To solve x²=3, use f(x)=x²−3 on [1,2]. Endpoint values are −2 and 1, so continuity gives a root. The first midpoint 1.5 has value −0.75; retain [1.5,2]. The next midpoint 1.75 has value 0.0625; retain [1.5,1.75]. Each decision removes half the interval while preserving a mathematical reason that a root remains. The method does not estimate a derivative.
Predict
Predict which is the certificate: a small residual or a narrow retained bracket. Under the hypotheses, the bracket directly bounds distance to at least one root. A small residual can coexist with a large root error for a flat function. A sign change alone does not establish uniqueness; here strict increase on [1,2] supplies that separate fact. Keep existence, uniqueness and numerical accuracy as distinct conclusions.
Experiment
Track completed halvings rather than vaguely counting steps.
Completed halvings
Retained interval
Midpoint error bound
0
[1,2]
1/2
1
[1.5,2]
1/4
2
[1.5,1.75]
1/8
Continue once: the new bracket is [1.625,1.75]. Its midpoint differs from the midpoint just evaluated to select it. Compute the new midpoint and error bound before inspecting further function values.
Understand
Assume f is continuous on [a,b] and its endpoint signs are opposite. If an evaluated midpoint is exactly a zero, finish. Otherwise retain the half with opposite signs. After k completed halvings the width is (b−a)/2^k. The nested closed intervals have one common point r, because their widths tend to zero. Both endpoint sequences converge to r. Their function values keep opposite weak signs, so continuity forces f(r)=0. The midpoint of the current interval has error at most (b−a)/2^(k+1). To guarantee error at most ε>0, choose k so this bound is at most ε. State whether the reported midpoint precedes or follows the kth halving; otherwise iteration counts can differ by one.
Apply
Finite arithmetic adds failure modes. A midpoint may round to an endpoint, preventing further shrinkage. Detect that and report stagnation, not successful accuracy unless the current width already suffices. Computing a+(b−a)/2 avoids overflow of a+b in some ranges, but b−a can overflow for opposite huge endpoints. Robust arithmetic must handle these cases explicitly. Compare signs without multiplying huge values, since their product can overflow. Reject nonfinite evaluations. A finite collection of samples cannot establish continuity, and a pole can mimic a sign change. An even-multiplicity root may produce no sign change at all.
Continue
Bisection is reliable inside a valid bracket but gains only a fixed factor in interval width per step. Derivative and secant proposals can be faster, but require safeguards or stronger local assumptions. A useful trace reports endpoints, function signs, evaluated midpoint, retained interval and stopping reason. Later hybrid methods will retain this certificate while trying faster proposals. The worked table assumes exact signs; uncertain function evaluations require additional error analysis before a sign can be treated as verified.
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
For f(x)=x³−3 on [1,2], find the first midpoint.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
An initial interval has width 8. After 4 completed halvings, bound the error of the remaining interval midpoint.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Prove bisection converges under continuity and opposite endpoint signs.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Why does 1/x on [−1,1] have no bisection root guarantee?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Specify stopping rules that distinguish accuracy from arithmetic stagnation.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Does a sign bracket ensure uniqueness? Give an example.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
