Average rate of change and secant lines
Problem 2.1626 · easy
Find the average rate of change of \( \displaystyle f(x) = 6 \sqrt{x} \) on \( \displaystyle [9, 16] \), and the equation of the secant line through the endpoints.
- \[ \left. 6 \sqrt{x} \right|_{\substack{ x=9 }} = 18 \]The value at the left endpoint.✓ Proved
- \[ \left. 6 \sqrt{x} \right|_{\substack{ x=16 }} = 24 \]The value at the right endpoint.✓ Proved
- \[ \frac{6}{7} \]Change in output over change in input.✓ Proved
- \[ \frac{6 x}{7} + \frac{72}{7} \]Point-slope form through the left endpoint.✓ Proved
Answer \( \frac{\Delta f}{\Delta x} = \frac{6}{7},\quad y = \frac{6 x}{7} + \frac{72}{7} \)
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 fails to explicitly calculate the average rate of change as (24-18)/(16-9) in the text, jumping straight to the result. More critically, it does not state the formula for the secant line or explain how the y-intercept was derived, making the logical connection between the slope and the final equation opaque.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to explicitly calculate the average rate of change as (24-18)/(16-9) in the text, jumping straight to the result. More critically, it does not state the formula for the secant line or explain how the y-intercept was derived, making the logical connection between the slope and the final equation opaque.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly calculates the function values, the average rate of change, and the equation of the secant line. The algebraic steps are verified and the logic is sound.gpt-oss:20b: pass 2026-10-05
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 2026-10-05 with SymPy 1.14.0.