Tangent lines
Problem 2.774 · medium Video
Find an equation of the tangent line to \( \displaystyle y = - 2 x^{2} + 2 x + 4 + \frac{2}{x} \) at \( \displaystyle x = 4 \).
- The tangent line passes through the point (a, f(a)) and has slope f'(a).Reviewed
- \[ \left. - 2 x^{2} + 2 x + 4 + \frac{2}{x} \right|_{\substack{ x=4 }} = - \frac{39}{2} \]The point of tangency.✓ Proved
- \[ \frac{d}{d x} \left(- 2 x^{2} + 2 x + 4 + \frac{2}{x}\right) = - 4 x + 2 - \frac{2}{x^{2}} \]Differentiate.✓ Proved
- \[ \left. - 4 x + 2 - \frac{2}{x^{2}} \right|_{\substack{ x=4 }} = - \frac{113}{8} \]The slope at the point.✓ Proved
- \[ 37 - \frac{113 x}{8} \]Point-slope form, then simplify.
Answer \( y = 37 - \frac{113 x}{8} \)
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 definition of the tangent line at x=a: it 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: passqwen3.6:27b-mlx: fail (error) — The solution claims the point of tangency is derived from substituting x=4 into the function, but the computed value -39/2 is incorrect (the correct value is 11/2). Consequently, the final equation uses an incorrect y-intercept (37 instead of 15/2).
Senior review claude-sonnet-5-5, 2026-10-12: pass — f(4) = -39/2 and f'(4) = -113/8. The tangent line is y = -39/2 - (113/8)(x-4) = 37 - 113x/8, which matches the stated answer.
qwen3.6:27b-mlx: dismiss — f(4) = -32 + 8 + 4 + 1/2 = -39/2, so line 2 is correct (SymPy proved it); the claimed value 11/2 is wrong, and the intercept is -39/2 + 113/2 = 37, so the stated answer is correct.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-12 — f(4) = -39/2 and f'(4) = -113/8. The tangent line is y = -39/2 - (113/8)(x-4) = 37 - 113x/8, which matches the stated answer.qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution claims the point of tangency is derived from substituting x=4 into the function, but the computed value -39/2 is incorrect (the correct value is 11/2). Consequently, the final equation uses an incorrect y-intercept (37 instead of 15/2).gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution contains a contradiction: Step 2 calculates the y-coordinate as -39/2, but Step 5 uses 37 in the point-slope form. The correct y-value is -19.5, so the final equation is incorrect.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution claims the point of tangency is at y = -39/2, but the final equation uses y-intercept 37, which corresponds to a y-value of 37/2. The sign of the constant term in the function evaluation was likely mishandled or the final equation does not match the calculated point.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.