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
An exponential model for the number of squirrels in a certain area estimates that there were 200 squirrels in the area in the year 2003, and that at the end of each 4-year period for the next 20 years, the number of squirrels in the area was 120% more than the number at the end of the previous 4-year period. Which of the following equations represents this model, where N is the estimated number of squirrels in this area t years after 2003, and \( 0\le t\le20 \)?
参考答案
指数增长・衰减与复利进阶
James purchased a certain baseball card on January 1. The function \( f(x)=85(1.03)^x \), where \( 0\le x\le10 \) , gives the predicted value, in dollars, of the baseball card x years after James purchased it. What is the best interpretation of the statement "\( f(8) \) is approximately equal to 108" in this context?
参考答案
指数增长・衰减与复利进阶
The function \( f \) is defined by \( f(x)=55(0.12)^x \). 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?
参考答案
指数增长・衰减与复利进阶
\( y=7,200(0.85)^x \)
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?
参考答案
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 ≤ 3. 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 120% of the estimated number of visitors 30 minutes earlier. Which of the following equations best represents this model?
参考答案
指数增长・衰减与复利进阶
\( P(t)=22.5(1.036)^t \)
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 this function, what is the best interpretation of the statement “P(8) is approximately equal to 29.86”?
参考答案
指数增长・衰减与复利挑战
An environmental scientist studies the decomposition of paper bags when the bags are placed in an environment with a certain type of bacteria. The exponential function \( f(x)=545(0.91)^x \) gives the estimated remaining mass, in grams, of the paper bags x days after they are placed in an environment with this bacteria, where \( 0\le x\le30 \). If this function is graphed in the xy-plane, where \( y=f(x) \), which statement is the best interpretation of the point (1, 495.95) on the graph?
参考答案
指数增长・衰减与复利进阶
For an online trivia game, 300 points are awarded for a correct answer if a question is answered in less than 4 seconds from the time the question is asked. Each 4 seconds after the question is asked, the number of points awarded for a correct answer decreases by 15% of the number of points awarded for a correct answer in the previous 4 seconds. Which function gives the number of points awarded for a correct answer x seconds from the time the question is asked, where x is a multiple of 4?
参考答案
指数增长・衰减与复利挑战
A scientist initially measures 6,000 bacteria in a growth medium. 4 hours later, the scientist measures 12,000 bacteria. Assuming exponential growth, the formula \( P=C2^{rt} \) 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?
参考答案
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?
参考答案
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?
参考答案
指数增长・衰减与复利进阶
A researcher observes a sample of a nuclide. An exponential model estimates that the mass, in grams, of the sample decreases by 21% every 19.72 minutes. Which of the following equations could represent this model, where M is the estimated mass, in grams, of the sample t minutes after the researcher began observing the sample?
参考答案
