Given f(x,y,z)=x^(2)y^(3)z^(6), in what direction is f(x,y,z) increasing the most rapidly at the point P(1,-1,1)? What is its rate of increase in that direction
Given f(x,y,z)=x^(2)y^(3)z^(6), in what direction is f(x,y,z) increasing the most rapidly at the point P(1,-1,1)? What is its rate of increase in that direction
so at the point in question:
,
so the unit vector in the direction that is increasing most rapidly is:
Given f(x,y,z)=x^(2)y^(3)z^(6), in what direction is f(x,y,z) increasing the most rapidly at the point P(1,-1,1)? What is its rate of increase in that direction
That is the gradient at the point.
Substitute, in the values of to get the optimal vector it increases by.
The norm of this vector is .
Thus, the unit vector is,