Tangent lines
Problem 2.1039 · medium
Find an equation of the tangent line to \( \displaystyle y = 4 x^{2} + \frac{2}{x} \) at \( \displaystyle x = 1 \).
- The tangent line passes through the point (a, f(a)) and has slope f'(a).
- \[ \left. 4 x^{2} + \frac{2}{x} \right|_{\substack{ x=1 }} = 6 \]The point of tangency.✓ Proved
- \[ \frac{d}{d x} \left(4 x^{2} + \frac{2}{x}\right) = 8 x - \frac{2}{x^{2}} \]Differentiate.✓ Proved
- \[ \left. 8 x - \frac{2}{x^{2}} \right|_{\substack{ x=1 }} = 6 \]The slope at the point.✓ Proved
- \[ 6 x \]Point-slope form, then simplify.✓ Proved
Answer \( y = 6 x \)
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 solution fails to explicitly state the point-slope equation y - 6 = 6(x - 1) before simplifying. It jumps from the slope and point directly to the simplified result 6*x, omitting the crucial step of substituting the point (1, 6) into the formula, which makes the derivation logically incomplete.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution fails to explicitly state the point-slope equation y - 6 = 6(x - 1) before simplifying. It jumps from the slope and point directly to the simplified result 6*x, omitting the crucial step of substituting the point (1, 6) into the formula, which makes the derivation logically incomplete.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution claims to use point-slope form but fails to include the y-intercept adjustment. The correct equation is y - 6 = 6(x - 1), which simplifies to y = 6x, but the step labeled 'Point-slope form' jumps directly to the final answer without showing the necessary algebraic manipulation, making the derivation logically incomplete and potentially confusing.gpt-oss:20b: pass 2026-09-27
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 2026-09-27 with SymPy 1.14.0.