Question

Identify, graph, and state the symmetries for the polar equation r=2+2sintheta.

Which are the critical points in the graph? please help

Answer 1

Graph = Cardioid

Axis of symmetry = y-axis

Critical points=
`(\pi)/(2) , (3\pi )/(2)`

Explanation:

General equation for this type of cardioid is:

a ± b sinθ

Condition for a cardioid =
`(a)/(b) = 1`

Axis of symmerty according to the graph of 2 + 2 sinθ is along y-axis.

Critical points:

r = 2 + 2 sinθ ⇒ r = 2(1 + sinθ) ⇒ r' = 2 cosθ

∵derivative of 1 + sinθ = cosθ

For finding critical point the derivative is equal to zero,

2 cosθ = 0 ⇒ cosθ = 0

the value of cosθ is equal to zero at intervals:
`(\pi)/(2) , (3\pi )/(2)`

So, critical points =
`(\pi)/(2) , (3\pi )/(2)`