Question

If g(x, y) = x² + y² - 6x, find the gradient vector ∇g(1,6).

∇g(1,6) = ?

Answer 1

To find the gradient vector ∇g(1,6), first calculate the partial derivative ∂g/∂x and ∂g/∂y of g(x, y). Then evaluate the partial derivatives at the point (1,6) to get the gradient vector (-4, 12).

Step-by-step explanation:

The gradient vector of a function represents the rate of change of the function in different directions. To find the gradient vector ∇g(1,6), we need to find the partial derivatives of g(x, y) with respect to x and y, and then evaluate them at the point (1,6).

First, calculate the partial derivative ∂g/∂x:

∂g/∂x = 2x - 6

Next, calculate the partial derivative ∂g/∂y:

∂g/∂y = 2y

Now evaluate the partial derivatives at the point (1,6):

∂g/∂x(1,6) = 2(1) - 6 = -4

∂g/∂y(1,6) = 2(6) = 12

Therefore, the gradient vector ∇g(1,6) is (-4, 12).