Home›Calculus 2›Tangent lines to parametric curves›Problem 8.61

Tangent lines to parametric curves

Problem 8.61 · hard

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

dy/dx = (dy/dt)/(dx/dt).
Reviewed
\[ \frac{d}{d t} e^{t} = e^{t} \]
dx/dt.✓ Proved
\[ \frac{d}{d t} \left(3 t + 1\right) = 3 \]
dy/dt.✓ Proved
\[ \frac{3}{e^{2}} \]
The slope at t = 2.✓ Proved
\[ \left[\begin{matrix}e^{2}\\7\end{matrix}\right] \]
The point.✓ Proved
\[ \frac{3 x - 3 e^{2}}{e^{2}} + 7 = \frac{3 x}{e^{2}} + 4 \]
Point-slope form.✓ Proved

Answer \( y = \frac{3 x}{e^{2}} + 4 \)

✓ Nihil obstat Lines: 5 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
5	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
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: pass — The solution correctly applies the chain rule for parametric equations, computes the derivatives and point coordinates accurately, and derives the correct tangent line equation.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the chain rule for parametric equations, computes the derivatives and point coordinates accurately, and derives the correct tangent line equation.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-27
gpt-oss:20b: pass 2026-09-27

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 2026-09-27 with SymPy 1.14.0.