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Tangent lines

Problem 2.1151 · medium Video

Find an equation of the tangent line to \( \displaystyle y = - x^{2} - 2 x - 3 + \frac{1}{x} \) at \( \displaystyle x = 1 \).

The tangent line passes through the point (a, f(a)) and has slope f'(a).
\[ \left. - x^{2} - 2 x - 3 + 1 \frac{1}{x} \right|_{\substack{ x=1 }} = -5 \]
The point of tangency.✓ Proved
\[ \frac{d}{d x} \left(- x^{2} - 2 x - 3 + \frac{1}{x}\right) = - 2 x - 2 - \frac{1}{x^{2}} \]
Differentiate.✓ Proved
\[ \left. - 2 x - 2 - 1 \frac{1}{x^{2}} \right|_{\substack{ x=1 }} = -5 \]
The slope at the point.✓ Proved
\[ - 5 x \]
Point-slope form, then simplify.

Answer \( y = - 5 x \)

The graph of y = -x² - 2x - 3 + 1/x near x = 1. A secant line through the point (1, -5) and a second point on the curve turns into the tangent line as the second point slides in; its slope settles at -5, giving y = -5x.

Lines: 3 proved, 2 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see the receipt. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Not checked	—	a sentence; read, not computed
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: fail (misleading) — The final sentence incorrectly states the point‑slope form as "-5*x = -5*x", which is an identity and does not represent the tangent line. The correct equation should be derived from y‑y₀ = m(x‑x₀) leading to y = -5x.
qwen3.6:27b-mlx: fail (error) — The solution fails to use the point of tangency (1, -5) in the point-slope formula. It incorrectly sets up the equation as y = -5x, which passes through the origin, rather than y - (-5) = -5(x - 1), which simplifies to y = -5x + 0? No, y + 5 = -5x + 5 => y = -5x. Wait, let me re-calculate. Point (1, -5), slope -5. y - (-5) = -5(x - 1) => y + 5 = -5x + 5 => y = -5x. The final answer is actually correct. However, line 5 says '-5*x = -5*x' which is a tautology and does not represent the equation of the line y = -5x. The sentence 'Point-slope form, then simplify' implies a derivation that is not shown and the equation shown is nonsensical as a representation of the line. A student would be confused by '-5*x = -5*x' being the result of simplifying the point-slope form. The correct simplification leads to y = -5x. The line provided is an identity, not the equation of the line.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution fails to use the point of tangency (1, -5) in the point-slope formula. It incorrectly sets up the equation as y = -5x, which passes through the origin, rather than y - (-5) = -5(x - 1), which simplifies to y = -5x + 0? No, y + 5 = -5x + 5 => y = -5x. Wait, let me re-calculate. Point (1, -5), slope -5. y - (-5) = -5(x - 1) => y + 5 = -5x + 5 => y = -5x. The final answer is actually correct. However, line 5 says '-5*x = -5*x' which is a tautology and does not represent the equation of the line y = -5x. The sentence 'Point-slope form, then simplify' implies a derivation that is not shown and the equation shown is nonsensical as a representation of the line. A student would be confused by '-5*x = -5*x' being the result of simplifying the point-slope form. The correct simplification leads to y = -5x. The line provided is an identity, not the equation of the line.
gpt-oss:20b: fail (misleading) 2026-10-03 — The final sentence incorrectly states the point‑slope form as "-5*x = -5*x", which is an identity and does not represent the tangent line. The correct equation should be derived from y‑y₀ = m(x‑x₀) leading to y = -5x.
qwen3.6:27b-mlx: inconclusive 2026-09-28 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The final equation is incorrect. The point-slope form y - (-5) = -5(x - 1) simplifies to y = -5x, but the line must pass through (1, -5). Substituti
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution incorrectly derives the tangent line equation as y = -5x, which does not pass through the point of tangency (1, -5). The correct equation using point-slope form y - (-5) = -5(x - 1) simplifies to y = -5x.
gpt-oss:20b: pass 2026-09-28

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.