Answer to Question #190415 in Calculus for Nikhil Rawat

Question

Find the second Taylor polynomial of

f(x,y) = xy+3y^2-2 at (1,2)

Answer 1

Second Taylor degree polynomial can be written as -

$Q(x,y)$ = $L(x,y)+\dfrac{fxx_{(1},_{2)}}{2}{(x-1)}^{2}+ {fxy_{(1},_{2)}}{(x-1)}{(y-2)}+\dfrac{fyy_{(1},_{2)}}{2}(y-2)^{2}.......A)$

$L(X,Y)=f(1,2)\ +fx(1,2)(x-1)+fy(1,2)(y-2).........B)$

$f(x,y)=$ $xy\ +3y^{2}$ $-2$

$f(1,2)=2$ .......1)

$fx(1,2)=y=2$ ......2)

$fy(1,2)\ =x+6y=13$ .........3)

Putting the value of 1, 2 and 3 in equation B)...

$L(X,Y)=2+2(x-1)\ +13(y-2)=2x\ +13y\ -26$

$fxx(1,2)=0$ ........4)

$fyy(1,2)=6$ ..........5)

$fxy(1,2)=$ $x+7y=15..........6)$

Putting 4, 5 and 6 and equation B in equation A , we get-

$Q(x,y)=2x\ +13y\ -26\ + 15(x-1)(x-2)\ +3(y-2)^{2}$ which is required second taylor polynomial.