Average rate of change and secant lines
Problem 2.2036 · easy
Find the average rate of change of \( \displaystyle f(x) = \sqrt{x} \) on \( \displaystyle [4, 9] \), and the equation of the secant line through the endpoints.
- \[ \left. \sqrt{x} \right|_{\substack{ x=4 }} = 2 \]The value at the left endpoint.✓ Proved
- \[ \left. \sqrt{x} \right|_{\substack{ x=9 }} = 3 \]The value at the right endpoint.✓ Proved
- \[ 1 \cdot \frac{1}{5} = \frac{1}{5} \]Change in output over change in input.✓ Proved
- \[ \frac{x}{5} + \frac{6}{5} \]Point-slope form through the left endpoint.✓ Proved
Answer \( \frac{\Delta f}{\Delta x} = \frac{1}{5},\quad y = \frac{x}{5} + \frac{6}{5} \)
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) — Step 3 claims the average rate of change equals 1/5 without showing the calculation (Δf/Δx = (3−2)/(9−4) = 1/5). The solution skips the essential derivation, so the reasoning is incomplete.qwen3.6:27b-mlx: pass — The solution correctly calculates the average rate of change and derives the secant line equation. The intermediate steps are algebraically sound and lead to the correct final answer.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-10 — Step 3 claims the average rate of change equals 1/5 without showing the calculation (Δf/Δx = (3−2)/(9−4) = 1/5). The solution skips the essential derivation, so the reasoning is incomplete.qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly calculates the average rate of change and derives the secant line equation. The intermediate steps are algebraically sound and lead to the correct final answer.gpt-oss:20b: fail (error) 2026-10-10 — The solution never actually derives the secant line; it merely states an identity x/5+6/5 = x/5+6/5. The correct line should come from y-2 = (1/5)(x-4) leading to y = x/5 + 6/5, but this step is missing and the reasoning is incomplete.qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly calculates the average rate of change and derives the equation of the secant line. The algebraic steps are verified, and the final result 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.