11-04 Runge-Kutta method (part 1 of section 5.4 - Taylor formula)
readlib(mtaylor):with(plots): with(student):f:=(t,y)->exp(-(t-2)^2/4-(y-3)^2/4)*cos(2*t+y-7);
plot3d(f(x,y),x=0..5,y=0..5,axes=normal);mtaylor(exp(-(x-2)^2/4-(y-3)^2/4)*cos(2*x+y-7),[x,y],3);
mtaylor(exp(-(x-2)^2/4-(y-3)^2/4)*cos(2*x+y-7),[x=2,y=3],3);plot3d(1-9/4*(x-2)^2-2*(x-2)*(y-3)-3/4*(y-3)^2,x=0..5,y=0..5,axes=normal);A:=plot3d(exp(-(x-2)^2/4-(y-3)^2/4)*cos(2*x+y-7),x=1..3,y=2..4,axes=normal):
B:=plot3d(1-9/4*(x-2)^2-2*(x-2)*(y-3)-3/4*(y-3)^2,x=1..3,y=2..4,axes=normal,transparency=0.5):
display([A,B]);