Relate a sinusoid to its own rate of change
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
- A sinusoid has a period, an amplitude and a midline. If the rate of change of a sinusoid is itself a sinusoid, which of those three would you expect the two to share?
- At the top of a Ferris wheel a rider is momentarily neither rising nor falling. What does that suggest about the rate of change at a maximum of a sinusoidal model?
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.
| f(x) at x = 0.79 | |
|---|---|
| Slope f′(x) at x = 0.79 | |
| f(x) at x = 1.57 | |
| Slope f′(x) at x = 1.57 | |
| f(x) at x = 3.14 | |
| Slope f′(x) at x = 3.14 | |
| f(x) at x = 3.93 | |
| Slope f′(x) at x = 3.93 | |
| Next x after 0 at which the slope returns to 1.00 |
Answer the free response
A sinusoidal model and its rate of change are both visible in this lab at once. (a) Report your readings at x = 1.57 and explain what the slope value there means physically for a Ferris-wheel or tide model. (b) Report your readings at x = 3.14 and state where in the cycle the slope is largest in magnitude. Explain why that location makes sense. (c) Use your recorded x value to state the period of the rate of change, and compare it with the period of sin x itself. (d) Using your four pairs of readings, state the relationship between the graph of the rate of change and the graph of sin x, expressing the horizontal offset as a fraction of a period. Then state one feature of a real tide record that this lab’s sin x cannot represent.
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