# Tema 11
11-3  Aproximació de funcions diferenciables.
          Fórmula de Taylor

T[1](x,y) = taylor(f(x,y), [x,y], [x0,y0] , [1,1]);

T[2](x,y) = taylor(f(x,y), [x,y], [x0,y0] , [2,2]);

Exemple 11.3.1

kill(all)$

f(x,y):=exp(-x+2*y);

T[1](x,y) = taylor(f(x,y), [x,y], [0,0] , [1,1]);

T[2](x,y) = taylor(f(x,y), [x,y], [0,0] , [2,2]);

ratsimp(%);

expand(%);

T[3](x,y) = taylor(f(x,y), [x,y], [0,0], [3,3]);

ratsimp(%);

expand(%);

Exemple 11.3.2

kill(all)$

f(x,y):=exp(-x+2*y);

T[1](x,y) = taylor(f(x,y), [x,y], [1,2] ,[1,1]);

ratsimp(%);

T[2](x,y) = taylor(f(x,y), [x,y], [1,2] ,[2,2] );

ratsimp(%);

expand(%);

T[3](x,y) = taylor(f(x,y), [x,y], [1,2] ,[3,3] );

ratsimp(%);

expand(%);

Exemple 11.3.3

kill(all)$

g(x,y,z):=exp(x+y+2*z);

T[1](x,y,z) = taylor(g(x,y,z), [x,y,z], [0,0,0] , [1,1,1]);

T[2](x,y,z) = taylor(g(x,y,z), [x,y,z], [0,0,0] , [2,2,2]);

ratsimp(%);

expand(%);

T[3](x,y,z) = taylor(g(x,y,z), [x,y,z], [0,0,0] , [3,3,3]);

ratsimp(%);

expand(%);