Tangent lines
Problem 2.1457 · medium
Find an equation of the tangent line to \( \displaystyle y = - x^{2} - 2 x - 3 + \frac{3}{x} \) at \( \displaystyle x = 1 \).
- The tangent line passes through the point (a, f(a)) and has slope f'(a).
- \[ \left. - x^{2} - 2 x - 3 + \frac{3}{x} \right|_{\substack{ x=1 }} = -3 \]The point of tangency.✓ Proved
- \[ \frac{d}{d x} \left(- x^{2} - 2 x - 3 + \frac{3}{x}\right) = - 2 x - 2 - \frac{3}{x^{2}} \]Differentiate.✓ Proved
- \[ \left. - 2 x - 2 - \frac{3}{x^{2}} \right|_{\substack{ x=1 }} = -7 \]The slope at the point.✓ Proved
- \[ 4 - 7 x \]Point-slope form, then simplify.✓ Proved
Answer \( y = 4 - 7 x \)
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: fail (error) — The final equation y = 4 - 7x is incorrect. Using the point (1, -3) and slope -7, the point-slope form is y - (-3) = -7(x - 1), which simplifies to y = -7x - 4. The solution incorrectly states the constant term is 4.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — The final equation y = 4 - 7x is incorrect. Using the point (1, -3) and slope -7, the point-slope form is y - (-3) = -7(x - 1), which simplifies to y = -7x - 4. The solution incorrectly states the constant term is 4.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly identifies the point of tangency and the slope, and derives the correct equation for the tangent line.gpt-oss:20b: pass 2026-10-03
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 2026-10-03 with SymPy 1.14.0.