Tangent lines to parametric curves
Problem 8.80 · hard
Find the equation of the tangent line to \( \displaystyle x = t^{2} + 1,\ y = \sin{\left(t \right)} \) at \( \displaystyle t = -1 \).
- dy/dx = (dy/dt)/(dx/dt).
- \[ \frac{d}{d t} \left(t^{2} + 1\right) = 2 t \]dx/dt.✓ Proved
- \[ \frac{d}{d t} \sin{\left(t \right)} = \cos{\left(t \right)} \]dy/dt.✓ Proved
- \[ \frac{\left(-1\right) \cos{\left(1 \right)}}{2} = - \frac{\cos{\left(1 \right)}}{2} \]The slope at t = -1.
- \[ \left[\begin{matrix}2\\- \sin{\left(1 \right)}\end{matrix}\right] \]The point.
- \[ \frac{\left(2 - x\right) \cos{\left(1 \right)}}{2} - \sin{\left(1 \right)} = - \frac{x \cos{\left(1 \right)}}{2} - \sin{\left(1 \right)} + \cos{\left(1 \right)} \]Point-slope form.✓ Proved
Answer \( y = - \frac{x \cos{\left(1 \right)}}{2} - \sin{\left(1 \right)} + \cos{\left(1 \right)} \)
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: passqwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the y-coordinate of the point of tangency as -sin(1) instead of sin(-1) = -sin(1). While -sin(1) is numerically e
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-10-04 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the y-coordinate of the point of tangency as -sin(1) instead of sin(-1) = -sin(1). While -sin(1) is numerically egpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies the chain rule for parametric equations, evaluates the derivatives and coordinates at t = -1, and uses the point-slope form to derive the 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, evaluates the derivatives and coordinates at t = -1, and uses the point-slope formula to derive the final 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.