Question

How do you find where the maximum r value occurs in a polar graph ?

Answer 1

To find the maximum r value , we should have an equation between r and (thete)

Then find the derivative between r and (theta)

which is

$\frac{dr}{d\theta}$

Then equating the previous value to zero

The derivative will be a function of (theta)

then solve the function of (theta) to find the values of theta

After finding these value of theta

Substitute with these value at the original function between r and theta.

For example:

let r = 1 + sin th

So,

dr/d th = cos th

so,

cos th = 0

th = pi/2 or 3pi/2

So, r will be maximum at th = pi/2 or 3pi/2

substitute with th = pi/2

r = 1 + sin pi/2 = 1 + 1 = 2

substitute with th = 3pi/2

r = 1 + sin (3pi/2) = 1 - 1 = 0

So, r will be maximum at th = pi/2 and the maximum r = 2

As a prove, see the following image: