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
For the function \( f \) , for every increase of 2 in the value of x, the value of \( f\left(x\right) \) increases by a factor of c, where c is a constant. Which of the following equivalent forms of function \( f \) displays the value of c as the base or the coefficient?
原库参考解析
指数增长・衰减与复利挑战
A model predicts that the population of a certain town was 5,000 in 2008. The model also predicts that each year for the next 5 years, the population p increased by 4% of the previous year's population. Which equation best represents this model, where x is the number of years after 2008, for \( x\le5 \)?
原库参考解析
指数增长・衰减与复利进阶
\( C\left(t\right)=240\left(\frac{53}{52}\right)^{t-15}+7 \)
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・衰减与复利进阶
The current value of a house is $174,000, and this value increases by 2.4% each year from the value of the previous year. Which function \( f \) gives the value, in dollars, of the house x years from now?
原库参考解析
指数增长・衰减与复利挑战
The number of bacteria in a liquid medium doubles every hour. There are 66,000 bacteria in the liquid medium at the start of an observation. Which of the following represents the number of bacteria, y, in the liquid medium t hours after the start of the observation?
原库参考解析
指数增长・衰减与复利挑战
The function \( f\left(n\right)=7\left(20.44\right)^{\frac{n}{4}} \) 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?
原库参考解析
指数增长・衰减与复利挑战
Each year, the value of an investment increases by 0.69% of its value the previous year. Which of the following functions best models how the value of the investment changes over time?
原库参考解析
指数增长・衰减与复利挑战
In a certain state, the population of pheasants, a type of bird, is estimated each year by counting the number of pheasants observed along certain roads in the state. On average, each year from 2005 to 2015 the number of pheasants counted per mile of road decreased by 3.5% of the number of pheasants per mile of road the previous year. Based on this average, if there were 6.22 pheasants per mile of road in this state in 2005, which of the following best approximates the number of pheasants per mile of road in 2015?
原库参考解析
指数增长・衰减与复利挑战
\( f\left(x\right)=2,000\left(1.005\right)^{3x} \)
The given function \( f \) models the balance of a bank account, in dollars, x years after it is opened. Which statement is the best interpretation of \( \left(1.005\right)^{3x} \)?
原库参考解析
指数增长・衰减与复利进阶
\( y=500\left(2\right)^{2x} \)
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 1997, 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
The relationship between the variables x and y is defined by an exponential equation. When \( x=0 \) , the value of y is 90, and for every increase in the value of x by 1, the corresponding value of y increases by 20% 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
\( z\left(w\right)=\left(0.829\right)^{2w} \)
The function z is defined by the equation shown. The value of \( z\left(w\right) \) decreases by p% for each increase by 1 in the value of w. Which of the following is closest to the value of p?
原库参考解析
