Tangent lines to parametric curves
Problem 8.54 · hard
Find the equation of the tangent line to \( \displaystyle x = t^{2} + 1,\ y = t^{3} - 3 t \) at \( \displaystyle t = 1 \).
- dy/dx = (dy/dt)/(dx/dt).
- \[ \frac{d}{d t} \left(t^{2} + 1\right) = 2 t \]dx/dt.✓ Proved
- \[ \frac{d}{d t} \left(t^{3} - 3 t\right) = 3 t^{2} - 3 \]dy/dt.✓ Proved
- \[ 0 \]The slope at t = 1.
- \[ \left[\begin{matrix}2\\-2\end{matrix}\right] \]The point.
- \[ -2 \]Point-slope form.
Answer \( y = -2 \)
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) — The solution misstates the slope as "0, 0" and repeats the point matrix unnecessarily. The slope at t=1 is 0, not a pair of zeros, and the point should be listed once as (2, –2).qwen3.6:27b-mlx: fail (error) — The solution incorrectly calculates the slope at t=1 as 0 instead of -1.5, leading to the wrong tangent line equation.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution incorrectly calculates the slope at t=1 as 0 instead of -1.5, leading to the wrong tangent line equation.gpt-oss:20b: fail (error) 2026-09-27 — The solution misstates the slope as "0, 0" and repeats the point matrix unnecessarily. The slope at t=1 is 0, not a pair of zeros, and the point should be listed once as (2, –2).qwen3.6:27b-mlx: fail (style) 2026-09-26 — [domain objection, downgraded to style] The solution incorrectly calculates the slope at t=1 as 0, whereas dy/dt = 0 and dx/dt = 2, implying a vertical tangent (undefined slope). Consequently, the final equation y = -2 is incorrect; the correct tangent line is x = 2.gpt-oss:20b: fail (error) 2026-09-26 — The solution incorrectly states the slope as "0, 0" and misidentifies the point and point‑slope form. The slope at t=1 is 0, the point is (2,‑2), and the tangent line is y=‑2. The presented equations do not follow from the correct calculations.qwen3.6:27b-mlx: inconclusive 2026-09-26 — reviewer returned a non-objectgpt-oss:20b: pass 2026-09-26
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.