Answer:
Max=3693.71
Direction=z
Explanation:
The maximum rate of change of f(x, y, z) = e^(3x) * sin(y + 2z) at (3, -1, 1) is given by the magnitude of the gradient vector at that point.
The gradient of f(x, y, z) is given by the partial derivatives with respect to x, y, and z:
∇f(x, y, z) = <3e^(3x) * sin(y + 2z), e^(3x) * cos(y + 2z), 2e^(3x) cos(y + 2z)>
Evaluating this at (3, -1, 1), we get:
∇f(3, -1, 1) = <3e^9 * sin(1), e^9 * cos(1), 2e^9 cos(1)>
The magnitude of this gradient vector is:
|∇f(3, -1, 1)| = sqrt( (3e^9*sin(1))^2 + (e^9*cos(1))^2 + (2e^9*cos(1))^2 )
|∇f(3, -1, 1)| = sqrt( 13e^18*cos^2(1) + 9e^18 * sin^2(1) )
|∇f(3, -1, 1)| = sqrt( 22e^18 - 4e^18 * sin^2(1) )
|∇f(3, -1, 1)| ≈ 3693.71
Therefore, the maximum rate of change of f(x,y,z) at (3,-1,1) is approximately 3693.71.
To find the direction in which this maximum rate of change occurs, we need to look at the direction of the gradient vector:
∇f(3, -1, 1) = <3e^9 * sin(1), e^9 * cos(1), 2e^9 cos(1)>
The direction of the gradient vector is the unit vector in the same direction, which is:
u = <3e^9 * sin(1)/|∇f(3, -1, 1)|, e^9 * cos(1)/|∇f(3, -1, 1)|, 2e^9 cos(1)/|∇f(3, -1, 1)|>
u ≈ <0.000810088, 0.000274316, 0.999999>
Therefore, the maximum rate of change of f(x,y,z) occurs in the direction of the vector u≈<0.000810088, 0.000274316, 0.999999>, which points in the z-direction.