# Greatest Rate of Angle Change

1. Feb 28, 2013

### Gackhammer

So I have been thinking of this problem... what is the greatest rate of angle change for a function? As in, what is the point in which a function achieves its greatest rate of angle change....

Well, the angle of a function can be determined by arctan(y')

The Rate of Angle change is (arctan(y'))', which equals $\frac{y'}{(y')^2 +1}$

So the greatest rate of angle change is the derivative of that set to zero, which is equal to

$\frac{y''' - 2(y')^2y' + y'''(y')^2}{((y')^2 +1)} = 0$

Which , you can simplify to....

$y''' - 2(y')^2y' + y'''(y')^2= 0$

Is there a way that this differential equation can be solved? (This is not for homework, this is just a general question that I would like to know the answer to)

2. Feb 28, 2013

### pwsnafu

Or you could use the fact that $\frac{x}{x^2+1}$ attains it's maximum at x=1.