Average rate of change and secant lines
Problem 2.1889 · easy
Find the average rate of change of \( \displaystyle f(x) = x^{2} - x + 4 \) on \( \displaystyle [-3, 3] \), and the equation of the secant line through the endpoints.
- \[ \left. x^{2} - x + 4 \right|_{\substack{ x=-3 }} = 16 \]The value at the left endpoint.✓ Proved
- \[ \left. x^{2} - x + 4 \right|_{\substack{ x=3 }} = 10 \]The value at the right endpoint.✓ Proved
- \[ -1 \]Change in output over change in input.✓ Proved
- \[ 13 - x \]Point-slope form through the left endpoint.✓ Proved
Answer \( \frac{\Delta f}{\Delta x} = -1,\quad y = 13 - x \)
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 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — 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 2026-10-08qwen3.6:27b-mlx: fail (error) 2026-10-08 — The setup is incorrect because the average rate of change is 14/6 = 7/3, not -1. The solution incorrectly calculates the change in output (10 - 16 = -6) but fails to divide by the change in input (3 - (-3) = 6), or simply states the wrong result. Consequently, the secant line equation is also wrong.
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.