What Is MVT?
The Mean Value Theorem states that if a function is continuous on [a,b] and differentiable on (a,b), there exists point c where instantaneous rate equals average rate of change.
Formula
f'(c) = [f(b) - f(a)] / (b - a). If you drive 100 miles in 2 hours at 50 mph average, MVT guarantees your speed matched 50 mph at some point.
Steps
1) Verify continuity and differentiability. 2) Calculate average rate. 3) Find derivative. 4) Solve for c. 5) Verify c is in interval.
Examples
Linear f(x)=2x+3 has f'(x)=2. Quadratic f(x)=x² on [1,3] has average rate 4, f'(2)=4.
Applications
Physics (motion), Economics (cost), Engineering (behavior), Optimization.
Requirements
Continuity on [a,b] and differentiability on (a,b). Fails at discontinuities, cusps, or vertical tangents.
Conclusion
MVT bridges instantaneous and average rates, connecting differential and integral calculus.