Section 4.7

The Mean Value Theorem

If a car averages 60 mph on a trip, at some specific moment the speedometer must have read exactly 60 mph. This simple intuition is the backbone of calculus existence theorems.

The Theorem

"If

is continuous on

and differentiable on

, then there exists a number

such that:"

Geometric Translation

There is at least one point where the Tangent Line is parallel to the Secant Line connecting endpoints.

Why it matters

MVT connects the "average" change over time to the "instant" change at a moment. It guarantees solutions exist.

Worked Example

Verify MVT for Square Root

Find such that for on .

The Parallel Scanner

Drag point to find matching slope.

Find where the blue line (Tangent) matches the slope of the dotted line.

1. Find Average Slope (Secant)

2. Find Instantaneous Slope (Tangent)

3. Solve for c

Is inside ? Yes.

Level Up Examples

Example A: Polynomial

Verify MVT for on .

1. Average Rate:

2. Derivative:

3. Solve:

Only positive root is in interval: .

Example B: Why Conditions Matter

Try MVT for on .

1. Calculate Average Slope:

2. Try to solve f'(c) = 1:

(Impossible!)

3. Why it failed:

Function is Discontinuous at .

MVT requires continuity on the whole closed interval.