Average rate of change and secant lines
Problem 2.2032 · easy
Find the average rate of change of \( \displaystyle f(x) = - 2 x^{2} - x - 2 \) on \( \displaystyle [-3, 1] \), and the equation of the secant line through the endpoints.
- \[ \left. - 2 x^{2} - x - 2 \right|_{\substack{ x=-3 }} = -17 \]The value at the left endpoint.✓ Proved
- \[ \left. - 2 x^{2} - x - 2 \right|_{\substack{ x=1 }} = -5 \]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: pass — The solution correctly calculates the average rate of change and derives the secant line equation. The intermediate steps are algebraically verified and logically sound.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly calculates the average rate of change and derives the secant line equation. The intermediate steps are algebraically verified and logically sound.gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution fails to explicitly calculate the average rate of change using the difference quotient formula, instead presenting a tautology (3=3) as the derivation. Additionally, the final equation is presented without showing the substitution of the point and slope into the point-slope formula.
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 2026-10-10 with SymPy 1.14.0.