check my answer please. by using the local linearization, i need to find the fun

Question

check my answer please.

by using the local linearization, i need to find the function that approximates the f(x,y) = ln(x^2+1)+y^2 around (0,3).

What i did:

1) i know that the general local linearization formula is L(x,y) = f (a,b)+fx (a,b)(x-a)+fy(a,b)(y-b)

2) f(0,3) = ln (02+1) + 9 = 9

3) fx (x,y) = [1/(x2+1)] (2x) ==> fx (0,3) = [1/(02+1)] [2(0)] = 1

4) fy (x,y) = 2y ==> fy (0,3) = 2(3) = 6

5) 9+1 (x-1) + 6(y-3)

=9+x-1+6y-18

=x+6y-10

please check if I did it right and tell me what i did wrong. Thanks!

Explanation / Answer

you did a mistake in step 3.

3) fx (x,y) = [1/(x2+1)] (2x) ==> fx (0,3) = [1/(02+1)] [2(0)] = 0 ( x is in the numerator , so when you substitute x=0 ,it gives 0. because anything multiplied by 0 is 0)

which gives

5) 9+0 (x-1) + 6(y-3)

ans is 6y-9

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