Home›Calculus 2›Tangent lines to parametric curves›Problem 8.39

Tangent lines to parametric curves

Problem 8.39 · hard

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

dy/dx = (dy/dt)/(dx/dt).
\[ \frac{d}{d t} e^{t} = e^{t} \]
dx/dt.✓ Proved
\[ \frac{d}{d t} t^{3} = 3 t^{2} \]
dy/dt.✓ Proved
\[ 3 e \]
The slope at t = -1.
\[ \left[\begin{matrix}e^{-1}\\-1\end{matrix}\right] \]
The point.
\[ 3 e \left(x - e^{-1}\right) - 1 = 3 e x - 4 \]
Point-slope form.✓ Proved

Answer \( y = 3 e x - 4 \)

Lines: 3 proved, 3 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	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
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 point-slope form is applied incorrectly: the equation should be y - y_0 = m(x - x_0), but the solution writes 3*E*(x - exp(-1)) - 1, which equals m(x - x_0) - y_0 rather than y. This results in the equation 3Ex - 4 = 0 (a vertical line) instead of the correct tangent line y = 3Ex - 4.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The point-slope form is applied incorrectly: the equation should be y - y_0 = m(x - x_0), but the solution writes 3*E*(x - exp(-1)) - 1, which equals m(x - x_0) - y_0 rather than y. This results in the equation 3Ex - 4 = 0 (a vertical line) instead of the correct tangent line y = 3Ex - 4.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the chain rule for parametric derivatives, evaluates the slope and point at t=-1, and uses the point-slope formula to derive the final equation. The algebraic simplification 3E*exp(-1) = 3 is correct.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The point-slope form is applied incorrectly. The equation should be y - y_0 = m(x - x_0), but the solution writes 3*E*(x - exp(-1)) - 1, which corresponds to m(x - x_0) + y_0 (or similar algebraic error), resulting in the constant term -4 instead of the correct value. Specifically, y - (-1) = 3e(x - 1/e) => y + 1 = 3ex - 3 => y = 3ex - 4. Wait, let me re-calculate. Slope m = 3e. Point (1/e, -1). Point-slope: y - (-1) = 3e(x - 1/e) y + 1 = 3ex - 3 y = 3ex - 4. The stated answer is Eq(y, 3*E*x - 4). This is correct. Let's look at line 6 again: "3*E*(x - exp(-1)) - 1, 3*E*x - 4". LHS: 3e(x - 1/e) - 1 = 3ex - 3 - 1 = 3ex - 4. RHS: 3ex - 4. So the equation in line 6 is mathematically true: 3ex - 4 = 3ex - 4. However, the sentence says "Point-slope form". The point-slope form is y - y_1 = m(x - x_1). The expression on the LHS of line 6 is NOT y - y_1. It is m(x - x_1) - 1? No, it's just an expression equal to the final answer. Actually, usually point-slope form is written as y - y_0 = m(x - x_0). Here, the solution seems to have skipped writing 'y =' or 'y - y_0 ='. Line 6 says: `3*E*(x - exp(-1)) - 1, 3*E*x - 4`. If this is meant to be the equation, it is missing the 'y'. If it is meant to be the derivation, it is confusing. But more importantly, look at the structure. Line 6 is marked [equation, proved]. The CAS checked that `3*E*(x - exp(-1)) - 1` equals `3*E*x - 4`. This is an identity, not the equation of the line. The equation of the line is `y = 3*E*x - 4`. Line 6 does not contain `y`. It just simplifies an expression. Therefore, the solution fails to actually state the equation of the tangent line in the final step, or rather, the final step is just an algebraic simplification of the RHS, not the full equation. Wait, the stated answer is `Eq(y, 3*E*x - 4)`. Line 6 produces `3*E*x - 4`. It does not produce `y = ...`. So the solution is incomplete/wrong because it doesn't equate y to the expression. Also, the label "Point-slope form" is misleading for the expression `3*E*(x - exp(-1)) - 1`. That expression is `m(x-x_0) + y_0`? No, `m(x-x_0) - 1`. Since `y_0 = -1`, `m(x-x_0) + y_0` would be `3e(x-1/e) - 1`. Yes. So the LHS is `m(x-x_0) + y_0`. This is equal to `y`. So the line is implicitly `y = ...`. But it's not written. Is this an error? Yes, the equation is not stated. The last line is just a simplification of the right-hand side. The variable `y` is missing from the final equation line.
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.