Mean Value Theorem
Understand the Mean Value Theorem: f'(c) = (f(b) - f(a)) / (b - a).
Guarantees a point where instantaneous rate equals average rate.
The Formula
The Mean Value Theorem (MVT) is one of the most important theorems in calculus. It states that if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in (a, b) where the instantaneous rate of change (the derivative) equals the average rate of change over the entire interval.
In simple, everyday terms: if you drive 150 km in 2 hours, your average speed is 75 km/h. The Mean Value Theorem guarantees that at some moment during that drive, your speedometer read exactly 75 km/h. You may have gone faster and slower at various points, but you must have passed through that average speed at least once.
Geometrically, the right side of the equation (f(b) - f(a)) / (b - a) represents the slope of the secant line connecting the two endpoints of the function's graph. The theorem guarantees there is at least one point where the tangent line to the curve is parallel to this secant line. The tangent line at point c has slope f'(c), and the theorem says these slopes are equal.
The two conditions (continuity on [a, b] and differentiability on (a, b)) are both essential. If the function has a jump or a sharp corner, the theorem may not hold. For example, the absolute value function f(x) = |x| is continuous everywhere but not differentiable at x = 0, so the MVT does not apply on intervals that include the origin in a way that would require differentiability there.
The Mean Value Theorem serves as the foundation for many other results in calculus. It is used to prove that a function with a zero derivative everywhere on an interval must be constant, that increasing functions have positive derivatives, and it forms the basis for L'Hopital's Rule and Taylor's theorem. It also has a generalization called Cauchy's Mean Value Theorem, which relates the derivatives of two functions.
Variables
| Symbol | Meaning |
|---|---|
| f'(c) | Derivative of f at the point c (instantaneous rate of change) |
| f(a), f(b) | Function values at the endpoints of the interval |
| a, b | Endpoints of the interval [a, b] |
| c | A point in (a, b) where the derivative equals the average rate |
Example 1
Find the value of c that satisfies the Mean Value Theorem for f(x) = x² on the interval [1, 3].
Average rate = (f(3) - f(1)) / (3 - 1) = (9 - 1) / 2 = 4
f'(x) = 2x, so we need f'(c) = 4
2c = 4 → c = 2
c = 2, which is in the interval (1, 3). The tangent at x = 2 is parallel to the secant from (1,1) to (3,9).
Example 2
A car's position is given by s(t) = t³ - 6t² + 9t + 2, where t is in hours and s is in km. Find c in [0, 4] where the instantaneous velocity equals the average velocity.
s(0) = 2, s(4) = 64 - 96 + 36 + 2 = 6
Average velocity = (6 - 2) / (4 - 0) = 1 km/h
v(t) = s'(t) = 3t² - 12t + 9, set equal to 1
3t² - 12t + 9 = 1 → 3t² - 12t + 8 = 0
t = (12 ± √(144 - 96)) / 6 = (12 ± √48) / 6
c ≈ 0.845 hours and c ≈ 3.155 hours (both are in (0, 4))
When to Use It
Use the Mean Value Theorem when you need to:
- Prove that a function must achieve a particular derivative value on an interval
- Establish bounds on a function's values given information about its derivative
- Show that a constant function has zero derivative (or vice versa)
- Justify the existence of solutions in applied problems (speed, growth rates)
- Build proofs for more advanced calculus results like Taylor's theorem
A special case is Rolle's Theorem, where f(a) = f(b). In that case, the MVT guarantees there is a point c where f'(c) = 0, meaning the function has a horizontal tangent somewhere between a and b.