Average rate of change and secant lines
Problem 2.1726 · easy
A ball's height after \( \displaystyle t \) seconds is \( \displaystyle s(t) = - 16 t^{2} + 36 t + 30 \) feet. Find its average velocity over \( \displaystyle 0 \le t \le 1 \).
- \[ \left. - 16 t^{2} + 36 t + 30 \right|_{\substack{ t=0 }} = 30 \]The value at the left endpoint.✓ Proved
- \[ \left. - 16 t^{2} + 36 t + 30 \right|_{\substack{ t=1 }} = 50 \]The value at the right endpoint.✓ Proved
- \[ 20 \]Change in output over change in input.✓ Proved
Answer \( 20\ \text{ft/s} \)
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 (misleading) — The solution provides the numerical result but fails to state the definition of average velocity or show the calculation (change in position divided by change in time). This omits the critical reasoning step required to solve the problem.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (misleading) 2026-10-06 — The solution provides the numerical result but fails to state the definition of average velocity or show the calculation (change in position divided by change in time). This omits the critical reasoning step required to solve the problem.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution fails to explicitly state the formula for average velocity, making the final equation '20 = 20' logically disconnected from the previous steps. A student cannot infer that 20 is the result of (50-30)/(1-0) without the explicit setup.
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 2026-10-06 with SymPy 1.14.0.