Test whether a secant slope equals the tangent slope at the midpoint
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
- Average rate of change on [a, b] is [f(b) − f(a)]/(b − a). Is that the slope of a line touching the curve at one point, or cutting it at two?
- A linear function has the same average rate of change on every interval. What does that tell you to expect for a function that is NOT linear?
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 = 1.00 on f(x) = x² | |
|---|---|
| f(x) at x = 3.00 on f(x) = x² | |
| Average rate of change of x² on [1, 3]([f(3) − f(1)] / (3 − 1)) | |
| Slope f′(x) at the midpoint x = 2.00 | |
| f(x) at x = 0.50 on f(x) = x³ − 3x | |
| f(x) at x = 2.50 on f(x) = x³ − 3x | |
| Average rate of change of x³ − 3x on [0.5, 2.5]([f(2.5) − f(0.5)] / 2) | |
| Slope f′(x) at the midpoint x = 1.50 |
Answer the free response
You have measured an average rate of change and a tangent slope for two functions. (a) Report both values for f(x) = x² on [1, 3] and state whether they agree. (b) Report both values for f(x) = x³ − 3x on [0.5, 2.5] and state whether they agree. (c) One of the two functions gave an exact match and the other did not. Using the fact that an average rate of change is the slope of the secant line through the endpoints, explain what is special about the quadratic that makes its match exact. (d) A classmate concludes from part (a) that the average rate of change on any interval always equals the tangent slope at the midpoint, and proposes using this as a shortcut. State whether the shortcut is valid, cite your own data, and explain what a student would have to check before using it on an unfamiliar function.
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