Calculate the change in a function f(x,y) = x^2 + 3xy - y^2 if x changes from 2 to 2.05 and y changes from 3 to 2.96

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justaguide's profile pic

Posted on

We need the change in the value of f(x, y) = x^2 + 3xy - y^2 when x changes from 2 to 2.05 and y changes from 3 to 2.96.

Initially f(x, y) = 2^2 + 3*2*3 - 3^2 = 13

When the value of x = 2.05 and y = 2.96

f(x, y) = 2.05^2 + 3*2.05*2.96 - 2.96^2

=> 4.2025 + 18.204 - 8.7616

=> 13.6449

The change is 13 - 13.6449 = 0.6449

When the value of x and y is changed, the value of the function increases by 0.6449.

giorgiana1976's profile pic

Posted on

The increment delta z could be calculated in this manner:

delta z = f(2.05 , 2.96) - f(2 , 3)

delta z = [(2.05)^2 + 3(2.05)*(2.96) - (2.96)^2] - [(2)^2 + 3*2*3 - (3)^2]

delta z = (4.2025 + 18.204 - 8.7616) - (4 + 18 - 9)

delta z = 13.6449 - 13

delta z = 0.6449

Another method to calculate the increment delta z is to determine dz

dz = [d(x^2 + 3xy - y^2)/dx]*dx + [d(x^2 + 3xy - y^2)/dy]*dy

The increment of z, in the given situation, is delta z = 0.6449.

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