00:01
In this problem we are given a function of function f of x and y as follows.
00:11
This function equal to 8 times x times y.
00:17
We are going to consider this point p given by 167.
00:27
With this we will find the maximum rate of change of this function at this point.
00:32
Point p and also the direction in which this maximum rate of change occurs.
00:40
So since we have, since we are in this two -dimensional xy space, when we say the maximum rate of change, it will be a directional derivative, indicated by the following notation like this.
01:01
So we have this directional derivative in direction of this unit vector u.
01:05
Of this function f it is given by u head dot the gradient of f evaluated at this point p like this so we know of the function we can compute the gradient and we can evaluate this part right away but we don't know the direction so there is this nice little lemma or theorem from calculus it says the maximum rate of change of this function occurs in the direction of this gradient at this point p so let's write this down maximum rate of change occurs in the direction of this this gradient of this function f evaluated at this point p so this is a nice theorem so with this in mind let us compute the gradient of this function f at this point and also normalize the vector so what is the gradient we have in the x component partial f or partial x and in the y component we have partial f over partial y.
02:44
We have this really nice enough function so we can compute the partial x derivative to be eight times y and partial y derivative is just eight times x so eight times y comma x...