Fundamental Theorem of Calculus

Core Sentence / 核心句

The Fundamental Theorem of Calculus links accumulation and rate of change.

中文理解：微积分基本定理说明积分和导数不是两个孤立工具，而是互相反向的过程。

F (x) = \int_{a}^{x} f (t) d t

then

F^{'} (x) = f (x)

If $G^{'} (x) = f (x)$ , then

\int_{a}^{b} f (x) d x = G (b) - G (a) .

Part 1 differentiates accumulation; Part 2 computes accumulation from antiderivative endpoint change.

看到 accumulated area function、derivative of integral、net change，就想到 FTC。

不要只记成“积分求反导”；它的概念核心是 accumulation 和 original function 的连接。

Chapter 9 · Area and Slope
- dividing FTC by interval width turns total accumulation into average slope
Chapter 8 · Area-Function Derivative
- the derivative of accumulated area equals the current integrand height
Chapter 8 · Endpoint Evaluation
- $\int_{a}^{b} f = F (b) - F (a)$ compresses continuous accumulation into two endpoint values
01 - The Essence of Calculus · Knowledge Point 6: Derivative and Integral Undo Each Other
- 这就是 Fundamental Theorem of Calculus 的直觉版。
第8课 · FTC Part 2
- continuous accumulation equals antiderivative endpoint change
第8课 · FTC Part 1
- the derivative of an accumulation function recovers the current integrand