Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Bound bisection error through interval contraction.
- Justify the conclusion "Ten bisections give width 1/1024 and midpoint error at most 1/2048" using the stated assumptions.
Before you start
Continuity and the intermediate-value theorem.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
For f(x)=x³−2 on [1,2], how many bisections make the bracket width at most 1/1024?
Why this math matters
Bound bisection error through interval contraction. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- f is continuous on the initial interval.
- Endpoint signs and subsequent evaluations are reliable.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Guarantee a root bracket after a fixed number of steps
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For f(x)=x³−2 on [1,2], how many bisections make the bracket width at most 1/1024?
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
f(1)=−1 and f(2)=6
Continuity and opposite signs ensure at least one root inside.
Work through the mathematics
After n bisections, width=1/2ⁿ
Each step preserves a sign-changing half interval.
Check the conclusion
Ten bisections give width 1/1024 and midpoint error at most 1/2048
The final midpoint lies within half the bracket width of any enclosed root.
The result
Ten bisections give width 1/1024 and midpoint error at most 1/2048
The final midpoint lies within half the bracket width of any enclosed root.
Common mistakes to catch
- A sign change does not imply uniqueness without another argument.
- Bracket width and midpoint error differ by a factor of two.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
How many steps give width at most 0.01?
Show a hint
Solve 2⁻ⁿ≤0.01.
Reveal answer and explanation
Seven
2⁻⁶ is too large and 2⁻⁷≈0.0078125.
Practice 2
Does bisection require a derivative?
Show a hint
Its decision only uses function signs.
Reveal answer and explanation
No
Continuity and a valid sign-changing bracket are the essential ingredients here.
Take the idea with you
Budget function evaluations when a certified root tolerance matters.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Balance truncation error against measurement noise
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