Mean Value Theorem Calculator
Find where the instantaneous rate of change equals the average rate of change over an interval.
Mean Value Theorem Calculator
Find where the instantaneous rate of change equals the average rate of change
Function Parameters
Mean Value Theorem
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)
This means the instantaneous rate of change at c equals the average rate of change over [a,b].
Results
Your MVT results will appear here
Common Functions
| Function | Valid Interval | Notes |
|---|---|---|
| x^2 | [0, 2] | c = 1 (exact solution) |
| sin(x) | [0, π/2] | c ≈ 0.8807 |
| x^3 - 2x | [-1, 2] | Two possible c values |
| 1/x | [-1, 1] | Invalid (not continuous at 0) |
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How It Works
Understand the Mean Value Theorem step-by-step
Enter Your Function
Input your function and interval [a,b]
Calculate MVT
The calculator finds f(a), f(b), and the average slope
Find c Value
Numerically approximates where f'(c) equals the average slope
Key Features
What makes our Mean Value Theorem Calculator special!
Numerical Approximation
Finds c values where the derivative equals the average slope
Detailed Results
Shows function values at endpoints and critical points
Common Examples
Includes template problems for quick testing
Who Can Benefit
Our Mean Value Theorem Calculator helps various users:
Calculus Students
Visualize and understand the Mean Value Theorem
Math Teachers
Demonstrate MVT concepts in classroom settings
Self-Learners
Check understanding of MVT applications
Tutors
Help students work through MVT problems
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"This calculator helped me finally understand the Mean Value Theorem."
Emily R.
Teaching Essential
"I use this tool to demonstrate MVT concepts to my calculus students."
Mr. Johnson, Math Teacher
Homework Helper
"Perfect for checking my MVT problem solutions."
Daniel T.
Visual Learning
"Seeing the numbers makes the theorem much clearer."
Sophia L.
Frequently Asked Questions
1Is this Mean Value Theorem Calculator free?
Is this Mean Value Theorem Calculator free?
Yes, our Mean Value Theorem Calculator is completely free to use.
2What is the Mean Value Theorem?
What is the Mean Value Theorem?
The Mean Value Theorem states that for a function continuous on [a,b] and differentiable on (a,b), there exists at least one point c in (a,b) where the instantaneous rate of change (derivative) equals the average rate of change over [a,b].
3How accurate are the results?
How accurate are the results?
The calculator uses numerical approximation, so results are close but not exact. For precise solutions, you would need to solve the equation f'(c) = (f(b)-f(a))/(b-a) symbolically.
4What functions work with this calculator?
What functions work with this calculator?
The calculator works with most differentiable functions of x. The function must be continuous on [a,b] and differentiable on (a,b) for the theorem to apply.
5Why might no c value be found?
Why might no c value be found?
If the function isn't differentiable on (a,b) or doesn't satisfy the MVT conditions, no valid c value will be found. The calculator will indicate this.
6Can I use trigonometric functions?
Can I use trigonometric functions?
Yes, functions like sin(x), cos(x), etc. work as long as they're continuous and differentiable on the interval.
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