Understanding Non-Function Polar Equations in Calculus II

If you know r, that does not determine \theta, since \theta can be pi/4, 3pi/4, 5pi/4, etc. And even if you know \theta is one of those values, it does not determine r, so it is not a function.
  • #1
lovelylila
17
0
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!
 
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  • #2
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.
 
  • #3
no, I'm looking for a polar equation that is not a function in the polar plane.
 
  • #4
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?
 
  • #5
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! :-)
 
  • #6
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.
 
  • #7
thank you very much! i have a much clearer idea of what is going on now! :-) a thousand thanks!
 
  • #8
lovelylila said:
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 [itex]\theta[/itex] 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 [itex]\theta[/itex], [itex]r^2= \theta^2[/itex] is an obvious choice.
 

FAQ: Understanding Non-Function Polar Equations in Calculus II

What is a non-function polar equation?

A non-function polar equation is an equation that relates the distance and angle of a point from the origin in a polar coordinate system. Unlike a function, which has only one output for each input, a non-function polar equation can have multiple outputs for the same input, resulting in a non-uniform curve.

How are non-function polar equations graphed?

Non-function polar equations are graphed by plotting points on a polar coordinate system. The angle and distance values from the equation are used to determine the coordinates of each point, and then these points are connected to create a curve.

What are some common non-function polar equations?

Some common non-function polar equations include cardioids, limaçons, roses, and lemniscates. These equations can be used to model various real-world phenomena, such as the shape of a satellite's orbit or the path of a swinging pendulum.

How do non-function polar equations differ from function polar equations?

Non-function polar equations differ from function polar equations in that they do not have a one-to-one relationship between the input and output values. In other words, for a given input, there can be multiple output values in a non-function equation, while there is only one output value for each input in a function equation.

What is the importance of understanding non-function polar equations in Calculus II?

Understanding non-function polar equations in Calculus II is important because they are used to model many real-world phenomena and can be solved using advanced calculus techniques. They also provide a deeper understanding of polar coordinates and can be used to graph complex curves and shapes.

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