Tangent lines
Problem 2.1463 · medium
Find an equation of the tangent line to \( \displaystyle y = x e^{x} \) at \( \displaystyle x = -1 \).
- The tangent line passes through the point (a, f(a)) and has slope f'(a).
- \[ \left. x e^{x} \right|_{\substack{ x=-1 }} = - \frac{1}{e} \]The point of tangency.✓ Proved
- \[ \frac{d}{d x} x e^{x} = \left(x + 1\right) e^{x} \]Differentiate.✓ Proved
- \[ \left. \left(x + 1\right) e^{x} \right|_{\substack{ x=-1 }} = 0 \]The slope at the point.✓ Proved
- \[ - \frac{1}{e} \]Point-slope form, then simplify.✓ Proved
Answer \( y = - \frac{1}{e} \)
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 explicitly state the equation of the tangent line (y = -1/e) despite the problem asking for it. Step 5 merely verifies that the point lies on a line with slope 0, but does not present the final answer as an equation.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution fails to explicitly state the equation of the tangent line (y = -1/e) despite the problem asking for it. Step 5 merely verifies that the point lies on a line with slope 0, but does not present the final answer as an equation.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution claims to use point-slope form in step 5, but the equation provided (-exp(-1) = -exp(-1)) is a tautology that does not represent the tangent line equation. The correct equation should be y = -1/e.gpt-oss:20b: fail (misleading) 2026-10-03 — The final sentence incorrectly states "-exp(-1) = -exp(-1)" as the point‑slope form; it does not produce the tangent line equation. The correct line is y = -1/e, but the solution fails to show this derivation.
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.