Average rate of change and secant lines
Problem 2.1880 · easy
Find the average rate of change of \( \displaystyle f(x) = 4 x^{2} + 4 x - 4 \) on \( \displaystyle [-1, 3] \), and the equation of the secant line through the endpoints.
- \[ \left. 4 x^{2} + 4 x - 4 \right|_{\substack{ x=-1 }} = -4 \]The value at the left endpoint.✓ Proved
- \[ \left. 4 x^{2} + 4 x - 4 \right|_{\substack{ x=3 }} = 44 \]The value at the right endpoint.✓ Proved
- \[ 12 \]Change in output over change in input.✓ Proved
- \[ 12 x + 8 \]Point-slope form through the left endpoint.✓ Proved
Answer \( \frac{\Delta f}{\Delta x} = 12,\quad y = 12 x + 8 \)
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 provides the correct numerical results but fails to show the calculation steps for the average rate of change or the derivation of the secant line equation. Specifically, it jumps from endpoint values to the final slope without showing the difference quotient, and presents the final line equation as a tautology rather than deriving it from the point-slope formula.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution provides the correct numerical results but fails to show the calculation steps for the average rate of change or the derivation of the secant line equation. Specifically, it jumps from endpoint values to the final slope without showing the difference quotient, and presents the final line equation as a tautology rather than deriving it from the point-slope formula.gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — 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. The algebraic steps are verified and the logic is sound.
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 2026-10-08 with SymPy 1.14.0.