Average rate of change and secant lines
Problem 2.1623 · easy
Find the average rate of change of \( \displaystyle f(x) = - x^{3} + x^{2} + 2 x + 2 \) on \( \displaystyle [-1, 1] \), and the equation of the secant line through the endpoints.
- \[ \left. - x^{3} + x^{2} + 2 x + 2 \right|_{\substack{ x=-1 }} = 2 \]The value at the left endpoint.✓ Proved
- \[ \left. - x^{3} + x^{2} + 2 x + 2 \right|_{\substack{ x=1 }} = 4 \]The value at the right endpoint.✓ Proved
- \[ 1 \]Change in output over change in input.✓ Proved
- \[ x + 3 \]Point-slope form through the left endpoint.✓ Proved
Answer \( \frac{\Delta f}{\Delta x} = 1,\quad y = x + 3 \)
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 fails to explicitly state the formula for the average rate of change or the secant line equation, making the logical connection between the endpoint values and the final results opaque. Specifically, it does not show the calculation (4-2)/(1-(-1)) = 1, nor does it derive y = x + 3 from the point-slope form y - 2 = 1(x - (-1)).
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to explicitly state the formula for the average rate of change or the secant line equation, making the logical connection between the endpoint values and the final results opaque. Specifically, it does not show the calculation (4-2)/(1-(-1)) = 1, nor does it derive y = x + 3 from the point-slope form y - 2 = 1(x - (-1)).gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly calculates the average rate of change and derives the correct secant line equation. The steps are logically sound and the final answer matches the stated answer.gpt-oss:20b: pass 2026-10-05
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 2026-10-05 with SymPy 1.14.0.