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Tangent lines

Problem 2.725 · medium Video

Find an equation of the tangent line to \( \displaystyle y = 2 x + \sin{\left(x \right)} \) at \( \displaystyle x = \pi \).

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

Answer \( y = x + \pi \)

The graph of y = 2x + sin(x) near x = pi. A secant line through the point (pi, 2pi) and a second point on the curve turns into the tangent line as the second point slides in; its slope settles at 1, giving y = x + pi.

✓ Nihil obstat Lines: 3 proved, 2 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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	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, and applies the point-slope formula accurately to derive the final equation.

Senior review claude-sonnet-5-5, 2026-10-12: pass — The point (π, 2π), slope 1, and tangent line y = x + π are all correct. Line 5 is a bare identity that merges point-slope and simplification, which is a minor presentational issue and not an error.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-12 — The point (π, 2π), slope 1, and tangent line y = x + π are all correct. Line 5 is a bare identity that merges point-slope and simplification, which is a minor presentational issue and not an error.
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the point of tangency and the slope, and applies the point-slope formula accurately to derive the final equation.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the point of tangency and the slope, and applies the point-slope formula accurately to derive the final equation.
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, and applies the point-slope formula accurately to derive the final equation.
gpt-oss:20b: fail (error) 2026-09-26 — The final sentence claims the tangent line is "x + pi" but omits the dependent variable y. The correct equation is y = x + π, so the solution as written is incomplete and would 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/tangent_line, checked 2026-10-13 with SymPy 1.14.0.