Tangent lines
Problem 2.1464 · easy
Find an equation of the tangent line to \( \displaystyle y = x^{3} + 3 x + 2 \) at \( \displaystyle x = 1 \).
- The tangent line passes through the point (a, f(a)) and has slope f'(a).
- \[ \left. x^{3} + 3 x + 2 \right|_{\substack{ x=1 }} = 6 \]The point of tangency.✓ Proved
- \[ \frac{d}{d x} \left(x^{3} + 3 x + 2\right) = 3 x^{2} + 3 \]Differentiate.✓ Proved
- \[ \left. 3 x^{2} + 3 \right|_{\substack{ x=1 }} = 6 \]The slope at the point.✓ Proved
- \[ 6 x \]Point-slope form, then simplify.✓ Proved
Answer \( y = 6 x \)
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 solution fails to use the point of tangency (1, 6) in the point-slope formula. It incorrectly equates the slope (6) with the y-intercept, resulting in y = 6x instead of the correct y = 6x - 6.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution fails to use the point of tangency (1, 6) in the point-slope formula. It incorrectly equates the slope (6) with the y-intercept, resulting in y = 6x instead of the correct y = 6x - 6.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution fails to use the point-slope formula correctly. It ignores the y-coordinate of the point of tangency (y=6) and the subtraction of the x-coordinate, resulting in the equation y=6x instead of the correct y=6x-6.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.