Average rate of change and secant lines
Problem 2.2210 · easy
A ball's height after \( \displaystyle t \) seconds is \( \displaystyle s(t) = - 16 t^{2} + 61 t + 50 \) feet. Find its average velocity over \( \displaystyle 0 \le t \le 1 \).
- \[ \left. - 16 t^{2} + 61 t + 50 \right|_{\substack{ t=0 }} = 50 \]The value at the left endpoint.✓ Proved
- \[ \left. - 16 t^{2} + 61 t + 50 \right|_{\substack{ t=1 }} = 95 \]The value at the right endpoint.✓ Proved
- \[ 45 \]Change in output over change in input.✓ Proved
Answer \( 45\ \text{ft/s} \)
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 formula for average velocity or show the calculation (95 - 50) / (1 - 0). Line 3 is a tautology that does not demonstrate the required division by the change in time.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: fail (error) 2026-10-13 — The solution fails to explicitly state the formula for average velocity or show the calculation (95 - 50) / (1 - 0). Line 3 is a tautology that does not demonstrate the required division by the change in time.gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: fail (error) 2026-10-13 — The solution fails to explicitly state the average velocity formula or show the division by the time interval (1-0). Line 3 presents '45 = 45' as 'Change in output over change in input', but does not show the calculation (95-50)/(1-0), making the derivation opaque and logically incomplete.
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 2026-10-13 with SymPy 1.14.0.