Consider the function f(x,y)=xy+xz+yz+4 at the point P=(2,-1,1)
Find the unit vector in direction of maximum increase of f in the direction of p.
I know the magnitude of p=sqrt6
how do i find this???
as noted at math dot stackechange,
the rate of change in a function f in the direction of the unit vector η is ∇f⋅η (This is simply the chain rule.). This quantity reaches its maximum when η shares the same direction as ∇f (to maximize the cosine of the angle that's part of the dot product formula). Thus the magnitude of ∇f gives the rate of change in that direction, as well.
∇f = (y+z)i + (x+z)j + (x+y)k
∇f•p = (0,3,1)•(2,-1,1) = (0,-3,1)
divide by magnitude to find the unit vector.
Hope I understood the question right.
a search on vector analysis gradient will turn up lots of articles.
oops - stackexchange
I believe its actually (0,3,1) not negative 3, right?
Nevermind about that last comment!!
To find the unit vector in the direction of maximum increase of the function f at the point P, you need to find the gradient of f at P and normalize it to obtain a unit vector.
Here's how you can find it step by step:
1. Find the gradient of f(x, y, z) by computing the partial derivatives with respect to each variable:
∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)
In this case, we have:
∂f/∂x = y + z
∂f/∂y = x + z
∂f/∂z = x + y
Therefore, the gradient is:
∇f = (y + z, x + z, x + y)
2. Evaluate the gradient at the point P(2, -1, 1):
Substitute x = 2, y = -1, and z = 1 into the gradient:
∇f(P) = (-1 + 1, 2 + 1, 2 - 1)
= (0, 3, 1)
3. Calculate the magnitude of the gradient vector ∇f(P) using the Euclidean norm formula:
||∇f(P)|| = √(∇f_x(P)^2 + ∇f_y(P)^2 + ∇f_z(P)^2)
Therefore, in this case:
||∇f(P)|| = √(0^2 + 3^2 + 1^2)
= √(0 + 9 + 1)
= √10
The magnitude of the gradient vector at the point P is √10.
4. Finally, divide the gradient vector by its magnitude to obtain the unit vector in the direction of maximum increase:
Unit vector = ∇f(P) / ||∇f(P)||
In this case:
Unit vector = (0, 3, 1) / √10
That's it! You have now found the unit vector in the direction of maximum increase of the function f at the point P.