Natural Exponential Function y=e^x: Operations and Properties | 自然指数函数 y=e^x 的运算与性质

📚 Natural Exponential Function y=e^x: Operations and Properties | 自然指数函数 y=e^x 的运算与性质

The natural exponential function, y = e^x, is a cornerstone of A-Level mathematics. It appears in differentiation, integration, differential equations, and real-world models such as population growth and radioactive decay. Understanding its operations and properties is essential for exam success.

自然指数函数 y = e^x 是 A-Level 数学的基石。它在微分、积分、微分方程以及人口增长、放射性衰变等现实世界建模中都会出现。理解其运算与性质对考试成功至关重要。

1. What is the Number e? | 什么是数 e？

The number e is an irrational constant, approximately equal to 2.71828. It is often defined by the following limit:

数 e 是一个无理常数，约等于 2.71828。它通常由以下极限定义：

e = lim_n→∞ (1 + 1/n)ⁿ = 2.71828…

e can also be written as an infinite series: e = 1 + 1/1! + 1/2! + 1/3! + … . This series converges quickly and is used in numerical approximations.

e 也可以用无穷级数表示：e = 1 + 1/1! + 1/2! + 1/3! + … 。这个级数收敛很快，常用于数值近似。

The function f(x) = e^x is called the natural exponential function because it is the unique exponential function whose derivative is equal to itself.

函数 f(x) = e^x 被称为自然指数函数，因为它是唯一一个导数等于自身的指数函数。

2. Graph and Transformations | 图像与变换

The graph of y = e^x passes through (0, 1) and (1, e). Its domain is all real numbers, and its range is all positive real numbers. The graph is always above the x-axis, strictly increasing, and concave upward.

y = e^x 的图像经过点 (0, 1) 和 (1, e)。其定义域为全体实数，值域为所有正实数。图像始终位于 x 轴上方，严格递增，并且凹向上。

• For all real x, e^x > 0.

  对所有实数 x，e^x > 0。

• e^0 = 1, so the y-intercept is 1.

  e^0 = 1，因此 y 截距为 1。

• As x → −∞, e^x → 0, so the x-axis is a horizontal asymptote.

  当 x → −∞ 时，e^x → 0，因此 x 轴是水平渐近线。

• As x → ∞, e^x → ∞.

  当 x → ∞ 时，e^x → ∞。

Transformations of e^x are also common. For example, y = e^(x+1) is the graph of e^x shifted left by 1 unit, while y = e^x + 2 is shifted up by 2 units. Similarly, y = −e^x is a reflection in the x-axis, and y = e^(−x) is a reflection in the y-axis.

e^x 的变换也很常见。例如，y = e^(x+1) 是 e^x 的图像向左平移 1 个单位，y = e^x + 2 是向上平移 2 个单位。类似地，y = −e^x 是关于 x 轴的反射，y = e^(−x) 是关于 y 轴的反射。

3. Laws of Exponents | 指数运算法则

The natural exponential function obeys the same laws as other exponential functions. For any real numbers a and b:

自然指数函数遵循与其他指数函数相同的运算法则。对于任意实数 a 和 b：

e^ae^b = e^a+b, e^a / e^b = e^a−b, (e^a)^b = e^ab

Also, e^0 = 1 and e^(−x) = 1 / e^x. These rules are frequently tested in simplification and equation-solving questions.

同时，e^0 = 1，e^(−x) = 1 / e^x。这些规则在化简和求解方程的问题中经常考查。

For example, e^(2x) / e^x simplifies to e^x, and e^(x)e^(−2x) simplifies to e^(−x).

例如，e^(2x) / e^x 化简为 e^x，e^(x)e^(−2x) 化简为 e^(−x)。

4. The Derivative of e^x | e^x 的导数

One of the most remarkable properties of e^x is that its derivative is itself:

e^x 最显著的性质之一是它的导数等于自身：

d/dx (e^x) = e^x

Using the chain rule, for a differentiable function u(x), we have:

根据链式法则，对于可微函数 u(x)，我们有：

d/dx (e^u) = e^u · du/dx

In particular, d/dx (e^(kx)) = k e^(kx), where k is a constant. For example, d/dx (e^(3x)) = 3e^(3x), and d/dx (e^(x²+1)) = 2x e^(x²+1).

特别地，d/d

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