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.
- 2 worked examples
- 15 in Applications of Derivatives
Worked examples
- 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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Related formulas
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Common questions
What is the 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. It is one of the applications of derivatives formulas in the CalcRef reference.
When does the Mean Value Theorem apply?
The Mean Value Theorem holds under this condition: f must be continuous on [a, b] and differentiable on (a, b).
How do you use the Mean Value Theorem?
Worked example. 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.
Practice
Quiz yourself on Applications of Derivatives
Mean Value Theorem is one of 15 Applications of Derivatives formulas in CalcRef, out of 136 in the library. Today's free round is 10 mixed multiple-choice questions drawn from the same dataset as this page, refreshed every day and replayable as often as you like - no account needed.