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Additional Polar Coordinates Practice Problems

— McGraw-Hill Professional
Updated on Sep 27, 2011

If necessary, review:

Additional Polar Coordinates Practice

PROBLEM 1

Provide an example of a graphical object that can be represented as a function in polar coordinates, but not in Cartesian coordinates.

SOLUTION 1

Recall the definitions of the terms relation and function from Chapter 6. When we talk about a function f , we can say that r = f ( θ ). A simple function of θ in polar coordinates is a constant function such as this:

f ( θ ) = 3

Because f ( θ ) is just another way of denoting r , the radius, this function tells us that r = 3. This is a circle with a radius of 3 units.

In Cartesian coordinates, the equation of the circle with radius of 3 units is more complicated. It looks like this:

x 2 + y 2 = 9

(Note that 9 = 3 2 , the square of the radius.) If we let y be the dependent variable and x be the independent variable, we can rearrange the equation of the circle to get:

y = ±(9 − x 2 ) 1/2

If we say that y = g ( x ) where g is a function of x in this case, we are mistaken. There are values of x (the independent variable) that produce two values of y (the dependent variable). For example, when x = 0, y = ± 3. If we want to say that g is a relation, that’s fine, but we cannot call it a function.

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