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.
Predict before you look
- A function has a local maximum where its derivative changes from positive to negative. What is the derivative's value at that point?
- An accumulation function is defined as an integral from a fixed point. What does its derivative equal?
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 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.
Sign in to have this graded and saved to your progress.