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・衰减与复利进阶
A sample of a certain isotope takes 29 years to decay to half its original mass. The functions \( s(t)=184(0.5)^{\frac{t}{29}} \) gives the approximate mass of this isotope, in grams, that remains t years after a 184-gram sample starts to decay. which statement is the best interpretation of s(87)= 23 in this context?
原库参考解析
指数增长・衰减与复利进阶
The number of bacteria in a liquid medium doubles every day. There are 48,000 bacteria in the liquid medium at the start of an observation. Which represents the number of bacteria, y, in the liquid medium t days after the start of the observation?
原库参考解析
指数增长・衰减与复利挑战
\( f\left(x\right)=75\left(0.3\right)^x \)
The function \( f\left(x\right) \) is a decreasing exponential function. By what percent is \( f\left(n\right) \)less than \( f\left(n-1\right) \)? (Disregard the % sign when entering your answer. For example, if your anwer is 4.97%, enter 4.97.)
原库参考解析
指数增长・衰减与复利进阶
A certain neighborhood had a population of 1340 in 2009. Each year for the next 5 years, the population of the neighborhood increased by approximately 3% of the population the previous year. Which of the following equations represents the population, N, of the neighborhood t years after 2009, where t≤5?
原库参考解析
指数增长・衰减与复利进阶
A certain neighborhood had a population of 1,340 in 2009. Each year for the next 5 years, the population of the neighborhood increased by approximately 3% of the population the previous year. Which of the following equations represents the population, N, of the neighborhood t years after 2009, where t \( \le \) 5?
原库参考解析
指数增长・衰减与复利进阶
In 2018,approximately 40,000 kilowatts of solar power capacity was newly installed in a certain area in the United States.In 2019, the amount of newly installed solar power capacity in this area increased by 11% from the amount of newly installed solar power capacity in 2018.If the amount of newly installed solar power capacity continues to increase each year by 11% of the previous year's newly installed solar power capacity, which of the following types of functions best describes how the amount of newly installed solar power capacity changes over time?
原库参考解析
指数增长・衰减与复利进阶
f(x) =7,000(1.003)2x
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 (1.003)2x?
原库参考解析
指数增长・衰减与复利挑战
The amount of money owed on a certain type of loan has two components: the principal balance of the loan and the amount of interest accrued on the loan. For a loan of this type, each payment was $298.00 and there were 60 payments total. Part of each payment was applied to the principal balance of the loan, and the rest was applied to the amount of interest accrued on the loan. The amount of the 1st payment that was applied to the principal balance was $220.93. For each additional payment, the amount that was applied to the principal balance was approximately 0.5% greater than the amount that was applied to the principal balance for the previous payment. Which function best approximates the amount of the x th payment, in dollars, that was applied to the amount of interest accrued, where x ≤ 60?
原库参考解析
指数增长・衰减与复利进阶
\( y=7,400\left(0.87\right)^x \)
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?
原库参考解析
指数增长・衰减与复利挑战
\( f\left(x\right)=k\left(1.85\right)^x \)
The function f is defined by the given equation, where k is a constant. The value of f(x) increases by p% for every increase of x by 1. For which of the following functions, where k is a constant, does the value of g(x) increase by p% for every increase of x by \( \frac{1}{3} \)?
原库参考解析
指数增长・衰减与复利挑战
\( f\left(x\right)=6,110\left(0.18\right)^{\frac{x}{12}} \)
The function f gives the value, in dollars, of a certain piece of equipment after x months of use. If the value of the equipment decreases each year by p% of its value the preceding year, what is the value of p?
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 day. There are 83,000 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 days after the start of the observation?
原库参考解析
