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・衰减与复利基础
\( y=3,600\left(a\right)^x \)
The given equation, where \( a \) is a positive constant, gives the predicted number of bacteria,\( y \), in a growth medium \( x \) hours after the number of bacteria was initially measured. According to the equation, what was the predicted number of bacteria initially measured in the growth medium?
原库参考解析
指数增长・衰减与复利进阶
The number of bacteria in a liquid medium doubles every day. There are 44,000 bacteria in the liquid medium at the start of an observation. Which 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
The function f(t) =
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
A model predicts that the population of Springfield was 15,000 in 2005. The model also predicts that each year for the next 5 years, the population p increased by 4% of the previous year's population. Which equation best represents this model, where x is the number of years after 2005, for x ≤ 5?
原库参考解析
指数增长・衰减与复利进阶
Each year, the value of an investment increases by 0. 49% of its value the previous year. Which of the following functions best models how the value of the investment changes over time?
原库参考解析
指数增长・衰减与复利进阶
A scientist initially measures 12,000 bacteria in a growth medium. 4 hours later, the scientist measures 24,000 bacteria. Assuming exponential growth, the formula P = C(2)rt gives the number of bacteria in the growth medium, where r and C are constants and P is the number of bacteria t hours after the initial measurement. What is the value of r?
原库参考解析
指数增长・衰减与复利基础
A company opens an account with an initial balance of $36,100.00. 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 13 years is $68,071.93. Which equation could define A?
原库参考解析
指数增长・衰减与复利挑战
For the function q, the value of q(x) decreases by 45% for every increase in the value of x by 1.If q(0) = 14, which equation defines q?
原库参考解析
指数增长・衰减与复利挑战
f(x) =
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
指数增长・衰减与复利进阶
Bacteria are growing in a liquid growth medium. There were 300,000 cells per milliliter during an initial observation. The number of cells per milliliter doubles every 3 hours. How many cells per milliliter will there be 15 hours after the initial observation?
原库参考解析
指数增长・衰减与复利进阶
The first term of a sequence is 9. Each term after the first is 4 times the preceding term. If w represents the nth term of the sequence, which equation gives w in terms of n?
原库参考解析
指数增长・衰减与复利挑战
P(t) =
The function P models the population, in thousands, of a certain city t years after 2005. According to the model, the population is predicted to increase by n% every 18 months. What is the value of n?
原库参考解析
