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Mean Value Theorem — Guaranteed Instantaneous Rate Equals Average Rate
If f is continuous on [a,b] and differentiable on (a,b), then there exists c in (a,b) with f'(c) = [f(b)−f(a)]/(b−a).
CalculusCommon Core FUN-1.BDERVerified correct ✓
Worked examples
The function f(x) = x³ is continuous on [0, 2] and differentiable on (0, 2). The Mean Value Theorem guarantees a c in (0, 2) with f'(c) = [f(2) - f(0)] / (2 - 0). Find c. Round to 4 decimal places. Answer: 1.1547
The function f(x) = x² + 0x - 2 is continuous on [0, 4] and differentiable on (0, 4). The Mean Value Theorem guarantees a c in (0, 4) with f'(c) = [f(4) - f(0)] / (4 - 0). Find c. Round to 4 decimal places. Answer: 2
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