化合物利息

播放段落

The Purpose of this Lesson
::本课程的目的

播放段落

In this lesson, you will apply to the problem of compound interest. You'll use this context to develop a model for continuous .
::在此教训中, 您将应用到复利问题 。 您将使用此背景来开发一个持续模式 。

---

播放段落

Activity 1 : Compounding Interest
::活动1:增加利息

播放段落

Work it Out
::工作出来

播放段落
1. A bank offers Maria  an investment opportunity. If she  gives the bank $1,000, they will return her   principal  with 12% in interest at the end of the year. The principal is the initial amount of money, and interest is the amount by which the principal is increased. How much money will Maria  have at the end of the year?
   ::一家银行为玛利亚提供了投资机会。 如果她给了银行1000美元,他们将在年底退还她的本金12%的利息。本金是初始金额,利息是本金增加的金额。 玛利亚在年底将有多少钱?
播放段落
2. The bank extends the above offer for 9 more years. This means that every year that Maria leaves the money in the account, she finds she has 112% of the amount she had in the previous year. She is accumulating compound interest, that is, interest on previously earned interest. How much money will she have after a total of 10 years?
   ::银行将上述提议再延期9年。 这意味着每年Maria离开账户时,她都会发现自己拥有上一年金额的112%。 她正在积累复合利息,即先前赚取利息的利息。 在总共10年之后,她将有多少钱?

播放段落

Interactive
::交互式互动

播放段落

Use the interactive below to compare two ways to calculate 12% interest on $1,000.
::使用下面的交互效果比较两种方法来计算1 000美元12%的利息。

 

 

 

播放段落

This is an exponential growth scenario. The initial value is $1,000 and the multiplier is 1.12. The amount of money in the account after  $\begin{array}{r}x\end{array}$  years is given by:
::这是一个指数增长假设。 初始值为 1 000 美元,乘数为 1. 12 。 账户中X 年后的金额由下列单位提供:

播放段落

::A=1000( 1.1. 12) x

播放段落

After 10 years:
::10年后:

播放段落

::A=1000(1.12)10A $3106

播放段落

Not bad, in 10 years, Maria will more than triple her money!
::不错 十年后 玛利亚的钱会翻三倍多

---

播放段落

Activity 2 : Compounding at Various Frequencies
::活动2:使各种原因的复合化

播放段落

Example 2 -1
::例2-1

播放段落

Maria's friend Kris  is interested in the same investment but approaches the bank with a proposition. She doesn't want the bank to wait until the end of each year to perform the multiplication that increases the amount of money. She wants the bank to multiply every 6 months, in case she wants to take the money out at, for example, 7 and half years. The bank agrees, but stipulates that naturally, they can't give her 12% every half year – they will give her 6% each half year. Kris  agrees to the plan and gives the bank $1,000. Determine the amount of money Kris   will have after 10 years.
::Maria的朋友Kris对同样的投资感兴趣,但向银行提出一个提议。她不希望银行等到每年年底才进行增加金额的倍增。她希望银行每6个月翻一番,如果她想在7年半的时间里把钱拿出来的话。 银行同意,但自然规定,他们不能每半年给她12 % — —每半年给她6 % 。 克里斯同意该计划,并给银行1 000美元。 确定10年后Kris的金额。

 

播放段落

Solution:  In every 6 month period, Kris will finds she has 106% of what she had previously. The multiplier is 1.06. This multiplication happens every half-year, so instead of happening just 10 times in 10 years, it will happen 20 times in 10 years:
::解决方案: 每6个月, Kris就会发现她拥有她以前拥有的106%。 乘数是1.06。 这种乘数每半年一次, 所以在10年中不会发生10次, 而是10年中会发生20次:

播放段落

::A=1000(1.06)20A=3207美元

播放段落

Kris will earn even more than Maria in the same time period!
::克里斯的收入会比玛丽亚 在同一时间段里还要多!

播放段落

Example 2 -2
::例2-2

播放段落

The previous two examples involved  compound interest calculated over different time periods. Maria's interest was compounded yearly, while Kris's  was compounded semi-annually. Raymond observes that increasing the frequency of compounding increases the money earned, even with the stipulation that the interest rate is divided into equal portions throughout the year. He approaches the bank and asks them for the same 12% yearly rate, but to compound monthly at 1% per month. Determine the amount of money Raymond will have after 10 years. 
::前两个例子涉及在不同时期计算的复合利息。 Maria的利息每年增加, Kris的利息每半年增加一次。 Raymond 指出, 增加复合频率会增加收入, 即使规定全年的利率均分。 他到银行要求年利率为12%,但每月复利为1%。 确定雷蒙德在10年后的金额。

 

播放段落

Solution:  Raymond will only earn 1% every month, but the interest will be compounded every month for 10 years. That's 120 times:
::解决方案:雷蒙德每月只挣1%,但利息将每月加起来,持续10年。这是120倍:

播放段落

::A=1000(1.01)120A=3300美元

播放段落

Raymond will have earned even more than Kris!
::雷蒙德赚的钱比克里斯还多!

 

 

播放段落

Interactive
::交互式互动

播放段落

Use the interactive below to experiment with increasing the frequency of compounding. What do you observe? The interest rate in the examples above was 12%. Experiment with changing the interest rate. What do you observe? The principal in the examples above was $1,000. Experiment with changing the principal. What do you observe? Based on your observations, is it possible to earn an infinite amount of money by increasing the frequency of compounding? Is there a limit to how much can be earned by increasing the frequency of compounding? Explain. 
::使用下面的互动来实验增加复合频率。 您观察什么? 上面的例子中的利率是 12 % 。 实验来改变利率。 您观察什么? 上面的例子中的本金是 1 000 美元 。 实验来改变本金 。 您观察什么? 根据你的观察, 是否可能通过增加复合频率来赚取无限的金钱? 增加复合频率能赚到多少? 解释一下 。

 

 

播放段落

 Work it Out
::工作出来

播放段落
1. If you could  convince the bank to compound weekly, do you think that the amount of money earned would  be substantially more than that earned by Maria, Kris, and Raymond? Why or why not? Use your calculator to experiment. Be careful to avoid excessive rounding. How about daily? Experiment.
   ::如果你能说服银行每周复工,你是否认为赚到的钱将大大超过Maria、Kris和Raymond的收入?为什么?为什么不?用你的计算器来试验?小心避免过度四舍五入。每天呢?实验呢?
播放段落
2. Let  $\begin{array}{r}P\end{array}$  be the principal,  $\begin{array}{r}r\end{array}$  be the interest rate,  $\begin{array}{r}t\end{array}$  be the time in years, and  $\begin{array}{r}n\end{array}$  be the number of times interest is compounded per year. Write a general formula for  $\begin{array}{r}A,\end{array}$  the amount in the account after  $\begin{array}{r}t\end{array}$  years.
   ::让 P 成为本金, r 成为利率, t 是年中的时间, n 成为年复数的利息倍数。 写入 A 的一般公式, 年后在账户中的数额 。
播放段落
3. $567 is invested at 5% interest, compounded quarterly (4 times per year). How much is in the account after 7 years? Use graphing technology to approximate when the amount in the account reaches $5,670. 
   ::567美元的投资利率为5%,每季5次(每年4次),7年后账户内有多少?使用图表技术来估计账户内金额达到5 670美元。

播放段落

PLIX Interactive
::PLIX 交互式互动

 

 

---

播放段落

Activity 3 : Compounding Interest  Continuously
::活动3:增加利息

播放段落

Interactive
::交互式互动

播放段落

In the previous examples, you saw that increasing the frequency of compounding increases the amount of money in the account, but only up to a certain value. In th e interactive below , you will explore this limit.
::在前几个例子中,您看到,增加复合频率会增加账户中的资金数额,但只能达到一定价值。在下文互动中,您将探索这一限制。

 

 

播放段落

The  amount  returned after 1 year from investing $1 at a 100% interest rate approaches a number as the number of times compounded increases. Write down this number, rounding to 3 decimal places. This number is  $\begin{array}{r}e.\end{array}$  Like  $\begin{array}{r}\pi ,\end{array}$   $\begin{array}{r}e\end{array}$  is an irrational number. It's not a root , it's a transcendental number, like  $\begin{array}{r}\pi .\end{array}$  Like  $\begin{array}{r}\pi ,\end{array}$   $\begin{array}{r}e\end{array}$  shows up everywhere in mathematics. The interactive shows you that $1 invested at 100% for 1 year, compounded  continuously  (in every single instant), returns  $\begin{array}{r}e\end{array}$  dollars.
::投资 1 美元, 利率为 100% , 利率为 100% , 投资 一年后, 回报的金额会随着复数的倍数增加而接近一个数字。 写下这个数字, 舍入到小数点后的三个位数。 这个数字是 . , 像 , e 是 不合理的数字。 它不是根, 它是一个超凡数字, 像 。 象 , e 在数学中无处不在。 互动显示您在 1 年投资 1 美元, 以 100% , 连续( 每瞬间) , 返回 e 美元 。

播放段落

Example 3 -1
::例3-1

播放段落

The fact above means that you  can create a much simpler exponential function to model a scenario where money (or anything) is being compounded continuously. Fill in the blanks for the statements below.
::以上事实意味着您可以创建一个简单得多的指数函数来模拟货币(或任何东西)不断复杂化的假想。填写以下语句的空白。

播放段落
1. If $1 invested at an interest rate of 100% for 1 year, compounded continuously, returns  $\begin{array}{r}e\end{array}$  dollars,  $\begin{array}{r}P\end{array}$  dollars similarly invested returns ___________.
   ::如果1美元投资,利率为1年的100%, 持续复算,回报e美元,P美元 类似的投资回报 。
播放段落
2. If $1 invested at an interest rate of 100% for 1 year, compounded continuously, returns $\begin{array}{r}e\end{array}$  dollars, $1 invested for 5 years returns __________.
   ::如果投资1美元,利率为1年的100%,且连续复算,回报e美元,投资1美元,5年回报_______________________________________________________。
播放段落
3. $\begin{array}{r}P\end{array}$  dollars invested at an interest rate of 100% for $\begin{array}{r}t\end{array}$  years, compounded continuously, returns __________.
   ::P 投资美元利率为 T 年100%, 且持续复数, 回报 。

 

播放段落

Solution:
::解决方案 :

播放段落
1. $\begin{array}{r}\mathrm{\$}Pe\end{array}$  
   ::美元Pe 美元
播放段落
2. $\begin{array}{r}\$e^5\end{array}$  
   ::5美元 5美元
播放段落
3. $\begin{array}{r}\mathrm{\$}P{e}^t\end{array}$  
   ::美元 美元

 

 

---

播放段落

Activity 4 :  Compare Continuous Compounding with Compounding in Various Intervals
::活动4:比较各种不同间隙中连续化合物与化合物化合物的比较

播放段落

In fact, most banks do not compound interest continuously. They would have to continually recalculate the amount in your account. Furthermore, as seen in the previous interactives, the difference between compounding, for example, daily, is not so different from compounding continuously. Banks compound weekly or daily so they don't have to perform continuous calculations.
::事实上,大多数银行并不持续增加利息。 它们必须不断重新计算您的账户中的金额。 此外,正如前几次互动所见,例如每天的复利差异与连续复利的差别并不大。 银行每周或每天会合并,因此他们不必进行连续计算。

播放段落

However, many natural phenomena feature continuous exponential growth and can be modeled with the continuous compounding formula. 
::然而,许多自然现象以持续指数增长为特点,可以与连续的复合公式为模范。

播放段落

The formula as written above only allows for a growth rate of 100%. In order to make the formula more versatile, it needs to accommodate other rates. The more general formula is given below without proof. The proof of this formula is a topic for future courses.
::上面写入的公式只允许100%的增长率。 为了让公式更加灵活, 它需要适应其他的增长率。 下面给出的公式比较一般, 但没有证据。 这个公式的证据是未来课程的主题 。

播放段落

Work it Out
::工作出来

播放段落
1. Create a scenario involving an interest rate of 100% to compare compounding interest annually, semi-annually, monthly, weekly, daily, and continuously. Use the previous formula for compound interest and the above formula for continuous compounding. Use your calculator to compare the amounts returned. Is there a substantial difference between compounding annually versus weekly? How about weekly versus continuously? Explain.
   ::创建包含100%利率的假设情景, 以比较每年、 半年、 半年、 半月、 月、 周、 每日和连续的复利 。 使用以前的复合利息公式和以上公式进行连续复利 。 使用您的计算器比较返回的金额 。 每年复利和每周复利之间是否有很大的差别? 每周还是连续的? 解释一下 。
播放段落
2. If $432 is invested at 11% compounded semi-annually, how much will the account contain after 6 years? Use graphing or solving technology to determine when the amount  will reach $4,320.
   ::如果每半年将432美元投资11%,则该账户在6年后将包含多少资金?使用图表或解决技术来确定何时将达到4 320美元。
播放段落
3. An algal bloom in the Gulf of Mexico covers 300 square miles. It's growing continuously at a relative growth rate of 1% per week . How big is the bloom after 100 days? Use graphing or solving technology to determine when the bloom will double in size.
   ::墨西哥湾的藻类开花面积为300平方英里。 它以每周1%的相对增长率持续增长。 100天后的开花有多大? 使用图形化或解决技术来确定开花何时会翻倍。
播放段落
4. A dollar is invested in an account at an interest rate of 100% per hour. It is compounded continuously for 1 hour . How much money is in the account after 1 hour?
   ::一美元投资在一个账户中,每小时利率为100%。它持续增加1小时。1小时后有多少钱在账户中?

 

 

播放段落

PLIX Interactive
::PLIX 交互式互动