Average rate of change and secant lines
Problem 2.1628 · easy
A ball's height after \( \displaystyle t \) seconds is \( \displaystyle s(t) = - 16 t^{2} + 45 t + 88 \) feet. Find its average velocity over \( \displaystyle 2 \le t \le 4 \).
- \[ \left. - 16 t^{2} + 45 t + 88 \right|_{\substack{ t=2 }} = 114 \]The value at the left endpoint.✓ Proved
- \[ \left. - 16 t^{2} + 45 t + 88 \right|_{\substack{ t=4 }} = 12 \]The value at the right endpoint.✓ Proved
- \[ -51 \]Change in output over change in input.✓ Proved
Answer \( -51\ \text{ft/s} \)
Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | recomputed in floating point |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The solution fails to explicitly state the average velocity formula or show the division by the time interval (4-2). It jumps from endpoint values to the final answer without demonstrating the calculation (12-114)/(4-2), making the derivation incomplete and potentially confusing.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to explicitly state the average velocity formula or show the division by the time interval (4-2). It jumps from endpoint values to the final answer without demonstrating the calculation (12-114)/(4-2), making the derivation incomplete and potentially confusing.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution lacks the explicit calculation of the average velocity formula (change in position divided by change in time). While the numerical result is correct, the third line merely asserts the final answer without showing the division by the time interval (4-2=2), which is the core of the problem.gpt-oss:20b: pass 2026-10-05
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/secant_slope, checked 2026-10-05 with SymPy 1.14.0.