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Tangent lines

Problem 2.784 · medium Video

Find an equation of the tangent line to \( \displaystyle y = - x + 2 \sin{\left(x \right)} \) at \( \displaystyle x = 0 \).

The tangent line passes through the point (a, f(a)) and has slope f'(a).
Reviewed
\[ \left. - x + 2 \sin{\left(x \right)} \right|_{\substack{ x=0 }} = 0 \]
The point of tangency.✓ Proved
\[ \frac{d}{d x} \left(- x + 2 \sin{\left(x \right)}\right) = 2 \cos{\left(x \right)} - 1 \]
Differentiate.✓ Proved
\[ \left. 2 \cos{\left(x \right)} - 1 \right|_{\substack{ x=0 }} = 1 \]
The slope at the point.✓ Proved
\[ x \]
Point-slope form, then simplify.

Answer \( y = x \)

The graph of y = -x + 2sin(x) near x = 0. A secant line through the point (0, 0) 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.

Lines: 3 proved, 1 reviewed, 1 not checked. 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	claude-sonnet-5-5	claude-sonnet-5-5: This is the standard statement that a tangent line at x=a passes through (a, f(a)) with slope f'(a). It is a sentence and not an algebraic equivalence, and it is correct.
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	Not checked	—	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: fail (error) — The final equation is incorrect. The point is (0,0) and the slope is 1, so the tangent line is y = x. The solution outputs 'x, x', which is not a valid equation format (likely missing 'y =') and fails to represent the line correctly.

Senior review claude-sonnet-5-5, 2026-10-12: pass — The point of tangency f(0)=0, the derivative 2cos x - 1 and the slope f'(0)=1 are all correct, so the tangent line is y = x, matching the stated answer. The final line's 'x = x' rendering is slightly awkward but not wrong.

qwen3.6:27b-mlx: dismiss — The point (0,0), the slope 1 and the line y = x are all correct, and SymPy proved line 5. The 'x = x' is only how the point-slope expression 0 + 1*(x - 0) is rendered after simplification, which is a formatting quirk and not a mathematical defect.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-12 — The point of tangency f(0)=0, the derivative 2cos x - 1 and the slope f'(0)=1 are all correct, so the tangent line is y = x, matching the stated answer. The final line's 'x = x' rendering is slightly awkward but not wrong.
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The final equation is incorrect. The point is (0,0) and the slope is 1, so the tangent line is y = x. The solution outputs 'x, x', which is not a valid equation format (likely missing 'y =') and fails to represent the line correctly.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The final equation is incorrect. The tangent line should be y = x + 0, or simply y = x, but the solution presents 'x, x' which is not a valid equation format (Eq(y, x) implies y=x, but the derivation step 5 is nonsensical and does not show the point-slope calculation y - 0 = 1*(x - 0)). More critically, step 5 claims to simplify point-slope form but outputs 'x, x', which is syntactically invalid as an equation and fails to represent the line y=x properly in the context of the stated answer format.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The final equation is incorrect. The tangent line has slope 1 and passes through (0,0), so the equation should be y = x. The solution lists 'x, x', which is algebraically nonsensical as an equation for y and fails to represent the line y=x.
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 2026-10-13 with SymPy 1.14.0.