Question

Calculate the directional derivative of g(x, y, z) = z^2 – xy + 4y^2 in the direction v = (1, -4,2) at the point P = (2,1,-4). Remember to use a unit vector in directional derivative computation. (Use symbolic notation and fractions where needed.) Dvg(2, 1, –4) =

Answer 1

The directional derivative of g(x, y, z) = z^2 - xy + 4y^2 in the direction v = (1, -4,2) at the point P = (2,1,-4) is 7/√21.

Step-by-step explanation:

To calculate the directional derivative, we need to find the gradient vector of the function and then take the dot product with the unit vector in the given direction. The gradient vector is given by:

∇g(x, y, z) = (∂g/∂x, ∂g/∂y, ∂g/∂z) = (-y, -x+8y, 2z)

At the point P(2, 1, -4), the gradient vector is ∇g(2, 1, -4) = (-1, -6, -8)

The unit vector in the direction v = (1, -4, 2) is u = v/|v|, where |v| is the magnitude of v:

|v| = √(1^2 + (-4)^2 + 2^2) = √21

Therefore, u = (1/√21, -4/√21, 2/√21)

The directional derivative Dv(g) at P is the dot product of the gradient vector with the unit vector:

Dv(g) = ∇g(2, 1, -4) · u = (-1, -6, -8) · (1/√21, -4/√21, 2/√21) = (-1/√21) + (24/√21) + (-16/√21) = 7/√21

Answer 2

The directional derivative of the function g(x, y, z) at the point (2,1,-4) in the direction of vector (1, -4,2) is 3/√21.

Step-by-step explanation:

The question is asking to calculate the directional derivative of the function g(x, y, z) = z^2 − xy + 4y^2 at the point P = (2,1,−4) in the direction of the vector v = (1, -4,2). To begin with, we need to find the gradient of g and then normalize the direction vector v.

First, we find the partial derivatives of g with respect to x, y, and z:

• gx = −y
• gy = −4y − x
• gz = 2z

The gradient vector of g at P is ∇g(2, 1, −4) = (gx, gy, gz) evaluated at P, which is (−1, −3, −8).

Next, we normalize the direction vector v. The unit vector in the direction of v is given by v/|v|, where |v| is the magnitude of v. The magnitude |v| = √(1^2 + (−4)^2 + 2^2) = √21. Hence, the unit vector is v/|v| = (1/√21, −4/√21, 2/√21).

Finally, the directional derivative Dvg(2, 1, −4) is the dot product of the gradient vector and the unit vector in the direction of v: Dvg(2, 1, −4) = ∇g ⋅ (v/|v|) = (−1, −3, −8) ⋅ (1/√21, −4/√21, 2/√21) = (−1/√21 − 12/√21 + 16/√21) = 3/√21.