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Tangent lines

Problem 2.735 · medium Video

Find an equation of the tangent line to \( \displaystyle y = x e^{2 x} \) at \( \displaystyle x = 4 \).

The tangent line passes through the point (a, f(a)) and has slope f'(a).
Reviewed
\[ \left. x e^{2 x} \right|_{\substack{ x=4 }} = 4 e^{8} \]
The point of tangency.✓ Proved
\[ \frac{d}{d x} x e^{2 x} = \left(2 x + 1\right) e^{2 x} \]
Differentiate.✓ Proved
\[ \left. \left(2 x + 1\right) e^{2 x} \right|_{\substack{ x=4 }} = 9 e^{8} \]
The slope at the point.✓ Proved
\[ \left(9 x - 36\right) e^{8} + 4 e^{8} = 9 x e^{8} - 32 e^{8} \]
Point-slope form, then simplify.✓ Proved

Answer \( y = 9 x e^{8} - 32 e^{8} \)

The graph of y = x·e^(2x) near x = 4. A secant line through the point (4, 4e^(8)) and a second point on the curve turns into the tangent line as the second point slides in; its slope settles at 9e^(8), giving y = 9x·e^(8) - 32e^(8).

✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

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
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	the line meets the curve at x = a, and its slope matches a central difference quotient of f there

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the point of tangency and the slope, applies the point-slope formula accurately, and simplifies the result correctly.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the point of tangency and the slope, applies the point-slope formula accurately, and simplifies the result correctly.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the point of tangency, computes the derivative and slope, and applies the point-slope formula accurately. The final algebraic simplification is correct.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the point of tangency and the slope, then applies the point-slope formula accurately. The algebraic simplification is correct.
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/tangent_line, checked 2026-09-28 with SymPy 1.14.0.