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 equation \( N(m)=64(Q)^{\frac{m}{6}} \) gives the predicted population N(m), in thousands, of a certain bacteria colony m minutes after the initial measurement, where Q is a constant greater than 1. The predicted population increases by p% every 120 seconds. What is the value of p in terms of Q?
参考答案
指数增长・衰减与复利进阶
The function \( f\left(x\right)=236\left(1.022\right)^x \) models the value, in dollars, of a certain bank account from 1958 through 1973, where x is the number of years after the end of 1958. Which of the following is the best interpretation of “\( f\left(5\right) \) is approximately equal to 263" in this context?
原库参考解析
指数增长・衰减与复利挑战
The exponential function \( f \) is defined by \( f\left(x\right)=ab^x \) , where a and b are positive constants. If \( f\left(s\right)=t \) and \( f\left(s+1\right)=t-0.87t \), where s and t are constants, what is the value of b?
原库参考解析
指数增长・衰减与复利进阶
\( C(t)=240(\frac{54}{53})^{t−14}+7 \)
The function C gives the estimated number of cephalopods, a class of marine animal, in a certain area, where t is the number of months since a study began. How many months after the study began was the number of cephalopods in the area estimated to be 247?
原库参考解析
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?
原库参考解析
指数增长・衰减与复利进阶
The predicted population of a city x years after 1982, where \( x\le10 \), can be modeled by the function \( f(x)=47,000(1.04)^x \). Which of the following is the best interpretation of the statement “\( f(6) \) is approximately equal to 59,470” in this context?
原库参考解析
指数增长・衰减与复利挑战
The function \( f(t)=40,000(2)^{\frac{t}{410}} \) 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
\( f\left(x\right)=3,000\left(0.75\right)^x \)
A conservation scientist implemented a program to reduce the population of a certain species in an area. The given function estimates this species' population x years after 2008, where \( x\le8 \). Which of the following is the best interpretation of 3,000 in this context?
原库参考解析
指数增长・衰减与复利进阶
Two variables, x and y, are related such that for each increase of 1 in the value of x, the value of y increases by a factor of 4. When \( x=0 \) , \( y=200 \) . Which equation represents this relationship?
原库参考解析
指数增长・衰减与复利挑战
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 \( P=C\left(2\right)^{rt} \) 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・衰减与复利进阶
The population of the town of Smithville doubled every 75 years from 1659 to 1959. The population of this town was 240,000 in 1959. What was the population of this town in 1659?
原库参考解析
指数增长・衰减与复利进阶
There are 240 players in a tennis competition that includes 4 rounds of matches. Each player in the competition will play a match against another player in round 1. At the end of each round, the player who loses the match is eliminated and the player who won the match advances to the next round to play a match against another winning player. Which equation gives the number of players, p, eliminated at the end of round r, where \( r\le4 \)?
原库参考解析
