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Tangent lines

Problem 2.1462 · medium

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

The tangent line passes through the point (a, f(a)) and has slope f'(a).
\[ \left. - x - 3 \sin{\left(x \right)} - 2 \right|_{\substack{ x=\frac{\pi}{2} }} = -5 - \frac{\pi}{2} \]
The point of tangency.✓ Proved
\[ \frac{d}{d x} \left(- x - 3 \sin{\left(x \right)} - 2\right) = - 3 \cos{\left(x \right)} - 1 \]
Differentiate.✓ Proved
\[ \left. - 3 \cos{\left(x \right)} - 1 \right|_{\substack{ x=\frac{\pi}{2} }} = -1 \]
The slope at the point.✓ Proved
\[ - x - 5 \]
Point-slope form, then simplify.✓ Proved

Answer \( y = - x - 5 \)

Lines: 4 proved, 1 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.

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	✓ 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: fail (error) — The solution claims the final equation is y = -x - 5, but the correct tangent line is y = -x - 5 - pi/2. The constant term in the final answer omits the y-coordinate of the point of tangency (-pi/2), likely due to an arithmetic error in the point-slope step.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution claims the final equation is y = -x - 5, but the correct tangent line is y = -x - 5 - pi/2. The constant term in the final answer omits the y-coordinate of the point of tangency (-pi/2), likely due to an arithmetic error in the point-slope step.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution calculates the correct slope (-1) and point ((pi/2, -5 - pi/2)), but fails to substitute these values into the point-slope formula. The final equation y = -x - 5 is incorrect; the correct tangent line is y = -x - 5 - pi/2 + pi/2 = -x - 5? No, y - (-5 - pi/2) = -1(x - pi/2) => y = -x + pi/2 - 5 - pi/2 => y = -x - 5. Wait, let me re-calculate. Point: (pi/2, -pi/2 - 5). Slope: -1. Equation: y - (-pi/2 - 5) = -1(x - pi/2). y + pi/2 + 5 = -x + pi/2. y = -x - 5. The stated answer is actually correct. However, step 5 says 'Point-slope form, then simplify' and shows '-x - 5 = -x - 5'. This is a tautology, not a derivation. It does not show the application of the point-slope formula. It asserts the result without showing the work, which is misleading about the process. But is it an error? The prompt asks to find what is wrong. The sentence 'Point-slope form, then simplify' implies the equation shown IS the result of that process. But '-x - 5 = -x - 5' is not an equation of a line in the form y = mx + b, nor is it the point-slope form. It's a true statement that provides no information. A student would not learn how to get the answer from this step. It's a 'style' or 'misleading' issue because it skips the actual algebraic step. Let's look closer. Step 5 is marked [equation, unchecked]. The text says '-x - 5 = -x - 5'. This is not the equation of the tangent line. The equation of the tangent line is y = -x - 5. The line shown is a tautology. This is a severe presentation error. It fails to state the final answer clearly as an equation for y. It presents a tautology as the result of simplification. This is misleading/error in presentation.
gpt-oss:20b: pass 2026-10-03

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 2026-10-03 with SymPy 1.14.0.