# Vector function

• Oct 23rd 2009, 06:18 AM
kodos
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.
• Oct 23rd 2009, 08:19 AM
HallsofIvy
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.
• Oct 23rd 2009, 10:33 AM
kodos
Thank you for clarifying that. One thing though, wouldn't it be (1/2,2) instead of (1/4, 4)?

$4x=1/x => x^2=1/4 => x=\pm1/2 => x=1/2$

I' missing one of the parts.

$
r(t)=\left\{\begin{array}{cc}(t,t),&\mbox{ if }
0\leq t\leq 1\\(1/t,t), & \mbox{ if } 1$

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...