Tangent lines to parametric curves
Problem 8.66 · hard
Find the equation of the tangent line to \( \displaystyle x = \cos{\left(t \right)},\ y = t^{3} - 3 t \) at \( \displaystyle t = \frac{\pi}{3} \).
- dy/dx = (dy/dt)/(dx/dt).
- \[ \frac{d}{d t} \cos{\left(t \right)} = - \sin{\left(t \right)} \]dx/dt.✓ Proved
- \[ \frac{d}{d t} \left(t^{3} - 3 t\right) = 3 t^{2} - 3 \]dy/dt.✓ Proved
- \[ - \frac{2 \sqrt{3} \left(-3 + \frac{\pi^{2}}{3}\right)}{3} = \frac{2 \sqrt{3} \left(9 - \pi^{2}\right)}{9} \]The slope at t = \frac{\pi}{3}.✓ Proved
- \[ \left[\begin{matrix}1 \cdot \frac{1}{2}\\- \pi + \frac{\pi^{3}}{27}\end{matrix}\right] = \left[\begin{matrix}\frac{1}{2}\\- \pi + \frac{\pi^{3}}{27}\end{matrix}\right] \]The point.✓ Proved
- \[ \frac{2 \sqrt{3} \left(9 - \pi^{2}\right) \left(x - \frac{1}{2}\right)}{9} - \pi + \frac{\pi^{3}}{27} = - \frac{2 \sqrt{3} \pi^{2} x}{9} + 2 \sqrt{3} x - \pi - \sqrt{3} + \frac{\pi^{3}}{27} + \frac{\sqrt{3} \pi^{2}}{9} \]Point-slope form.✓ Proved
Answer \( y = - \frac{2 \sqrt{3} \pi^{2} x}{9} + 2 \sqrt{3} x - \pi - \sqrt{3} + \frac{\pi^{3}}{27} + \frac{\sqrt{3} \pi^{2}}{9} \)
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 (error) — The slope calculation is incorrect: dy/dx = (π²/3−3)/(−√3/2) ≠ −2√3(−3+π²/3)/3. The point is correct, but the tangent line equation derived from the wrong slope is invalid.qwen3.6:27b-mlx: pass — The solution correctly applies the chain rule for parametric derivatives, evaluates the derivatives and coordinates at the specified parameter value, and constructs the tangent line equation using the point-slope form. All steps are logically sound and algebraically verified.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the chain rule for parametric derivatives, evaluates the derivatives and coordinates at the specified parameter value, and constructs the tangent line equation using the point-slope form. All steps are logically sound and algebraically verified.gpt-oss:20b: fail (error) 2026-09-27 — The slope calculation is incorrect: dy/dx = (π²/3−3)/(−√3/2) ≠ −2√3(−3+π²/3)/3. The point is correct, but the tangent line equation derived from the wrong slope is invalid.qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the parametric derivative formula, computes the derivatives and slope accurately, identifies the point on the curve, and uses the point-slope form to derive the final equation.gpt-oss:20b: pass 2026-09-27
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 2026-09-27 with SymPy 1.14.0.