Let $C$ be the curve which connects (0,0) and (1,1) along $y^2=x^3$, the straight line from (1,1) to (0,2) and the semi circle with radius 1 and center (0,1) which connects (0,2) and (0,0)
So I drew this at first. So I think the curve is closed and only piecewise smooth. would this be correct?
Now I need the parameterization What I have until now
$\phi(t)=\begin{cases} <t, \sqrt{t^3}> \hspace{0.5cm} 0\leq t \leq 1 \\ <2-t, t> \hspace{0.5cm} 1\leq t \leq 2 \\ .... \end{cases}$
but how to do the circle part?
the equation of the full circle would be $x^2+(y-1)^2=1$
I tried: $y=t$ and then $x^2 = 1-(t-1)^2$
$x=-\sqrt{1-(t-1)^2}$ since we are in the second quadrant, we would be having the minus in front?) but what would the boundaries be ? $(1-t)^2$ has be to smaller than 1.
I also tried it with trig functions,
I have $<cost, sint+1>$ for $t \in [\pi/2, 3\pi/2]$ . would this be right?
I further need to determine whether the curve has a tangential vector in (0,0). Could please someone help me here?