Mean Value Theorem

Mean Value Theorem (MVT)

The MVT connects the derivative of a function to its average rate of change.

Statement

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

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

Interpretation

There is at least one point where the instantaneous rate of change equals the average rate of change over the interval.

Applications

• If (f'(x) = 0) for all (x) in an interval, then (f) is constant
• If (f'(x) > 0) for all (x), then (f) is strictly increasing
• Rolle's Theorem (special case): if (f(a) = f(b)), then (f'(c) = 0) for some (c \in (a, b))

Finding the MVT Point Numerically

We can search for the point (c) where the derivative equals the average slope:

def find_mvt_point(f, a, b, n=10000, h=1e-7):
    avg_slope = (f(b) - f(a)) / (b - a)
    best_c = a
    best_diff = float('inf')
    for i in range(1, n):
        c = a + (b - a) * i / n
        deriv = (f(c + h) - f(c - h)) / (2 * h)
        diff = abs(deriv - avg_slope)
        if diff < best_diff:
            best_diff = diff
            best_c = c
    return best_c

Your Task

Implement:

1. average_slope(f, a, b) -- returns ((f(b) - f(a)) / (b - a))
2. find_mvt_point(f, a, b, n) -- searches (n) evenly spaced interior points for the one where the numerical derivative is closest to the average slope. Uses central difference with (h = 10^{-7}).