Applications of Derivatives · Theorems
Mean Value Theorem
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).
- 0 variables
- 2 worked examples
- 15 in Applications of Derivatives
Worked examples
Find c satisfying MVT for f(x) = x² on [1, 3].
- Average rate = (f(3)-f(1))/(3-1) = (9-1)/2 = 4
- f'(x) = 2x. Set 2c = 4 → c = 2
- c = 2 is in (1, 3) ✓
Answer: c = 2
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Find c satisfying MVT for f(x) = x³ on [0, 2].
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