Average rate of change and secant lines
Problem 2.1729 · easy
Find the average rate of change of \( \displaystyle f(x) = - \frac{5}{x} \) on \( \displaystyle [1, 5] \), and the equation of the secant line through the endpoints.
- \[ \left. - \frac{5}{x} \right|_{\substack{ x=1 }} = -5 \]The value at the left endpoint.✓ Proved
- \[ \left. - \frac{5}{x} \right|_{\substack{ x=5 }} = -1 \]The value at the right endpoint.✓ Proved
- \[ 1 \]Change in output over change in input.✓ Proved
- \[ x - 6 \]Point-slope form through the left endpoint.✓ Proved
Answer \( \frac{\Delta f}{\Delta x} = 1,\quad y = x - 6 \)
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: fail (error) — The solution never derives the secant line equation; it merely states the trivial identity x-6=x-6. The correct line is y=x-6, obtained from the point (1,-5) and slope 1. The missing step misleads the reader about how to form the equation.qwen3.6:27b-mlx: fail (error) — The solution provides tautological equations (e.g., '1 = 1') instead of showing the calculation steps for the average rate of change and the secant line equation. It fails to demonstrate the method required to solve the problem.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-06 — The solution never derives the secant line equation; it merely states the trivial identity x-6=x-6. The correct line is y=x-6, obtained from the point (1,-5) and slope 1. The missing step misleads the reader about how to form the equation.qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution provides tautological equations (e.g., '1 = 1') instead of showing the calculation steps for the average rate of change and the secant line equation. It fails to demonstrate the method required to solve the problem.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution provides no explanatory sentences to justify the calculations or connect the steps to the problem statement. It merely lists algebraic identities and results without showing the derivation of the average rate of change or the secant line equation.
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 2026-10-06 with SymPy 1.14.0.