Tangent lines to parametric curves
Problem 8.51 · hard
Find the equation of the tangent line to \( \displaystyle x = 2 t,\ y = t^{3} - 3 t \) at \( \displaystyle t = 1 \).
- dy/dx = (dy/dt)/(dx/dt).
- \[ \frac{d}{d t} 2 t = 2 \]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 (misleading) — The solution incorrectly lists the slope as "0, 0" and then produces a point‑slope expression of "-2, -2" instead of the correct tangent line equation y = –2. This misleads a student into thinking the slope is a pair of values and that the point‑slope form is simply two numbers, rather than the proper linear equation.qwen3.6:27b-mlx: fail (error) — The solution incorrectly calculates the slope at t=1 as 0 instead of -1 (since dy/dt = 0 but dx/dt = 2, so dy/dx = 0/2 = 0? Wait. dy/dt = 3(1)^2 - 3 = 0. dx/dt = 2. So slope is 0. The tangent line is horizontal. The point is (2, -2). Equation is y = -2. The stated answer is Eq(y, -2). The solution says slope is 0. Then it says 'Point-slope form' results in '-2, -2'. This is confusing notation but the final answer is correct. However, line 4 says '0, 0 -- The slope at t = 1'. This is ambiguous. Does it mean slope is 0? Or is it a pair? Line 6 says '-2, -2 -- Point-slope form'. This is not a standard representation of a line equation. It seems to imply the line is y = -2. But the derivation is opaque. Let's look closer. Slope m = 0. Point (2, -2). Equation: y - (-2) = 0(x - 2) => y = -2. The solution outputs '-2, -2' as the result of point-slope form. This is not an equation. It's just two numbers. The stated answer is Eq(y, -2). The solution does not clearly derive the equation y = -2. It just outputs numbers. This is misleading/wrong format. Also, line 4 '0, 0' is weird. Is it (slope, something)? If slope is 0, why is there a second 0? Maybe it's (dy/dt, dx/dt)? No, dy/dt=0, dx/dt=2. So (0, 2) would be correct for derivatives. (0, 0) is wrong for derivatives. If it means slope is 0, why the second 0? It's likely an error in the intermediate representation or a hallucination of values. Given the ambiguity and potential error in line 4 (claiming 0,0 when dx/dt is 2), it's an error.
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 (since dy/dt = 0 but dx/dt = 2, so dy/dx = 0/2 = 0? Wait. dy/dt = 3(1)^2 - 3 = 0. dx/dt = 2. So slope is 0. The tangent line is horizontal. The point is (2, -2). Equation is y = -2. The stated answer is Eq(y, -2). The solution says slope is 0. Then it says 'Point-slope form' results in '-2, -2'. This is confusing notation but the final answer is correct. However, line 4 says '0, 0 -- The slope at t = 1'. This is ambiguous. Does it mean slope is 0? Or is it a pair? Line 6 says '-2, -2 -- Point-slope form'. This is not a standard representation of a line equation. It seems to imply the line is y = -2. But the derivation is opaque. Let's look closer. Slope m = 0. Point (2, -2). Equation: y - (-2) = 0(x - 2) => y = -2. The solution outputs '-2, -2' as the result of point-slope form. This is not an equation. It's just two numbers. The stated answer is Eq(y, -2). The solution does not clearly derive the equation y = -2. It just outputs numbers. This is misleading/wrong format. Also, line 4 '0, 0' is weird. Is it (slope, something)? If slope is 0, why is there a second 0? Maybe it's (dy/dt, dx/dt)? No, dy/dt=0, dx/dt=2. So (0, 2) would be correct for derivatives. (0, 0) is wrong for derivatives. If it means slope is 0, why the second 0? It's likely an error in the intermediate representation or a hallucination of values. Given the ambiguity and potential error in line 4 (claiming 0,0 when dx/dt is 2), it's an error.gpt-oss:20b: fail (misleading) 2026-09-27 — The solution incorrectly lists the slope as "0, 0" and then produces a point‑slope expression of "-2, -2" instead of the correct tangent line equation y = –2. This misleads a student into thinking the slope is a pair of values and that the point‑slope form is simply two numbers, rather than the proper linear equation.qwen3.6:27b-mlx: fail (error) 2026-09-26 — The slope calculation is incorrect; dy/dt at t=1 is 0, so the slope is 0, not -2. The final equation y = -2 is wrong (it should be y = -2, wait, the point is (2, -2) and slope is 0, so y = -2 is actually correct? Let me re-read. Point: x=2(1)=2, y=1-3=-2. Slope: dy/dt = 3(1)^2-3=0. dx/dt=2. Slope = 0/2 = 0. Tangent line: y - (-2) = 0(x - 2) => y = -2. The final answer is correct. But line 4 says '0, 0 -- The slope at t=1'. This is confusing notation. It likely means dy/dt=0 and dx/dt=2? No, it just says 0, 0. And line 6 says '-2, -2 -- Point-slope form'. This is not point-slope form. The reasoning is garbled and misleading, even if the final answer happens to be correct. Specifically, claiming the slope is derived from '0, 0' is wrong. The slope is 0. The '0, 0' likely refers to dy/dt=0 and... something else? It's unclear. But more importantly, line 4 is labeled 'The slope at t=1' but contains '0, 0'. If it means slope=0, why two zeros? If it means dy/dt=0 and dx/dt=0, that's wrong. It's a mess. Let's look closer. Line 4: '0, 0'. Line 6: '-2, -2'. The solution seems to be outputting pairs of numbers for some reason. It's not a standard derivation. The sentence 'dy/dx = (dy/dt)/(dx/dt)' is correct. But the subsequent steps are not explained as sentences, they are equations. The prompt says 'Each line... is either an EQUATION... or a SENTENCE'. Lines 2-6 are marked [equation, proved]. I should not re-check the algebra. But I must check if the *sentences* are correct. There are no sentences between 1 and the end except line 1. Line 1 is correct. The equations are 'proved'. So I assume the algebra is correct. Wait, if the algebra is proved, then line 4 '0, 0' is a proven fact? What fact? It's not a standard equation. It looks like a tuple. This is likely a format error in the solution generation, but as a reviewer, if the algebra is 'proved', I must accept it. However, line 6 '-2, -2' is labeled 'Point-slope form'. Point-slope form is y - y1 = m(x - x1). '-2, -2' is not an equation. This is a severe formatting/logic error in the solution structure. But the prompt says 'do not re-check the algebra of a line marked proved'. It does not say 'do not check if the label matches the content'. Line 6 is labeled 'Point-slope form' but contains '-2, -2'. This is misleading/wrong. Also, the final answer 'Eq(y, -2)' is correct. But the derivation is nonsensical. Is it 'error' or 'misleading'? A student would learn that '-2, -2' is point-slope form. That is false. So 'misleading' or 'error'. I'll go with 'error' because the step is fundamentally broken.gpt-oss:20b: fail (misleading) 2026-09-26 — The solution incorrectly lists the slope as "0, 0" in step 4, which is confusing and could mislead a student into thinking the slope is a pair of values. The rest of the reasoning is correct, but the wording of the slope and the point‑slope step is unclear.qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution incorrectly calculates the slope at t=1 as 0 (it should be 0/2 = 0, wait, dy/dt = 3(1)^2 - 3 = 0, so slope is 0. Let me re-read. Ah, line 4 says '0, 0 -- The slope at t = 1'. The slope is indeed 0. But the final answer is y = -2. The point is (2, -2). The tangent line is y - (-2) = 0(x - 2) => y = -2. So the math is actually correct? Let me check line 4 again. '0, 0'. Usually this format implies (slope, something) or just the value. If it means slope is 0, that's correct. Line 6 says '-2, -2 -- Point-slope form'. This is ambiguous. Does it mean y = -2? The stated answer is Eq(y, -2). The derivation seems to skip the actual point-slope equation setup. However, looking closely at line 4: '0, 0'. If this is meant to be the slope, it is 0. But usually, a line like this in these datasets represents a calculation result. Let's look at the structure. Line 4 is marked [equation, unchecked]. The content is '0, 0'. This is not a standard equation. It looks like a tuple. If it represents the slope, it's 0. If it represents (dx/dt, dy/dt) at t=1, dx/dt=2, dy/dt=0. So (2,0) would be correct. (0,0) is wrong for the derivatives. If it represents the slope, it's just 0. Why two zeros? Maybe it's (x, y) of the point? No, line 5 is the point. Maybe it's (slope, intercept)? Slope 0, intercept -2? No, it says 0, 0. This is likely an error in the intermediate step representation or calculation. Let's assume the '0, 0' is a mistake for the slope calculation or vector. Actually, if line 4 is meant to be the slope, it should be a single number. If it's meant to be the derivative vector, it's (2, 0). The text says 'The slope at t = 1'. The slope is 0. Writing '0, 0' is confusing/wrong notation for a scalar slope. Furthermore, line 6 '-2, -2' is labeled 'Point-slope form'. Point-slope form is an equation like y - y1 = m(x - x1). '-2, -2' is not an equation. It looks like coordinates. The final answer y = -2 is correct, but the intermediate steps are nonsensical or incorrectly labeled. Specifically, line 4 claims the slope is '0, 0' which is not a standard representation for a scalar, and if interpreted as a vector, it's wrong. Line 6 is not point-slope form. The reasoning is flawed.gpt-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.