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
A company has a customer loyalty program. In January 2018, there were 200 customers enrolled in the loyalty program. For the next 24 months after January 2018, the total number of customers enrolled in the loyalty program each month was 4% greater than the total number enrolled the previous month. Which equation gives the total number of customers, c, enrolled in the company's loyalty program m months after January 2018, where \( m\le24 \) ?
原库参考解析
指数增长・衰减与复利进阶
The function \( f(t)=8,000(2)^{\frac{t}{300}} \) 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
The growth of the number of bacteria in a certain population is modeled by a function that increases exponentially over time. Which of the following could describe how the number of bacteria in the population changes each hour?
原库参考解析
指数增长・衰减与复利挑战
According to a model, the estimated volume of a certain Douglas fir tree increases by 14.3% for every increase of 100 centimeters in the tree's height. The equation \( y=ab^{\frac{x}{100}} \) represents this model, where y represents the estimated volume, in cubic centimeters, of this Douglas fir tree, x represents the tree's height, in centimeters, and a and b are constants. In this equation, what is the value of b?
原库参考解析
指数增长・衰减与复利进阶
Harper has $110 in an account. Each year she expects to have 2.7% more money in the account than she had the previous year. Which of the following models best describes how Harper expects the money in her account to change over time?
原库参考解析
指数增长・衰减与复利进阶
A researcher observes a sample of a nuclide. An exponential model estimates that the mass, in grams, of the sample decreases by 23% every 11.69 minutes. Which of the following equations could represent this model, where M is the estimated mass, in grams, of the sample t minutes after the researcher began observing the sample?
原库参考解析
指数增长・衰减与复利挑战
A researcher observes a sample of a nuclide. An exponential model estimates that the mass, in grams, of the sample decreases by 22% every 11.11 minutes. Which of the following equations could represent this model, where M is the estimated mass, in grams, of the sample t minutes after the researcher began observing the sample?
原库参考解析
指数增长・衰减与复利进阶
A sample of a certain isotope takes 29 years to decay to half its original mass. The function \( s\left(t\right)=136\left(0.5\right)^{\frac{t}{29}} \) gives the approximate mass of this isotope, in grams, that remains t years after a 136-gram sample starts to decay. Which statement is the best interpretation of \( s\left(58\right)=34 \) in this context?
原库参考解析
指数增长・衰减与复利进阶
Antonia has $130 in an account. Each year she expects to have 2.5% more money in the account than she had the previous year. Which of the following models best describes how Antonia expects the money in her account to change over time?
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 rabbits in a population was 170% more than the number of rabbits in the population at the end of the previous year. The model estimates that at the end of 2016, there were 216 rabbits in the population. Which of the following equations represents this model, where n is the estimated number of rabbits in the population t years after the end of 2015 and \( t\le5 \) ?
原库参考解析
指数增长・衰减与复利挑战
The potential difference across a capacitor in an electric circuit is initially 16 volts and decreases with time, as shown in the graph. Which of the following is closest to the time, in seconds, from the initial measurement at which the potential difference across the capacitor is 15% of its initial value?
原库参考解析
指数增长・衰减与复利挑战
A sample of a certain isotope takes 29 years to decay to half its original mass. The function \( s(t)=128(0.5)^{\frac{t}{29}} \) gives the approximate mass of this isotope, in grams, that remains t years after a 128-gram sample starts to decay. Which statement is the best interpretation of \( s\left(58\right)=32 \) in this context?
原库参考解析
