mean value theorem for real functions

微分中值定理

Analysis, Measure & PDE · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

If a real function f is continuous on [a,b], differentiable on (a,b), and a<b, some c in (a,b) satisfies f'(c)=(f(b)-f(a))/(b-a).

实函数在非退化闭区间连续且在内部可导，则内部有一点的导数等于两端点函数值之差除以区间长度。

Example

For f(x)=x2f(x)=x^2 on [1,3][1,3], the secant slope is 44 and f′(2)=4f'(2)=4.

Usage and distinctions

The real scalar conclusion does not automatically extend to a vector-valued function with a single common c.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Michael Taylor — Introduction to Analysis in One Variable§4.1, Theorem 4.1.2 and equation (4.1.15) · Checked 2026-10-14

Translation sources

  1. 北京大学数学科学学院 — 数学分析（实验班）课程介绍第一部分 一元微积分学，五、导数的应用：微分中值定理 · Checked 2026-10-14
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