Tangent lines to parametric curves
Problem 8.77 · hard
Find the equation of the tangent line to \( \displaystyle x = 2 t,\ y = t^{2} - 3 t \) at \( \displaystyle t = 2 \).
- dy/dx = (dy/dt)/(dx/dt).
- \[ \frac{d}{d t} 2 t = 2 \]dx/dt.✓ Proved
- \[ \frac{d}{d t} \left(t^{2} - 3 t\right) = 2 t - 3 \]dy/dt.✓ Proved
- \[ 1 \cdot \frac{1}{2} = \frac{1}{2} \]The slope at t = 2.
- \[ \left[\begin{matrix}4\\-2\end{matrix}\right] \]The point.
- \[ \frac{x}{2} - 4 \]Point-slope form.
Answer \( y = \frac{x}{2} - 4 \)
Lines: 2 proved, 4 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. How we verify
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 | Not checked | — | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 5 | Not checked | — | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 6 | Not checked | — | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 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 (error) — Sentence 6 incorrectly states the point‑slope form as an identity; it should produce y = x/2 – 4.qwen3.6:27b-mlx: pass — The solution correctly applies the chain rule for parametric equations, computes the derivatives and point coordinates accurately, and derives the correct tangent line equation.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the chain rule for parametric equations, computes the derivatives and point coordinates accurately, and derives the correct tangent line equation.gpt-oss:20b: fail (error) 2026-10-04 — Sentence 6 incorrectly states the point‑slope form as an identity; it should produce y = x/2 – 4.qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies the chain rule for parametric equations, calculates the slope and point at t=2, and derives the correct tangent line equation.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies the chain rule for parametric equations, calculates the derivative and point at t=2, and derives the correct tangent line equation.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/parametric_tangent, checked 2026-10-13 with SymPy 1.14.0.