Q.2 - Calculus

Question

Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) An equation in polar coordinates whose graph is a cardioid.

(b) An equation in polar coordinates whose graph is a limacon. 

(c) An equation in polar coordinates whose graph is a lemniscate. 

Step-by-Step Solution

Answer

a)  The equation of cardioid in polar coordinate is:$r=1-\cos \theta$.

b)The equation of limacon in polar coordinate is: $r=1-2\sin \theta$.

c)The equation of lemniscate in polar coordinate is:  $r^2=4\sin \left(2\theta \right)$.

Step 1. Given information

In polar for function is $r=f\left(\theta \right)$

Part (a). Step 1. Example of cardioid.

The general form of cardioid is:

$r=a\left(1\pm \cos \theta \right)$

So if $a=1$then equation of cardioid is $r=1-\cos \theta$

Part (b). Step 1. Example of limacon.

The general equation of a limacon is $r=a\pm b\sin \theta$

When $a=1,b=-2$

Then  equation of limacon is $r=1-2\sin \theta$

Part (c). Step 1. Example of lemniscate.

The general equation of lemniscate is $r^2=\pm a\sin \left(2\theta \right)$

If $r=4$

Then equation of lemniscate is $r^2=4\sin \left(2\theta \right)$