Tangent lines
Problem 2.916 · medium
Find an equation of the tangent line to \( \displaystyle y = 2 \sqrt{x} - x + 3 \) at \( \displaystyle x = 1 \).
- The tangent line passes through the point (a, f(a)) and has slope f'(a).
- \[ \left. 2 \sqrt{x} - x + 3 \right|_{\substack{ x=1 }} = 4 \]The point of tangency.✓ Proved
- \[ \frac{d}{d x} \left(2 \sqrt{x} - x + 3\right) = -1 + \frac{1}{\sqrt{x}} \]Differentiate.✓ Proved
- \[ \left. -1 + \frac{1}{\sqrt{x}} \right|_{\substack{ x=1 }} = 0 \]The slope at the point.✓ Proved
- \[ 4 \]Point-slope form, then simplify.✓ Proved
Answer \( y = 4 \)
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: passqwen3.6:27b-mlx: fail (error) — The calculated slope is 0, so the tangent line should be y = 4. However, the solution claims to use 'Point-slope form' to arrive at '4, 4', which is not an equation of a line but rather a coordinate pair or a tautology. The final answer format 'Eq(y, 4)' is also non-standard for an equation of a line.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The calculated slope is 0, so the tangent line should be y = 4. However, the solution claims to use 'Point-slope form' to arrive at '4, 4', which is not an equation of a line but rather a coordinate pair or a tautology. The final answer format 'Eq(y, 4)' is also non-standard for an equation of a line.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution calculates the slope as 0, which is correct, but then fails to write the point-slope equation y - 4 = 0(x - 1). Instead, it jumps to '4, 4', which is not an equation of a line and does not follow logically from the previous steps.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 2026-09-26 with SymPy 1.14.0.