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Tangent lines

Problem 2.1153 · medium

Find an equation of the tangent line to \( \displaystyle y = 2 x - 3 \sin{\left(x \right)} - 2 \) at \( \displaystyle x = \pi \).

The tangent line passes through the point (a, f(a)) and has slope f'(a).
Reviewed
\[ \left. 2 x - 3 \sin{\left(x \right)} - 2 \right|_{\substack{ x=\pi }} = -2 + 2 \pi \]
The point of tangency.✓ Proved
\[ \frac{d}{d x} \left(2 x - 3 \sin{\left(x \right)} - 2\right) = 2 - 3 \cos{\left(x \right)} \]
Differentiate.✓ Proved
\[ \left. 2 - 3 \cos{\left(x \right)} \right|_{\substack{ x=\pi }} = 5 \]
The slope at the point.✓ Proved
\[ 5 x - 3 \pi - 2 \]
Point-slope form, then simplify.✓ Proved

Answer \( y = 5 x - 3 \pi - 2 \)

Lines: 4 proved, 1 reviewed. 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.

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, and the final equation is algebraically correct.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the point of tangency and the slope, and the final equation is algebraically correct.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The final equation is incorrect. The point-slope form y - f(a) = m(x - a) yields y - (2pi - 2) = 5(x - pi), which simplifies to y = 5x - 3pi - 2. However, the stated answer is Eq(y, 5*x - 3*pi - 2), which implies y = 5x - 3pi - 2. Wait, let me re-calculate. f(pi) = 2pi - 3sin(pi) - 2 = 2pi - 2. Slope m = 2 - 3cos(pi) = 2 - 3(-1) = 5. Equation: y - (2pi - 2) = 5(x - pi) => y = 5x - 5pi + 2pi - 2 => y = 5x - 3pi - 2. The stated answer matches this. Why did I think it was wrong? Let me check the solution steps again. Step 5 says '5*x - 3*pi - 2, 5*x - 3*pi - 2'. This is just an identity, not a derivation. The sentence in step 5 is missing or implied. The prompt asks to review the solution. Step 5 is marked as an equation and 'unchecked' in the prompt description but the prompt says 'equation, unchecked' for steps 2-5. Actually, looking closely at the provided text: '5. [equation, unchecked] 5*x - 3*pi - 2, 5*x - 3*pi - 2 -- Point-slope form, then simplify.' This line is nonsensical as a derivation step. It asserts an identity rather than showing the work from point-slope form to the final equation. A student would not learn how to get from the point and slope to the final equation. It skips the actual algebraic manipulation. Is this an error or misleading? It's misleading because it presents the final answer without showing the connection to the previous steps (point and slope). It looks like a copy-paste error in the solution generation. The step does not follow from the previous lines logically; it just states the answer. Therefore, the solution is incomplete/misleading.
gpt-oss:20b: pass 2026-09-28

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.