Tangent lines
Problem 2.1151 · medium
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).
- \[ \left. - x^{2} - 2 x - 3 + \frac{1}{x} \right|_{\substack{ x=1 }} = -5 \]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
- \[ \left. - 2 x - 2 - \frac{1}{x^{2}} \right|_{\substack{ x=1 }} = -5 \]The slope at the point.✓ Proved
- \[ - 5 x \]Point-slope form, then simplify.✓ Proved
Answer \( y = - 5 x \)
Lines: 4 proved, 1 not checked. 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 | 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 | ✓ 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 | the line meets the curve at x = a, and its slope matches a central difference quotient of f there |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: inconclusive — 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
Every verdict on record (4)
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). Substitutigpt-oss:20b: pass 2026-09-28qwen3.6:27b-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 2026-09-28 with SymPy 1.14.0.