Premed · Premed · Calculus 1
Lecture 13: The Mean Value Theorem
Calculus I — Differential Calculus
Learning Objectives
By the end of this lecture, students will be able to:
- State and understand Rolle's Theorem
- State and apply the Mean Value Theorem (MVT)
- Interpret the MVT geometrically and physically
- Use the MVT to prove inequalities and establish function behavior
- Apply corollaries: constant functions, increasing/decreasing functions
Lecture Content
I. Rolle's Theorem
Rolle's Theorem states that if f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists at least one c in (a, b) such that f'(c) = 0. Geometrically, this means that if a continuous, differentiable curve starts and ends at the same height, there must be at least one point along the way where the tangent line is horizontal. Rolle's Theorem is a special case of the Mean Value Theorem, corresponding to the situation where the endpoint values are equal.
All three hypotheses are necessary. Without continuity on [a, b], the theorem can fail. Without differentiability on (a, b), consider f(x) = |x| on [-1, 1]: the function satisfies f(-1) = f(1) = 1, but f'(0) does not exist and there is no point with a horizontal tangent. Without the condition f(a) = f(b), there is no guarantee of a horizontal tangent anywhere.
<image>A graph illustrating Rolle's Theorem. A smooth curve starts at point (a, f(a)) and ends at point (b, f(b)) where f(a) = f(b). The curve rises, reaches a peak, then descends back to the same height. A horizontal tangent line is drawn at the peak, at the point (c, f(c)) where f'(c) = 0. The point c is marked on the x-axis. Title: "Rolle's Theorem: equal endpoint values guarantee a horizontal tangent."</image>
II. The Mean Value Theorem
The Mean Value Theorem states that if f is continuous on [a, b] and 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 right-hand side is the slope of the secant line connecting (a, f(a)) and (b, f(b)). The theorem guarantees that somewhere between a and b, the tangent line is parallel to the secant line. Rolle's Theorem is recovered as the special case where f(a) = f(b), making the secant slope zero.
III. Geometric Interpretation
Visually, draw the secant line from (a, f(a)) to (b, f(b)). The Mean Value Theorem asserts that somewhere between a and b, there is a tangent line with exactly the same slope. You can imagine "sliding" the secant line vertically until it becomes tangent to the curve -- the point of tangency is the c guaranteed by the theorem. There may be more than one such point, but the MVT guarantees at least one.
<image>A graph showing a smooth curve y = f(x) on [a, b]. The secant line connecting (a, f(a)) to (b, f(b)) is drawn as a dashed line. At a point c between a and b, a tangent line is drawn parallel to the secant line (same slope). Both lines are clearly parallel, with double-arrow annotations showing equal slopes. The formula f'(c) = [f(b) - f(a)]/(b - a) is displayed. Title: "The Mean Value Theorem: the tangent is parallel to the secant."</image>
IV. Physical Interpretation
If s(t) represents position at time t, then [s(b) - s(a)] / (b - a) is the average velocity over the interval [a, b], and s'(c) is the instantaneous velocity at time c. The MVT says that at some moment during the trip, the instantaneous velocity equals the average velocity. For example, if you drive 200 km in 2 hours (average speed 100 km/h), then at some instant during the trip your speedometer must have read exactly 100 km/h.
V. Consequences and Corollaries
The Mean Value Theorem yields several powerful corollaries. Corollary 1 states that if f'(x) = 0 for all x in an interval (a, b), then f is constant on (a, b). The proof is straightforward: for any two points x_1 and x_2 in the interval, the MVT gives f(x_2) - f(x_1) = f'(c)(x_2 - x_1) = 0. Corollary 2 follows immediately: if f'(x) = g'(x) for all x in (a, b), then f(x) = g(x) + C for some constant C, since the function h(x) = f(x) - g(x) has h'(x) = 0 everywhere.
Corollary 3, the Increasing/Decreasing Test, states that if f'(x) > 0 on (a, b), then f is strictly increasing on that interval, and if f'(x) < 0 on (a, b), then f is strictly decreasing. The proof again uses the MVT: for x_1 < x_2, f(x_2) - f(x_1) = f'(c)(x_2 - x_1), which is positive when f'(c) > 0.
VI. Using the MVT to Prove Inequalities
The MVT is a powerful tool for establishing bounds and inequalities. To show that sin x < x for all x > 0, define f(x) = x - sin x. Then f(0) = 0 and f'(x) = 1 - cos x >= 0 for all x, with equality only at isolated points. Since f is increasing, f(x) > f(0) = 0 for x > 0, which gives x - sin x > 0, or equivalently sin x < x.
To show that sqrt(1 + x) < 1 + x/2 for x > 0, let f(x) = sqrt(1 + x) with a = 0. By the MVT, f(x) - f(0) = f'(c) x for some c in (0, x). Since f'(c) = 1/(2sqrt(1 + c)) < 1/2 when c > 0, it follows that sqrt(1 + x) - 1 < x/2, which gives the desired inequality.
VII. Worked Examples
To verify the MVT for f(x) = x^3 - x on [0, 2], note that f is a polynomial and therefore continuous and differentiable everywhere. Computing f(0) = 0 and f(2) = 6, the secant slope is (6 - 0)/(2 - 0) = 3. The derivative is f'(x) = 3x^2 - 1. Setting 3x^2 - 1 = 3 gives x^2 = 4/3, so x = 2/sqrt(3), approximately 1.155. This value lies in (0, 2), confirming the MVT.
<image>The graph of f(x) = x^3 - x on [0, 2]. The secant line from (0, 0) to (2, 6) is drawn with slope 3. At the point c = 2/sqrt(3) ≈ 1.155, the tangent line is drawn parallel to the secant. Both lines have the same slope (parallel), visually confirming the MVT. The point c is marked on the x-axis, and the tangent point on the curve is labeled. Title: "Verifying the MVT for f(x) = x^3 - x on [0, 2]."</image>
