Tangent lines
Problem 2.1459 · medium
Find an equation of the tangent line to \( \displaystyle y = x^{2} - x - 2 - \frac{3}{x} \) at \( \displaystyle x = 4 \).
- The tangent line passes through the point (a, f(a)) and has slope f'(a).Reviewed
- \[ \left. x^{2} - x - 2 - \frac{3}{x} \right|_{\substack{ x=4 }} = \frac{37}{4} \]The point of tangency.✓ Proved
- \[ \frac{d}{d x} \left(x^{2} - x - 2 - \frac{3}{x}\right) = 2 x - 1 + \frac{3}{x^{2}} \]Differentiate.✓ Proved
- \[ \left. 2 x - 1 + \frac{3}{x^{2}} \right|_{\substack{ x=4 }} = \frac{115}{16} \]The slope at the point.✓ Proved
- \[ \frac{115 x}{16} - \frac{39}{2} \]Point-slope form, then simplify.✓ Proved
Answer \( y = \frac{115 x}{16} - \frac{39}{2} \)
✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly identifies the point of tangency and the slope, and the final equation is algebraically equivalent to the point-slope form derived from these values.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly identifies the point of tangency and the slope, and the final equation is algebraically equivalent to the point-slope form derived from these values.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 the final equation is algebraically correct.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.