The bisection method iteratively does what to an interval in order to optimise for a particular value?

Question

BlackTom AI · Accepted Answer

First, let's restate the question and the available choices to ground our analysis: The bisection method iteratively does what to an interval in order to optimise for a particular value?
Option 1: Divides an interval, choosing a the new interval which contains the function value.
Option 2: Widens an interval until it is a factor 2 of the function value.
Option 3: Divides the interval, selecting the interval which is wider to next bisect and continue optimising.
Option 4: Widens the interval until the interval contains a wanted function value.
Now, evaluating each option one by one:
Option 2 suggests widening the interval by a factor of 2 to reach the function value. This is contrary to the core idea of the bisection method, which is about narrowing the search interval to zero in on a root (or target value) rather than expanding it. Therefore, this option is incorrect because it contradicts the fundamental shrinking principle behind bisection.
Option 3 talks about dividing the interval but then choosing the wider subinterval for the next step. The bisection method specifically halves the interval and uses the subinterval that brackets the target value (where the sign or value change indicates the root or target lies). Selecting the wider subinterval would defeat the purpose of converging to a value and is not how bisection operates. Hence, this option is incorrect as well due to mischaracterization of the selection criterion.
Option 4 states that we widen the interval until it contains the wanted function value. This again implies expansion rather than contraction. Bisection relies on maintaining a bracketing interval where the target lies between its endpoints, and with each step the interval is halved, not widened. Therefore, this option is incorrect for the same reason: it describes widening, not narrowing.
Option 1 describes dividing an interval and choosing the new subinterval that contains the function value. This aligns with the essence of bisection: evaluate the function at the midpoint, determine which subinterval brackets the target (e.g., a root or value where the function crosses a threshold), and replace the interval with that subinterval. This gradual halving and bracketing is the hallmark of the method, making this option the correct one in terms of the underlying procedure.
In summary, Options 2, 3, and 4 each depict widening or inappropriate selection criteria that contradict the bracketing-and-halving approach. Option 1 accurately captures the iterative narrowing and bracketing step central to the bisection method, where the interval is divided and the subinterval containing the target value becomes the new interval.

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