Question

Find the linearization L(x,y) of the function f(x,y)= e 3x cos (y) at the points (0,0) and RIN The linearization at (0,0) is L(x,y)= (Type an exact answer, using a as needed.) The linearization at 0. is L(x,y)= 0 (Type an exact answer, using a as needed.)

Answer 1

30/ Given multivariable function is f(x,y) = e37c05cy) at the points (0,0) and (0,3) (1) L'x, y) = Pla, b) + [alca, b)]x-a) + Loy Cajbo]Cy-) 37 (1) The linearization at 10,0) = (a,b) From the formula, ce calculate step by step f(x,y) = concosly) f(0,0) = 360) cosco) = (101) f(0,0)=1 [-:e°-1; L050ʻ=17 of 2010) d dx (032 da. [QUV - CoDo v'+ 0.600] = 3e0%cosy +2340) "casy) d. f әх 31 3² cosy o f10,0) (0,0) = 3e310 c05 (0) = 3.(1).(1) Of (0,0) = 3 dx of 10,0) of les oy ca "Cosy) 034/-siny) +csly).00) eersing of JX 12 :360) sin(0) of (0,0) =-e of (0,0) = 0 дх

Νοω, L(x,y)= $1,0 +fea (09.03.Chea) + = 1+3(2-0)+ oly-o) +(10)(y-2) 460,0) = 1737 90) The linearization at :(0,7) f(x,y) = e coscy) PLO, 5) – @360) cos() = (120) [ f(0, 1) =0 df.(0,5) of - 3eBlcoscy? [": from 1] dx ox of дх 10,5) (0,2) = 3e%%c514) = 311) (0) 010,4)=0 Әх 3% of 10,5) of (0,2) = -siny.e ду sy 310) of (0,5) =-8; Sin (3) dy =-(1) (1) 0:10,3)=-1 og Now, from the formula L(x,y) = flor£)+L(013)(x-a3+loof 10145 ]y-b) 0 +00-0) + (-1)(y-2) 110,3)= 1-41,
Hence the linearization of the given function at the given points (0,0) and (0,π/2) are found.