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
The function \( f \) is defined by \( f\left(x\right)=53\left(0.15\right)^x \) . For any positive integer n, the value of \( f\left(n\right) \) is p% less than the value of \( f\left(n-1\right) \). What is the value of p?
原库参考解析
指数增长・衰减与复利挑战
The number of bacteria in a growth medium is expected to increase by 170% every 2 hours during a period of observation. The number of bacteria in the growth medium was estimated to be 5,000 when the period of observation began. Which function P gives the expected number of bacteria in this growth medium t hours after the period of observation began?
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 \( s\left(t\right)=184\left(0.5\right)^{\frac{t}{29}} \) gives the approximate mass of this isotope, in grams, that remains t years after a 184-gram sample starts to decay. Which statement is the best interpretation of \( s\left(58\right)=46 \) in this context?
原库参考解析
指数增长・衰减与复利进阶
A researcher observe 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
For the function \( f \) , for every increase of 2 in the value of x, the value of \( f\left(x\right) \) increases by a factor of c, where c is a constant. Which of the following equivalent forms of function \( f \) displays the value of c as the base or the coefficient?
原库参考解析
指数增长・衰减与复利挑战
Luis has $150 in an account. Each year he expects to have 1.1% more money in the account than he had the previous year. Which of the following models best describes how Luis expects the money in his account to change over time?
原库参考解析
指数增长・衰减与复利挑战
The number of bacteria in a liquid medium doubles every day. There are 48,000 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・衰减与复利进阶
A company opens an account with an initial balance of $35,600.00. 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 15 years is $129,672.38. Which equation could define A?
原库参考解析
指数增长・衰减与复利进阶
The value of a share of stock was $240 on the day it was purchased. Every year, the value of the share of stock increases by 1% of its value the previous year. Which function V gives the value, in dollars, of the share of stock t years after the day it was purchased?
原库参考解析
指数增长・衰减与复利挑战
For the function f, for each increase in the value of x by c, where c is a positive constant, the value of \( f\left(x\right) \) increases by a factor of 27. Which of the following equivalent forms of the function f displays \( \frac{1}{c} \) as a coefficient of x?
原库参考解析
指数增长・衰减与复利进阶
32 teams are competing in a baseball tournament. The tournament has 5 rounds. After each round, half the teams competing are eliminated from the tournament. lf \( r \) teams remain in the tournament after \( x \) rounds of play, which of the following gives \( r \) in terms of \( x \)?
原库参考解析
指数增长・衰减与复利挑战
At different times on a certain day, a manager at a theme park recorded the number of visitors. An exponential model gives the estimated number of visitors, \( V \), at the theme park \( t \) hours after the theme park opened, where \( t\le3 \). Based on the model, \( 80 \) visitors were estimated to be at the theme park when it opened, and every \( 30 \) minutes the estimated number of visitors increased by \( 150\% \) of the estimated number of visitors \( 30 \) minutes earlier. Which of the following equations best represents this model?
原库参考解析
