Linear Approximation of f(x, y, z) = 2x^2 − xy + y^2 − 3x − 4y + 1 about the point (-1, 1)

To find the linear approximation of the given function f(x, y, z), we will use the Taylor series expansion. The linear approximation of a function f(x, y, z) about a point (a, b, c) can be represented as:

L(x, y, z) = f(a, b, c) + ∑(∂f/∂x)(a, b, c)(x - a) + ∑(∂f/∂y)(a, b, c)(y - b) + ∑(∂f/∂z)(a, b, c)(z - c)

In our case, the point (a, b, c) is (-1, 1, 0), and the function f(x, y, z) is 2x^2 − xy + y^2 − 3x − 4y + 1. Let's calculate the partial derivatives of f(x, y, z) with respect to x, y, and z:

∂f/∂x = 4x - y - 3
∂f/∂y = -x + 2y - 4
∂f/∂z = 0

Now, let's substitute these values into the linear approximation formula:

L(x, y, z) = f(-1, 1, 0) + (4(-1) - 1 - 3)(x - (-1)) + (-1 + 2(1) - 4)(y - 1) + 0(z - 0)
= f(-1, 1, 0) + (-4 - 1 - 3)(x + 1) + (-1 + 2 - 4)(y - 1)
= f(-1, 1, 0) - 8(x + 1) - 3(y - 1)

Now, let's calculate f(-1, 1, 0) using the original function:

f(-1, 1, 0) = 2(-1)^2 - (-1)(1) + (1)^2 - 3(-1) - 4(1) + 1
= 2 + 1 + 1 + 3 - 4 + 1
= 4

Substituting this value into the linear approximation equation, we get:

L(x, y, z) = 4 - 8(x + 1) - 3(y - 1)
= 4 - 8x - 8 - 3y + 3
= -8x - 3y - 1

Maximum Error in the region |x - 1| < 0.1,="" |y="" -="" 1|="" />< />

To find the maximum error in the given region, we need to find the maximum value of the absolute difference between the original function f(x, y, z) and its linear approximation L(x, y, z) in that region.

Let's substitute the given region values