Given z=x²+3xy-2y² Estimate the change in z

[Doesnt Matter]

1. Given z=x²+3xy-2y² Estimate the change in z when x changes from 2 to 2,5 and y changes from 3 to 2,52. I am stuck with the math and can't figure it out as what I need to do further.3.
a=2
b=3
Δx= +0,5
Δy= -0,5

dz/dx = 2x+3y
=2(2)+3(3)
=13
dz/dy = 3x-4y
= 3(2)-4(3)
=-6
Δz=f(a+Δx ; b+Δy) - f(a,b)
=f [2+0,5 ; 3+(-0,5)] - f(2,-0,5)
=f(2,3)(0,5) + f(2,3)(-0,5)
=** This is where I am stuck.

=13(0.5) + (-6)(-0,5)
=9,5?

 

[stevendaryl]

1. Given z=x²+3xy-2y² Estimate the change in z when x changes from 2 to 2,5 and y changes from 3 to 2,52. I am stuck with the math and can't figure it out as what I need to do further.3.
a=2
b=3
Δx= +0,5
Δy= -0,5

dz/dx = 2x+3y
=2(2)+3(3)
=13
dz/dy = 3x-4y
= 3(2)-4(3)
=-6

You're almost there: To first order, you can approximate: $\delta z = \frac{\partial z}{\partial x} \delta x + \frac{\partial z}{\partial y} \delta y$

You know that:
$\delta x = + 0.5$ (in America, we use "." for decimals, rather than ",")
$\delta y = - 0.5$
$\frac{\partial z}{\partial x} = 13$
$\frac{\partial z}{\partial y} = -6$

So you just plug in what you know to compute $\delta z$

 

[HallsofIvy]

That's a plausible way to estimate the change in z but the exact value is z(2.5, 2.5)- z(2, 3).

 

[stevendaryl]

That's a plausible way to estimate the change in z but the exact value is z(2.5, 2.5)- z(2, 3).

But the homework asked for an estimate.