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
\( P\left(t\right)=24.8\left(1.036\right)^t \)
The function P gives the predicted population, in millions, of a certain country for the period from 1984 to 2018, where t is the number of years after 1984. According to the model, what is the best interpretation of the statement “P(8) is approximately equal to 32.91”?
原库参考解析
指数增长・衰减与复利进阶
A company has a newsletter. In January 2018, there were 1,300 customers subscribed to the newsletter. For the next 24 months after January 2018, the total number of customers subscribed to the newsletter each month was 7% greater than the total number subscribed the previous month. Which equation gives the total number of customers, c, subscribed to the company's newsletter m months after January 2018, where \( m\le24 \) ?
原库参考解析
指数增长・衰减与复利挑战
Jeremy deposited x dollars in his investment account on January 1, 2001. The amount of money in the account doubled each year until Jeremy had 480 dollars in his investment account on January 1, 2005. What is the value of x ?
原库参考解析
指数增长・衰减与复利进阶
The function ƒ (t)=60,000\( \left(2\right)^{\frac{t}{410}} \) 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・衰减与复利挑战
p(t) =90,000(1.06)t
The given function p models the population of Lowell t years after a census. Which of the following functions best models the population of Lowell m months after the census?
原库参考解析
指数增长・衰减与复利挑战
Two variables, x and y, are related such that for each increase of 1 in the value of x, the value of y increases by a factor of 4. When x =0, y =200. Which equation represents this relationship?
原库参考解析
指数增长・衰减与复利进阶
The function f(x) =206(1.034)x models the value, in dollars, of a certain bank account by the end of each year from 1957 through 1972, where x is the number of years after 1957. Which of the following is the best interpretation of “f (5) is approximately equal to 243” in this context?
原库参考解析
指数增长・衰减与复利挑战
f(x) =\( 5,470(0.64)^{\frac{x}{12}} \)
The function f gives the value, in dollars, of a certain piece of equipment after x months of use. If the value of the equipment decreases each year by p% of its value the preceding year, what is the value of p ?
原库参考解析
