Exponential Growth and Decay | 指数增长与衰减

📚 Exponential Growth and Decay | 指数增长与衰减

Exponential growth and decay are mathematical models that describe how quantities change over time at a rate proportional to their current value. These models appear throughout IGCSE Mathematics, from compound interest to radioactive decay, and are essential for understanding real-world processes such as population growth, cooling, and depreciation.

指数增长与衰减是描述一个量以与其当前值成正比的速率随时间变化的数学模型。这些模型贯穿IGCSE数学课程，从复利到放射性衰变，都是理解人口增长、冷却、折旧等现实过程的关键工具。

1. The Exponential Function Form | 指数函数的基本形式

The general exponential function can be written as $ y = ab^x $, where $ a $ is the initial value, $ b $ is the growth or decay factor, and $ x $ represents time or number of periods. When $ b > 1 $, the function models exponential growth; when $ 0 < b < 1 $, it models exponential decay.

一般的指数函数可以写成 y = a·bˣ，其中 a 是初始值，b 是增长或衰减因子，x 表示时间或期数。当 b > 1 时，该函数表示指数增长；当 0 < b < 1 时，表示指数衰减。

y = a·bˣ

For example, if a population of bacteria doubles every hour, the model is $ N = N_0 \cdot 2^t $, where $ N_0 $ is the initial population and $ t $ is time in hours. Here $ b = 2 $, indicating growth.

例如，如果一个细菌种群每小时翻倍，模型为 N = N₀·2ᵗ，其中 N₀ 是初始种群数量，t 是以小时为单位的时间。此时 b = 2，表示增长。

2. Growth Factor and Percentage Increase | 增长因子与百分增长率

In exponential growth problems, the growth factor $ b $ is related to the percentage increase per period. If a quantity increases by $ r\% $ each period, then $ b = 1 + \frac{r}{100} $. For example, a 5% annual increase gives $ b = 1.05 $.

在指数增长问题中，增长因子 b 与每期的百分增长率有关。如果一个量每期增加 r%，则 b = 1 + r/100。例如，每年增长5%，则 b = 1.05。

If an investment grows by 8% per year, the multiplier is 1.08.

如果一项投资每年增长8%，乘数就是1.08。
If a population increases by 2.5% per year, the multiplier is 1.025.

如果一个人口每年增长2.5%，乘数就是1.025。
To find the growth factor from a percentage, divide by 100 and add 1.

要从百分比求增长因子，先除以100再加1。

3. Decay Factor and Percentage Decrease | 衰减因子与百分衰减率

For exponential decay, the decay factor $ b $ is related to the percentage decrease per period. If a quantity decreases by $ r\% $ each period, then $ b = 1 – \frac{r}{100} $. For example, a 12% annual depreciation gives $ b = 0.88 $.

对于指数衰减，衰减因子 b 与每期的百分衰减率有关。如果一个量每期减少 r%，则 b = 1 − r/100。例如，年折旧12%，则 b = 0.88。

If a radioactive substance decays by 3% per day, the multiplier is 0.97.

如果一种放射性物质每天衰减3%，乘数就是0.97。
If a car loses 15% of its value each year, the multiplier is 0.85.

如果一辆汽车每年贬值15%，乘数就是0.85。
The decay factor must always be between 0 and 1 for a decreasing quantity.

对于递减的量，衰减因子必须始终在0和1之间。

4. The Formula for Repeated Growth or Decay | 重复增长或衰减的公式

When a quantity grows or decays by a fixed percentage over $ n $ equal time periods, the final value is given by:

当一个量在 n 个相等的时间段内按固定百分比增长或衰减时，最终值由下式给出：

Final Value = Initial Value × (1 ± r/100)ⁿ

Here $ r $ is the percentage rate per period, and the plus sign is used for growth, the minus sign for decay. This formula is fundamental in IGCSE questions involving compound interest, population changes, and depreciation.

这里 r 是每期的百分率，增长用加号，衰减用减号。这个公式在IGCSE涉及复利、人口变化和折旧的题目中至关重要。

5. Worked Example: Population Growth | 例题：人口增长

A town has a population of 20,000. The population increases by 3.5% each year. Find the population after 6 years.

一个小镇有20,000人口。人口每年增长3.5%。求6年后的人口。

P = 20000 × (1 + 3.5/100)⁶ = 20000 × (1.035)⁶

Using a calculator, $ (1.035)^6 \approx 1.2293 $. Therefore $ P \approx 20000 \times 1.2293 = 24586 $. The population is approximately 24,600.

使用计算器，(1.035)⁶ ≈ 1.2293。因此 P ≈ 20000 × 1.2293 = 24586。人口约为24,600。

6. Worked Example: Radioactive Decay | 例题：放射性衰变

A radioactive sample has an initial mass of 80 grams. It decays at a rate of 4% per day. Find the mass remaining after 10 days.

一份放射性样品的初始质量为80克。它以每天4%的速率衰减。求10天后剩余的质量。

M = 80 × (1 − 4/100)¹⁰ = 80 × (0.96)¹⁰

Using a calculator, $ (0.96)^{10} \approx 0.6648 $. Therefore $ M \approx 80 \times 0.6648 = 53.18 $ grams. The mass remaining is about 53.2 grams.

使用计算器，(0.96)¹⁰ ≈ 0.6648。因此 M ≈ 80 × 0.6648 = 53.18 克。剩余质量约为53.2克。

7. Compound Interest as Exponential Growth | 复利作为指数增长

Compound interest is a classic example of exponential growth. The formula for compound interest is:

复利是指数增长的经典例子。复利公式为：

A = P(1 + r/100)ⁿ

where $ A $ is the final amount, $ P $ is the principal, $ r $ is the annual interest rate, and $ n $ is the number of compounding periods. When interest is compounded annually, $ n $ equals the number of years.

其中 A 是最终金额，P 是本金，r 是年利率，n 是复利期数。当利息按年复利时，n 等于年数。

If the interest is compounded quarterly, $ n $ is multiplied by 4 and $ r $ is divided by 4.

如果利息按季度复利，n 乘以4，r 除以4。
If the interest is compounded monthly, $ n $ is multiplied by 12 and $ r $ is divided by 12.

如果利息按月复利，n 乘以12，r 除以12。

8. Depreciation: Exponential Decay in Real Life | 折旧：现实中的指数衰减

Depreciation is the reduction in the value of an asset over time. It follows the same structure as exponential decay. The value of an asset after $ n $ years is:

折旧是资产随时间价值减少的过程。它与指数衰减遵循相同的结构。一项资产在 n 年后的价值为：

V = V₀(1 − r/100)ⁿ

For example, a machine bought for $ \$5000 $ depreciates at 10% per year. After 3 years, its value is $ V = 5000 \times (0.90)^3 = 5000 \times 0.729 = 3645 $ dollars.

例如，一台机器以5000美元购买，每年折旧10%。3年后，其价值为 V = 5000 × (0.90)³ = 5000 × 0.729 = 3645 美元。

9. Comparing Growth and Decay | 增长与衰减的比较

The table below summarises the key differences between exponential growth and exponential decay.

下表总结了指指数增长与指数衰减的主要区别。

Feature	Growth	Decay
Multiplier	b > 1	0 < b < 1
Percentage change	Increase	Decrease
Graph shape	Rises sharply	Falls towards zero
Examples	Population, investment	Radioactive decay, depreciation

Notice that in both cases the quantity changes by the same percentage each period, but the direction is different. This is why the graph of growth curves upward, while the graph of decay curves downward.

注意在两种情况下，每个时期数量变化的百分比相同，但方向不同。这就是为什么增长曲线向上弯曲，而衰减曲线向下弯曲。

10. Finding the Rate or Time | 求利率或时间

Sometimes you are given the initial and final values and need to find the rate or the time. For example, if a population grows from 5000 to 6000 in 4 years at a constant percentage rate, you can set up the equation:

有时我们已知初始值和最终值，需要求利率或时间。例如，如果一个人口在4年内从5000增长到6000，按固定百分率增长，可以建立方程：

6000 = 5000 × b⁴

Dividing both sides by 5000 gives $ b^4 = 1.2 $. Taking the fourth root, $ b = 1.2^{1/4} \approx 1.0466 $. Therefore the annual growth rate is approximately 4.66%.

两边除以5000得 b⁴ = 1.2。取四次方根，b = 1.2^(1/4) ≈ 1.0466。因此年增长率约为4.66%。

11. Exponential Graphs and Key Features | 指数图像与关键特征

Understanding the graph of $ y = ab^x $ is essential. For growth ($ b > 1 $), the graph starts at $ y = a $ when $ x = 0 $ and increases rapidly as $ x $ increases. For decay ($ 0 < b < 1 $), the graph also starts at $ y = a $ but decreases towards zero, never actually reaching zero.

理解 y = a·bˣ 的图像至关重要。对于增长（b > 1），图像在 x = 0 时从 y = a 开始，随着 x 增大而迅速上升。对于衰减（0 < b < 1），图像同样从 y = a 开始，但逐渐趋近于零，永远不会真正到达零。

The y-intercept is always $ a $.

y 截距始终是 a。
The x-axis is a horizontal asymptote for decay graphs.

x 轴是衰减图像的水平渐近线。
The graph is always above the x-axis for $ a > 0 $.

当 a > 0 时，图像始终位于 x 轴上方。

12. Common Mistakes and Exam Tips | 常见错误与考试技巧

Many students make errors by confusing the base $ b $ with the percentage rate. Remember that $ b = 1 \pm r/100 $, so a 20% decrease means $ b = 0.80 $, not $ b = -0.20 $. Always convert percentages to decimals before substituting.

许多学生容易将底数 b 与百分率 r 混淆。记住 b = 1 ± r/100，所以减少20%意味着 b = 0.80，而不是 b = −0.20。代入前务必先将百分数转换为小数。

Another common mistake is forgetting to use the correct time period. If the rate is annual but the time is given in months, convert the time to years or adjust the rate accordingly. Always check units carefully.

另一个常见错误是忘记使用正确的时间单位。如果利率是年利率但时间以月给出，需要将时间转换为年或相应调整利率。始终仔细检查单位。

In exam questions, show all working steps clearly. When using a calculator, do not round intermediate values too early; round only at the final step. For growth and decay problems, always write the formula first, substitute the values, then compute.

在考试中，要清晰地写出所有步骤。使用计算器时，不要过早四舍五入中间值；只在最后一步四舍五入。对于增长与衰减问题，先写出公式，再代入数值，最后计算。

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Exponential Growth and Decay | 指数增长与衰减