Watch an exponential lose a race it is guaranteed to win
Three steps, the way the exam actually works: work through the lab, write down your own measurements, then answer a 8-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
- State the growth hierarchy for large x, ordering ln x, powers of x, and exponentials. You are about to test it against measurements.
- Decide what it would take to DISPROVE the claim that eˣ eventually outgrows x², and what would merely fail to confirm it.
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.
| x² at x = 1.00 | |
|---|---|
| x² at x = 2.00 | |
| x² at x = 3.00 | |
| eˣ/4 at x = 1.00 | |
| eˣ/4 at x = 2.00 | |
| eˣ/4 at x = 3.00 | |
| Ratio (eˣ/4) ÷ x² at x = 2 | |
| Ratio (eˣ/4) ÷ x² at x = 3 |
Answer the free response
You have raced an exponential against a power function inside the lab window. (a) Report both functions at all three values and state which is larger at each. (b) The growth hierarchy says lim(x→∞) x²/eˣ = 0, so the exponential eventually dominates. Does your data contradict that? Explain carefully. (c) Your ratio column is not monotone. Explain what that shows about where the crossover lies, and whether it is inside or outside the window. (d) Explain what the constant 1/4 does and does not affect, and state the general principle your measurements illustrate about limits at infinity.
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