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Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Derive the accuracy of a centered derivative approximation.
- Justify the conclusion "The leading truncation error is order h² for a sufficiently smooth f" using the stated assumptions.
Before you start
Taylor expansions and numerical differentiation.
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
Approximate f′(x) using [f(x+h)−f(x−h)]/(2h).
Why this math matters
Derive the accuracy of a centered derivative approximation. 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:
- h is nonzero and small.
- The function has enough derivatives for the stated Taylor estimate.
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.
Balance truncation error against measurement noise
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Approximate f′(x) using [f(x+h)−f(x−h)]/(2h).
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(x±h)=f(x)±hf′(x)+h²f″(x)/2±h³f‴(x)/6+…
Symmetric expansions reveal which terms cancel.
Work through the mathematics
Subtract and divide by 2h: f′(x)+h²f‴(x)/6+…
Even-power terms disappear from the numerator.
Check the conclusion
The leading truncation error is order h² for a sufficiently smooth f
Smaller h reduces this idealized error, but subtracting noisy nearly equal values can amplify data error.
The result
The leading truncation error is order h² for a sufficiently smooth f
Smaller h reduces this idealized error, but subtracting noisy nearly equal values can amplify data error.
Common mistakes to catch
- Taking h extremely small can worsen total numerical error.
- The order claim concerns truncation, not every error source.
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
Apply the formula to f(x)=x².
Show a hint
Expand the two squares.
Reveal answer and explanation
Exactly 2x for nonzero h
Quadratic terms cancel with no higher odd-order correction.
Practice 2
If each sampled value has error at most ε, how large can the derivative's data-error contribution be?
Show a hint
Bound the difference error by 2ε.
Reveal answer and explanation
At most ε/|h|
Dividing by 2|h| magnifies small absolute sample errors.
Take the idea with you
Choose a derivative step by considering both smoothness and data precision.
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.
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