评分及理由

（1）得分及理由（满分4分）

学生答案：\(\frac{4\sqrt{2}}{3}\)，标准答案：\(\frac{2}{3}\)。

曲率计算公式为 \(k = \frac{|x'y'' - x''y'|}{(x'^2 + y'^2)^{3/2}}\)。计算过程如下：

• \(x = \cos^3 t,\ y = \sin^3 t\)
• \(x' = -3\cos^2 t \sin t,\ y' = 3\sin^2 t \cos t\)
• \(x'' = -3\cos^3 t + 6\cos t \sin^2 t,\ y'' = -3\sin^3 t + 6\sin t \cos^2 t\)
• 在 \(t = \frac{\pi}{4}\) 时：
  • \(\cos t = \sin t = \frac{\sqrt{2}}{2}\)
  • \(x' = -\frac{3\sqrt{2}}{4},\ y' = \frac{3\sqrt{2}}{4}\)
  • \(x'' = \frac{3\sqrt{2}}{4},\ y'' = \frac{3\sqrt{2}}{4}\)
  • 分子：\(|x'y'' - x''y'| = \left|(-\frac{3\sqrt{2}}{4})(\frac{3\sqrt{2}}{4}) - (\frac{3\sqrt{2}}{4})(\frac{3\sqrt{2}}{4})\right| = \left|-\frac{18}{16} - \frac{18}{16}\right| = \frac{36}{16} = \frac{9}{4}\)
  • 分母：\((x'^2 + y'^2)^{3/2} = \left(\frac{18}{16}\right)^{3/2} = \left(\frac{9}{8}\right)^{3/2} = \frac{27}{16\sqrt{2}}\)
  • 曲率：\(k = \frac{9/4}{27/(16\sqrt{2})} = \frac{9}{4} \times \frac{16\sqrt{2}}{27} = \frac{4\sqrt{2}}{3}\)

学生的计算过程正确，但标...