Mean Value & Rolle's Theorems Cheat Sheet

Key Facts

• Continuity on $[a,b]$ means the graph has no breaks, holes, or jumps anywhere from $a$ to $b$, including the endpoints.
• Differentiability on $(a,b)$ means $f'(x)$ exists for every interior point, so the graph has no corners, cusps, vertical tangents, or discontinuities inside the interval.
• Rolle's Theorem states that if $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)$, then there is at least one $c\in(a,b)$ such that $f'(c)=0$.
• The Mean Value Theorem states that if $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there is at least one $c\in(a,b)$ such that $f'(c)=\frac{f(b)-f(a)}{b-a}$.
• The secant slope between $a$ and $b$ is $m_{sec}=\frac{f(b)-f(a)}{b-a}$, which represents the average rate of change on the interval.
• The tangent slope at $x=c$ is $m_{tan}=f'(c)$, which represents the instantaneous rate of change at that point.
• Rolle's Theorem is the special case of the Mean Value Theorem when $f(a)=f(b)$, so $\frac{f(b)-f(a)}{b-a}=0$.
• The theorems guarantee at least one value of $c$, but they do not guarantee that the value is unique.

Vocabulary

Continuity

A function is continuous on an interval when its graph can be drawn there without breaks, holes, or jumps.

Differentiability

A function is differentiable at a point when its derivative exists at that point.

Secant Slope

The secant slope is the average rate of change between two points, given by $\frac{f(b)-f(a)}{b-a}$.

Tangent Slope

The tangent slope is the instantaneous rate of change at a point, given by $f'(c)$.

Rolle's Theorem

Rolle's Theorem guarantees a point where $f'(c)=0$ when a function is continuous, differentiable, and has equal endpoint values.

Mean Value Theorem

The Mean Value Theorem guarantees a point where the tangent slope equals the secant slope over an interval.