Connect a derivative, its zeros, and an accumulation function
Three steps, the way the exam actually works: work through the lab, write down your own measurements, then answer a 6-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.
Run the investigation
- 1Select the function f(x) = x³ − 3x.
- 2Move the point along the curve and record the slope the lab reports at x = −1, x = 0, and x = 1.
- 3Record the slope at x = 2 as well.
- 4Turn on the area display and move the point to x = 2, then record the accumulated area from 0 to 2.
Booting the lab…
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 at x = −1 | |
|---|---|
| Slope at x = 0 | |
| Slope at x = 1 | |
| Slope at x = 2 | |
| Accumulated area from 0 to 2 |
Answer the free response
Let f(x) = x³ − 3x. (a) Using your recorded slopes, find f′(x) algebraically and identify every value of x where f′(x) = 0. Confirm your answer against your data. (b) Classify each of those critical points as a local maximum or a local minimum, and justify your classification with a test rather than by looking at the picture. (c) Your accumulated area from 0 to 2 should be negative. Explain how a definite integral can be negative and what that says about f on the interval. (d) Define F(x) = ∫₀ˣ f(t) dt. State F′(x), name the theorem that guarantees it, and determine whether F is increasing or decreasing at x = 1.
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