Mean Value Theorem Calculator
Calculate the point c guaranteed by the Mean Value Theorem (MVT), with step-by-step logic showing where the instantaneous rate of change equals the average rate of change.
Supported: +, -, *, /, **, sin, cos, tan, sqrt, log, ln, exp, abs, pi, e
Mean Value Theorem Formula:
f'(c) = rac{f(b) - f(a)}{b - a}where a < c < b
Enter function and interval, then click Calculate to see result.
Enter function and interval, then click Calculate to see result.
The Mean Value Theorem (MVT) is a key result in differential calculus. It states that if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in (a, b) such that the derivative f'(c) equals the average rate of change over [a, b].
Geometrically, this means there is at least one point where the tangent to the curve is parallel to the secant line joining (a, f(a)) and (b, f(b)).
Mean Value Theorem Formula:
f'(c) = rac{f(b) - f(a)}{b - a}where a < c < b
Step-by-Step Logic:
- Verify the function is continuous on [a, b] and differentiable on (a, b).
- Compute the average rate of change: rac{f(b) - f(a)}{b - a}.
- Find the derivative f'(x).
- Solve f'(c) = rac{f(b) - f(a)}{b - a} for c in (a, b).
Conditions for MVT:
- f(x) must be continuous on the closed interval [a,b]
- f(x) must be differentiable on the open interval (a,b)
- The theorem guarantees at least one such c, but there may be more.
- Enter your function f(x) using standard mathematical notation (e.g., x**2, sin(x), -4*x**3 + 6*x - 2).
- Specify the interval endpoints a and b (with a < b).
- Click "Calculate MVT Point" to find the value(s) of c.
- Review the step-by-step solution showing all calculations.
- Verify that each c lies in the open interval (a, b).
Example 1: f(x) = x², [1, 3]
Step-by-step solution:
• f(1) = 1, f(3) = 9
• AROC = (9-1)/(3-1) = 4
• f'(x) = 2x
• Solve: 2c = 4 ⇒ c = 2
• Verify: 2 ∈ (1,3) ✅
Example 2: f(x) = x³, [0, 2]
Step-by-step solution:
• f(0) = 0, f(2) = 8
• AROC = (8-0)/(2-0) = 4
• f'(x) = 3x²
• Solve: 3c² = 4 ⇒ c² = 4/3 ⇒ c = ±√(4/3) ≈ ±1.154 (only positive in interval)
• Verify: 1.154 ∈ (0,2) ✅
Example 3: f(x) = -4x³ + 6x - 2, [-4, 2] (Corrected Calculation)
Step-by-step solution:
• f(-4) = 230, f(2) = -22
• AROC = (-22 - 230)/(2 - (-4)) = -252/6 = -42
• f'(x) = -12x² + 6
• Solve: -12c² + 6 = -42 ⇒ -12c² = -48 ⇒ c² = 4 ⇒ c = ±2
• Verify: -2 and 2 ∈ (-4,2)? -2 ✅, 2 ❌ (2 is endpoint, but open interval)
What is the Mean Value Theorem in simple words?
The Mean Value Theorem states that for any smooth curve between two points, there's at least one point where the slope of the tangent line equals the slope of the line connecting the endpoints. It's like saying "somewhere on your journey, your instantaneous speed equaled your average speed."
How does this calculator work?
The calculator evaluates your function at the endpoints, computes the average rate of change (AROC), finds the derivative (symbolically for simple cases, numerically otherwise), and solves the equation f'(c) = AROC to find the MVT point(s) c in the given interval using numerical root-finding methods.
What are the conditions for applying the Mean Value Theorem?
The function must be: (1) continuous on the closed interval [a,b], and (2) differentiable on the open interval (a,b). If these conditions aren't met, the theorem doesn't guarantee the existence of point c.
Can there be more than one value of c?
Yes! The Mean Value Theorem guarantees at least one point c, but there can be multiple points where f'(c) equals the average rate of change. For example, higher-degree polynomials may have multiple solutions.
What is the difference between Mean Value Theorem and Mean Value Theorem for Integrals?
The standard MVT relates to derivatives and rates of change. The MVT for Integrals (also called the Average Value Theorem) states that there exists c where f(c) equals the average value of f over [a, b], i.e., f(c) = (1/(b-a)) ∫_a^b f(x) dx.