Locate a maximum of an accumulation function from the graph of f
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
- If g(x) = ∫₀ˣ f(t)dt, write down what g′ and g″ are in terms of f. You will need all three levels.
- State the condition on f that makes g have a local maximum at x = c, and say why f(c) = 0 alone is not sufficient.
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 about 1.57 | |
|---|---|
| Accumulated area from 0 to about 3.14 | |
| f(x) readout at x ≈ 3.14 | |
| f′(x) readout at x ≈ 3.14 | |
| Accumulated area from 0 to about 4.71 | |
| Accumulated area from 0 to about 6.28 | |
| Value of x at which the accumulated area is greatest |
Answer the free response
Let g(x) = ∫₀ˣ sin t dt, which is what the accumulated-area readout displays. (a) Report your four area measurements and describe in words what g does across [0, 2π]. (b) State the x at which g attains its maximum on (0, 2π) and justify it using the behavior of f rather than by inspecting your area column. (c) Your measurement at x ≈ 6.28 should be about zero even though sine spends the first half of that interval well above the axis. Explain what that tells you about g, and why an accumulated area can return to zero. (d) Using your f′ reading at x ≈ 3.14, state whether g is concave up or concave down there, and explain the chain of reasoning from f′ to the concavity of g.
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