Partial derivatives

Question

If f(x, y) = 16 − 4x² − y²

find f_x(8, 6)

 and f_y(8, 6)

 and interpret these numbers as slopes. 

Answer 1

To find fx(8, 6) and fy(8, 6), we need to take the partial derivatives of f(x, y) with respect to x and y, respectively, and then evaluate them at (8, 6).

So, we have:

fx(x, y) = d/dx [16 − 4x^2 − y^2] = -8x

fy(x, y) = d/dy [16 − 4x^2 − y^2] = -2y

Therefore,

fx(8, 6) = -8(8) = -64 fy(8, 6) = -2(6) = -12

Interpretation:

The value of fx(8, 6) = -64 means that when x increases by 1 unit, keeping y fixed, the value of f(x, y) decreases by 64 units. This indicates that the surface represented by the function f(x, y) is decreasing rapidly in the x-direction at the point (8, 6).

Similarly, the value of fy(8, 6) = -12 means that when y increases by 1 unit, keeping x fixed, the value of f(x, y) decreases by 12 units. This indicates that the surface represented by the function f(x, y) is decreasing less rapidly in the y-direction at the point (8, 6) compared to the x-direction.