What is the slope of the polar curve #f(theta) = theta + cottheta+thetasin^2theta # at #theta = (3pi)/8#?

2 Answers
Jul 5, 2016

#=1.51#

Explanation:

  • The slope of any curve/function at a certain point is always the function's first derivative.
    Hence, the slope of #f(theta)=f'(theta)#
  • #f'(theta)=1-csc^2theta+theta*2sintheta*costheta+sin^2theta#
    #=1-csc^2theta+thetasin2theta+sin^2theta#
  • Therefore, the slope at #theta=(3pi)/8 #is
    #f'((3pi)/8)=1-csc^2((3pi)/8)+(3pi)/8sin2((3pi)/8)+sin^2((3pi)/8)#
    #=1-1.17+0.83+0.85#
    #=1.51#
Jul 5, 2016

Added graph

Explanation:

The graphing package I am using must have a slight error in the coding as the point does not sit exactly on the plotted line.
However; the uploaded image should give you a rough guide.

enter image source here