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Tangent lines

Problem 2.736 · medium Video

Find an equation of the tangent line to \( \displaystyle y = 2 \sqrt{x} + x - 3 \) at \( \displaystyle x = 9 \).

The tangent line passes through the point (a, f(a)) and has slope f'(a).
Reviewed
\[ \left. 2 \sqrt{x} + x - 3 \right|_{\substack{ x=9 }} = 12 \]
The point of tangency.✓ Proved
\[ \frac{d}{d x} \left(2 \sqrt{x} + x - 3\right) = 1 + \frac{1}{\sqrt{x}} \]
Differentiate.✓ Proved
\[ \left. 1 + 1 \frac{1}{\sqrt{x}} \right|_{\substack{ x=9 }} = \frac{4}{3} \]
The slope at the point.✓ Proved
\[ \frac{4 x}{3} \]
Point-slope form, then simplify.

Answer \( y = \frac{4 x}{3} \)

The graph of y = 2sqrt(x) + x - 3 near x = 9. A secant line through the point (9, 12) and a second point on the curve turns into the tangent line as the second point slides in; its slope settles at 4/3, giving y = 4x/3.

Lines: 3 proved, 1 reviewed, 1 not checked. 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	Reviewed	claude-sonnet-5-5	claude-sonnet-5-5: This is the standard statement that a tangent line at x = a passes through (a, f(a)) with slope f'(a). It is not an algebraic equivalence, so no script is needed.
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
5	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	the line meets the curve at x = a, and its slope matches a central difference quotient of f there

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution fails to use the point of tangency (9, 12) when constructing the tangent line equation. It incorrectly outputs y = 4x/3, which passes through the origin, instead of the correct line y = 4/3(x - 9) + 12.

Senior review claude-sonnet-5-5, 2026-10-12: fail — The final answer y = 4x/3 is mathematically correct. Line 5 is a tautology (4x/3 = 4x/3) labelled 'point-slope form, then simplify', which hides two steps. It should show y = 12 + (4/3)(x-9) first and then simplify to 4x/3.

qwen3.6:27b-mlx: dismiss — The answer is correct: 12 + (4/3)(x-9) = 4x/3 + 12 - 12 = 4x/3, so the line does use (9,12) and passes through both the origin and (9,12).

Every verdict on record (7)

claude-sonnet-5-5: fail (style) 2026-10-12 — The final answer y = 4x/3 is mathematically correct. Line 5 is a tautology (4x/3 = 4x/3) labelled 'point-slope form, then simplify', which hides two steps. It should show y = 12 + (4/3)(x-9) first and then simplify to 4x/3.
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to use the point of tangency (9, 12) when constructing the tangent line equation. It incorrectly outputs y = 4x/3, which passes through the origin, instead of the correct line y = 4/3(x - 9) + 12.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution fails to use the point of tangency (9, 12) when constructing the line equation. It incorrectly outputs y = 4x/3, which passes through the origin, instead of the correct tangent line y = 4x/3 - 8.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution fails to use the point of tangency (9, 12) when constructing the tangent line equation. It incorrectly outputs y = 4x/3, which passes through the origin, instead of the correct line y = 4x/3 - 2.
gpt-oss:20b: pass 2026-09-26

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/tangent_line, checked 2026-10-13 with SymPy 1.14.0.