Determine concavity and locate an inflection point from the tangent slope
Three steps, the way the exam actually works: work through the lab, write down your own measurements, then answer a 9-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 the rate of change of a function is increasing, is the graph concave up or concave down? Answer before you open the lab.
- A quantity is decreasing, and the amount it drops each year is getting smaller. Is that graph concave up or concave down? This is the case students most often get backward.
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.
| Slope of x² at x = −2.00 | |
|---|---|
| Slope of x² at x = −1.00 | |
| Slope of x² at x = 1.00 | |
| Slope of x² at x = 2.00 | |
| Slope of ln x at x = 1.00 | |
| Slope of ln x at x = 2.00 | |
| Slope of ln x at x = 4.00 | |
| Slope of x³ − 3x at x = −2.00 | |
| Slope of x³ − 3x at x = −0.50 | |
| Slope of x³ − 3x at x = 0.50 | |
| Slope of x³ − 3x at x = 2.00 | |
| x value where the cubic’s slope stops falling and starts rising |
Answer the free response
AP Precalculus describes concavity entirely through rates of change, with no second derivative. (a) List your four slopes for f(x) = x² in order of increasing x, state whether they are increasing or decreasing, and give the concavity with justification. (b) Do the same for f(x) = ln x. Note that ln x is increasing on your whole interval, and explain how it can be increasing and concave down at the same time. (c) List your four slopes for f(x) = x³ − 3x, describe the pattern, and state the x value of the point of inflection together with the evidence that locates it. (d) A student argues that because the slopes of x³ − 3x at x = −2 and x = 2 are both 9, the function must have the same concavity at those two points, and therefore no inflection point lies between them. Identify the flaw in that argument.
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