Differentiability and the Mean Value Theorems: Exam Tips

A function f is differentiable at c when the limit f′(c) = limh→0 (f(c+h) − f(c))/h exists, meaning the graph has a well-defined tangent line there. The mean value theorems then connect the average rate of change over an interval to the instantaneous rate at some interior point: Rolle's theorem, Lagrange's mean value theorem, and Cauchy's mean value theorem.

These theorems are the workhorses of calculus. They prove that a function with zero derivative is constant, that increasing functions have positive derivatives, and they justify error bounds for approximations. Almost every "obvious" fact about derivatives secretly depends on the mean value theorem.

Key Concepts

• Differentiability: f′(c) exists as a limit. Differentiable implies continuous, but not conversely — f(x) = |x| is continuous at 0 but not differentiable there.
• Rolle's theorem: if f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then some c in (a, b) has f′(c) = 0.
• Lagrange's mean value theorem: under the same hypotheses without f(a) = f(b), some c in (a, b) satisfies f′(c) = (f(b) − f(a))/(b − a).
• Cauchy's mean value theorem: for f and g with g′(x) ≠ 0, some c satisfies (f(b) − f(a))/(g(b) − g(a)) = f′(c)/g′(c). This is the engine behind L'Hôpital's rule.
• Geometric meaning: somewhere inside the interval, the tangent is parallel to the secant line joining the endpoints.

Formulas

• Derivative: f′(c) = limh→0 (f(c+h) − f(c))/h, units of "output units per input unit". Requires the two-sided limit to exist.
• Lagrange MVT: ∃c ∈ (a, b): f′(c) = (f(b) − f(a))/(b − a). Needs continuity on the closed interval and differentiability on the open interval.
• Cauchy MVT: ∃c ∈ (a, b): (f(b) − f(a))/(g(b) − g(a)) = f′(c)/g′(c), with g′(x) ≠ 0 on (a, b).
• Consequences: f′ = 0 on an interval ⇒ f constant; f′ > 0 ⇒ f strictly increasing. Both are proved with the MVT.

Worked Example

Apply Lagrange's mean value theorem to f(x) = x³ on [0, 2] and find the point c it guarantees.

1. Check the hypotheses: x³ is continuous on [0, 2] and differentiable on (0, 2), with f′(x) = 3x².
2. Compute the average rate of change: (f(2) − f(0))/(2 − 0) = (8 − 0)/2 = 4.
3. The MVT promises a c in (0, 2) with f′(c) = 4, i.e. 3c² = 4.
4. Solve: c² = 4/3, so c = 2/√3 ≈ 1.1547, which indeed lies inside (0, 2). At this point the tangent slope 3c² = 4 matches the secant slope.

Common Misconceptions

• "Differentiable means the derivative is continuous." False: f(x) = x²sin(1/x) for x ≠ 0, f(0) = 0, is differentiable everywhere but f′ is discontinuous at 0.
• "The MVT works if the function is differentiable at the endpoints too." The theorem only needs differentiability on the open interval; f(x) = √x on [0, 1] qualifies even though it is not differentiable at 0.
• "The point c is unique or easy to find." The theorem guarantees existence, not uniqueness or a formula; finding c explicitly, as above, is a bonus of simple examples.

Real-World Applications

• Speed traps and speed cameras: average speed over a stretch forces some instant where the speedometer showed exactly that value.
• Error bounds for linear approximation: the MVT bounds |f(x) − f(a)| ≤ (max |f′|)·|x − a|, used throughout numerical analysis.
• Proving monotonicity and uniqueness of solutions: showing an equation has exactly one root by combining Rolle's theorem with sign analysis.

Differentiability gives you a tangent line at each point, and the mean value theorems turn those local slopes into global information: Rolle's theorem finds flat tangents, Lagrange's matches a tangent to the secant slope, and Cauchy's extends the idea to pairs of functions. For exams, always verify the continuity and differentiability hypotheses first, then translate "there exists c" into an equation you can solve.