Update 2024-03-01-数学学习.md

_posts/2024-03-01-数学学习.md

@@ -311,9 +311,37 @@ dx = 1/10, dy = 1/48 dx = 1/480
 
 ---
 
+### 中值定理
+
+费马定理
+
+f在极值点x0 ∈ (a, b)处可导 => f'(x0) = 0
+
+罗尔中值定理
+
+f在[a, b]连续，(a, b)可导
+
+f(a) = f(b) => $\exist \zeta \in $(a,b), f'($\zeta$) = 0
+
+拉格朗日中值定理
+
+f在[a, b]连续，(a, b)可导
+
+=> $\exist \zeta \in (a, b), s.t. f'(\zeta) = \frac {f(b) - f(a)} {b - a}$ 
+
+科西中值定理
+
+f，g在[a, b]连续，(a, b)可导，g'(x) != 0, $\forall x \in (a,b)$ 
+
+=> $\exist \zeta \in (a,b), s.t. \frac {f(b) - f(a)} {g(b) - g(a)} = \frac {f'(\zeta)} {g'(\zeta)}$ 
+
+---
+
 ### $G(x) = \int g(x)dx $
 
+积分的唯一性
 
+if F' = G', F(x) = G(x) + c