Calculus I · Limits and Continuity · lesson
Bisection Method After the IVT
Learn Bisection Method After the IVT with plain-language explanations, guided examples, worked homework methods, interactive checks, and exam-style practice.
Where this chapter fits
Chapter 5: Continuity
Connect limits to function values, classify discontinuities, repair piecewise definitions, and use the Intermediate Value Theorem.
Reading lens: Do the limit, the function value, and the surrounding domain fit together at the point or across the interval? Keep that question in view while reading Bisection Method After the IVT; the worked mathematics is evidence for the idea, not a substitute for it.
This page connects Intermediate Value Theorem to Epsilon-Delta Definition: An Introduction. Read the explanation first, predict each example’s next move, and only then compare the written solution.
Learning objectives
Use repeated sign changes on smaller intervals to approximate a root guaranteed by the Intermediate Value Theorem.
Bisection: Turning Existence Into an Approximation
For
we know a root lies in . Test the midpoint :
Since and , a root lies in .
Test the new midpoint :
Now the root lies in .
Continue:
| \text{interval} | \text{midpoint} | \text{sign at midpoint} |
|---|---|---|
| (0,1) | 0.5 | - |
| (0.5,1) | 0.75 | + |
| (0.5,0.75) | 0.625 | - |
| (0.625,0.75) | 0.6875 | + |
| (0.625,0.6875) | 0.65625 | - |
The root is being trapped in shorter intervals. This is the bisection method.
The Intermediate Value Theorem says a root is in the box. Bisection keeps cutting the box in half and throwing away the half that cannot contain the sign change.
Source & rights
Original instruction with traceable references.
The exposition is original. No Active Calculus exercise is reproduced verbatim. Public-domain examples were modernized and recomposed when used as inspiration.
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