Average rate of change and secant lines
Problem 2.2041 · easy
Find the average rate of change of \( \displaystyle f(x) = 5 \sqrt{x} \) on \( \displaystyle [4, 16] \), and the equation of the secant line through the endpoints.
- \[ \left. 5 \sqrt{x} \right|_{\substack{ x=4 }} = 10 \]The value at the left endpoint.✓ Proved
- \[ \left. 5 \sqrt{x} \right|_{\substack{ x=16 }} = 20 \]The value at the right endpoint.✓ Proved
- \[ \frac{5}{6} \]Change in output over change in input.✓ Proved
- \[ \frac{5 x}{6} + \frac{20}{3} \]Point-slope form through the left endpoint.✓ Proved
Answer \( \frac{\Delta f}{\Delta x} = \frac{5}{6},\quad y = \frac{5 x}{6} + \frac{20}{3} \)
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 skips the crucial computation of the average rate of change: it states “5/6 = 5/6” without showing that (f(16)-f(4))/(16-4) = (20-10)/12 = 5/6. Likewise, the secant line is written without explicitly applying the point‑slope formula or confirming that the slope used is indeed the average rate of change. These omissions mean the reasoning is incomplete and could mislead a student.qwen3.6:27b-mlx: pass — The solution correctly calculates the average rate of change and derives the secant line equation. The algebraic steps are verified and the logic is sound.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-10 — The solution skips the crucial computation of the average rate of change: it states “5/6 = 5/6” without showing that (f(16)-f(4))/(16-4) = (20-10)/12 = 5/6. Likewise, the secant line is written without explicitly applying the point‑slope formula or confirming that the slope used is indeed the average rate of change. These omissions mean the reasoning is incomplete and could mislead a student.qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly calculates the average rate of change and derives the secant line equation. The algebraic steps are verified and the logic is sound.gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly computes the average rate of change and the secant line equation. The steps are logically sound and the final answer matches the stated answer.
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 2026-10-10 with SymPy 1.14.0.