Tangent lines to parametric curves
Problem 8.89 · hard
Find the equation of the tangent line to \( \displaystyle x = t^{2} + 1,\ y = t^{3} \) at \( \displaystyle t = 1 \).
- dy/dx = (dy/dt)/(dx/dt).
- \[ \frac{d}{d t} \left(t^{2} + 1\right) = 2 t \]dx/dt.✓ Proved
- \[ \frac{d}{d t} t^{3} = 3 t^{2} \]dy/dt.✓ Proved
- \[ \frac{3}{2} \]The slope at t = 1.✓ Proved
- \[ \left[\begin{matrix}2\\1\end{matrix}\right] \]The point.✓ Proved
- \[ \frac{3 x}{2} - 2 \]Point-slope form.✓ Proved
Answer \( y = \frac{3 x}{2} - 2 \)
Lines: 5 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 |
| 6 | ✓ 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 | a secant through nearby points of the curve has the same slope; the line passes through the point |
Reviewers
gpt-oss:20b: fail (misleading) — The final sentence states a tautology "3*x/2 - 2 = 3*x/2 - 2" instead of the tangent line equation. It does not correctly apply the point‑slope formula and would mislead a student into thinking the line is already given.qwen3.6:27b-mlx: pass — The solution correctly applies the chain rule for parametric equations, calculates the derivatives and slope at t=1, identifies the point, and derives the correct tangent line equation.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies the chain rule for parametric equations, calculates the derivatives and slope at t=1, identifies the point, and derives the correct tangent line equation.gpt-oss:20b: fail (misleading) 2026-10-03 — The final sentence states a tautology "3*x/2 - 2 = 3*x/2 - 2" instead of the tangent line equation. It does not correctly apply the point‑slope formula and would mislead a student into thinking the line is already given.qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies the chain rule for parametric equations, calculates the derivatives and slope accurately, identifies the correct point, and derives the correct tangent line equation.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/parametric_tangent, checked 2026-10-03 with SymPy 1.14.0.