Lesson 4: The Mean Value Theorem
Estimated time: 30-40 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- State Rolle's Theorem and the Mean Value Theorem
- Verify that hypotheses are satisfied before applying these theorems
- Find the value of c guaranteed by the MVT
- Apply the MVT to draw conclusions about function behavior
Rolle's Theorem
Rolle's Theorem
If f is (1) continuous on [a, b], (2) differentiable on (a, b), and (3) f(a) = f(b), then there exists at least one c in (a, b) such that f'(c) = 0.
Rolle's Theorem says: if a smooth curve starts and ends at the same height, it must have at least one horizontal tangent in between.
Example 1: Applying Rolle's Theorem
Show that f(x) = x² − 4x + 3 has a point where f'(c) = 0 in [1, 3].
Check: f is a polynomial (continuous and differentiable everywhere). f(1) = 0, f(3) = 0. So f(1) = f(3).
Find c: f'(x) = 2x − 4 = 0 when x = 2. Indeed, c = 2 is in (1, 3).
The Mean Value Theorem
Mean Value Theorem (MVT)
If f is (1) continuous on [a, b] and (2) differentiable on (a, b), then there exists at least one c in (a, b) such that:
f'(c) = [f(b) − f(a)] / (b − a)
The MVT says the instantaneous rate of change equals the average rate of change at some point. Geometrically, there is a tangent line parallel to the secant line through (a, f(a)) and (b, f(b)).
Example 2: Finding c in the MVT
Find the value of c guaranteed by the MVT for f(x) = x³ on [0, 2].
Average rate: [f(2) − f(0)]/(2 − 0) = (8 − 0)/2 = 4.
Set f'(c) = 4: 3c² = 4, so c = 2/√3 = 2√3/3 ≈ 1.155, which is in (0, 2).
Example 3: MVT Application - Speed Limit
A car travels 150 miles in 2 hours. Can we conclude it was going 75 mph at some moment?
Yes! Let s(t) = position. Average speed = 150/2 = 75 mph. If s is differentiable, the MVT guarantees some time c where s'(c) = 75 mph.
Consequences of the MVT
Key Consequence
If f'(x) = 0 for all x in an interval, then f is constant on that interval.
If f'(x) > 0 on an interval, then f is increasing. If f'(x) < 0, then f is decreasing.
Example 4: Proving Two Functions Differ by a Constant
If f'(x) = g'(x) for all x, what can we conclude?
Let h(x) = f(x) − g(x). Then h'(x) = 0 for all x. By the MVT consequence, h(x) = C (a constant). So f(x) = g(x) + C.
Key Takeaways
- Rolle's Theorem: same start and end values guarantee a horizontal tangent.
- MVT: instantaneous rate equals average rate at some interior point.
- Always check hypotheses: continuity on [a, b] and differentiability on (a, b).
- The MVT justifies that f' > 0 implies f is increasing, and f' = 0 implies f is constant.
Check Your Understanding
1. Find c satisfying the MVT for f(x) = x² + 2x on [1, 3].
2. Can we apply the MVT to f(x) = |x| on [−1, 1]?
3. If f is differentiable and f'(x) ≥ 2 for all x, and f(0) = 5, what is the minimum possible value of f(3)?
4. Apply Rolle's Theorem to show sin x = 0 has no more than one root in [π/4, 3π/4].