0 1
0
-1
0
1
0
-1
0
1
and then find the points where the tangent line is vertical and
where it is horizontal.
Solution: By plotting points
or by using your graphing calculator, you find the curve is a four leaf rose:
0 1
The values tabulated above indicate how the curve is traced out: As
goes from 0 to
the top half of the leaf in the first quadrant is traced out. As
goes from
to
the left half of the leaf in the third quadrant is traced out, etc.
The horizontal tangent lines
As for the horizontal tangent lines, by the graph, there appear to be 6. To being a bit more analytical:
The tangent lines are horizontal when
.
By the above, this will happen when
or when
.
So, if we set
,
then the vertical tangent lines occur
when
Since
.
The values of r
corresponding to these angles are 2/3, -1, 2/3, 2/3, -1, and 2/3, respectively
The vertical tangent lines
We have:
Thus the tangent line is vertical when
or when
.
Next,
since
,
we have r = -2/3 when
.
Finally, if we
set
then the angles for the vertical tangent line are
The relation between
and
Since
and
are in the first quadrant we have
That is
