We are asked to graph the polar function `r=sqrt(theta), 0<= theta <= 2pi ` , along with its vertical and horizontal tangents.
We can graph by plotting points:
`theta: ` r:
We can find the horizontal and vertical tangents by using:
`(dy)/(dx)=((dy)/(d theta))/((dx)/(d theta))=(1/(2sqrt(theta)) sin theta + sqrt(theta) cos theta)/(1/(2sqrt(theta))cos theta-sqrt(theta) sin theta) `
The horizontal tangents occur when the numerator (`(dy)/(d theta) ` ) is zero, while the vertical tangents occur when the denominator is zero.
Solving numerically we get the horizontal tangents when `x ~~ +- .653 `
and the vertical tangents when `x ~~ -1.83 `