3. Mean Value Theorem Verifier — Theory
Mean Value Theorem (MVT)
\[
\textbf{If } f \text{ is continuous on } [a,b] \text{ and differentiable on } (a,b),
\textbf{ then there exists } c\in(a,b) \textbf{ such that}
\]
\[
f'(c)=\frac{f(b)-f(a)}{b-a}.
\]
Geometrically, the MVT says: the slope of the secant line through
\((a,f(a))\) and \((b,f(b))\) is matched by the slope of at least one tangent line somewhere in between.
What the calculator checks
- Secant slope:
\[
m=\frac{f(b)-f(a)}{b-a}.
\]
- Differentiate: compute \(f'(x)\).
- Solve:
\[
f'(x)=m \quad \text{on } (a,b).
\]
The solver can return multiple solutions \(c\) if they exist.
- Condition warnings (heuristic): the tool samples points on the interval to detect likely domain problems
(holes, vertical asymptotes, log/sqrt invalid inputs, etc.).
This is not a formal proof of continuity/differentiability, but it is very helpful for spotting issues.
Rolle’s Theorem as a special case
If the endpoints have the same function value, then the secant slope is zero.
\[
f(a)=f(b)\quad \Rightarrow\quad \frac{f(b)-f(a)}{b-a}=0.
\]
\[
\text{So MVT becomes: }\exists c\in(a,b)\text{ such that } f'(c)=0.
\]
This is exactly Rolle’s Theorem.
Worked example
Example: \(f(x)=x^2\) on \([0,2]\).
\[
m=\frac{f(2)-f(0)}{2-0}=\frac{4-0}{2}=2.
\]
\[
f'(x)=2x.
\]
\[
2x=2 \Rightarrow x=1.
\]
Thus, \(c=1\). On the graph, you will see the secant line from \((0,0)\) to \((2,4)\),
and a tangent at \(x=1\) with the same slope.
Common reasons MVT can fail
- Discontinuity on \([a,b]\): e.g. \(f(x)=\frac{1}{x-1}\) on \([0,2]\) (undefined at \(x=1\)).
- Not differentiable inside \((a,b)\): e.g. \(f(x)=|x|\) on \([-1,1]\) (corner at 0).
- Domain restrictions: \(\sqrt{x}\) needs \(x\ge0\); \(\ln(x)\) needs \(x>0\).
Graph interpretation
- Blue curve: \(y=f(x)\)
- Purple line: secant on \([a,b]\) with slope \(m\)
- Green line(s): tangent line(s) at each \(c\) found where \(f'(c)=m\)
- Yellow point: the point \((c,f(c))\)
Use zoom/pan to see how the tangent aligns with the secant slope.
University note: vector MVT (preview)
In multivariable settings there are related results (mean value inequalities, Jacobians, line integrals),
but the classical single-variable MVT is the foundation. A key takeaway:
average rate of change over an interval is achieved as an instantaneous rate of change somewhere inside.
