Question

how to sketch polar curves?

Answer 1

By ploting points and understanding what are the basic relations between radius and angle(2).

The derivative of the radius function still tells you how the radius changes, but you must understand it like that the origin is a compressed version of the positive x-axle(1).

So if the derivative is always zero, eg. r is a constant, you get circle, which corresponds to a vertical line on x-y-plane. If the derivative is always positive, you get outward spiraling curve, if negative inward spiraling curve. The second derivative tells how much wobble there is on the line when it's spiralling.

(1) Of course you can handle negative angles also, but I'd say that one should try to focus on positive angles first as the derivative tells you what happens on negative angles anyways. (3)

(2) Or make co-ordinate change to x-y-plane and use what you know about sketching curves on that co-ordinate-system.

(3) Normally one uses polar co-ordinates only on functions that are easier to handle in them. Like point symmetrical curves and symmetry around origin means that negative angles give you same curve as positive angles.

Answer 2

And also check the session 84 of 18.01.