Find the directional derivative of the function f(x,y,z) =z^2 -xy^2
at the point P(2,1,3) in the direction of the vector v=< -1, 2, 2 >.
df/du=2z dz/du -y^2 dx/du -2xy dy/du and at P
df/du=6 dz/du -dx/du -4 dy/du check that.
du/du= - dx/du -4 dy/du + 6 dz/du by puting them in order.
Now applying the directional v,
df/du= 1+8 +12=21 check that.
To find the directional derivative of a function at a point in a specific direction, we can use the formula:
D_vf(x,y,z) = ∇f(x,y,z) · v
Where ∇f(x,y,z) represents the gradient of the function f(x,y,z) and v is the direction vector.
First, let's find the gradient of the function f(x,y,z) = z^2 - xy^2 by taking the partial derivatives with respect to each variable:
∂f/∂x = -y^2
∂f/∂y = -2xy
∂f/∂z = 2z
Now, we can evaluate the gradient at the given point P(2, 1, 3):
∇f(2, 1, 3) = (-1, -4, 6)
Next, we need to calculate the dot product of the gradient and the given direction vector v = < -1, 2, 2 >:
D_vf(2, 1, 3) = ∇f(2, 1, 3) · v
= (-1, -4, 6) · < -1, 2, 2 >
To calculate the dot product, we multiply corresponding components and sum them:
= (-1)(-1) + (-4)(2) + (6)(2)
= 1 - 8 + 12
= 5
Therefore, the directional derivative of the function f(x, y, z) = z^2 - xy^2 at the point P(2, 1, 3) in the direction of the vector v = < -1, 2, 2 > is 5.