Intermediate Value Theorem Calculator
Verify the intermediate value theorem on an interval and estimate where a target value is reached.
Table of Contents
How to Use
- Enter the left endpoint a of the interval
- Enter the right endpoint b of the interval
- Provide the function values f(a) and f(b)
- Set the target value k to check (k = 0 for root finding) and calculate
What the intermediate value theorem guarantees
If a function is continuous on [a, b], it must take every value between f(a) and f(b). That means any target k between those two outputs occurs at least once inside the interval.
- Check that f is continuous on the closed interval.
- Confirm the target k lies between f(a) and f(b).
- If f(a) and f(b) have opposite signs, a root exists between them.
The theorem is a guarantee of existence, not uniqueness. It does not locate the exact point on its own.
Estimating the point c
A straight-line interpolation between (a, f(a)) and (b, f(b)) provides a quick approximation for where f(c) = k may occur.
- Compute the secant slope m = (f(b) - f(a)) / (b - a).
- Solve a + (k - f(a)) / (f(b) - f(a)) · (b - a) to estimate c.
- Use this as a starting point for methods like bisection or Newton's method.
Frequently Asked Questions
- What counts as proof from this calculator?
- The calculator checks the value range f(a) to f(b). If the target lies between them and continuity is assumed, the theorem guarantees at least one point c with f(c) = k on (a, b).
- Does the result give me the exact solution?
- No. The intermediate value theorem only proves existence. The linear estimate shown here is a convenient starting guess, not the exact location.
- What if f(a) = f(b) = k?
- Then every point in the interval satisfies f(x) = k. If f(a) = f(b) but not equal to k, the theorem does not guarantee that k is attained.