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
Sonja purchased a laptop in September 2018 when the laptop had a value of $1,000. Each month for the next 50 months, the value of the laptop decreased by 2% of the previous month's value. Which equation represents this situation, where v is the value of the laptop, in dollars, m is the number of months since Sonja purchased the laptop, and \( m\le50 \) ?
原库参考解析
指数增长・衰减与复利挑战
On January 1, there were 233 views of a video posted on a social media site. Every 2 days after January 1, the number of views of the video had increased by 60% of the number of views 2 days earlier. The function \( f \) gives the predicted number of views x days after January 1. Which equation defines \( f \)?
原库参考解析
指数增长・衰减与复利挑战
\( f\left(x\right)=3,950\left(0.38\right)^{\frac{x}{12}} \)
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 exponential function \( f \) is defined by \( f(x)=ab^x \), where a and b are positive constants. If \( f(s)=t \) and \( f(s+1)=t-0.87t \), 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
James purchased a certain baseball card on January 1. The function \( f(x)=55(1.04)^x \), 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 “\( f(7) \) is approximately equal to 72” in this context?
原库参考解析
指数增长・衰减与复利进阶
\( P(t)=260(1.03)^{\left(\frac{3}{2}\right)t} \)
The function P models the population, in thousands, of a certain city t years after 2009. According to the model, the population is predicted to increase by n% every 8 months. What is the value of n?
原库参考解析
指数增长・衰减与复利进阶
Anton is selling toys from his toy collection. When he started selling the toys, there were 2,800 toys in the collection. At the end of each week for 3 weeks after Anton started selling the toys, the number of toys in the collection had decreased by \( \frac{1}{2} \) of the toys in the collection at the end of the previous week. Which equation gives the number of toys, c, in the collection w weeks after Anton started selling the toys, where \( 0≤w<3 \) ?
原库参考解析
指数增长・衰减与复利进阶
The number of bacteria in a growth medium is expected to increase by 150% every 2 hours during a period of observation. The number of bacteria in the growth medium was estimated to be 7,000 when the period of observation began. Which function \( P \) gives the expected number of bacteria in this growth medium \( t \) hours after the period of observation began?
原库参考解析
指数增长・衰减与复利进阶
A savings account is opened with an initial deposit of $7,000. The amount of money in the account t years after the initial deposit is given by the function \( f\left(t\right)=7,000\left(1.01\right)^{2t} \). Which of the following is the best interpretation of the statement "f(3) is approximately equal to 7,430.64" in this context?
原库参考解析
指数增长・衰减与复利挑战
\( f\left(x\right)=3,600\left(0.28\right)^{\frac{x}{12}} \)
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
A researcher observes a sample of a nuclide. An exponential model estimates that the mass, in grams, of the sample decreases by 27% every 14.07 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?
原库参考解析
指数增长・衰减与复利挑战
A certain investment account offers a special interest rate for the first 4 months the account is open followed by a lower interest rate for the remainder of the time the account is open. Bennett opened one of these accounts with an original account balance of $900 and did not make any other deposits or withdrawals. 4 months after Bennett opened the account, the balance had increased by 0.6% of the original balance. 6 months after Bennett opened the account, the balance had increased by an additional 0.1% of the balance at the end of the first 4 months. Every 2 months after the first 6 months, the balance had increased by an additional 0.1% of the balance 2 months before. Which of the following equations could represent the account balance \( B(x) \), in dollars, x months after the account was opened, where \( x\ge4 \)?
原库参考解析
