Measure what "increasing at a decreasing rate" looks like
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 two sign conditions on f′ and f″ that together mean "increasing at a decreasing rate".
- Before measuring, decide how you could tell from the lab's readouts alone whether f″ is negative, given that the lab displays f and f′ but not f″.
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.
| ln x: f(x) at x = 1.00 | |
|---|---|
| ln x: f′(x) at x = 1.00 | |
| ln x: f(x) at x = 2.00 | |
| ln x: f′(x) at x = 2.00 | |
| ln x: f(x) at x = 4.00 | |
| ln x: f′(x) at x = 4.00 | |
| eˣ/4: f(x) at x = 1.00 | |
| eˣ/4: f′(x) at x = 1.00 |
Answer the free response
Using your measurements: (a) Report your three f and three f′ values for ln x and state whether each list is increasing or decreasing. (b) Explain which of the four standard shape descriptions applies to ln x on [1, 4], and justify it from your data rather than from memory. (c) The lab does not display f″. Explain how your f′ column nonetheless establishes the sign of f″, and state that sign. (d) Report your two readings for eˣ/4 at x = 1 and explain what relationship they show. Then explain why that relationship means eˣ/4 is increasing at an increasing rate, and contrast it with ln x.
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