Tangent lines
Problem 2.739 · medium Video
Find an equation of the tangent line to \( \displaystyle y = - 3 x^{2} - 4 x - 2 + \frac{2}{x} \) at \( \displaystyle x = -1 \).
- The tangent line passes through the point (a, f(a)) and has slope f'(a).Reviewed
- \[ \left. - 3 x^{2} - 4 x - 2 + \frac{2}{x} \right|_{\substack{ x=-1 }} = -3 \]The point of tangency.✓ Proved
- \[ \frac{d}{d x} \left(- 3 x^{2} - 4 x - 2 + \frac{2}{x}\right) = - 6 x - 4 - \frac{2}{x^{2}} \]Differentiate.✓ Proved
- \[ \left. - 6 x - 4 - \frac{2}{x^{2}} \right|_{\substack{ x=-1 }} = 0 \]The slope at the point.✓ Proved
- \[ -3 \]Point-slope form, then simplify.
Answer \( y = -3 \)
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 a sentence stating the standard tangent-line setup: the line passes through (a, f(a)) with slope f'(a). It is true and is not an algebraic equivalence. |
| 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: passqwen3.6:27b-mlx: fail (error) — The final equation is incorrect. The point of tangency is (-1, -3) and the slope is 0, so the tangent line is y = -3. The solution states the equation as y = -3, but line 5 is labeled 'Point-slope form, then simplify' and shows '-3, -3', which is not an equation of a line. More critically, the stated answer in the problem description is 'Eq(y, -3)', but the solution steps do not explicitly form the equation y - (-3) = 0(x - (-1)) => y = -3. Step 5 is just a number comparison or value check, not the derivation of the line equation. However, looking closer, step 5 says '-3, -3'. This likely represents the LHS and RHS of the final equation y = -3? No, it's ambiguous. Let's look at the result. The tangent line is horizontal at y=-3. The solution claims the slope is 0. The point is (-1, -3). The equation is y = -3. The stated answer is Eq(y, -3). The solution steps are: 1. Theory. 2. Point y-coord is -3. 3. Derivative. 4. Slope is 0. 5. '-3, -3'. This step is nonsensical as a derivation of the line equation. It doesn't show y = -3. It just shows two -3s. This is a severe error in the presentation/logic of the final step, failing to actually produce the equation.
Senior review claude-sonnet-5-5, 2026-10-12: pass — The point (-1, -3) and slope 0 are both correct, and they give y = -3. The 'Point-slope form, then simplify' label on the last line is slightly compound, and the line renders oddly as '-3 = -3', but neither is a mathematical defect.
qwen3.6:27b-mlx: dismiss — The mathematics is correct: f(-1) = -3, f'(-1) = 0, so the tangent is y = -3. The complaint is only that line 5 is displayed as '-3 = -3', which looks like a rendering of y = -3 with y evaluated and is not a false statement or wrong rule. The model's own reasoning agrees with the stated answer.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-12 — The point (-1, -3) and slope 0 are both correct, and they give y = -3. The 'Point-slope form, then simplify' label on the last line is slightly compound, and the line renders oddly as '-3 = -3', but neither is a mathematical defect.qwen3.6:27b-mlx: fail (error) 2026-09-28 — The final equation is incorrect. The point of tangency is (-1, -3) and the slope is 0, so the tangent line is y = -3. The solution states the equation as y = -3, but line 5 is labeled 'Point-slope form, then simplify' and shows '-3, -3', which is not an equation of a line. More critically, the stated answer in the problem description is 'Eq(y, -3)', but the solution steps do not explicitly form the equation y - (-3) = 0(x - (-1)) => y = -3. Step 5 is just a number comparison or value check, not the derivation of the line equation. However, looking closer, step 5 says '-3, -3'. This likely represents the LHS and RHS of the final equation y = -3? No, it's ambiguous. Let's look at the result. The tangent line is horizontal at y=-3. The solution claims the slope is 0. The point is (-1, -3). The equation is y = -3. The stated answer is Eq(y, -3). The solution steps are: 1. Theory. 2. Point y-coord is -3. 3. Derivative. 4. Slope is 0. 5. '-3, -3'. This step is nonsensical as a derivation of the line equation. It doesn't show y = -3. It just shows two -3s. This is a severe error in the presentation/logic of the final step, failing to actually produce the equation.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution claims to use point-slope form in step 5 but provides no equation, only the y-coordinate. The correct tangent line is y = -3, but the derivation is missing the explicit equation y - (-3) = 0(x - (-1)).gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution claims to use point-slope form in step 5 but provides no equation, only the y-value. It fails to explicitly state the equation of the line (y = -3) derived from the point (-1, -3) and slope 0.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.