Average rate of change and secant lines
Problem 2.1885 · easy
Find the average rate of change of \( \displaystyle f(x) = - 3 x^{2} - 2 \) on \( \displaystyle [-2, 1] \), and the equation of the secant line through the endpoints.
- \[ \left. - 3 x^{2} - 2 \right|_{\substack{ x=-2 }} = -14 \]The value at the left endpoint.✓ Proved
- \[ \left. - 3 x^{2} - 2 \right|_{\substack{ x=1 }} = -5 \]The value at the right endpoint.✓ Proved
- \[ 3 \]Change in output over change in input.✓ Proved
- \[ 3 x - 8 \]Point-slope form through the left endpoint.✓ Proved
Answer \( \frac{\Delta f}{\Delta x} = 3,\quad y = 3 x - 8 \)
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 |
| 4 | ✓ 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 lacks the explicit calculation of the average rate of change (slope) from the endpoint values, jumping directly to the result. It also fails to explicitly state that the secant line equation is derived using the calculated slope and one of the endpoints, making the logical connection between the steps opaque.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution lacks the explicit calculation of the average rate of change (slope) from the endpoint values, jumping directly to the result. It also fails to explicitly state that the secant line equation is derived using the calculated slope and one of the endpoints, making the logical connection between the steps opaque.gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution fails to show the calculation of the average rate of change, presenting only the final result '3 = 3' without deriving it from the endpoints. Additionally, the justification for the secant line equation is vague and does not explicitly verify that the line passes through both endpoints.
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.