Tangent lines to parametric curves
Problem 8.9 · medium
Find the equation of the tangent line to \( \displaystyle x = e^{t},\ y = t^{3} - 3 t \) at \( \displaystyle t = -1 \).
- dy/dx = (dy/dt)/(dx/dt).
- \[ \frac{d}{d t} e^{t} = e^{t} \]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.
- \[ \left[\begin{matrix}e^{-1}\\2\end{matrix}\right] \]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: passqwen3.6:27b-mlx: fail (error) — Line 4 incorrectly states the slope is 0; dy/dt at t=-1 is 0, but dx/dt is e^{-1}, so the slope dy/dx is 0. However, Line 6 uses a slope of 2 in the point-slope form, which is inconsistent with the calculated slope of 0 and the stated answer y=2. The stated answer y=2 implies a slope of 0, but the solution text claims the point-slope form yields 2,2 which is nonsensical notation for a line equation and contradicts the slope calculation if interpreted as m=2.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — Line 4 incorrectly states the slope is 0; dy/dt at t=-1 is 0, but dx/dt is e^{-1}, so the slope dy/dx is 0. However, Line 6 uses a slope of 2 in the point-slope form, which is inconsistent with the calculated slope of 0 and the stated answer y=2. The stated answer y=2 implies a slope of 0, but the solution text claims the point-slope form yields 2,2 which is nonsensical notation for a line equation and contradicts the slope calculation if interpreted as m=2.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (style) 2026-09-27 — [domain objection, downgraded to style] The slope calculation is incorrect; dy/dt at t=-1 is 0, but dx/dt is e^{-1}, so the slope is 0, not undefined or derived from a 0/0 error as implied by line 4's label 'The slope'. Line 4 claims the slope is 0 (which is correct) but the label and context suggest a confusion with the derivatives themselves. More critically, line 6 is nonsensical: '2, 2' is not an equation, nor does it represent the point-slope form correctly. The final answer 'Eq(y, 2)' is correct (horizontal line y=2), but the derivation steps 4 and 6 are garbled and do not logically lead to the result in a way that teaches the method.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (style) 2026-09-26 — [domain objection, downgraded to style] The slope calculation is incorrect; dy/dt at t=-1 is 0, but dx/dt is e^{-1}, so the slope is 0, not undefined or derived from a 0/0 indeterminate form as implied by the context of line 4. However, the main error is that the final answer 'Eq(y, 2)' implies a horizontal line y=2, but the point is (e^{-1}, 2). A horizontal line through (e^{-1}, 2) is indeed y=2. Wait, let's re-read carefully. dy/dt = 3(-1)^2 - 3 = 0. dx/dt = e^{-1} != 0. So dy/dx = 0. The tangent line is horizontal. The point is (e^{-1}, 2). The equation is y - 2 = 0(x - e^{-1}) => y = 2. The stated answer is Eq(y, 2), which is y=2. This is correct. Why did I think it was wrong? Let's look at line 4. It says '0, 0 -- The slope at t = -1'. This is confusing notation. It likely means the numerator is 0 and the denominator is 0? No, dx/dt is not 0. If line 4 claims the slope is 0/0, that's an error. If it claims the slope is 0, the notation '0, 0' is very strange. Let's look at line 6. '2, 2 -- Point-slope form'. This is also nonsensical notation. The solution steps are opaque and potentially misleading about the calculation of the slope. Specifically, line 4 '0, 0' suggests a 0/0 indeterminate form, which is false since dx/dt = e^{-1} != 0. This is a significant conceptual error in the explanation.gpt-oss:20b: fail (misleading) 2026-09-26 — The solution incorrectly states the slope as "0, 0" and repeats the point matrix, which could confuse a student. The correct slope is 0 and the point is (e^{-1},2).
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.