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Average rate of change and secant lines

Problem 2.1627 · easy

Find the average rate of change of \( \displaystyle f(x) = - x^{3} + 2 x^{2} - x + 1 \) on \( \displaystyle [-1, 1] \), and the equation of the secant line through the endpoints.

\[ \left. - x^{3} + 2 x^{2} - x + 1 \right|_{\substack{ x=-1 }} = 5 \]
The value at the left endpoint.✓ Proved
\[ \left. - x^{3} + 2 x^{2} - x + 1 \right|_{\substack{ x=1 }} = 1 \]
The value at the right endpoint.✓ Proved
\[ -2 \]
Change in output over change in input.✓ Proved
\[ 3 - 2 x \]
Point-slope form through the left endpoint.✓ Proved

Answer \( \frac{\Delta f}{\Delta x} = -2,\quad y = 3 - 2 x \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly calculates the average rate of change and derives the secant line equation. The steps are logically sound and the algebraic results are verified.

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 steps are logically sound and the algebraic results are verified.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly calculates the function values at the endpoints, determines the average rate of change (slope of the secant line), and derives the correct equation for the secant line.
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.