Average rate of change and secant lines
Problem 2.2550 · easy
Find the average rate of change of \( \displaystyle f(x) = 3 \sqrt{x} \) on \( \displaystyle [1, 9] \), and the equation of the secant line through the endpoints.
- \[ \left. 3 \sqrt{x} \right|_{\substack{ x=1 }} = 3 \]The value at the left endpoint.✓ Proved
- \[ \left. 3 \sqrt{x} \right|_{\substack{ x=9 }} = 9 \]The value at the right endpoint.✓ Proved
- \[ \frac{- \left. 3 \sqrt{x} \right|_{\substack{ x=1 }} + \left. 3 \sqrt{x} \right|_{\substack{ x=9 }}}{-1 + 9} = \frac{3}{4} \]Change in output over change in input.✓ Proved
- \[ \frac{3 \left(x - 1\right)}{4} + \left. 3 \sqrt{x} \right|_{\substack{ x=1 }} = \frac{3 x}{4} + \frac{9}{4} \]Point-slope form through the left endpoint.✓ Proved
Answer \( \frac{\Delta f}{\Delta x} = \frac{3}{4},\quad y = \frac{3 x}{4} + \frac{9}{4} \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation. How we verify
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: pass — The solution correctly calculates the average rate of change and derives the secant line equation using valid algebraic steps and the point-slope formula.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-11-07 — The solution correctly calculates the average rate of change and derives the secant line equation using valid algebraic steps and the point-slope formula.gpt-oss:20b: pass 2026-11-07qwen3.6:27b-mlx: pass 2026-11-06 — The solution correctly calculates the average rate of change and derives the secant line equation using valid algebraic steps.gpt-oss:20b: pass 2026-11-06qwen3.6:27b-mlx: pass 2026-11-06 — The solution correctly calculates the average rate of change and derives the equation of the secant line using the point-slope form. All steps are logically sound and algebraically correct.gpt-oss:20b: pass 2026-11-06
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 2026-11-07 with SymPy 1.14.0.