Final answer:
To find the directional derivative of F(x,y,z) = 4xyz at point P(2,1,2) in the direction of Q(4,2,4), calculate the gradient of F at point P, find the unit vector in the direction of Q, and compute their dot product. The directional derivative is 16.
Step-by-step explanation:
To find the directional derivative of the function F(x,y,z) = 4xyz at point P(2,1,2) in the direction of Q(4,2,4), we can use the formula:
DuF(2,1,2) = ∇F(2,1,2) · u
where ∇F(2,1,2) is the gradient of F at point P and u is the unit vector in the direction of Q. First, find the gradient of F by taking the partial derivatives of F with respect to x, y, and z:
∇F = (4yz, 4xz, 4xy)
Substitute the coordinates of P into the gradient:
∇F(2,1,2) = (4(1)(2), 4(2)(2), 4(2)(1)) = (8, 16, 8)
Next, find the unit vector u:
u = QP / |QP|
where QP is the vector from P to Q. Calculate QP and |QP|:
QP = (4-2, 2-1, 4-2) = (2, 1, 2)
|QP| = sqrt((2^2)+(1^2)+(2^2)) = sqrt(9) = 3
Finally, calculate the directional derivative:
DuF(2,1,2) = (8, 16, 8) · (2/3, 1/3, 2/3)
DuF(2,1,2) = (16/3) + (16/3) + (16/3) = 16