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
\( f(x)=6,000(1.003)^{2x} \)
The given function \( f \) models the balance of a bank account, in dollars, x years after it is opened. Which statement is the best interpretation of \( (1.003)^{2x} \) ?
原库参考解析
指数增长・衰减与复利进阶
On January 1, there were 236 views of a video posted on a social media site. Every 3 days after January 1, the number of views of the video had increased by 60% of the number of views 3 days earlier. The function \( f \) gives the predicted number of views x days after January 1. Which equation defines \( f \)?
原库参考解析
指数增长・衰减与复利挑战
\( N(t)=1,000\left(\frac{3}{2}\right)^{\frac{t}{3}} \)
The function N gives the estimated number of bacteria in a growth medium t hours after a study began. According to the function, the number of bacteria is estimated to increase by 50% every k minutes. What is the value of k?
原库参考解析
指数增长・衰减与复利进阶
The function \( P(x)=685(1.128)^x \) models the number of gray seal pups each year from 1967 through 1997 on Sable Island, Nova Scotia, where x is the number of years after 1967. Which of the following is the best interpretation of " P(9) is approximately equal to 2,025" in this context?
原库参考解析
指数增长・衰减与复利进阶
On January 1, there were 231 views of a video posted on a social media site. Every 2 days after January 1, the number of views of the video had increased by 60% of the number of views 2 days earlier. The function \( f \) gives the predicted number of views x days after January 1. Which equation defines \( f \)?
原库参考解析
指数增长・衰减与复利挑战
The function \( P(x)=685(1.128)^x \) models the number of gray seal pups each year from 1967 through 1997 on Sable Island, Nova Scotia, where x is the number of years after 1967. Which of the following is the best interpretation of " P(28) is approximately equal to 19,968" in this context?
原库参考解析
指数增长・衰减与复利挑战
On January 1, there were 231 views of a video posted on a social media site. Every 3 days after January 1, the number of views of the video had increased by 55% of the number of views 3 days earlier. The function \( f \) gives the predicted number of views x days after January 1. Which equation defines \( f \)?
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 15 pounds per square inch (psi). The pressure inside the container doubled every 24 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・衰减与复利挑战
The function \( f(t)=90,000(2)^{\frac{t}{600}} \) 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
When a scientist began observing a sample of an isotope, the mass of the sample was 530,000 milligrams. The mass of the sample decreases by half approximately every 5 days. Which of the following is closest to the mass, in milligrams, of the sample 35 days after the scientist began observing the sample?
原库参考解析
指数增长・衰减与复利基础
The number of cells of the bacteria Bacillus megaterium doubles every 25 minutes during its growth period when placed in a medium of sucrose-salts in a controlled laboratory environment. If a sample of 530 cells of Bacillus megaterium is placed in sucrose-salts in a controlled laboratory environment, which function \( f \) best represents the number of cells of the bacteria x minutes after the growth period begins, where x is less than the number of minutes in the growth period?
原库参考解析
指数增长・衰减与复利挑战
\( f(x)=23(1.20)^{\frac{x}{4}} \)
For the given function \( f \), the value of \( f(x) \)increases by p% for every increase of x by 8. What is the value of p ?
原库参考解析
