We're going to determine a piecewise smooth curve
for the curve shown below
where if the curve is the curve C,
it's equal to curve C sub one
plus curve C sub two
plus curve C sub three.
So to make the curve C piecewise smooth,
we'll write parametric equations for these three curves.
Looking at the orientation,
we'll first go from the point zero comma zero
to the point two comma five along this segment.
Notice how the x-coordinate increases from zero to two
and the y-coordinate increases from zero to five.
So one way to parameterize curve one
or curve C sub one
would be to let x or x sub t
be equal to zero
plus two times t
and let y of t
be equal to zero plus five t,
where t would be on the closed interval from zero
to one.
And, of course, we can simplify x sub t to just two t
and y of t to just five t.
But the reason I included the zero here,
because notice how these were the starting coordinates
of this point here, zero comma zero.
Now let's focus on curve C sub two
which goes from the point two comma five
to the point five comma two along this segment here.
So notice how the x-coordinate starts at two
and increases by three to a value of five,
and the y-coordinate starts at a value of five
and decreases by three to a value of two.
Which means for curve C sub two,
we can let x or x sub t
be equal to two plus,
now, the x-coordinate increases by three,
so you might be thinking it should be plus three the.
But that would only be true
if t was on the interval from zero to one,
but not if t is going to be on the integral from
one to two.
Which means it's not going to be three times t,
it's going to be three times the quantity t minus one.
And then for y of t,
we would have five,
and the y-coordinate decreases by three,
so we'll have minus three,
but then not times t,
we'll have times the quantity t minus one.
And, of course, we can simplify here.
If we distribute, we'd have two plus three t minus three,
which would three t minus one.
And here, we'd have five minus three t plus three,
which should be eight minus three t.
Notice how if we substitute one in for t,
the x value would be two
and the y value would be five,
which would give us the point two comma five.
And when t is two, x is six minus one or five
and y is equal to eight minus six,
which does equal two.
And then finally for C sub three,
we want to go from the point five comma two
to the point zero comma zero along this segment.
So the x-coordinate starts at five
and decrease by five to zero,
and the y-coordinate starts at two
and decreases by two
back to a value of zero.
So x or x sub t
is going to be equal to five
minus five times not t
because now t
is going to be on the closed interval from two to three.
So it's going to be five times the quantity t minus two.
And for y of t,
the y-coordinate starts at two, so we have two,
and it decreases by two back to zero,
so we'll have minus two times the quantity t minus two.
And, of course, I can go ahead and simplify.
Here, we have five minus five t plus 10,
which should be 15 minus five t.
And here, we'd have two minus two t plus four,
which would give us six minus two t.
So here we have the parametric equations.
To make the curve C piecewise smooth,
we could also write this piecewise smooth curve
as a vector-valued function
where we'd have r of t
is equal to,
for curve C sub one
the x-component would be two t,
the y-component would be five t.
This would be when t is on the closed interval
from zero to one.
For curve C sub two, we'd have
an x-component of three t minus one
and y-component of eight minus three t.
This is when t is on the closed interval from one to two.
And finally, for curve C sub three,
the x-component would be 15 minus five t
and the y-component would be six minus two t.
And t would be on the closed interval
from two to three.
I hope you found this helpful.