(Solved): 1. Consider the path c:RR2 defined by c(t)=(cost,sin2t). (a) What curve is traced out by c(t) fo ...

1. Consider the path $$\vec{c}:\mathbb{R}?{\mathbb{R}}^2$$ defined by $$\vec{c}(t)=\left(\cos t,{\sin }^{2}t\right)$$. (a) What curve is traced out by $$\vec{c}(t)$$ for $$t?\mathbb{R}$$ ? [Careful! $$?\sin t?,?\cos t??1$$ for all $$t$$.] (b) Veriff that the path is smooth (or regular) for $$0<t<?$$, but not for any larger interval. What goes wrong at $$t=0,?$$ ? (c) Find a smooth reparametrization $$\vec{r}(u),u?[?1,1]$$, which traces out the entire curve.