Verify the Fundamental Theorem 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
- FTC Part 2 says that if g(x) = ∫ₐˣ f(t)dt then g′(x) = f(x). State in one sentence what that predicts about how the accumulated area changes as you drag x.
- Before opening the lab, decide what quantity you would compute from two area readings in order to estimate the rate at which area is accumulating.
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 0 to 1.00 | |
|---|---|
| Accumulated area from 0 to 2.00 | |
| f(x) readout at x = 1.00 | |
| f(x) readout at x = 2.00 | |
| Accumulated area from 0 to 1.10 | |
| (area at 1.10 − area at 1.00) ÷ 0.10(This estimates the rate at which area accumulates near x = 1.05.) |
Answer the free response
You have measured an accumulated area and a function value for f(x) = x². (a) Report the accumulated areas at x = 1 and x = 2 and state what quantity each represents. (b) Report your difference quotient from the small interval [1, 1.10] and the f(x) readout near x = 1.05, and state how closely they agree. (c) Explain what result of the Fundamental Theorem of Calculus your comparison in part (b) provides evidence for, being precise about which function equals which derivative. (d) A classmate proposes checking the same claim by comparing your difference quotient against the f′(x) readout instead. Explain why that comparison would fail, and what it would actually be testing.
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