这部分需要会什么？
- 将增长率或衰减率转换为乘数
- 匹配时间单位与指数周期
- 处理复利、倍增和半衰模型
核心概念与公式
每期增长率为 \(r\) 时，\(P(t)=P_0(1+r)^t\)；每期减少比例 \(r\) 时，\(P(t)=P_0(1-r)^t\)。这里 \(r\) 用小数表示，\(t\) 是经历的期数。
若每 \(d\) 个时间单位翻倍，可写 \(P(t)=P_0\,2^{t/d}\)；每 \(d\) 个时间单位减半，则为 \(P_0(1/2)^{t/d}\)。名义年利率 \(r\)、每年复利 \(n\) 次的模型为 \(P_0(1+r/n)^{nt}\)，其中 \(t\) 以年计。
做题时怎样使用
- 标出初始量、每期变化率和每期有多长
- 将百分率改为乘数，并把时间换成对应期数
- 比较两时刻时可求比值，消去共同的初始量
容易混淆的地方
增长 5% 的乘数是 1.05；衰减 5% 的乘数是 0.95。每两小时增长 10%，指数应为小时数除以 2。
用一个例题串起方法
知识点教学例题某种数量初始为 500，每 3 小时增长 20%。求 6 小时后的数量。
展开解题思路 ＋
6 小时包含 \(6/3=2\) 期，所以数量为 \(500(1.2)^2=720\)。不是在初值上只增加一次 40%。
本知识点题库
先独立作答，再查看答案与解析。
指数增长・衰减与复利进阶
\( f(t)=500(0.5)^{\frac{t}{12}} \)
The function f models the intensity of an X-ray beam, in number of particles in the X-ray beam, t millimeters below the surface of a sample of iron. According to the model, what is the estimated number of particles in the X-ray beam when it is at the surface of the sample of iron?
原库参考解析
指数增长・衰减与复利进阶
A function p estimates that there were 2,000 animals in a population in 1998. Each year from 1998 to 2010, the function estimates that the number of animals in this population increased by 3% of the number of animals in the population the previous year. Which equation defines this function, where p(x) is the estimated number of animals in the population x years after 1998?
原库参考解析
指数增长・衰减与复利进阶
The function \( f\left(t\right)=40,000\left(2\right)^{\frac{t}{790}} \) gives the number of bacteria in a population t minutes after an initial observation. How much time, in minutes, does it take for the number of bacteria in the population to double?
原库参考解析
指数增长・衰减与复利基础
An investment account was opened with an initial value of $890. The value of the account doubled every 10 years. Which equation represents the value of the account M(t), in dollars, t years after the account was opened?
原库参考解析
指数增长・衰减与复利进阶
The function \( f\left(t\right)=60,000\left(2\right)^{\frac{t}{410}} \) gives the number of bacteria in a population t minutes after an initial observation. How much time, in minutes, does it take for the number of bacteria in the population to double?
原库参考解析
指数增长・衰减与复利进阶
Each year, the value of an investment increases by 0.49% of its value the previous year. Which of the following functions best models how the value of the investment changes over time?
原库参考解析
指数增长・衰减与复利基础
A company opens an account with an initial balance of $36,100.00. The account earns interest, and no additional deposits or withdrawals are made. The account balance is given by an exponential function A, where A(t) is the account balance, in dollars, t years after the account is opened. The account balance after 13 years is $68,071.93. Which equation could define A?
原库参考解析
指数增长・衰减与复利挑战
For the function q, the value of q(x) decreases by 45% for every increase in the value of x by 1. If \( q\left(0\right)=14 \) , which equation defines q?
原库参考解析
指数增长・衰减与复利基础
The number of bacteria in a liquid medium doubles every day. There are 44,000 bacteria in the liquid medium at the start of an observation. Which of the following represents the number of bacteria, y, in the liquid medium t days after the start of the observation?
原库参考解析
指数增长・衰减与复利挑战
A model predicts that the population of Bergen was 15,000 in 2005. The model also predicts that each year for the next 5 years, the population p increased by 4% of the previous year's population. Which equation best represents this model, where x is the number of years after 2005, for \( x\le5 \)?
原库参考解析
指数增长・衰减与复利挑战
Two investments were made as shown in the table above. The interest in Account A is compounded once per year. Which of the following is true about the investments?
原库参考解析
指数增长・衰减与复利挑战
A model estimates that at the end of each year from 2015 to 2020, the number of squirrels in a population was 150% more than the number of squirrels in the population at the end of the previous year. The model estimates that at the end of 2016 , there were 180 squirrels in the population. Which of the following equations represents this model, where n is the estimated number of squirrels in the population t years after the end of 2015 and t\( ≤ 5 \)?
原库参考解析
