Home›Calculus 1›Average rate of change and secant lines›Problem 2.2553

Average rate of change and secant lines

Problem 2.2553 · easy

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

\[ \left. - 2 x^{2} - 1 \right|_{\substack{ x=-3 }} = -19 \]
The value at the left endpoint.✓ Proved
\[ \left. - 2 x^{2} - 1 \right|_{\substack{ x=2 }} = -9 \]
The value at the right endpoint.✓ Proved
\[ \frac{- \left. - 2 x^{2} - 1 \right|_{\substack{ x=-3 }} + \left. - 2 x^{2} - 1 \right|_{\substack{ x=2 }}}{2 + 3} = 2 \]
Change in output over change in input.✓ Proved
\[ 2 \left(x + 3\right) + \left. - 2 x^{2} - 1 \right|_{\substack{ x=-3 }} = 2 x - 13 \]
Point-slope form through the left endpoint.✓ Proved

Answer \( \frac{\Delta f}{\Delta x} = 2,\quad y = 2 x - 13 \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly calculates the average rate of change and derives the secant line equation using the point-slope form. All steps are logically sound and algebraically verified.

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 secant line equation using the point-slope form. All steps are logically sound and algebraically verified.
gpt-oss:20b: pass 2026-11-07
qwen3.6:27b-mlx: pass 2026-11-06 — The solution correctly calculates the average rate of change and derives the secant line equation using valid algebraic steps.
gpt-oss:20b: pass 2026-11-06
qwen3.6:27b-mlx: pass 2026-11-06 — The solution correctly calculates the average rate of change and derives the secant line equation using valid algebraic steps.
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.