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Tangent lines

Problem 2.1149 · medium Video

Find an equation of the tangent line to \( \displaystyle y = 2 x - \sin{\left(x \right)} + 1 \) at \( \displaystyle x = \frac{\pi}{2} \).

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

Answer \( y = 2 x \)

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

Lines: 3 proved, 2 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. How we verify

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	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 solution fails to use the point of tangency (pi/2, pi) in the point-slope formula, resulting in an incorrect equation y = 2x instead of y = 2x - pi + 1.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution fails to use the point of tangency (pi/2, pi) in the point-slope formula, resulting in an incorrect equation y = 2x instead of y = 2x - pi + 1.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to apply the point-slope formula using the calculated point (pi/2, pi) and slope 2. It incorrectly jumps to y = 2x, which does not pass through the point of tangency.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to use the point-slope formula correctly. It ignores the y-coordinate of the point of tangency (which is \pi, not 0) and the x-coordinate (\pi/2), resulting in a line y=2x that does not pass through the point (\pi/2, \pi).
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 2026-10-13 with SymPy 1.14.0.