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・衰减与复利挑战
\( f\left(x\right)=66\left(b\right)^x \)
In the given function, b is a positive constant. For every increase in the value of x by 1, the value of \( f\left(x\right) \) increases by c%, where \( 0<c<100 \). Which expression gives the value of c in terms of b?
原库参考解析
指数增长・衰减与复利挑战
The population of a certain city doubled every 75 years from 1652 to 1952. The population of this city was 160,000 in 1952. What was the population of this city in 1652?
原库参考解析
指数增长・衰减与复利挑战
The equation \(N(m)=65(Q)^{\frac{m}{4}}\) gives the predicted population \(N(m)\), in thousands, of a certain bacteria colony \(m\) minutes after the initial measurement, where \(Q\) is a constant greater than 1. The predicted population increases by \(p\)% every 120 seconds. What is the value of \(p\) in terms of \(Q\)?
原库参考解析
指数增长・衰减与复利进阶
\( f\left(x\right)=1,300\left(0.5\right)^{\frac{x}{14}} \)
The function f models the intensity of a beam, f (x), in number of photons in the beam, x millimeters below the surface of a certain material. According to the model, what is the number of photons in the beam when it is at the surface of the material?
原库参考解析
指数增长・衰减与复利进阶
\( f\left(x\right)=3,000\left(0.75\right)^x \)
A conservation scientist implemented a program to reduce a certain invasive insect population in an area. The given function estimates this insect species' population x years after 2008, where x\( \le \) 8. Which of the following is the best interpretation of 3,000 in this context?
原库参考解析
指数增长・衰减与复利进阶
In a certain state, the population of pheasants, a type of bird, is estimated each year by counting the number of pheasants observed along certain roads in the state. On average, each year from 2005 to 2015 the number of pheasants counted per mile of road decreased by 3.5% of the number of pheasants per mile of road the previous year. Based on this average, if there were 6.29 pheasants per mile of road in this state in 2005, which of the following best approximates the number of pheasants per mile of road in 2015?
原库参考解析
指数增长・衰减与复利挑战
At the start of an experiment, the pressure inside a container was 18 pounds per square inch (psi). The pressure inside the container doubled every 29 minutes after the start of the experiment. Which equation represents this situation, where y is the pressure, in psi, inside the container x minutes after the start of the experiment?
原库参考解析
指数增长・衰减与复利进阶
The function f is defined by \( f\left(x\right)=44\left(0.21\right)^x \) . For any positive integer n, the value of f (n) is p% less than the value of f (n- 1). What is the value of p?
原库参考解析
指数增长・衰减与复利进阶
The function \( f\left(x\right)=240,000\left(1.22\right)^x \) gives a company's predicted annual revenue, in dollars, x years after the company started selling clothes online, where \( 0<x≤10 \). What is the best interpretation of the statement "f (5) is approximately equal to 648,650" in this context?
原库参考解析
指数增长・衰减与复利挑战
At the time of posting a video, a social media channel had 67 subscribers. Each day for five days after the video was posted, the number of subscribers doubled from the number the previous day. Which equation gives the total number of subscribers, n, to the channel d days after the video was posted?
原库参考解析
指数增长・衰减与复利进阶
The function \( f(m)=125(5)^{\frac{m}{30}} \) represents the population of elephants m months after the initial observation. How many months since the initial observation does it take for the population of elephants to be 625 ?
原库参考解析
指数增长・衰减与复利进阶
By the end of March, the water level at the reservoir was 250 meters. It was projected that each month for the next 6 months, the water level, w, will decrease by 25% from the previous month. Which equation best represents this scenario, where m is the number of months after March, for m ≤ 6 ?
原库参考解析
