Average rate of change and secant lines
Problem 2.2552 · easy
Find the average rate of change of \( \displaystyle f(x) = 2 x^{2} - 1 \) on \( \displaystyle [-1, 1] \), and the equation of the secant line through the endpoints.
- \[ \left. 2 x^{2} - 1 \right|_{\substack{ x=-1 }} = 1 \]The value at the left endpoint.✓ Proved
- \[ \left. 2 x^{2} - 1 \right|_{\substack{ x=1 }} = 1 \]The value at the right endpoint.✓ Proved
- \[ \frac{- \left. 2 x^{2} - 1 \right|_{\substack{ x=-1 }} + \left. 2 x^{2} - 1 \right|_{\substack{ x=1 }}}{1 + 1} = 0 \]Change in output over change in input.✓ Proved
- \[ 0 \left(x + 1\right) + \left. 2 x^{2} - 1 \right|_{\substack{ x=-1 }} = 1 \]Point-slope form through the left endpoint.✓ Proved
Answer \( \frac{\Delta f}{\Delta x} = 0,\quad y = 1 \)
✓ 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. How we verify
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 equation of the secant line using the point-slope form. All steps are logically sound and algebraically correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-11-07 — The solution correctly calculates the average rate of change and derives the equation of the secant line using the point-slope form. All steps are logically sound and algebraically correct.gpt-oss:20b: pass 2026-11-07qwen3.6:27b-mlx: pass 2026-11-06 — The solution correctly calculates the average rate of change and derives the equation of the secant line. The steps are logically sound and algebraically correct.gpt-oss:20b: pass 2026-11-06qwen3.6:27b-mlx: pass 2026-11-06 — The solution correctly calculates the average rate of change and derives the equation of the secant line using the point-slope form. The algebraic steps are verified and the logic is sound.gpt-oss:20b: pass 2026-11-06
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 2026-11-07 with SymPy 1.14.0.