Rolle’s Theorem and the Mean Value Theorem: When Does a Secant Become a Tangent?

This lesson explains why a tangent line can share the slope of an entire secant segment. You will learn Rolle’s Theorem as the zero-slope case of the Mean Value Theorem, then apply the hypotheses and solve for interior points where the derivative matches the target slope. 1

The lecture works through Rolle’s Theorem for on $[-2,2]$, and the Mean Value Theorem for on $[-1,2]$. It also shows why continuity alone does not guarantee a differentiable interior point using , and ends with an exam-ready order: state the theorem, check its conditions, compute the target slope, solve, keep only interior roots, and conclude. 2