Question

Calculate ∂f ∂x , ∂f ∂y , ∂f ∂x (1, −1) , and ∂f ∂y (1, −1) when defined. (If an answer is undefined, enter UNDEFINED.) f(x, y) = 9,000 − 70x + 12y +7xy ∂f ∂x = Incorrect: Your answer is incorrect. ∂f ∂y = Incorrect: Your answer is incorrect. ∂f ∂x (1, −1) = Incorrect: Your answer is incorrect. ∂f ∂y (1, −1) = Incorrect: Your answer is incorrect.

Answer 1

To calculate the partial derivatives of the given function, differentiate it with respect to x and y separately. Substituting the given values of x and y into the expressions for the partial derivatives gives the values of ∂f/∂x (1, -1) and ∂f/∂y (1, -1).

Step-by-step explanation:

To calculate the partial derivatives of the function f(x, y) = 9,000 - 70x + 12y + 7xy, we need to find the derivative of the function with respect to each variable separately.

To find ∂f/∂x, we differentiate the function with respect to x, treating y as a constant. The derivative of 9000 with respect to x is 0, the derivative of -70x is -70, and the derivative of 7xy is 7y. Therefore, ∂f/∂x = -70 + 7y.

To find ∂f/∂y, we differentiate the function with respect to y, treating x as a constant. The derivative of 12y with respect to y is 12, and the derivative of 7xy with respect to y is 7x. Therefore, ∂f/∂y = 12 + 7x.

To find ∂f/∂x (1, -1), we substitute x = 1 and y = -1 into the expression for ∂f/∂x. This gives us ∂f/∂x (1, -1) = -70 + 7(-1) = -77.

To find ∂f/∂y (1, -1), we substitute x = 1 and y = -1 into the expression for ∂f/∂y. This gives us ∂f/∂y (1, -1) = 12 + 7(1) = 19.

Answer 2

`(\partial f)/(\partial x) = -70+7\cdot y`,
`(\partial f)/(\partial y) = 12+7\cdot x`,
`(\partial f)/(\partial x) (1,-1) = -77`,
`(\partial f)/(\partial y)(1,-1) = 19`

Step-by-step explanation:

Let
`f(x,y) = 9000-70\cdot x +12\cdot y +7\cdot x \cdot y`, then the first partial derivatives of this multivariate function are, respectively:

`(\partial f)/(\partial x) = -70+7\cdot y` (1)

`(\partial f)/(\partial y) = 12+7\cdot x` (2)

Now we evaluate the partial derivatives at
`(x,y) = (1, -1)`:

`(\partial f)/(\partial x)(1,-1) = -70+ 7\cdot (-1)`

`(\partial f)/(\partial x) (1,-1) = -77`

`(\partial f)/(\partial y)(1,-1) = 12+7\cdot (1)`

`(\partial f)/(\partial y)(1,-1) = 19`