包括发展抗生素防御和抗抗抗药性。当细菌突变以抗抗抗生素的方式出现时,它们不能再被抗生素杀死。这些抗生素抗生素的细菌可以自由地繁殖和繁殖。细菌不仅能够通过繁殖通过抗生素抗体,还可以通过特殊信息系统将抗体转移到其他细菌。

When healthcare providers prescribe a course of antibiotics to a patient with a bacterial infection, they consider how long the antibiotics need to be in the body to kill off all the bacteria. If this time period is not long enough, the strongest bacteria may survive because the antibiotics were not in the system long enough to kill them. That is why patients are prescribed three days of antibiotics for a simple urinary tract infection, a 14-day course of antibiotics for strep throat, and six to nine months of antibiotics for tuberculosis. The rate of decay of bacteria due to antibiotic treatment is typically exponential. Because more and more bacteria are developing resistance to antibiotics and there are only a finite number of antibiotics available, it is important to take antibiotics for the entire course prescribed to ensure that all the bacteria are killed and new antibiotic-resistant bacteria do not develop.
::当保健提供者为细菌感染病人开抗生素时,他们会考虑抗生素在身体里需要多久才能杀死所有细菌。如果这个时间段不够长,最强的细菌可能存活下来,因为抗生素在系统里的时间不够长,无法杀死他们。这就是为什么为病人开为期三天的抗生素进行简单的尿道感染、为期14天的喉咙抗生素和六至九个月的结核病抗生素。抗生素治疗导致细菌衰减的速度一般是指数指数化的。由于越来越多的细菌正在对抗生素产生抗生素抗药性,而且只有有限的抗生素数量,因此必须在规定的整个过程中服用抗生素,以确保所有细菌都被杀死,新的抗生素抗菌菌不会发展。

by WinchPharmaGroup discusses antibiotics and how they work.
::WinchPharma Group讨论抗生素及其工作方式。

Summary
::摘要

The Euler number $\begin{aligned} e \end{aligned}$ is approximately 2.72.
::Euler e 的编号约为2.72。

To graph a function of the form $\begin{aligned} e^{x}, \end{aligned}$ approximate it with 3.
::图示表单的函数, 接近于 3 。

Review
::回顾

Determine if the functions below are exponential growth, decay, or neither. Give a reason for your answer.
::确定下面的函数是指数增长、衰变还是两者都不是。请说明答案的理由。

1. a. $\begin{aligned} y = \frac{4}{3} e^{x} \end{aligned}$ b. $\begin{aligned} y = - e^{- x} + 3 \end{aligned}$ c. $\begin{aligned} y = {(\frac{1}{e})}^{x} + 2 \end{aligned}$ d. $\begin{aligned} y = {(\frac{3}{e})}^{- x} - 5 \end{aligned}$
::1. a.y=43ex b.y=-e-x+3 c.y=(1e)x+2 d.y=(3e)-x-5

Simplify the following expressions with $\begin{aligned} e \end{aligned}$ :
::以 e 简化以下表达式:

2. $\begin{aligned} e^{- 3} \cdot e^{12} \end{aligned}$
::2. e-3e12

3. $\begin{aligned} \frac{4 e^{6}}{16 e^{2}} \end{aligned}$
::3. 4e616e2

4. $\begin{aligned} \frac{5 e^{- 4}}{e^{3}} \end{aligned}$
::4. 5e-4e3

5. $\begin{aligned} 6 e^{5} e^{- 4} \end{aligned}$
::5. 6e5e-4

6. $\begin{aligned} {(\frac{4 e^{4}}{3 e^{- 2} e^{3}})}^{- 2} \end{aligned}$
::6. (4e43e-2e3)-2

Sketch a graph for the following functions:
::绘制下列函数的图表:

7. $\begin{aligned} y = \frac{1}{2} e^{x} \end{aligned}$
::7. y=12ex

8. $\begin{aligned} y = - 4 e^{x} \end{aligned}$
::8.y=-4ex y=-4ex

9. $\begin{aligned} y = e^{- x} \end{aligned}$
::9. y=e-x

10. $\begin{aligned} y = 2 {(\frac{1}{e})}^{- x} + 1 \end{aligned}$
::10. y=2(1e)-x+1

Explore More
::探索更多

1. The population of Springfield is growing exponentially. The growth can be modeled by the function $\begin{aligned} P = I e^{0.055 t} \end{aligned}$ , where $\begin{aligned} P \end{aligned}$ represents the projected population, $\begin{aligned} I \end{aligned}$ represents the current population of 100,000 in 2012, and $\begin{aligned} t \end{aligned}$ represents the number of years after 2012.
::1. 斯普林菲尔德的人口正在成倍增长,增长可以P=Ie0.055t的函数作为模型,P代表预计人口,我代表2012年目前的10万人口, t代表2012年之后的年数。

a. To the nearest person, what will the population be in 2022?
::a. 对最近的一个人来说,2022年的人口将是什么?

b. In what year will the population double in size if this growth rate continues?
::b. 如果这一增长率继续下去,人口规模将在哪一年翻一番?

2. The value of Steve's car decreases in value according to the exponential decay function: $\begin{aligned} V = P e^{- 0.12 t} \end{aligned}$ , where $\begin{aligned} V \end{aligned}$ is the current value of the vehicle, $\begin{aligned} t \end{aligned}$ is the number of years Steve has owned the car, and $\begin{aligned} P \end{aligned}$ is the purchase price of the car, $25,000.
::2. 根据指数衰变函数,Steve的汽车价值下降:V=Pe-0.12t, V是车辆的现值, t是Steve拥有汽车的年数,P是汽车的购买价格,25 000美元。

a. To the nearest dollar, what will be the value of Steve's car in 2 years?
::a. 对于最近的美元,史蒂夫的汽车在两年内的价值是多少?

b. To the nearest dollar, what will be the value in 10 years?
::b. 最接近的美元,十年内的价值是多少?

3. Naya invests $7,500 in an account that accrues interest continuously at a rate of 4.5%.
::3. Naya投资7 500美元,用于一个以4.5%的利率不断累积利息的账户。

a. Write an exponential growth function to model the value of her investment after $\begin{aligned} t \end{aligned}$ years.
::a. 写出指数增长函数,以模拟她在T年之后投资的价值。

b. How much interest does Naya earn in the first 6 months to the nearest dollar?
::b. 纳亚在头6个月到最近的美元赚取多少利息?

c. How much money, to the nearest dollar, is in the account after 8 years?
::c. 8年后,账户中有多少钱,最接近的美元?

4. The rate of radioactive decay of radium is modeled by $\begin{aligned} R = P e^{- 0.00043 t} \end{aligned}$ , where $\begin{aligned} R \end{aligned}$ is the amount (in grams) of radium present after $\begin{aligned} t \end{aligned}$ years, and $\begin{aligned} P \end{aligned}$ is the initial amount (also in grams). If 698.9 grams of radium are present after 5,000 years, what was the initial amount?
::4. R=Pe-0.00043t以放射性衰减率为模型,以R为模型,其R为年之后的放射性衰减率(克),P为初始值(也以克计),如果在5 000年后有698.9克的放射性衰减率,初始值是多少?

5. The interest on a sum of money that compounds continuously can be calculated with the formula $\begin{aligned} I = P e^{r t} - P \end{aligned}$ , where P is the amount invested (the principal), r is the interest rate, and t is the amount of time the money is invested. If you invest $1,000 in a bank account that pays 2.5% interest compounded continuously, and you leave the money in that account for 4 years, how much interest will you earn?
::5. 不断混合的金额的利息,可以按照公式I=Pert-P计算,P是投资金额(本金),r是利率, t是投资时间,如果在持续支付2.5%利息的银行账户上投资1 000美元,而你将资金留在该账户4年,你将赚取多少利息?

6. There are other formulas for $\begin{aligned} e \end{aligned}$ . Calculate these formulas until $\begin{aligned} n = 5 \end{aligned}$ .
::6. e. 在n=5之前计算这些公式。

\begin{aligned} 2 & + \frac{1}{2!} + \frac{1}{3!} + . . . + \frac{1}{n!} & 2 + \frac{1}{1 + \frac{1}{2 + \frac{2}{3 + \frac{3}{4 + \frac{4}{5 + 5}}}}} \\ n! & = n \cdot (n - 1) \cdot (n - 2) \cdot \cdot \cdot 2 \cdot 1 \end{aligned}

::2+12!+13!+...+1n!2+11+12+23+34+45+5n!=n%(n-1)__(n-2)________________________________________________________________________________________

7. The curvature in a power line is the shape of what is called a "catenary curve." Catenary curves can be modeled by the function $\begin{aligned} y = c (e^{k x} + e^{- k x}), \end{aligned}$ where $\begin{aligned} y \end{aligned}$ is the vertical height and $\begin{aligned} x \end{aligned}$ is the horizontal distance ³ . Find the function when $\begin{aligned} c = 1.5 \end{aligned}$ and $\begin{aligned} k = \frac{1}{3} \end{aligned}$ . Find the height of the cable when $\begin{aligned} x = 10 \end{aligned}$ .
::7. 功率线的曲率是所谓的“碳酸曲线”的形状。可按函数 y=c(ekx+e-kx) 模拟催化曲线, y 是垂直高度, x 是水平距离。 3 在 c=1.5 和 k=13 时查找函数。在 x=10 时查找电缆的高度。

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

Please see the Appendix.
::请参看附录。

References
::参考参考资料

1. "Tortoise," last edited May 14, 2017,
::1. 2017年5月14日编辑的《乌龟》

2. "e (mathematical constant)," last edited June 2, 2017,
::2. “e(数学常数)”,2017年6月2日编辑,

3. "Catenary," last edited March 28, 2017,
::3. 2017年3月28日最后一次编辑

章节大纲

常规

编号e

The Number e ::编号e

The Value of e ::e 值的e值

Investigation: Finding the values of ( 1 + 1 n ) n as n gets larger ::调查:1+1nn)n值随n值增加而查找

Example 1 ::例1

The Function y = e x ::函数 y=ex

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Feature: The Bacteria Battle ::特色:细菌战斗

Why It Matters ::为何重要

Summary ::摘要

Review ::回顾

Explore More ::探索更多

Answers for Review and Explore More Problems ::回顾和探讨更多问题的答复

References ::参考参考资料

The Number e
::编号e

The Value of $\begin{aligned} e \end{aligned}$
::e 值的e值

Investigation: Finding the values of $\begin{aligned} {(1 + \frac{1}{n})}^{n} \end{aligned}$ as $\begin{aligned} n \end{aligned}$ gets larger
::调查:1+1nn)n值随n值增加而查找

Example 1
::例1

The Function $\begin{aligned} y = e^{x} \end{aligned}$
::函数 y=ex

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Feature: The Bacteria Battle
::特色:细菌战斗

Why It Matters
::为何重要

Summary
::摘要

Review
::回顾

Explore More
::探索更多

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

References
::参考参考资料