Average rate of change and secant lines
Problem 2.1888 · easy
A ball's height after \( \displaystyle t \) seconds is \( \displaystyle s(t) = - 16 t^{2} + 52 t + 21 \) feet. Find its average velocity over \( \displaystyle 2 \le t \le 4 \).
- \[ \left. - 16 t^{2} + 52 t + 21 \right|_{\substack{ t=2 }} = 61 \]The value at the left endpoint.✓ Proved
- \[ \left. - 16 t^{2} + 52 t + 21 \right|_{\substack{ t=4 }} = -27 \]The value at the right endpoint.✓ Proved
- \[ -44 \]Change in output over change in input.✓ Proved
Answer \( -44\ \text{ft/s} \)
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 calculate the average velocity using the formula (s(4) - s(2)) / (4 - 2). It lists the endpoint values and the final answer but skips the division by the time interval, making the derivation incomplete and logically disconnected.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution fails to explicitly calculate the average velocity using the formula (s(4) - s(2)) / (4 - 2). It lists the endpoint values and the final answer but skips the division by the time interval, making the derivation incomplete and logically disconnected.gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly computes the average velocity as the change in position divided by the change in time. The arithmetic is correct: (s(4) - s(2)) / (4 - 2) = (-27 - 61) / 2 = -88 / 2 = -44.
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 2026-10-08 with SymPy 1.14.0.