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
A computer program models the population of a certain insect in an environment where the insect has no natural predators. According to the model, the estimated total mass of the population of insects at the end of every 5-week period is 158% greater than the estimated total mass of the population of insects at the end of the previous 5-week period. The estimated total mass of the population of insects at the end of 15 weeks is 627.7 grams. Which equation best represents this model, where M is the estimated total mass, in grams, of the population of insects at the end of t weeks?
知序中文精讲
指数增长・衰减与复利挑战
For a certain position, an employer offers an hourly wage of $16.00 at the start of employment. One year after the start of employment, the hourly wage is increased by 9% of the hourly wage at the start of employment. For the next five years, at the end of every 6-month interval, the hourly wage is increased by 3% of the hourly wage at the end of the previous 6-month interval. Which of the following functions H models the hourly wage, in dollars, x years after the start of employment, where 1 ≤ x ≤ 6?
知序中文精讲
指数增长・衰减与复利进阶
The population of a certain city doubled every 75 years from 1666 to 1966. The population of this city was 240,000 in 1966. What was the population of this city in 1666?
原库参考解析
指数增长・衰减与复利进阶
\( y=7,700\left(0.85\right)^x \)
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・衰减与复利基础
\( y=500\left(2\right)^{2x} \)
The equation gives the estimated number of online newsletter subscribers at the end of every six-month period, x, starting from the end of January 1995, where \( 0\le x\le4 \) . Which statement is the best interpretation of the y-intercept of the graph of this equation in the xy-plane?
原库参考解析
指数增长・衰减与复利进阶
James purchased a certain baseball card on January 1. The function \( f\left(x\right)=55\left(1.04\right)^x \), where \( 0\le x\le10 \), gives the predicted value, in dollars, of the baseball card \( x \)x years after James purchased it. What is the best interpretation of the statement “\( f\left(7\right) \) is approximately equal to 72” in this context?
原库参考解析
指数增长・衰减与复利进阶
\( P\left(t\right)=260\left(1.03\right)^{\left(\frac{3}{2}\right)t} \)
The function P models the population, in thousands, of a certain city t years after 2009. 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
\( f\left(x\right)=7,000\left(0.90\right)^x \)
A conservation scientist implemented a program to reduce a certain invasive insect population in an area. The given function estimates this insect species' population x years after 2008, where x \( \le \) 8. Which of the following is the best interpretation of 7,000 in this context?
原库参考解析
指数增长・衰减与复利挑战
The function f is defined by \( f\left(x\right)=41\left(0.28\right)^x \) . For any positive integer n, the value of f (n) is p% less than the value of f (n -1). What is the value of p?
原库参考解析
指数增长・衰减与复利进阶
A certain neighborhood had a population of 1,170 in 2006. Each year for the next 5 years, the population of the neighborhood increased by approximately 3% of the population the previous year. Which of the following equations represents the population, N , of the neighborhood t years after 2006, where t ≤5?
原库参考解析
指数增长・衰减与复利进阶
The predicted number, B(t), of a certain type of bacterial cell in a growth medium is given by the function B(t) = 90(2)t, where t is the time, in days,since the number of bacterial cells was first measured. Which of the following is the best interpretation of B(4)= 1,440 in this context?
原库参考解析
指数增长・衰减与复利进阶
\( f\left(x\right)=8,000\left(0.75\right)^x \)
A conservation scientist implemented a program to reduce a certain invasive insect population in an area. The given function estimates this insect species' population x years after 2008, where x \( \le \) 8. Which of the following is the best interpretation of 8,000 in this context?
原库参考解析
