CalcRef

Applications of Derivatives · Theorems

Mean Value Theorem

f′(c)=f(b)−f(a)b−a for some c∈(a,b)f'(c) = \frac{f(b) - f(a)}{b - a} \text{ for some } c \in (a, b)

If f is continuous on [a,b] and differentiable on (a,b), then there exists at least one c in (a,b) where the instantaneous rate of change equals the average rate of change.

Conditions. f must be continuous on [a, b] and differentiable on (a, b).

Worked examples

Find c satisfying MVT for f(x) = x² on [1, 3].
  1. Average rate = (f(3)-f(1))/(3-1) = (9-1)/2 = 4
  2. f'(x) = 2x. Set 2c = 4 → c = 2
  3. c = 2 is in (1, 3) ✓

Answer: c = 2

Find c satisfying MVT for f(x) = x³ on [0, 2].
  1. Average rate = (8-0)/2 = 4
  2. f'(x) = 3x². Set 3c² = 4 → c = 2/√3 ≈ 1.155
  3. c = 2/√3 is in (0, 2) ✓

Answer: c = 2/√3 ≈ 1.155

Related formulas

Rolle's Theorem

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