Measure how a Taylor error grows away from the center
Three steps, the way the exam actually works: work through the lab, write down your own measurements, then answer a 8-point free response. What you recorded goes to the grader with your writing, so a conclusion that does not follow from your own numbers will cost you the point — exactly as it would with a real reader.
Predict before you look
- Write the third-degree Maclaurin polynomial for sin x from memory. You will compute with it by hand while the lab supplies the true values.
- Predict whether a Taylor error grows linearly, quadratically, or faster as you move away from the center. Commit to one before measuring.
Nothing to submit here — these are to think through, so the prediction below is an informed one rather than a guess.
Commit to an answer now. It is not graded and being wrong costs nothing — the point is to have something specific to reconcile against once you have the data.
Answer every prediction to unlock the lab. A sentence is enough.
Run the investigation
Predictions first
The procedure and the simulation unlock once you have committed above. Observing before predicting is how a wrong intuition survives a lab intact.
Record what you measured
These are your numbers, not ours. The grader sees them, so your conclusions have to follow from what you actually recorded.
| f(x) readout at x = 0.50 | |
|---|---|
| P₃(0.5), computed by hand | |
| f(x) readout at x = 1.00 | |
| P₃(1), computed by hand | |
| f(x) readout at x = 2.00 | |
| P₃(2), computed by hand | |
| Error at x = 2: |sin 2 − P₃(2)| |
Answer the free response
You have compared a third-degree Maclaurin polynomial with true values of sin x. (a) Report the three errors and describe how the error changes as x moves away from 0. (b) The Lagrange bound for P₃ is M|x|⁴/4!. Every derivative of sine is bounded by 1, so take M = 1 and compute the bound at x = 1 and at x = 2, comparing each with your measured error. (c) Explain why the error grows so much faster than the distance itself, referring to the form of the bound. (d) The Maclaurin series for sin x converges for ALL real x. Reconcile that with your finding that P₃ is nearly useless at x = 2, being precise about what converges and what does not.
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