5.2 曲線

1.

(a) $${(x,y,z) = (1,-1,2) + (2,-3,1)t}$$ (b) $${(x,y,z) = (3,1,0) + (3,-1,1)t}$$

(c) $${(x,y,z) = (1,0,3) + (1,-1,1)t}$$

2.

(a) $${{\bf r}^{\prime}(t) = (0,1,2t), (x,y,z) = (1,1,1) + (0,1,2)t}$$

(b) $${{\bf r}^{\prime}(t) = (4t,-1,4t), (x,y,z) = (2,0,2) + (4,-1,4)t}$$

3.

(a) $${\frac{\sqrt{2}\pi}{4}}$$ (b) $${\frac{1}{27}(13^{\frac{3}{2}} - 8)}$$

4.

(a) $${\kappa = \frac{\vert 6x\vert}{(1 + 9x^{4})^{3/2}}}$$ (b) $${\kappa = \frac{2}{(4x^{2} - 4x + 2)^{3/2}}}$$ (c) $${\kappa = \frac{\vert 2\sec^{2}{x}\tan{x}\vert}{(1 + \sec^{4}{x})^{3/2}}}$$