We are asked to fill in each component
of r prime of t with plus minus or zero
based upon the graph of the
vector valued function r of t
shown below in red.
Notice how in the red
space curve we have points
at t equals one, t equals
two, t equals three and so on.
Which gives us the orientation
shown by the red arrows.
The orientation is in this direction.
The components of r prime of t
are dx dt comma dy dt comma dz dt.
And the sign of these derivatives indicate
whether the x y and z components of r of t
are increasing decreasing
or remain constant.
To help us determine this
and these derivatives
we will look at a particular
point on this space curve
based upon the value of t,
then determine whether
the x y and z coordinates
are increasing, decreasing
or remain constant.
Looking at the graph
focusing on the x axis
if the point moves in
the positive x direction
or this direction,
the x coordinate is increasing
and therefore dx dt is positive.
If the point moves in
the negative x direction
which is this direction,
dx dt is negative because the
x coordinate is decreasing.
Focusing on the y axis,
if the point moves in this direction,
the positive y direction,
dy dt is positive.
If the point moves in
the negative y direction
or this direction,
dy dt is negative.
Focusing on the z axis,
if the point moves in
the positive z direction,
the z coordinate is
increasing, dz dt is positive.
If the point moves in
the negative z direction
or this direction, dz dt is negative.
And then finally to help us
determine the sign of dx dt
and dy dt it will be helpful
to focus on the projection
of the space curve onto the x
y plane shown here in black.
So we first have r prime of one
where the first component is dx dt.
So looking at the point on the space curve
where t equals one and then the projection
onto the x y plane,
it appears at this location
the point is moving
in the negative x direction
or closer to the y axis
and therefore the x coordinate
is decreasing slightly
which means dx dt must be negative.
The next component is
dy dt at t equals one
the point is moving in this direction
which is the positive y direction
and therefore the y
coordinate is increasing,
dy dt is positive.
The third component is dz dt
looking at the point where t
equals one on the space curve,
we can see the point is moving downhill
and therefore the z
coordinate is decreasing,
dz dt is negative.
Next we have r prime of three.
Focusing on the point where t equals three
on the space curve.
Notice how the projection
onto the x y plane
would be this point here.
The first component is
dx dt at t equals three,
the point is moving in
the negative x direction
which is this direction here,
and therefore the x
coordinate is decreasing,
dx dt is negative.
Next we have dy dt.
Notice how when t equals three
this is the right most
point on the positive y axis
which means when t equals three,
this is where the y coordinate changes
from increasing to decreasing
and therefore right at t equals three,
the y coordinate remains constant,
and therefore dy dt is zero.
We can also say that at this point,
we have a local maximum
for the y coordinate.
The third component is dz dt.
At t equals three the
point is moving uphill.
We're moving in the positive z direction
and therefore z is
increasing, dz dt is positive.
And finally we have r prime of five.
Focusing on the point where t equals five
and then the projection
onto the x y plane.
At t equals five the point is moving
in the positive x direction,
meaning the x coordinate is increasing
and therefore dx dt is positive.
Not by a lot but in this direction
the x coordinate is increasing.
Next we have dy dt,
at t equals five moving in this direction,
the y coordinate is decreasing,
or the point is moving in
the negative y direction,
and therefore dy dt is negative.
Looking back up at the space
curve at t equals five,
the point is moving downhill
and therefore the z
coordinate is decreasing,
dz dt is negative.
I hope you found this helpful.
