
Vector function
I'm having trouble with this
Find a vector function that describes the border of the region in the first quadrant limited by y=4x, y=x,y=1/x, parametrized counterclockwise and find a tangent vector in (3/4, 4/3). In what points can you not define a tangent vector to the given curve? Justify.
I'm still stuck on the first part. I thought maybe giving a function by parts but I can't figure out how to do it.

Because there are points where you cannot define a tangent vector to the given curve (there are "corners" at, (0, 0), (1/4, 4), and (1, 1)), one formula will not work. Because smooth functions are so nice, our ways of writing formulas has developed to give smooth functions and this boundary cannot be described "smoothly".
Yes, you will need to use a "piecewise" function, with separate pieces for y= 4x, y= x, and y= 1/x. You will have to be careful to make the "pieces" join at the same values of the parameter.

Thank you for clarifying that. One thing though, wouldn't it be (1/2,2) instead of (1/4, 4)?
$\displaystyle 4x=1/x => x^2=1/4 => x=\pm1/2 => x=1/2$
I' missing one of the parts.
$\displaystyle
r(t)=\left\{\begin{array}{cc}(t,t),&\mbox{ if }
0\leq t\leq 1\\(1/t,t), & \mbox{ if } 1<x\leq2\\(?,?), & \mbox{ if } 2<x\leq3\end{array}\right.
$
Let's see. I need a function that gives (1/2,2) when t=2, and (0,0) when t=3 but I can't come up with anything...