Average rate of change and secant lines
Problem 2.1629 · easy
Find the average rate of change of \( \displaystyle f(x) = - x^{3} - x^{2} - x + 1 \) on \( \displaystyle [-3, 5] \), and the equation of the secant line through the endpoints.
- \[ \left. - x^{3} - x^{2} - x + 1 \right|_{\substack{ x=-3 }} = 22 \]The value at the left endpoint.✓ Proved
- \[ \left. - x^{3} - x^{2} - x + 1 \right|_{\substack{ x=5 }} = -154 \]The value at the right endpoint.✓ Proved
- \[ -22 \]Change in output over change in input.✓ Proved
- \[ - 22 x - 44 \]Point-slope form through the left endpoint.✓ Proved
Answer \( \frac{\Delta f}{\Delta x} = -22,\quad y = - 22 x - 44 \)
✓ 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.
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 algebraic steps are verified, and the logic flows correctly from the endpoints to the final equation.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly calculates the average rate of change and derives the secant line equation. The algebraic steps are verified, and the logic flows correctly from the endpoints to the final equation.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 secant line equation. The intermediate steps are algebraically correct and consistent with the final 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.