Question

how to find the vector valued function forming the boundaries by the closed curve defined by connected points (0,0), (2,1), and (1,4)

Answer 1

You can define the function piecewise, and just let the vector vary along each of the straight line segments connecting the points.

Answer 2

So for the first segment
\[ \vec{f_1}(t)=(2t,t) \]

Answer 3

so the second is (t,4t) right?

Answer 4

Not really, at t=1 we are at the point (2,1), so you need to trace the function from there. You think correctly, but you need a term to align the end-points of the segments. With your definition we would jump from (2,1) to (1,4) instantaneously at t=1.

Answer 5

So you basically want
\[ \vec{f_1}(1)=\vec{f_2}(1) \\
\vec{f_2}(2)=\vec{f_3}(2) \\
\vec{f_3}(3)=(0,0) \]
If each segment should take 1 t-unit to trace.

Answer 6

Then the full function is just
\[ \vec{f}(t)=
\begin{cases}
\vec{f_1}(t) & \text{if } 0≤t≤1 \\
\vec{f_2}(t) & \text{if } 1<t≤2 \\
\vec{f_3}(t) & \text{if } 2<t≤3
\end{cases} \]

Answer 7

ar.....sorry dude I don't really get it
can u give me a final answer
i will figure it out myself

Answer 8

The second function could be
\[ \vec{f_2}(t)=(-t+3,\,3t-2) \]
and the third
\[ \vec{f_3}(t)=(-t+3,\,-4t+12)\]

Answer 9

thanks i will try to figure it out

Answer 10

Here is a picture I think will help a lot, if you think visually