Tangent lines to parametric curves
Problem 8.86 · hard
Find the equation of the tangent line to \( \displaystyle x = t^{2},\ y = t^{3} \) at \( \displaystyle t = 1 \).
- dy/dx = (dy/dt)/(dx/dt).Reviewed
- \[ \frac{d}{d t} t^{2} = 2 t \]dx/dt.✓ Proved
- \[ \frac{d}{d t} t^{3} = 3 t^{2} \]dy/dt.✓ Proved
- \[ \frac{3}{2} \]The slope at t = 1.✓ Proved
- \[ \left[\begin{matrix}1\\1\end{matrix}\right] \]The point.✓ Proved
- \[ \frac{3 x}{2} - 1 \cdot \frac{1}{2} = \frac{3 x}{2} - \frac{1}{2} \]Point-slope form.✓ Proved
Answer \( y = \frac{3 x}{2} - \frac{1}{2} \)
Lines: 5 proved, 1 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: passqwen3.6:27b-mlx: pass — 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.
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 accurately, identifies the correct point, and derives the correct tangent line equation.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies the chain rule for parametric equations, computes the derivatives and slope accurately, and derives the correct tangent line equation.gpt-oss:20b: fail (error) 2026-10-03 — The final step incorrectly states the point‑slope form as a tautology instead of y‑minus‑y₀ = m(x‑x₀). It should read y‑1 = (3/2)(x‑1), which simplifies to y = (3/2)x – 1/2.
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.