Average rate of change and secant lines
Problem 2.2549 · easy
A ball's height after \( \displaystyle t \) seconds is \( \displaystyle s(t) = - 16 t^{2} + 67 t + 98 \) feet. Find its average velocity over \( \displaystyle 1 \le t \le 4 \).
- \[ \left. - 16 t^{2} + 67 t + 98 \right|_{\substack{ t=1 }} = 149 \]The value at the left endpoint.✓ Proved
- \[ \left. - 16 t^{2} + 67 t + 98 \right|_{\substack{ t=4 }} = 110 \]The value at the right endpoint.✓ Proved
- \[ \frac{- \left. - 16 t^{2} + 67 t + 98 \right|_{\substack{ t=1 }} + \left. - 16 t^{2} + 67 t + 98 \right|_{\substack{ t=4 }}}{-1 + 4} = -13 \]Change in output over change in input.✓ Proved
Answer \( -13\ \text{ft/s} \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation. How we verify
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: pass — The solution correctly applies the definition of average velocity as the change in position divided by the change in time. The calculations are verified and the result matches the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-11-07 — The solution correctly applies the definition of average velocity as the change in position divided by the change in time. The calculations are verified and the result matches the stated answer.gpt-oss:20b: pass 2026-11-07qwen3.6:27b-mlx: pass 2026-11-06 — The solution correctly applies the definition of average velocity as the change in position divided by the change in time. The calculations are verified as correct.gpt-oss:20b: pass 2026-11-06qwen3.6:27b-mlx: pass 2026-11-06 — The solution correctly applies the definition of average velocity as the change in position divided by the change in time. The arithmetic is correct and the result matches the stated answer.gpt-oss:20b: pass 2026-11-06
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 2026-11-07 with SymPy 1.14.0.