PRIKLAD G > restart; > > f:=(x+3*y)/(y-3*x); dx:=0.5; dy:=-0.5; x0:=2; y0:=4; x + 3 y f := ------- y - 3 x dx := 0.5 dy := -0.5 x0 := 2 y0 := 4 > df1:=eval(diff(f,x),{x=x0,y=y0}); df1 := 10 > diff(f,x); 1 3 (x + 3 y) ------- + ----------- y - 3 x 2 (y - 3 x) > df2:=eval(diff(f,y),{x=x0,y=y0}); df2 := -5 > diff(f,y); 3 x + 3 y ------- - ---------- y - 3 x 2 (y - 3 x) > df3:=eval(diff(f,x,x),{x=x0,y=y0}); df3 := -30 > diff(f,x,x); 6 18 (x + 3 y) ---------- + ------------ 2 3 (y - 3 x) (y - 3 x) > df4:=eval(diff(f,x,y),{x=x0,y=y0}); df4 := 25/2 > diff(f,x,y); 8 6 (x + 3 y) ---------- - ----------- 2 3 (y - 3 x) (y - 3 x) > df5:=eval(diff(f,y,y),{x=x0,y=y0}); df5 := -5 > diff(f,y,y); 6 2 (x + 3 y) - ---------- + ----------- 2 3 (y - 3 x) (y - 3 x) > vysl2:=eval(f,{x=x0,y=y0})+df1*(dx)+df2*(dy)+(1/2)*(df3*(dx)^2+2*df4*(dx)*(dy)+df5*(dy)^2); vysl2 := -7.000000000 > vy:=eval(f,{x=x0+dx,y=y0+dy}); vy := -3.250000000 > vysl1:=eval(f,{x=x0,y=y0})+df1*(dx)+df2*(dy); vysl1 := 0.5 > vysl:=[vy,vysl1,vysl2]; vysl := [-3.250000000, 0.5, -7.000000000] PRIKLA F > restart; > > f:=x*sin(x+y); dx:=pi/180; dy:=2*pi/180; x0:=pi/6; y0:=pi/3; f := x sin(x + y) pi dx := --- 180 pi dy := ---- 90 pi x0 := ---- 6 pi y0 := ---- 3 > df1:=real(eval(diff(f,x),{x=x0,y=y0})); pi pi df1 := sin(----) + 1/6 pi cos(----) 2 2 > diff(f,x); sin(x + y) + x cos(x + y) > df2:=eval(diff(f,y),{x=x0,y=y0}); pi df2 := 1/6 pi cos(----) 2 > diff(f,y); x cos(x + y) > df3:=eval(diff(f,x,x),{x=x0,y=y0}); pi pi df3 := 2 cos(----) - 1/6 pi sin(----) 2 2 > diff(f,x,x); 2 cos(x + y) - x sin(x + y) > df4:=eval(diff(f,x,y),{x=x0,y=y0}); pi pi df4 := cos(----) - 1/6 pi sin(----) 2 2 > diff(f,x,y); cos(x + y) - x sin(x + y) > df5:=eval(diff(f,y,y),{x=x0,y=y0}); pi df5 := -1/6 pi sin(----) 2 > diff(f,y,y); -x sin(x + y) > vysl2:=eval(f,{x=x0,y=y0})+df1*(dx)+df2*(dy)+(1/2)*(df3*(dx)^2+2*df4*(dx)*(dy)+df5*(dy)^2); pi / pi pi \ vysl2 := 1/6 pi sin(----) + 1/180 |sin(----) + 1/6 pi cos(----)| pi 2 \ 2 2 / 2 pi + 1/540 pi cos(----) 2 / pi pi \ 2 + 1/64800 |2 cos(----) - 1/6 pi sin(----)| pi \ 2 2 / / pi pi \ 2 + 1/16200 |cos(----) - 1/6 pi sin(----)| pi \ 2 2 / 3 pi - 1/97200 pi sin(----) 2 > vy:=eval(f,{x=x0+dx,y=y0+dy}); 31 31 pi vy := --- pi sin(-----) 180 60 > vysl1:=eval(f,{x=x0,y=y0})+df1*(dx)+df2*(dy); pi / pi pi \ vysl1 := 1/6 pi sin(----) + 1/180 |sin(----) + 1/6 pi cos(----)| pi 2 \ 2 2 / 2 pi + 1/540 pi cos(----) 2 > vysl:=[vy,vysl1,vysl2]; 31 31 pi pi vysl := [--- pi sin(-----), 1/6 pi sin(----) 180 60 2 / pi pi \ + 1/180 |sin(----) + 1/6 pi cos(----)| pi \ 2 2 / 2 pi pi + 1/540 pi cos(----), 1/6 pi sin(----) 2 2 / pi pi \ + 1/180 |sin(----) + 1/6 pi cos(----)| pi \ 2 2 / 2 pi + 1/540 pi cos(----) 2 / pi pi \ 2 + 1/64800 |2 cos(----) - 1/6 pi sin(----)| pi \ 2 2 / / pi pi \ 2 + 1/16200 |cos(----) - 1/6 pi sin(----)| pi \ 2 2 / 3 pi - 1/97200 pi sin(----)] 2 > PRIKLAD E restart; > f:=(tan(y))*exp(x); dx:=-0.1; dy:=0.01; x0:=0; y0:=0; f := exp(x)tan(y) dx := -0.1 dy := 0.01 x0 := 0 y0 := 0 > df1:=eval(diff(f,x),{x=x0,y=y0}); df1 := 0 > diff(f,x); tan(y) exp(x) > df2:=eval(diff(f,y),{x=x0,y=y0}); df2 := 1 > diff(f,y); 2 (1 + tan(y) ) exp(x) > df3:=eval(diff(f,x,x),{x=x0,y=y0}); df3 := 0 > diff(f,x,x); tan(y) exp(x) > df4:=eval(diff(f,x,y),{x=x0,y=y0}); df4 := 1 > diff(f,x,y); 2 (1 + tan(y) ) exp(x) > df5:=eval(diff(f,y,y),{x=x0,y=y0}); df5 := 0 > diff(f,y,y); 2 2 tan(y) (1 + tan(y) ) exp(x) > vysl2:=eval(f,{x=x0,y=y0})+df1*(dx)+df2*(dy)+(1/2)*(df3*(dx)^2+2*df4*(dx)*(dy)+df5*(dy)^2); vysl2 := 0.009 > vy:=eval(f,{x=x0+dx,y=y0+dy}); vy := 0.009048675808 > vysl1:=eval(f,{x=x0,y=y0})+df1*(dx)+df2*(dy); vysl1 := 0.01 > vysl:=[vy,vysl1,vysl2]; vysl := [0.009048675808, 0.01, 0.009] > > PRIKLAD B > restart; > > f:=sqrt(x^3+y^3); dx:=0.03; dy:=-0.02; x0:=2; y0:=1; 3 3 1/2 f := (x + y ) dx := 0.03 dy := -0.02 x0 := 2 y0 := 1 > df1:=eval(diff(f,x),{x=x0,y=y0}); 1/2 2 9 df1 := ------ 3 > diff(f,x); 2 3 x -------------- 3 3 1/2 2 (x + y ) > df2:=eval(diff(f,y),{x=x0,y=y0}); 1/2 9 df2 := ---- 6 > diff(f,y); 2 3 y -------------- 3 3 1/2 2 (x + y ) > df3:=eval(diff(f,x,x),{x=x0,y=y0}); 1/2 2 9 df3 := ------ 9 > diff(f,x,x); 4 9 x 3 x - -------------- + ------------ 3 3 3/2 3 3 1/2 4 (x + y ) (x + y ) > df4:=eval(diff(f,x,y),{x=x0,y=y0}); 1/2 9 df4 := - ---- 9 > diff(f,x,y); 2 2 9 x y - -------------- 3 3 3/2 4 (x + y ) > df5:=eval(diff(f,y,y),{x=x0,y=y0}); 1/2 11 9 df5 := ------- 36 > diff(f,y,y); 4 9 y 3 y - -------------- + ------------ 3 3 3/2 3 3 1/2 4 (x + y ) (x + y ) > vysl2:=eval(f,{x=x0,y=y0})+df1*(dx)+df2*(dy)+(1/2)*(df3*(dx)^2+2*df4*(dx)*(dy)+df5*(dy)^2); 1/2 vysl2 := 1.016894445 9 > vy:=eval(f,{x=x0+dx,y=y0+dy}); vy := 3.050675171 > vysl1:=eval(f,{x=x0,y=y0})+df1*(dx)+df2*(dy); 1/2 vysl1 := 1.016666667 9 > vysl:=[vy,vysl1,vysl2]; 1/2 1/2 vysl := [3.050675171, 1.016666667 9 , 1.016894445 9 ] > > PRIKLAD III restart; > f:=(log(x))*sin(y); dx:=0.1; dy:=0.1; x0:=1; y0:=0; f := ln(x) sin(y) dx := 0.1 dy := 0.1 x0 := 1 y0 := 0 > df1:=eval(diff(f,x),{x=x0,y=y0}); df1 := 0 > diff(f,x); sin(y) ------ x > df2:=eval(diff(f,y),{x=x0,y=y0}); df2 := 0 > diff(f,y); ln(x) cos(y) > df3:=eval(diff(f,x,x),{x=x0,y=y0}); df3 := 0 > diff(f,x,x); sin(y) - ------ 2 x > df4:=eval(diff(f,x,y),{x=x0,y=y0}); df4 := 1 > diff(f,x,y); cos(y) ------ x > df5:=eval(diff(f,y,y),{x=x0,y=y0}); df5 := 0 > diff(f,y,y); -ln(x) sin(y) > vysl2:=eval(f,{x=x0,y=y0})+df1*(dx)+df2*(dy)+(1/2)*(df3*(dx)^2+2*df4*(dx)*(dy)+df5*(dy)^2); vysl2 := 0.01 > vy:=eval(f,{x=x0+dx,y=y0+dy}); vy := 0.009515140891 > vysl1:=eval(f,{x=x0,y=y0})+df1*(dx)+df2*(dy); vysl1 := 0. > vysl:=[vy,vysl1,vysl2]; vysl := [0.009515140891, 0., 0.01] > >