Maximum gradient

Quote:

Originally Posted by Prodigious

Is there a way to calculate the maximum gradient of a graph? :confused: I have a graph of:

y=sin2(pi)x<---------pi being the number 3.1415

I was wondering if there was a way to calculate when the gradient of this graph is highest and what that gradient is.

Any suggestions will be appreciated.

Welcome to these forums.

You want the points where the derivative (slope of the curve) is a maximum. The derivative is:

y' = 2*Pi*cos(2*Pi*x)

It will be maximum (or minimum!) when its own derivative is 0, i.e. when:

y'' = -4*Pi²sin(2*Pi*x) = 0

The solution is any of the values of this set of (infinite) points:

sin(2*Pi*x) = 0 -> x = +/-{0, 0.5, 1, 1.5, 2, 2.5, 3, ...}

Now, at half of these points, the derivative will be maximum and at the other half it will be minimum. You can easily verify that for the integer values of x, the gradient is a maximum and it's a minimum (negative) for the rest.

The value of the gradient at a specific point is found by substituting the value of x in the equation for the derivative. For example, at x = 1.5:

y'(1.5) = 2*Pi*cos(2*Pi*1.3) = 2*Pi*cos(3*Pi) = -2*Pi = -6.28... (minimum)

And at x= 2:

y'(2) = 2*Pi*cos(2*Pi*2) = 2*Pi*cos(4*Pi) = 2*Pi = 6.28...