Home›Calculus 1›Average rate of change and secant lines›Problem 2.2551

Average rate of change and secant lines

Problem 2.2551 · easy

Find the average rate of change of \( \displaystyle f(x) = 2 \sqrt{x} \) on \( \displaystyle [9, 16] \), and the equation of the secant line through the endpoints.

\[ \left. 2 \sqrt{x} \right|_{\substack{ x=9 }} = 6 \]
The value at the left endpoint.✓ Proved
\[ \left. 2 \sqrt{x} \right|_{\substack{ x=16 }} = 8 \]
The value at the right endpoint.✓ Proved
\[ \frac{- \left. 2 \sqrt{x} \right|_{\substack{ x=9 }} + \left. 2 \sqrt{x} \right|_{\substack{ x=16 }}}{-9 + 16} = \frac{2}{7} \]
Change in output over change in input.✓ Proved
\[ \frac{2 \left(x - 9\right)}{7} + \left. 2 \sqrt{x} \right|_{\substack{ x=9 }} = \frac{2 x}{7} + \frac{24}{7} \]
Point-slope form through the left endpoint.✓ Proved

Answer \( \frac{\Delta f}{\Delta x} = \frac{2}{7},\quad y = \frac{2 x}{7} + \frac{24}{7} \)

✓ 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: pass
qwen3.6:27b-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 2026-11-07
qwen3.6:27b-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 2026-11-06
qwen3.6:27b-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.