Tangent lines
Problem 2.914 · medium
Find an equation of the tangent line to \( \displaystyle y = x e^{- 2 x} \) at \( \displaystyle x = 0 \).
- The tangent line passes through the point (a, f(a)) and has slope f'(a).
- \[ \left. x e^{- 2 x} \right|_{\substack{ x=0 }} = 0 \]The point of tangency.✓ Proved
- \[ \frac{d}{d x} x e^{- 2 x} = \left(1 - 2 x\right) e^{- 2 x} \]Differentiate.✓ Proved
- \[ \left. \left(1 - 2 x\right) e^{- 2 x} \right|_{\substack{ x=0 }} = 1 \]The slope at the point.✓ Proved
- \[ x \]Point-slope form, then simplify.✓ Proved
Answer \( y = x \)
Lines: 4 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see 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: fail (misleading) — The final sentence claims "x, x" as the point‑slope form, which omits the dependent variable and does not clearly state the resulting line y = x. This could mislead a student into thinking the equation is simply x = x rather than y = x.qwen3.6:27b-mlx: fail (error) — The final equation is incorrect. The tangent line at x=0 with slope 1 passing through (0,0) is y=x, but the solution presents 'x=x' as the result, which is an identity rather than the equation of the line y=x.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The final equation is incorrect. The tangent line at x=0 with slope 1 passing through (0,0) is y=x, but the solution presents 'x=x' as the result, which is an identity rather than the equation of the line y=x.gpt-oss:20b: fail (misleading) 2026-09-26 — The final sentence claims "x, x" as the point‑slope form, which omits the dependent variable and does not clearly state the resulting line y = x. This could mislead a student into thinking the equation is simply x = x rather than y = x.qwen3.6:27b-mlx: fail (error) 2026-09-26 — Line 5 is mathematically incorrect; it claims the tangent line equation simplifies to x=x, whereas the correct equation is y=x. The solution fails to actually write the equation of the line using the point (0,0) and slope 1.gpt-oss:20b: fail (error) 2026-09-26 — The final step does not produce the tangent line equation; it merely repeats the variable x. The solution should state the line y = x (or y - 0 = 1(x - 0)).
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 2026-09-26 with SymPy 1.14.0.