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 population of a certain city doubled every 75 years from 1663 to 1963. The population of this city was 240,000 in 1963. What was the population of this city in 1663?
原库参考解析
指数增长・衰减与复利进阶
A computer program models the total mass of the population of a certain type of algae after the algae was placed in an environment where it has no natural predators. According to the model, the estimated total mass of this population of algae at the end of every 6-hour period is 129% greater than the estimated total mass of this population of algae at the end of the previous 6-hour period, and the estimated total mass of this population of algae is 613.90 grams after 18 hours. Which equation best represents this model, where A is the estimated total mass, in grams, of the population of algae after x hours, and \( x<50 \)?
原库参考解析
指数增长・衰减与复利挑战
A sum of 9,000 dollars was invested in a certificate of deposit (CD) that earns interest, and no other deposits or withdrawals were made after that. The function \( f(x)=9,000(1+0.03)^x \) gives the value of the CD, in dollars, x years after the investment was initially made, where \( x\le5 \). Which of the following is the best interpretation of the statement “\( f(2) \) is approximately equal to 9,474.08” in this context?
原库参考解析
指数增长・衰减与复利挑战
\( f(x)=18(2.80)^{\frac{x}{2}} \)
The function \( f \) is defined by the given equation. The value of \( f(x) \) increases by p% for every increase of x by 4. For which of the following functions, where n is a positive constant, does the value of \( g(x) \) increase by p% for every increase of x by 1?
原库参考解析
指数增长・衰减与复利挑战
At an initial observation, the number of bacteria in a population is 600. If the number of bacteria in the population doubles every 11 minutes, how many minutes after the initial observation will the number of bacteria in the population be 4,800?
原库参考解析
指数增长・衰减与复利挑战
In a computer application, the area of a figure on the screen increases over time. This area, in pixels, is given by the function \( A(t)=16(2)^t \), where t is the number of seconds after the area of the figure begins increasing. Which statement is the best interpretation of \( A\left(7\right)=2,048 \) in this context?
原库参考解析
指数增长・衰减与复利进阶
A model estimates that the initial number of photons in an X-ray beam is 800 when the beam reaches the surface of a certain material. The model also estimates that when the beam passes through this material, the number of photons in the beam decreases by 50% for each 7 millimeters of the material the beam passes through. Which equation represents this model, where I is the estimated number of photons in the beam when the beam reaches a point x millimeters from the surface?
原库参考解析
指数增长・衰减与复利进阶
The relationship between the variables x and y is defined by an exponential equation. When \( x=0 \), the value of y is 40, and for every increase in the value of x by 1, the corresponding value of y increases by 50% of its previous value. Which equation represents this relationship?
原库参考解析
指数增长・衰减与复利挑战
The function \( f \) is defined by \( f(x)=56(0.19)^x \). 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
\( y=2,600\left(a\right)^x \)
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, 85 visitors were estimated to be at the theme park when it opened, and every 30 minutes the estimated number of visitors increased by 140% of the estimated number of visitors 30 minutes earlier. Which of the following equations best represents this model?
原库参考解析
指数增长・衰减与复利进阶
On January 1, there were 237 views of a video posted on a social media site. Every 2 days after January 1, the number of views of the video had increased by 70% of the number of views 2 days earlier. The function j gives the predicted number of views x days after January 1. Which equation defines j?
原库参考解析
