Verify an integration-by-parts result numerically
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
- Derive ∫ln x dx by parts with u = ln x and dv = dx. Write the antiderivative before opening the lab.
- The lab accumulates area for ln x from a = 1 rather than 0. Say in one sentence why the base point could not be 0 for this function.
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.
| Accumulated area from 1 to 2.00 | |
|---|---|
| f(x) readout at x = 2.00 | |
| Accumulated area from 1 to 3.00 | |
| Accumulated area from 1 to 4.00 | |
| f(x) readout at x = 4.00 | |
| Your hand value of F(2) − F(1)(F(x) = x·ln x − x) | |
| Your hand value of F(4) − F(1) |
Answer the free response
You have compared a by-parts antiderivative against measured accumulated area. (a) State your antiderivative of ln x and show the derivation in two lines. (b) Report your lab areas and hand-computed values and state how closely they agree. (c) Explain why agreement is evidence the by-parts result is correct, and state precisely what it does NOT prove. (d) Explain why the lab accumulates from a = 1 for this function rather than from 0, and say what would happen to the accumulated area if the base point moved toward 0.
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