Find the linearization L(x,y)of the function f(x,y) at P_0. Then find an upper bound for the magnitude |E| of the error in the approximation f(x,y)=L(x,y) over the rectangle R.
f(x,y) = (1/2)x^2 + xy + (1/4)y^2 + 3x - 3y + 4 at P_0(2,2),R:|x-2|<=0.1, |y-2|<= 0.1
To find the linearization of the function f(x, y) at point P_0(2, 2), we need to find its linear approximation by using its tangent plane at P_0.
The partial derivatives of f(x, y) with respect to x and y are:
∂f/∂x = x + 3
∂f/∂y = y - 3
Evaluating these partial derivatives at P_0(2, 2):
∂f/∂x = 2 + 3 = 5
∂f/∂y = 2 - 3 = -1
The equation of the tangent plane at P_0 is given by:
L(x, y) = f(P_0) + ∂f/∂x (x - x_0) + ∂f/∂y (y - y_0)
Plugging in the values:
L(x, y) = f(2, 2) + 5(x - 2) - 1(y - 2)
To find an upper bound for the magnitude of the error |E| over the rectangle R = {|x - 2| ≤ 0.1, |y - 2| ≤ 0.1}, we need to find the maximum values of the second-order partial derivatives of f(x, y) within this rectangle.
The second-order partial derivatives are:
∂²f/∂x² = 1
∂²f/∂y² = 1
∂²f/∂x∂y = 1
Since these second-order partial derivatives are constant, we can find their maximum values over the rectangle R by evaluating them at any point within the rectangle. Let's evaluate them at the center of the rectangle, which is (2, 2).
∂²f/∂x² = 1
∂²f/∂y² = 1
∂²f/∂x∂y = 1
Thus, the maximum values of the second-order partial derivatives within the rectangle R are 1, 1, and 1 respectively.
The error E(x, y) in the linear approximation f(x, y) = L(x, y) can be bounded by using the formula:
|E(x, y)| ≤ C (|x - x_0|^2 + |y - y_0|^2)
In this case, C is the maximum value of the second-order partial derivatives within the rectangle R, which is 1.
Rearranging the formula:
|E(x, y)| ≤ 1 (|x - 2|^2 + |y - 2|^2)
Since the rectangle R is defined as {|x - 2| ≤ 0.1, |y - 2| ≤ 0.1}, we can substitute these values into the above formula to get an upper bound for the magnitude of the error |E|:
|E(x, y)| ≤ 1 (0.1^2 + 0.1^2)
|E(x, y)| ≤ 1 (0.02)
|E(x, y)| ≤ 0.02
Therefore, an upper bound for the magnitude of the error |E| in the approximation f(x, y) = L(x, y) over the rectangle R is 0.02.