Question

14.7. Taylor's theorem and Max/Min values. A statement of Taylor's theorem for functions of two variables and an example are in Part I (section 7) of my online notes if you didn't get it in class. H. Compute the Hessian of the function f(x,y) = y?e evaluated at the point (0,2), ans (lo 8 I. Use the formula involving the gradient and Hessian for z = Q(x, y) to determine the second order Tavlor polynomial for the functions. You should check your answers using the standard Taylor series (r, y) =(0,0) a) g(x, y) = 5+ x2 +e2+y, centered at (,y) = (0,0) b) f(x, y) = 2 y +cos(x 2y), centered at f(x, y)V 2y+1 J. Let z = a) Compute the local linearization (z = L(x, y)) of f at (0,0).(Answ: L(x,y) =1+}+y) b) Quadratic approximation (z = Q(x,y)) to f centered at (0,0) (Answ: Q(x, y) = 1+ y ry)

Answer 1

H) f(xI)2e 2M (012) 2fe2x 4ye2 2 E 4e2 Re t«y»] = f012) i tyy = 2;txy= dyn= H2)

I Lo,0) a) - 2 4 e 2x + e22+y e2xty 2 e2+ g(0,0)2 gyl0,0)= at = 2. Co,o)6 L2 =o, 0) + 9z(0,0) + 92z (0,0)x2 A (ot07 AB 2 + 2 ) 2! 2. y (o10) (y ) 2. 6 2x t 9 + 6x^ 2y 2x 72 322+-날 LTe Y1 n/2 6+ 2x-+ 3x2 +_y +2xy b) i) t -sin 2); f«u= - cos (x+2y) fy=1-2sin (+2y); -fyy= fyty-2 cos (x+2y) 2 Cosr+ 27) i (ey) lo,o) } 4- cos(x+2y 3 t O fxx- 1 4 ty= fy = -2 . (010) at

L2 ) o,0) + x(0,0) x +ty 0,0) [ +oalos0) x2 + 2txy (o,o)y+fyyl0,) y7 2 1 -427 3 t 4xy -2 21 2 3 + y 2xy-2y2 +21- L2cos(2y= 2+ tcose4y)= 2 y+-2 2xy -2y L2 2 2. = 3 +y 3) x,y) xt2y+/ 2 x+2y+ fy (iy) +2yt! y(o0) tx (o0) ;tex (o,) b) f (y 4 n t2yt)3/2 Ay(0,0)= -/ t 2 +13/2 0,0)-- 2(+2 +13/2 21 xy - y 4 2 2