Question

let f(x,y,z)=x2 + y2 - 4z, P(2,-1,1), and u = 4/5j - 3/5k a) find the gradient vector of f(x,y,z) b) find the direction of maximum increase at P c) find the rate of change of f at P in the direction o

Answer 1

The gradient vector of
`\( f(x, y, z) = x^2 + y^2 - 4z \)` is (2x, 2y, -4) . The direction of maximum increase at the point P(2, -1, 1) is
`\( \left( (2)/(3), (-1)/(3), (-2)/(3) \right) \)`. The rate of change of f at P in the direction of
`\( (4)/(5)\mathbf{j} - (3)/(5)\mathbf{k} \)` is 6.

Step-by-step explanation:

To solve this problem, we'll go through each part step by step.

a) Find the gradient vector of
`\( f(x, y, z) = x^2 + y^2 - 4z \)`:

The gradient vector
`\( \\abla f \)` is a vector containing the partial derivatives of f with respect to each variable. In this case:

`\[ \\abla f = \left( (\partial f)/(\partial x), (\partial f)/(\partial y), (\partial f)/(\partial z) \right) \]`

So, let's find the partial derivatives:

`\[ (\partial f)/(\partial x) = 2x \]`

`\[ (\partial f)/(\partial y) = 2y \]`

`\[ (\partial f)/(\partial z) = -4 \]`

Therefore, the gradient vector
`\( \\abla f \)` is (2x, 2y, -4).

b) Find the direction of maximum increase at P(2, -1, 1):

To find the direction of maximum increase, we use the gradient vector. The direction of maximum increase is the direction of the gradient vector. At the point
`\( P(2, -1, 1) \)`, the gradient vector is
`\( \\abla f = (4, -2, -4) \)`.

So, the direction of maximum increase at P is
`\( \mathbf{u} = (\\abla f)/(|\\abla f|) \)`, where
`\( |\\abla f| \)`is the magnitude of the gradient vector.

`\[ |\\abla f| = √(4^2 + (-2)^2 + (-4)^2) = √(16 + 4 + 16) = √(36) = 6 \]`

`\[ \mathbf{u} = \left( (4)/(6), (-2)/(6), (-4)/(6) \right) = \left( (2)/(3), (-1)/(3), (-2)/(3) \right) \]`

So, the direction of maximum increase at P is
`\( \left( (2)/(3), (-1)/(3), (-2)/(3) \right) \)`.

c) Find the rate of change of f at P in the direction of
`\( \mathbf{u} = (4)/(5)\mathbf{j} - (3)/(5)\mathbf{k} \)`:

The rate of change of f at P in the direction of
`\( \mathbf{u} \)` is given by the dot product of the gradient vector and the unit vector
`\( \mathbf{u} \)`.

`\[ \text{Rate of change} = \\abla f \cdot \mathbf{u} \]`

`\[ \text{Rate of change} = (4, -2, -4) \cdot \left( (2)/(3), (-1)/(3), (-2)/(3) \right) \]`

`\[ \text{Rate of change} = (8)/(3) + (2)/(3) + (8)/(3)`