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Tangent lines

Problem 2.771 · medium Video

Find an equation of the tangent line to \( \displaystyle y = \sqrt{x} - 2 x + 1 \) at \( \displaystyle x = 4 \).

The tangent line passes through the point (a, f(a)) and has slope f'(a).
Reviewed
\[ \left. \sqrt{x} - 2 x + 1 \right|_{\substack{ x=4 }} = -5 \]
The point of tangency.✓ Proved
\[ \frac{d}{d x} \left(\sqrt{x} - 2 x + 1\right) = -2 + \frac{1}{2 \sqrt{x}} \]
Differentiate.✓ Proved
\[ \left. -2 + 1 \frac{1}{2 \sqrt{x}} \right|_{\substack{ x=4 }} = - \frac{7}{4} \]
The slope at the point.✓ Proved
\[ 2 - \frac{7 x}{4} \]
Point-slope form, then simplify.

Answer \( y = 2 - \frac{7 x}{4} \)

The graph of y = sqrt(x) - 2x + 1 near x = 4. A secant line through the point (4, -5) and a second point on the curve turns into the tangent line as the second point slides in; its slope settles at -7/4, giving y = 2 - 7x/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 sentence states the standard fact that the tangent line at x=a passes through (a, f(a)) with slope f'(a). It is not an algebraic equivalence, so no script applies.
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 claims the final equation is 2 - 7*x/4, but the point-slope form using the point (4, -5) and slope -7/4 yields y = -5 - 7/4(x - 4) = 3 - 7x/4. The constant term is incorrect.

Senior review claude-sonnet-5-5, 2026-10-12: pass — Checking the numbers: f(4) = 2 - 8 + 1 = -5, f'(4) = 1/4 - 2 = -7/4, and the tangent line is y = -5 - (7/4)(x-4) = 2 - 7x/4, which matches the stated answer. Line 5 only shows the final form and skips the intermediate point-slope line, but that is a presentation point and not a defect.

qwen3.6:27b-mlx: dismiss — The objection's arithmetic is wrong: -5 - (7/4)(x-4) = -5 - 7x/4 + 7 = 2 - 7x/4, so the stated constant 2 is correct, not 3.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-12 — Checking the numbers: f(4) = 2 - 8 + 1 = -5, f'(4) = 1/4 - 2 = -7/4, and the tangent line is y = -5 - (7/4)(x-4) = 2 - 7x/4, which matches the stated answer. Line 5 only shows the final form and skips the intermediate point-slope line, but that is a presentation point and not a defect.
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution claims the final equation is 2 - 7*x/4, but the point-slope form using the point (4, -5) and slope -7/4 yields y = -5 - 7/4(x - 4) = 3 - 7x/4. The constant term is incorrect.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution claims the point of tangency is (4, -5), but f(4) = sqrt(4) - 2(4) + 1 = 2 - 8 + 1 = -5. Wait, the calculation is correct. Let me re-read carefully. f(4) = -5. Slope is -7/4. Point-slope: y - (-5) = -7/4(x - 4) => y + 5 = -7/4 x + 7 => y = -7/4 x + 2. The stated answer is y = 2 - 7x/4. This matches. Why did I think it was wrong? Let me check the algebra again. y = -7/4 x + 7 - 5 = -7/4 x + 2. Yes. The solution is actually correct. Let me look for subtle errors. Line 1 is generic. Line 2 calculates f(4). Line 3 calculates f'(x). Line 4 calculates f'(4). Line 5 states the final equation. The transition from lines 4 and 2 to 5 skips the explicit point-slope setup, but the result is correct. Is there a hidden error? 'Subs(..., -5)' implies the result is -5. 'Subs(..., -7/4)' implies the result is -7/4. The final line just asserts the equation. It doesn't show the work of combining them. However, the prompt says 'equation... proved'. If the CAS proved it, it's fine. But wait, is the point (4, -5)? sqrt(4)=2. 2 - 8 + 1 = -5. Correct. Slope: 1/(2*2) - 2 = 1/4 - 8/4 = -7/4. Correct. Equation: y = -7/4(x-4) - 5 = -7/4 x + 7 - 5 = -7/4 x + 2. Correct. The solution is sound. Why would it fail? Maybe the format? No. Maybe the sentence in line 1 is too vague? No. I will mark it pass.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution claims the point of tangency is -5, but f(4) = sqrt(4) - 2(4) + 1 = 2 - 8 + 1 = -5. Wait, let me re-calculate. f(4) = 2 - 8 + 1 = -5. The slope is f'(4) = 1/(2*2) - 2 = 1/4 - 2 = -7/4. The point-slope form is y - (-5) = -7/4(x - 4), which simplifies to y + 5 = -7/4 x + 7, so y = -7/4 x + 2. The stated answer is y = 2 - 7x/4. This matches. However, line 5 says '2 - 7*x/4, 2 - 7*x/4' and labels it 'Point-slope form, then simplify'. It skips the actual derivation from the point-slope formula y - y1 = m(x - x1). While the algebraic result is correct, the sentence describing the step is misleading because it doesn't show the substitution of the point (-5) and slope (-7/4) into the point-slope formula. It just asserts the final simplified equation. A student might not see how -5 and -7/4 lead to 2 - 7x/4. Specifically, the 'sentence' part of line 5 is missing the crucial step of writing out y - (-5) = -7/4(x-4). It jumps straight to the answer. Is this an error or misleading? It's misleading because it hides the arithmetic that connects the previous steps to the final answer. But wait, looking closer at line 2: `Subs(sqrt(x) - 2*x + 1, x, 4), -5`. This is correct. Line 4: `Subs(-2 + 1/(2*sqrt(x)), x, 4), -7/4`. This is correct. Line 5: `2 - 7*x/4, 2 - 7*x/4`. The label says 'Point-slope form, then simplify'. The equation shown is just the final answer equated to itself. It does not show the point-slope form. This is a 'style' or 'misleading' issue because the step is skipped entirely in the display, though the label claims it happens. However, usually 'error' is reserved for incorrect math. The math is correct. The explanation is poor. Let's look for a harder error. Did I miss something? f(x) = sqrt(x) - 2x + 1. f(4) = 2 - 8 + 1 = -5. f'(x) = 1/(2sqrt(x)) - 2. f'(4) = 1/4 - 2 = -7/4. Tangent: y - (-5) = -7/4(x - 4) => y + 5 = -7/4 x + 7 => y = -7/4 x + 2. The answer is correct. The issue is purely that line 5 does not actually display the point-slope form, making the sentence 'Point-slope form, then simplify' unsupported by the visible equation. This is misleading.
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.