Home›Calculus 2›Tangent lines to parametric curves›Problem 8.36

Tangent lines to parametric curves

Problem 8.36 · hard

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.
\[ \left[\begin{matrix}e\\-2\end{matrix}\right] \]
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: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly calculates the slope as 0 instead of -2. At t=1, dy/dt = 0 and dx/dt = e, so the slope dy/dx is 0/e = 0. However, the final

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-09-28 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly calculates the slope as 0 instead of -2. At t=1, dy/dt = 0 and dx/dt = e, so the slope dy/dx is 0/e = 0. However, the final
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution incorrectly calculates the slope at t=1 as 0, whereas dy/dt = 0 but dx/dt = e, so the slope dy/dx is 0/e = 0. Wait, 0/e is 0. Let me re-read. dy/dt = 3(1)^2 - 3 = 0. dx/dt = e^1 = e. Slope = 0/e = 0. The tangent line is horizontal. The point is (e, -2). The equation is y = -2. The stated answer is Eq(y, -2). The solution says '0, 0 -- The slope at t=1'. This is technically correct (slope is 0). Then 'Point-slope form' yields -2, -2? Point slope is y - y1 = m(x - x1). y - (-2) = 0(x - e) => y + 2 = 0 => y = -2. The final line '-2, -2' is ambiguous but likely represents the constant term or the y-value. However, line 4 says '0, 0'. Is the slope 0? Yes. Is the second 0 something else? It's unclear. But the main issue is line 4: '0, 0 -- The slope at t = 1'. The slope is a single number, 0. Why two zeros? And line 6 '-2, -2' is also unclear. But more importantly, does the solution actually derive y=-2? It seems to skip the algebra of point-slope form. But the biggest error is likely in the interpretation of the output format or the intermediate steps. Let's look closer. Line 4: '0, 0'. If this means slope=0, it's correct. Line 6: '-2, -2'. If this means y-intercept=-2 and something else? Or y=-2? The stated answer is Eq(y, -2). The solution seems to produce the correct final answer but the intermediate steps are poorly formatted or confusing. However, is there a mathematical error? Slope is 0. Point is (e, -2). Equation y = -2. The solution claims slope is 0. It claims the point is (e, -2). It claims the result is -2, -2. This is not an equation. It's a pair of numbers. The student might be confused about what '-2, -2' means. But is it 'wrong'? It's misleading. But wait, look at line 4 again. '0, 0'. Did it calculate dx/dt and dy/dt separately? No, line 2 and 3 did that. Line 4 is the slope. Slope is 0. Why is there a second 0? Maybe it's a typo for just 0. Line 6 '-2, -2'. Maybe it's y = -2. The format is weird. But is there a hard error? No, the math is correct. Slope 0, y-intercept -2. The answer y=-2 is correct. The steps are just poorly labeled. Is it 'misleading'? Yes. Is it 'error'? No. But wait, look at the prompt's instruction: 'find what is wrong'. If the solution is just ugly but correct, it's style. If it teaches something false, it's misleading. If it's wrong, it's error. The slope is 0. The line is y=-2. The solution outputs '-2, -2'. This is not an equation. The stated answer is 'Eq(y, -2)'. The solution does not produce an equation. It produces a tuple. This is a failure to answer the question 'Find the equation'. It provides numbers. This is an error in the final step's representation. Also, line 4 '0, 0' is suspicious. If it meant slope=0, why two zeros? If it meant (dx/dt, dy/dt) evaluated, that would be (e, 0). It says (0, 0). This is factually wrong if it represents derivatives. If it represents slope and something else, it's unclear. Given the ambiguity and the likely incorrect representation of the slope calculation (if the second 0 is a mistake for e, or if it's just a typo), it's risky. But actually, looking at line 4: '0, 0'. The slope is 0. The second 0 might be a placeholder or error. If the student thinks the slope is represented by two numbers, that's bad. But the biggest issue is that the final answer is not an equation. It's a pair of numbers. The question asks for an equation. The solution fails to state the equation. It states '-2, -2'. This is not an equation. Therefore, the solution is incomplete/incorrect in its final output format. Verdict: fail. Severity: error (failed to provide the requested equation).
gpt-oss:20b: fail (error) 2026-09-27 — The solution incorrectly states the slope as "0, 0" and misapplies the point‑slope form. The correct slope is 0, and the tangent line is simply y = –2. The written steps do not accurately reflect the computation.
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution incorrectly calculates the slope at t=1 as 0, whereas dy/dt = 0 and dx/dt = e, so the slope is 0. However, the final answer provided is y = -2, which is a horizontal line passing through the point (e, -2). While the slope calculation is technically correct (0/e = 0), the presentation in step 4 is confusing ('0, 0' likely implies slope 0 and y-intercept 0 or similar confusion), and step 6 '-2, -2' is nonsensical notation for point-slope form. More critically, the final answer 'Eq(y, -2)' implies y = -2. Let's check the point: x(1) = e, y(1) = 1-3 = -2. The slope is 0. The tangent line is y - (-2) = 0(x - e), which simplifies to y = -2. So the final answer is actually correct. But the intermediate steps are garbled. Step 4 says '0, 0 -- The slope at t=1'. This is ambiguous. Step 6 says '-2, -2 -- Point-slope form'. This is not standard notation. The main issue is that the solution steps are poorly explained and potentially misleading, but the final result is correct. Wait, looking closer at step 4: '0, 0'. If this means slope=0, that's correct. Step 6: '-2, -2'. If this means the equation is y = -2, that's correct. But the notation is terrible. Is it an error? The prompt asks to find what is wrong. The steps are not clearly derived. Step 4 just asserts '0, 0'. Step 6 asserts '-2, -2'. A student wouldn't learn how to get there. However, is it 'wrong'? The slope IS 0. The line IS y=-2. The steps are just bad notation. Let's look for a harder error. Did it calculate the point correctly? Step 5: Matrix([E], [-2]). Yes. Did it calculate slope correctly? dy/dt = 3(1)^2 - 3 = 0. dx/dt = e. Slope = 0/e = 0. Yes. So the math is right. The 'error' is in the clarity/notation of the sentences/steps. But the instructions say 'find what is wrong'. If the math is right, is it a fail? 'nothing would teach a student something false'. The notation '0, 0' for slope and '-2, -2' for point-slope form is confusing but not necessarily 'false' if interpreted charitably. However, usually 'point-slope form' is y - y1 = m(x - x1). Writing '-2, -2' is not that. It's likely a 'style' or 'misleading' issue. But wait, look at step 4 again. '0, 0'. Does it mean slope is 0? Or does it mean the point is (0,0)? No, it says 'The slope at t=1'. So it claims the slope is 0. That is correct. Step 6: '-2, -2'. It says 'Point-slope form'. This is not a form. It's just two numbers. This is misleading. I will mark it as misleading because the notation is non-standard and confusing, potentially teaching bad habits, even if the final answer is correct.
gpt-oss:20b: fail (misleading) 2026-09-26 — The solution incorrectly states the slope as "0, 0" and misrepresents the point‑slope form as "-2, -2". While the final equation y = -2 is correct, the intermediate sentences give false information that could mislead a student.

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.