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 function \( f(x)=217(1.030)^x \) models the value, in dollars, of a certain bank account from 1957 through 1972, where x is the number of years after the end of 1957. Which of the following is the best interpretation of "\( f(5) \) is approximately equal to 252" in this context?
参考答案
指数增长・衰减与复利进阶
The function \( f\left(x\right)\ =\ 900\left(0.5\right)^{\frac{x}{12}} \) gives the predicted intensity of a beam \( f\left(x\right)\ \) , in number of photons in the beam, \( x\ \) millimeters below the surface of a certain material. What is the best interpretation of \( f\left(12\right)=450\ \) ?
参考答案
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(7)=2,048 in this context?
参考答案
指数增长・衰减与复利进阶
A scientist initially measures 9,000 bacteria in a growth medium. 8 hours later, the scientist measures 18,000 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?
参考答案
指数增长・衰减与复利进阶
For a certain isotope, at the end of every 29-year period, the mass of a sample of the isotope has decayed to half its mass at the beginning of the 29-year period. The function \( s(t)=264(0.5)^{\frac{t}{29}} \) gives the approximate mass of this isotope, in grams, that remains t years after a 264-gram sample starts to decay. Which statement is the best interpretation of \( s(3·29)=33 \) in this context?
参考答案
指数增长・衰减与复利挑战
The function \( f(n)=7(20.59)^{\frac{n}{4}} \) gives each term of a sequence as a function of the term’s position, n, in the sequence, where n is a whole number. The value of the term in position 14 is p% more than the value of the term in position 10. What is the value of p?
参考答案
指数增长・衰减与复利进阶
For a certain isotope, at the end of every 29-year period, the mass of a sample of the isotope has decayed to half its mass at the beginning of the 29-year period. The function \( s(t)=192(0.5)^{\frac{t}{29}} \) gives the approximate mass of this isotope, in grams, that remains t years after a 192-gram sample starts to decay. Which statement is the best interpretation of \( s(2·29)=48 \) in this context?
参考答案
指数增长・衰减与复利挑战
The function \( f(n)=7(20.43)^{\frac{n}{5}} \) gives each term of a sequence as a function of the term's position, n, in the sequence, where n is a whole number. The value of the term in position 15 is p% more than the value of the term in position 10. What is the value of p?
参考答案
指数增长・衰减与复利挑战
\( f(x)=2,000(0.80)^x \)
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 \( x\le8 \). Which of the following is the best interpretation of 2,000 in this context?
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 \( x\le8 \). Which of the following is the best interpretation of 2,000 in this context?
参考答案
指数增长・衰减与复利进阶
The function \( f(x)=230,000(1.23)^x \) gives a company’s predicted annual revenue, in dollars, x years after the company started selling clothes online, where \( 0<x≤10 \). What is the best interpretation of the statement “\( f(6) \) is approximately equal to 796,450” in this context?
参考答案
指数增长・衰减与复利基础
The graph shows the estimated concentration of a certain strain of bacteria, in millions of cells per mL of nutrient medium, over time x, in minutes since the initial measurement. According to the graph, which of the following is closest to the number of minutes it took for the estimated concentration of the bacteria to increase from the initial concentration to 20 million cells per mL of nutrient medium?
参考答案
指数增长・衰减与复利基础
A fully charged battery has an initial charge of 100 units. When this battery is in use, the remaining charge of the battery at any given moment is approximately 80% of the charge from one hour earlier. Which function \( f \) gives the approximate remaining charge, in units, of this battery when it is in use for x hours?
参考答案
