Mean Value Theorem

Mean Value Theorem

The Mean Value Theorem (MVT) is one of the most important theorems in calculus:

If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists at least one point $c \in (a, b)$ such that:

$f'(c) = \frac{f(b) - f(a)}{b - a}$

In plain English: at some point, the instantaneous rate of change equals the average rate of change over the interval.

Geometric Interpretation

Draw the secant line through $(a, f(a))$ and $(b, f(b))$. The MVT says there is at least one point where the tangent line is parallel to that secant line.

Finding the MVT Point Numerically

We want $c$ where $f'(c) = \frac{f(b) - f(a)}{b - a}$. This is a root-finding problem on $g(x) = f'(x) - \text{slope}$, solved by bisection.

Example

For $f(x) = x^2$ on $[0, 4]$:

• Average slope: $\frac{16 - 0}{4 - 0} = 4$
• $f'(c) = 2c = 4$ → $c = 2$ ✓ (which is the midpoint — always true for parabolas)

Applications

• Speed enforcement: if your average speed between two cameras was 80 mph, at some moment you were going exactly 80 mph
• Proof of L'Hôpital's rule
• Error estimation in numerical integration
• Rolle's Theorem: special case where $f(a) = f(b)$, implying $f'(c) = 0$

Your Task

Implement double mvt_point(double (*f)(double), double a, double b, int n, double h) that finds the MVT point $c$ using bisection on $g(x) = f'(x) - \text{average\_slope}$.