Mean Value Theorem Calculator
Function and Interval
Use ^ for power, * for multiplication. Supports sin, cos, tan, log, sqrt, exp
Understanding the Mean Value Theorem
The Mean Value Theorem (MVT) is a fundamental result in calculus that connects the average rate of change of a function to its instantaneous rate of change.
Theorem Statement
If f is continuous on [a, b] and differentiable on (a, b), then there exists at least one point c in (a, b) such that:
f'(c) = [f(b) - f(a)] / (b - a)
Geometric Interpretation
There is at least one point on the curve where the tangent line is parallel to the secant line connecting (a, f(a)) and (b, f(b)).
Conditions Required
Continuity on [a, b]
The function must have no breaks, jumps, or holes on the closed interval
Differentiability on (a, b)
The function must have a derivative (smooth curve) on the open interval
Applications
- •Physics: If a car travels 100 miles in 2 hours, at some point it was going exactly 50 mph
- •Economics: Connecting average and marginal rates of change
- •Proofs: Foundation for many important calculus theorems
Related Theorems
Rolle's Theorem
Special case where f(a) = f(b), guaranteeing some c where f'(c) = 0
Cauchy's MVT
Generalization for ratios of two functions
About Mean Value Theorem Calculator - Find c Value
Apply the Mean Value Theorem to find where instantaneous rate equals average rate. Find the value c with step-by-step solutions.
Our **Mean Value Theorem Calculator** finds the value c where the instantaneous rate of change (derivative) equals the average rate of change over an interval. Enter your function and interval endpoints to verify MVT and find c. For implicit functions, see our Implicit Differentiation Calculator.
The Mean Value Theorem states: if f is continuous on [a,b] and differentiable on (a,b), there exists c in (a,b) where f'(c) = [f(b) - f(a)]/(b - a). Our calculator finds this c numerically and shows the geometric interpretation.
Understand one of calculus's most important theorems. See how the tangent line at c parallels the secant line through the endpoints. Essential for understanding derivatives, optimization, and advanced calculus proofs.
Key Features
Why Use This Tool?
Common Use Cases
Calculus: Verify and apply MVT.
Physics: Connect average and instantaneous velocity.
Proofs: Foundation for many calculus theorems.
Analysis: Bound function behavior.
Related Tools
How to Use
Enter your function f(x)
Enter interval [a, b]
Click Calculate
View c value(s) and verification