Definition and Derivative Property

After the area intuition, we formalize the upper-limit integral function and prove its key property: its derivative equals the integrand.

定义

基本性质

This is the “first part” of the Fundamental Theorem of Calculus, linking differentiation and integration.

证明的详细过程

步骤 1：利用导数的定义

$F'(x) = \lim_{h \to 0} \frac{F(x+h) - F(x)}{h} = \lim_{h \to 0} \frac{\int_a^{x+h} f(t) \, dt - \int_a^x f(t) \, dt}{h}$

步骤 2：利用积分的区间可加性

$F'(x) = \lim_{h \to 0} \frac{\int_x^{x+h} f(t) \, dt}{h}$

步骤 3：利用积分中值定理

There exists $\xi \in [x, x+h]$ (or $[x+h, x]$) such that

$\int_x^{x+h} f(t) \, dt = f(\xi) \cdot h$

步骤 4：取极限

$F'(x) = \lim_{h \to 0} \frac{f(\xi) \cdot h}{h} = \lim_{h \to 0} f(\xi) = f(x)$

The proof is complete. The upper-limit integral function has clear geometric meaning and is differentiable, enabling the applications that follow.

总结

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