Home›Calculus 2›Tangent lines to parametric curves›Problem 8.32

Tangent lines to parametric curves

Problem 8.32 · hard

Find the equation of the tangent line to \( \displaystyle x = t^{2},\ y = t^{3} - 3 t \) at \( \displaystyle t = 1 \).

dy/dx = (dy/dt)/(dx/dt).
\[ \frac{d}{d t} t^{2} = 2 t \]
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.
\[ \left[\begin{matrix}1\\-2\end{matrix}\right] \]
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: pass
qwen3.6:27b-mlx: fail (error) — The solution claims the slope is 0, but dy/dt = 0 and dx/dt = 2 at t=1, so the slope is 0/2 = 0. However, the final answer given is y = -2, which implies a slope of 0. Wait, let's re-read. Line 4 says '0, 0 -- The slope at t = 1'. This is ambiguous. Does it mean slope is 0? Yes. Line 6 says '-2, -2 -- Point-slope form'. This is not an equation. The stated answer is 'Eq(y, -2)'. The point is (1, -2). Slope is 0. Tangent line is y - (-2) = 0(x - 1) => y = -2. The math is actually correct. Let me re-evaluate. 1. dy/dx formula is correct. 2. dx/dt = 2t. At t=1, dx/dt = 2. 3. dy/dt = 3t^2 - 3. At t=1, dy/dt = 0. 4. Slope m = 0/2 = 0. Line 4 says '0, 0'. This likely represents the slope value 0. It's poorly labeled but the value is correct. 5. Point: x(1)=1, y(1)=-2. Correct. 6. Point-slope: y - (-2) = 0(x - 1) => y = -2. Line 6 says '-2, -2'. This is just the point again? Or the constant term? It's very unclear. But the stated answer is y = -2. Is there an error? Line 4: '0, 0 -- The slope at t = 1'. The slope is 0. The second 0 might be a typo or referring to something else. Line 6: '-2, -2 -- Point-slope form'. This is not an equation. It looks like it's just repeating the point or the y-intercept. However, the prompt asks to review the *sentences* and logic. Sentence 1 is correct. The equations are marked 'proved'. Line 4 is an equation '0, 0'. This is not a standard equation format, but if it means 'slope = 0', it's correct. Line 6 is '-2, -2'. This is not an equation. The main issue is that the solution steps are extremely opaque and potentially misleading in their labeling. Line 4 labels '0, 0' as 'The slope'. A slope is a single number. Line 6 labels '-2, -2' as 'Point-slope form'. Point-slope form is an equation like y - y1 = m(x - x1). '-2, -2' is not that. But is it an *error* in the math? No. The math leads to y = -2. Is it *misleading*? Yes, a student would not learn how to write the point-slope form from this. Is it an *error*? The label 'Point-slope form' for the tuple (-2, -2) is factually wrong. Point-slope form is an equation. Let's look closer at Line 4. '0, 0'. Maybe it means (dy/dt, dx/dt) at t=1? No, dx/dt is 2. Maybe it means (slope, something)? Actually, looking at the pattern, Line 5 is the point vector. Line 6 is likely the result of plugging into point-slope? If the system checks algebra, maybe Line 6 is just the y-intercept? The verdict should be fail because the explanation is nonsensical/misleading. Specifically, calling '-2, -2' the 'Point-slope form' is incorrect terminology and confusing. Also, '0, 0' for the slope is ambiguous. Severity: misleading. A student would be confused by the notation and labels.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution claims the slope is 0, but dy/dt = 0 and dx/dt = 2 at t=1, so the slope is 0/2 = 0. However, the final answer given is y = -2, which implies a slope of 0. Wait, let's re-read. Line 4 says '0, 0 -- The slope at t = 1'. This is ambiguous. Does it mean slope is 0? Yes. Line 6 says '-2, -2 -- Point-slope form'. This is not an equation. The stated answer is 'Eq(y, -2)'. The point is (1, -2). Slope is 0. Tangent line is y - (-2) = 0(x - 1) => y = -2. The math is actually correct. Let me re-evaluate. 1. dy/dx formula is correct. 2. dx/dt = 2t. At t=1, dx/dt = 2. 3. dy/dt = 3t^2 - 3. At t=1, dy/dt = 0. 4. Slope m = 0/2 = 0. Line 4 says '0, 0'. This likely represents the slope value 0. It's poorly labeled but the value is correct. 5. Point: x(1)=1, y(1)=-2. Correct. 6. Point-slope: y - (-2) = 0(x - 1) => y = -2. Line 6 says '-2, -2'. This is just the point again? Or the constant term? It's very unclear. But the stated answer is y = -2. Is there an error? Line 4: '0, 0 -- The slope at t = 1'. The slope is 0. The second 0 might be a typo or referring to something else. Line 6: '-2, -2 -- Point-slope form'. This is not an equation. It looks like it's just repeating the point or the y-intercept. However, the prompt asks to review the *sentences* and logic. Sentence 1 is correct. The equations are marked 'proved'. Line 4 is an equation '0, 0'. This is not a standard equation format, but if it means 'slope = 0', it's correct. Line 6 is '-2, -2'. This is not an equation. The main issue is that the solution steps are extremely opaque and potentially misleading in their labeling. Line 4 labels '0, 0' as 'The slope'. A slope is a single number. Line 6 labels '-2, -2' as 'Point-slope form'. Point-slope form is an equation like y - y1 = m(x - x1). '-2, -2' is not that. But is it an *error* in the math? No. The math leads to y = -2. Is it *misleading*? Yes, a student would not learn how to write the point-slope form from this. Is it an *error*? The label 'Point-slope form' for the tuple (-2, -2) is factually wrong. Point-slope form is an equation. Let's look closer at Line 4. '0, 0'. Maybe it means (dy/dt, dx/dt) at t=1? No, dx/dt is 2. Maybe it means (slope, something)? Actually, looking at the pattern, Line 5 is the point vector. Line 6 is likely the result of plugging into point-slope? If the system checks algebra, maybe Line 6 is just the y-intercept? The verdict should be fail because the explanation is nonsensical/misleading. Specifically, calling '-2, -2' the 'Point-slope form' is incorrect terminology and confusing. Also, '0, 0' for the slope is ambiguous. Severity: misleading. A student would be confused by the notation and labels.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution incorrectly calculates the slope as 0 instead of -1/2, leading to the wrong tangent line equation y = -2 instead of y = -1/2(x - 1) - 2.
gpt-oss:20b: fail (misleading) 2026-09-27 — The solution incorrectly states the slope as "0, 0" and mislabels the point‑slope step as giving "-2, -2". These sentences could mislead a student into thinking the slope is a pair of numbers or that the point‑slope form yields a point, rather than the correct horizontal line y = –2.
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The slope calculation is incorrect; dy/dt at t=1 is 0 and dx/dt is 2, so the slope is 0, not -2. The final equation y = -2 is wrong; it should be y = -2 (the point) with slope 0, so y = -2 is actually the correct line equation, but the derivation of the slope as -2 in step 6 is inconsistent with step 4 which claims slope 0. Wait, step 4 says slope is 0. Step 6 says -2, -2 which likely implies slope -2. The final answer y=-2 is correct for a horizontal line through (1, -2), but the solution text claims the slope is -2 in step 6, contradicting step 4. This is a contradiction/error in the reasoning.
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.