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Tangent lines

Problem 2.781 · medium Video

Find an equation of the tangent line to \( \displaystyle y = 2 \sqrt{x} + 3 x + 2 \) at \( \displaystyle x = 4 \).

The tangent line passes through the point (a, f(a)) and has slope f'(a).
\[ \left. 2 \sqrt{x} + 3 x + 2 \right|_{\substack{ x=4 }} = 18 \]
The point of tangency.✓ Proved
\[ \frac{d}{d x} \left(2 \sqrt{x} + 3 x + 2\right) = 3 + \frac{1}{\sqrt{x}} \]
Differentiate.✓ Proved
\[ \left. 3 + 1 \frac{1}{\sqrt{x}} \right|_{\substack{ x=4 }} = \frac{7}{2} \]
The slope at the point.✓ Proved
\[ \frac{7 x}{2} + 4 \]
Point-slope form, then simplify.

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

The graph of y = 2sqrt(x) + 3x + 2 near x = 4. A secant line through the point (4, 18) and a second point on the curve turns into the tangent line as the second point slides in; its slope settles at 7/2, giving y = 7x/2 + 4.

Lines: 3 proved, 2 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	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 (style) — The sentence in step 1 is vague; it should explicitly state the point-slope formula y - f(a) = f'(a)(x - a) to justify the transition to the final equation. Step 5 lacks a descriptive sentence explaining how the point-slope form was applied and simplified.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-09-28 — The sentence in step 1 is vague; it should explicitly state the point-slope formula y - f(a) = f'(a)(x - a) to justify the transition to the final equation. Step 5 lacks a descriptive sentence explaining how the point-slope form was applied and simplified.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the point of tangency and the slope, and the final equation is algebraically correct.
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.