Understanding Non-Function Polar Equations in Calculus II

[lovelylila]

We're currently working on polar equations in Calculus II. I'm very confused about one point of our discussion today, finding a polar equation that is not a function (ie, one value plugged in will give you two output values). My teacher mentioned how you can't use the vertical line test for polar equations, and that some equations that are not functions in the Cartesian plane are functions in the polar plane? I'd appreciate any help or guidance regarding an example of a polar equation that is not a function because I'm completely bewildered!

 

[Dick]

I assume you want a polar function that's not a function in the x-y plane? Try r=1. Can't get much simpler than that.

 

[lovelylila]

no, I'm looking for a polar equation that is not a function in the polar plane.

 

[Dick]

Well, x^2=y^2 is not a function in the cartesian plane. Why not? Does that help you think of a polar equation that would have similar problems?

 

[lovelylila]

its not a function because say if you plug 2 in for x, y can be either -2 or +2. i understand that. well, in the polar plane that equation would be (rcostheta)^2= (rsintheta)^2. i guess that works, since the rs would cancel out leaving you with (costheta)^2= (sintheta)^2 but it seems too simple. is this right? i really appreciate your help! :-)

 

[Dick]

I was thinking of r^2=theta^2, but that works, too. If you know r, that doesn't determine theta since theta can be pi/4, 3pi/4, 5pi/4... And even you know theta is one of those values, it doesn't determine r. So, no. Definitely not a function. It is simple.

 

[lovelylila]

thank you very much! i have a much clearer idea of what is going on now! :-) a thousand thanks!

 

[HallsofIvy]

We're currently working on polar equations in Calculus II. I'm very confused about one point of our discussion today, finding a polar equation that is not a function (ie, one value plugged in will give you two output values). My teacher mentioned how you can't use the vertical line test for polar equations, and that some equations that are not functions in the Cartesian plane are functions in the polar plane? I'd appreciate any help or guidance regarding an example of a polar equation that is not a function because I'm completely bewildered!

Are you trying to find a curve such that r is a function of \theta but y is not a function of x? If so, r= 1, as Dick said, works. If you are looking for some equation in which r is not a function of \theta, r^2= \theta^2 is an obvious choice.