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
A model estimates that in a particular forest, the number of trees with any given diameter measured at shoulder height is 21% less for each 1-inch increase in tree diameter measured at shoulder height. The model can be written in the form \( f(x)=ab^x \), where a and b are constants and x is the tree's diameter, in inches, measured at shoulder height, and \( x\ge5 \). The model estimates that 3,100 trees in this forest have a diameter of 13 inches measured at shoulder height. Which function best represents this model?
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 1984, where \( x\le10 \), can be modeled by the function \( f(x)=47,000(1.03)^{x} \) . Which of the following is the best interpretation of the statement "\( f(6) \) is approximately equal to 56,120" in this context?
原库参考解析
指数增长・衰减与复利挑战
The value of a collectible item increased by the same percentage every year between 2000 and 2007. In 2000, the value of the collectible item was $40, and in 2002, the value of the collectible item was $48.40. What was the value, in dollars, of this collectible item in 2007?
原库参考解析
指数增长・衰减与复利进阶
\( P(t)=67\left(\frac{5}{4}\right)^t \)
A reindeer population was introduced into an area and researched for 19 years. Function P models this reindeer population t years after the population was introduced into the area. Which statement is the best interpretation of \( \frac{5}{4} \) in this context?
原库参考解析
指数增长・衰减与复利进阶
\( C(t)=220\left(\frac{52}{51}\right)^{t-12}+8 \)
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 228?
原库参考解析
指数增长・衰减与复利挑战
\( f(x)=28(1.40)^{\frac{x}{3}} \)
For the given function \( f \), the value of \( f(x) \) increases by p% for every increase of x by 6. What is the value of p?
原库参考解析
指数增长・衰减与复利挑战
For the exponential function g, the value of \( g(x) \) doubles for every increase of 2 in the value of x. If \( g(4)=9 \), what is the value of \( g(6) \)?
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 $700 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 0.2% 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 0.2% 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
指数增长・衰减与复利进阶
An exponential function \( f \) gives the estimated amount of an X-ray beam's initial intensity remaining after passing through a w-centimeter-thick window made of beryllium. The function estimates that after passing through a 1.2-centimeter-thick window, the amount of the X-ray beam's initial intensity remaining is 0.74. Which equation could define ?
原库参考解析
指数增长・衰减与复利挑战
The equation \( E(t)=3(1.2)^t \) gives the estimated number of employees at a certain business, where t is the number of years since the business opened. Which of the following is the best interpretation of the number 3 in this context?
原库参考解析
指数增长・衰减与复利挑战
The equation \( E(t)=3(1.7)^t \) gives the estimated number of employees at a certain business, where t is the number of years since the business opened. Which of the following is the best interpretation of the number 3 in this context?
原库参考解析
指数增长・衰减与复利进阶
The function \( f(x)=900(0.5)^{\frac{x}{12}} \) gives the predicted intensity of a beam \( f(x) \) , in number of photons in the beam, x millimeters below the surface of a certain material. What is the best interpretation of \( f(12)=450 \)?
参考答案
