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Tangent lines

Problem 2.737 · medium Video

Find an equation of the tangent line to \( \displaystyle y = 2 \sqrt{x} + 2 x + 2 \) at \( \displaystyle x = 4 \).

The tangent line passes through the point (a, f(a)) and has slope f'(a).
Reviewed
\[ \left. 2 \sqrt{x} + 2 x + 2 \right|_{\substack{ x=4 }} = 14 \]
The point of tangency.✓ Proved
\[ \frac{d}{d x} \left(2 \sqrt{x} + 2 x + 2\right) = 2 + \frac{1}{\sqrt{x}} \]
Differentiate.✓ Proved
\[ \left. 2 + 1 \frac{1}{\sqrt{x}} \right|_{\substack{ x=4 }} = \frac{5}{2} \]
The slope at the point.✓ Proved
\[ \frac{5 x}{2} + 4 \]
Point-slope form, then simplify.

Answer \( y = \frac{5 x}{2} + 4 \)

The graph of y = 2sqrt(x) + 2x + 2 near x = 4. A secant line through the point (4, 14) and a second point on the curve turns into the tangent line as the second point slides in; its slope settles at 5/2, giving y = 5x/2 + 4.

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 of the tangent line: it passes through (a, f(a)) with slope f'(a). It is a sentence, not an algebraic equivalence, so there is nothing for a script to prove.
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 (misleading) — The solution skips the explicit application of the point-slope formula, jumping directly from the slope and point to the final equation. This obscures the critical step of substituting (x_1, y_1) = (4, 14) into y - y_1 = m(x - x_1), which is the core concept being tested.

Senior review claude-sonnet-5-5, 2026-10-12: fail — All the mathematics is correct: f(4)=14, f'(4)=5/2, and y=14+(5/2)(x-4)=5x/2+4. The last step should be split into writing y-14=(5/2)(x-4) and then expanding. This is a presentation issue only, not an error.

qwen3.6:27b-mlx: uphold — Line 5 is labelled 'Point-slope form, then simplify', so it substitutes (4,14) and slope 5/2 and also expands in one step, which breaks the one-change-per-step rule, although the final answer is correct.

Every verdict on record (7)

claude-sonnet-5-5: fail (style) 2026-10-12 — All the mathematics is correct: f(4)=14, f'(4)=5/2, and y=14+(5/2)(x-4)=5x/2+4. The last step should be split into writing y-14=(5/2)(x-4) and then expanding. This is a presentation issue only, not an error.
qwen3.6:27b-mlx: fail (misleading) 2026-09-28 — The solution skips the explicit application of the point-slope formula, jumping directly from the slope and point to the final equation. This obscures the critical step of substituting (x_1, y_1) = (4, 14) into y - y_1 = m(x - x_1), which is the core concept being tested.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The final equation is incorrect. Using point-slope form with point (4, 14) and slope 5/2 yields y - 14 = 5/2(x - 4), which simplifies to y = 5/2 x + 4. However, checking the point (4, 14) in the stated answer y = 5/2 x + 4 gives 14 = 10 + 4, which is correct. Wait, let me re-evaluate. y = 2.5(4) + 4 = 10 + 4 = 14. The equation is actually correct. Let me re-read the prompt's stated answer: Eq(y, 5*x/2 + 4). This is y = 2.5x + 4. At x=4, y=14. Slope is 2.5. This matches. Why did I think it was wrong? Ah, I need to check the algebra in step 5. Step 5 says '5*x/2 + 4, 5*x/2 + 4'. This implies the result of the point-slope simplification is 5x/2 + 4. Let's check: y - 14 = 5/2(x - 4) => y = 5/2 x - 10 + 14 => y = 5/2 x + 4. The algebra is correct. The solution is sound.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the point of tangency and the slope, and the final equation is algebraically correct.
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.