Question

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

Answer 1

`=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`

Answer 2

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.