Tangent lines to parametric curves
Problem 8.72 · hard
Find the equation of the tangent line to \( \displaystyle x = t^{3} - t,\ y = \sin{\left(t \right)} \) at \( \displaystyle t = 2 \).
- dy/dx = (dy/dt)/(dx/dt).
- \[ \frac{d}{d t} \left(t^{3} - t\right) = 3 t^{2} - 1 \]dx/dt.✓ Proved
- \[ \frac{d}{d t} \sin{\left(t \right)} = \cos{\left(t \right)} \]dy/dt.✓ Proved
- \[ \frac{\cos{\left(2 \right)}}{11} \]The slope at t = 2.
- \[ \left[\begin{matrix}6\\\sin{\left(2 \right)}\end{matrix}\right] \]The point.
- \[ \frac{\left(x - 6\right) \cos{\left(2 \right)}}{11} + \sin{\left(2 \right)} = \frac{x \cos{\left(2 \right)}}{11} - \frac{6 \cos{\left(2 \right)}}{11} + \sin{\left(2 \right)} \]Point-slope form.✓ Proved
Answer \( y = \frac{x \cos{\left(2 \right)}}{11} - \frac{6 \cos{\left(2 \right)}}{11} + \sin{\left(2 \right)} \)
Lines: 3 proved, 3 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 | ✓ 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 point‑slope step omits the dependent variable y; the correct form is y - sin(2) = (cos(2)/11)(x - 6).qwen3.6:27b-mlx: pass — The solution correctly applies the chain rule for parametric equations, evaluates the derivatives and coordinates at t=2, and uses the point-slope form to derive the final equation.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies the chain rule for parametric equations, evaluates the derivatives and coordinates at t=2, and uses the point-slope form to derive the final equation.gpt-oss:20b: fail (misleading) 2026-10-03 — The point‑slope step omits the dependent variable y; the correct form is y - sin(2) = (cos(2)/11)(x - 6).qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the chain rule for parametric equations, evaluates the derivatives and coordinates at t=2, and constructs the tangent line equation using the point-slope form.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the chain rule for parametric derivatives, evaluates the derivatives and coordinates at t=2, and uses the point-slope formula to derive the correct equation.gpt-oss:20b: pass 2026-09-28
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.