Tangent lines
Problem 2.1265 · easy
Find an equation of the tangent line to \( \displaystyle y = 2 x^{3} - 3 x^{2} - 1 \) at \( \displaystyle x = 0 \).
- The tangent line passes through the point (a, f(a)) and has slope f'(a).
- \[ \left. 2 x^{3} - 3 x^{2} - 1 \right|_{\substack{ x=0 }} = -1 \]The point of tangency.✓ Proved
- \[ \frac{d}{d x} \left(2 x^{3} - 3 x^{2} - 1\right) = 6 x \left(x - 1\right) \]Differentiate.✓ Proved
- \[ \left. 6 x \left(x - 1\right) \right|_{\substack{ x=0 }} = 0 \]The slope at the point.✓ Proved
- \[ -1 \]Point-slope form, then simplify.✓ Proved
Answer \( y = -1 \)
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) — Line 5 claims to use point-slope form but provides no equation, only the y-intercept value. It fails to state the actual equation of the line (y = -1) derived from the point (0, -1) and slope 0.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-29 — Line 5 claims to use point-slope form but provides no equation, only the y-intercept value. It fails to state the actual equation of the line (y = -1) derived from the point (0, -1) and slope 0.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 5 is mathematically incorrect and misleading. It claims that '-1, -1' represents the point-slope form, but point-slope form is an equation (e.g., y - y1 = m(x - x1)), not a pair of numbers. Furthermore, the derivation from the previous steps to the final answer is missing the actual equation construction.gpt-oss:20b: pass 2026-09-29
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 2026-09-29 with SymPy 1.14.0.