Question

Find The Linearization L(X,Y) Of The Function At Each Point. F(X,Y)=X2+Y2+1

A. (0,4)
B. (4,1)
A. L(X,Y)=
B. L(X,Y)=

Answer 1

The linearizations of the function f(x,y) = x^2 + y^2 + 1 at the points (0,4) and (4,1) are L(x,y) = 8y - 15 and L(x,y) = 8x + 2y - 16, respectively.

Step-by-step explanation:

Finding the Linearization L(x,y) of a Function

To find the linearization of the function f(x,y) = x^2 + y^2 + 1 near a point, we need to calculate the first-order Taylor expansion around that point. We will do this for the two given points: A. (0,4) and B. (4,1).

First, we find the partial derivatives of f with respect to x and y:

• fx = 2x
• fy = 2y

Next, we evaluate these derivatives at each point and use them to compose the linearization:

• A. At the point (0,4), fx(0,4) = 0 and fy(0,4) = 8. The function value is f(0,4) = 1 + 16 = 17. So, L(x,y) = 17 + 0(x - 0) + 8(y - 4) which simplifies to L(x,y) = 17 + 8y - 32 or L(x,y) = 8y - 15.
• B. At the point (4,1), fx(4,1) = 8 and fy(4,1) = 2. The function value is f(4,1) = 16 + 1 + 1 = 18. So, L(x,y) = 18 + 8(x - 4) + 2(y - 1) which simplifies to L(x,y) = 18 + 8x - 32 + 2y - 2 or L(x,y) = 8x + 2y - 16.