Mean Value Theorem

Core Sentence / 核心句

The Mean Value Theorem guarantees that some instantaneous rate inside an interval equals the average rate over the whole interval.

中文理解：只要函数在闭区间连续、内部可导，区间内至少有一点的切线斜率等于端点割线斜率。

Hypotheses / 前提

• continuous on $[a,b]$
• differentiable on $(a,b)$

Conclusion / 结论

There exists $c\in (a,b)$ such that

$$f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}.$$

Visual Meaning / 图像含义

至少一条 interior tangent line 平行于 endpoint secant line。

Watch Out / 易错点

• continuity alone is insufficient.
• theorem guarantees existence, not uniqueness.
• $c$ lies in the open interval.
• MVT concerns a derivative value; Intermediate Value Theorem concerns an attained function value.

Search and Source Mentions / 搜索与出现位置

Mentioned In / 出现位置

• 微积分BC第5次正课 · Mean Value Theorem
  • continuity and differentiability force an interior tangent slope to match the endpoint secant slope

Connected Notes / 连接

• 微积分BC第6次正课-导数符号极值与凹凸性
• Extreme Value Theorem
• 微积分BC第5次正课-相关变化率线性近似与中值定理
• Derivatives
• Average Rate of Change
• Secant Line
• Tangent Line
• Intermediate Value Theorem
• Linear Approximation