#vec r = theta^2-theta - 3sin^2theta + tan^2theta = r \ hat r#
the velocity vector is the tangent vector and is....
#vec v = dot r \ hat r + dot theta r \ hat theta#
#= ((2 theta dot theta - dot theta - 6 sin theta cos theta dot theta + 2 tan theta sec^2 theta dot theta ),(dot theta ( theta^2-theta - 3sin^2theta + tan^2theta ))) ((hat r),(hat theta))#
#= dot theta ((2 theta - 1 - 3 sin 2 theta + 2 tan theta sec^2 theta ),( theta^2-theta - 3sin^2theta + tan^2theta )) ((hat r),(hat theta))#
#= dot theta ((pi/2),( 1/16 (-8-4 pi+pi^2))) ((hat r),(hat theta))#
We can discard the #dot theta # term because it is merely a scalar multiple. It is the direction of this vector that we are interested in
To convert to Cartesian, we know that
#hat r = cos theta hat x + sin theta hat y# and #hat theta = - sin theta hat x + cos theta hat y#
or #((hat r),(hat theta)) = ((cos theta, sin theta),(- sin theta, cos theta)) ((hat x),(hat y))#
So #((hat x),(hat y)) = ((cos theta, -sin theta),( sin theta, cos theta)) ((hat r),(hat theta))#
#vec bar v = 1/sqrt2 ((1, -1),( 1, 1)) ((pi/2),( 1/16 (-8-4 pi+pi^2)))#
#= 1/sqrt2 ((pi/2 - 1/16 (-8-4 pi+pi^2) ),(pi/2 + 1/16 (-8-4 pi+pi^2)))#
Again we can drop the #1/sqrt2# and so the slope is
#(pi/2 + 1/16 (-8-4 pi+pi^2))/(pi/2 - 1/16 (-8-4 pi+pi^2) )#
#approx 0.4029#
