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
The number of bacteria in a liquid medium doubles every day. There are 72,000 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
According to a model, the estimated volume of a certain Douglas fir tree increases by 15.5% 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
A sample of a certain isotope takes 29 years to decay to half its original mass. The function\( s\left(t\right)=208\left(0.5\right)^{\frac{t}{29}} \) gives the approximate mass of this isotope, in grams, that remains t years after a 208-gram sample starts to decay. Which statement is the best interpretation of \( s\left(87\right)=26 \) in this context?
原库参考解析
指数增长・衰减与复利进阶
\( P\left(t\right)=260\left(1.03\right)^{\left(\frac{6}{4}\right)t} \)
The function P models the population, in thousands, of a certain city t years after 2001. According to the model, the population is predicted to increase by n % every 8 months. What is the value of n?
原库参考解析
指数增长・衰减与复利进阶
\( P\left(t\right)=230\left(1.03\right)^{\frac{3}{2}t} \)
The function P models the population, in thousands, of a certain city t years after 2001. According to the model, the population is predicted to increase by n% every 8 months. What is the value of n?
原库参考解析
指数增长・衰减与复利挑战
For a function \( f \), the value of \( f\left(x\right) \)is multiplied by 8 for every increase of 1 in the value of x . When x = 0, \( f\left(x\right)=5 \).Which of the following equations defines \( f \)?
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 posftive 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・衰减与复利挑战
A certain investment account offers a special interest rate for the first 4 months the account is open followed by a lower interest rate for the remainder of the time the account is open. Akeyla opened one of these accounts with an original account balance of $700.00 and did not make any other deposits or withdrawals. 4 months after Akeyla opened the account, the balance had increased by 0.6% of the original balance. 6 months after Akeyla opened the account, the balance had increased by an additional 0.1% of the balance at the end of the first 4 months. Every 2 months after the first 6 months, the balance had increased by an additional 0.1% of the balance 2 months before. Which of the following equations could represent the account balance \( B\left(x\right) \), in dollars, x months after the account was opened, where \( x\ge4 \)?
原库参考解析
指数增长・衰减与复利挑战
\( f\left(x\right)=25\left(1.40\right)^{\frac{x}{3}} \)
For the given function \( f \), the value of \( f\left(x\right) \) increases by p% for every increase of x by 6. What is the value of p?
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 25 years from 1660 to 1760. The population of this city was 208,000 in 1760. What was the population of this city in 1660?
原库参考解析
指数增长・衰减与复利进阶
\( y=7,800\left(0.85\right)^x \)
The given equation estimates the value, in dollars, of a certain car x years after it was purchased. According to the given equation, what was the estimated value, in dollars, of the car at the time it was purchased?
原库参考解析
指数增长・衰减与复利进阶
The function \( f\left(t\right)=30,000\left(2\right)^{\frac{t}{370}} \) 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
