请教一题高等微积分For each of the following functions f(t),determine whether the setS={f(t):t属於R}is a smooth curve.Draw a sketch of S.Examine the nature of S near any points f(t) where f'(t)=0.(a)f(t)=(t^2-1,t+1)(b)f(t)=(t^2-1,t^2+1)(c)f
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请教一题高等微积分For each of the following functions f(t),determine whether the setS={f(t):t属於R}is a smooth curve.Draw a sketch of S.Examine the nature of S near any points f(t) where f'(t)=0.(a)f(t)=(t^2-1,t+1)(b)f(t)=(t^2-1,t^2+1)(c)f
请教一题高等微积分
For each of the following functions f(t),determine whether the set
S={f(t):t属於R}is a smooth curve.Draw a sketch of S.
Examine the nature of S near any points f(t) where f'(t)=0.
(a)f(t)=(t^2-1,t+1)
(b)f(t)=(t^2-1,t^2+1)
(c)f(t)=((cost)^3,(sint)^3)
(d)f(t)=(cost+cos2t,sint+sin2t)
请教一题高等微积分For each of the following functions f(t),determine whether the setS={f(t):t属於R}is a smooth curve.Draw a sketch of S.Examine the nature of S near any points f(t) where f'(t)=0.(a)f(t)=(t^2-1,t+1)(b)f(t)=(t^2-1,t^2+1)(c)f
对下面的函数f(t)而言,判断集合S={f(t):t属於R}是否是光滑曲线,并作出曲线S的草图.检验曲线S在驻点 f'(t)=0的性质.
解法：由光滑曲线的定义,可知如果沿曲线有连续的切矢量,则该曲线就是光滑曲线.
如果导数X'(t),Y'(t)都是t的连续的函数,且切矢量(X'(t),Y'(t))非0（非奇异）,f'(t)=0处得到的是切矢量的奇点.
(a)f(t)=(t^2-1,t+1),切矢量为（2t,1）,不存在f'(t)=0的奇点,可知对于(a),S一定是光滑
(b)f(t)=(t^2-1,t^2+1),切矢量为（2t,2t）,当t=0是奇点,所以S不光滑
(c)f(t)=((cost)^3,(sint)^3),切矢量(-3(cost)^2*sint,3(sint)^2*cost),令f'(t)=0有解（例如t=0）,S不光滑
(d)也可以做类似的判断
直观上看b可知曲线在XY坐标下为（-1,1）到正无穷大的射线,但t是从负无穷大到正无穷大的变化,顺着t的方向在曲线上运动,可以看到过t=0点时切矢量方向发生了180度的反转
而c在t=0是奇点,当t过t=0时,切矢量的横坐标符号反转,纵坐标符号相同,所以切矢量方向也变了,直观上可判断有尖点存在.