1 Grafy funkcí
> f1:=(x,y)->2*x-3*y+1;
f1 := (x, y)  2 x - 3 y + 1
> plot3d(f1(x,y), x=-1..1, y=-1..1, axes=boxed, grid=[5,5]);
-1
-0,5 y
0
0,5
1
-1
-0,5
0
x0,5
1
> f2:=(x,y)->sin(x);
f2 := (x, y)  sin (x)
> plot3d(f2(x,y), x=0..2*Pi, y=-1..1, axes=boxed, grid=[25,10]);
-1
-0,5 y
0
0,5
1
0
1
2
3
4 x
5
6
> f3:=(x,y)->sqrt(y);
f3 := (x, y) 

y
> plot3d(f3(x,y), x=-1..1, y=0..1, axes=boxed, grid=[10,25]);
0
0,2
y0,4
0,6
0,8
1
-1
-0,5
0
x0,5
1
> f4:=(x,y)->sin(x)+cos(y);
f4 := (x, y)  sin (x) + cos (y)
> plot3d(f4(x,y), x=0..2*Pi, y=0..2*Pi, axes=boxed);
0
1
2 y
3
4
5
6
0
1
2
3
x4
5
6
> f5:=(x,y)->sin(x)*cos(y);
f5 := (x, y)  sin (x) cos (y)
> plot3d(f5(x,y), x=0..2*Pi, y=0..2*Pi, axes=boxed);
0
1
2 y
3
4
5
6
0
1
2
3
x4
5
6
> f6:=(x,y)->exp(-(x^2+y^2));
f6 := (x, y)  e-x2
-y2
> plot3d(f6(x,y), x=-1..1, y=-sqrt(1-x^2)..sqrt(1-x^2), axes=boxed);
-1
-0,5 y
0
0,5
1
-1
-0,5
0
x0,5
1
> f7:=(x,y)->sqrt(x^2+y^2)*sin(x^2+y^2);
f7 := (x, y)  x2 + y2 sin x2
+ y2
> plot3d(f7(x,y), x=-Pi/4..Pi/4, y=-sqrt(Pi^2-x^2)..sqrt(Pi^2-x^2), axes=boxed,
grid=[60,60], style=patchcontour, scaling=constrained);
-3
y
-2
-2
-1
-1
0
0
1
2
1
3
-0,8
2
-0,400,40,8
x
> f8:=(x,y)->2-sqrt(x^2+y^2);
f8 := (x, y)  2 - x2 + y2
> plot3d(f8(x,y), x=-2..2, y=-sqrt(4-x^2)..sqrt(4-x^2), axes=boxed, style=patchcontour,
grid=[40,40]);
-2
-1
y
0
1
2
-2
-1
0
x
1
> f9:=(x,y)->(x^2)/4+(y^2)/9;
f9 := (x, y)  1/4 x2
+ 1/9 y2
> plot3d(f9(x,y), x=-2..2, y=-3..3, axes=boxed);
-3
-2
-1 y
0
1
2
3
-2
-1
0
x
1
2
> f10:=(x,y)->x^2-x*y+y^2;
f10 := (x, y)  x2
- xy + y2
> plot3d(f10(x,y), x=-1..1, y=-1..1, axes=boxed);
-1
-0,5 y
0
0,5
1
-1
-0,5
0
x0,5
1
> f11:=(x,y)->x^2/4-y^2/9;
f11 := (x, y)  1/4 x2
- 1/9 y2
> plot3d(f11(x,y), x=-2..2, y=-3..3, axes=boxed);
-3
-2
-1 y
0
1
2
3
-2
-1
0
x
1
2
> f12:=(x,y)->x^2+4*x*y+y^2;
f12 := (x, y)  x2
+ 4 xy + y2
> plot3d(f12(x,y), x=-2..2, y=-2..2, axes=boxed);
-2
-1
y
0
1
2
-2
-1
0
x
1
2
> f13:=(x,y)->x^2-2*x*y+y^2;
f13 := (x, y)  x2
- 2 xy + y2
> plot3d(f13(x,y), x=-1..1, y=-1..1, axes=boxed);
-1
-0,5 y
0
0,5
1
-1
-0,5
0
x0,5
1
> f14:=(x,y)->1-x*y;
f14 := (x, y)  1 - xy
> plot3d(f14(x,y), x=-1..1, y=-1..1, axes=boxed);
-1
-0,5 y
0
0,5
1
-1
-0,5
0
x0,5
1
2 Limity funkcí
> restart;
> with(plots):setoptions3d(axes=boxed);
Warning, the name changecoords has been redefined
> f1:=(x,y)->x-x^3-x*y^2+x^3*y^2;
f1 := (x, y)  x - x3
- xy2
+ x3
y2
> plot3d(f1(x,y), x=-1.4..1.4, y=-1.4..1.4, view=-1..1);
-1
-0,5 y
0
0,5
1
-1
-0,5
0
x0,5
1
> f2:=(x,y)->(x^2*y)/(x^2+y^2);
f2 := (x, y)  x2
y
x2+y2
> plot3d(f2(x,y), x=-3..3, y=-3..3, orientation=[-57,38], axes=framed, grid=[26,26]);
3
-1,5
2
-1
1
0
-0,5
-1
y
0
-2
-3
-3
0,5
-2
-1
1
0
1x
2
1,53
> f3:=(x,y)->((x^2-y^2)/(x^2+y^2))^2;
f3 := (x, y) 
(x2
-y2
)
2
(x2+y2)2
> plot3d(f3(x,y), x=-3..3, y=-3..3, grid=[51,39], axes=framed);
-3
-2
-1 y
0
1
2
3
-3
-2
-1
0
x1
2
3
> f4:=(x,y)->(x*y)/(x^2+y^2);
f4 := (x, y)  xy
x2+y2
> f4 := proc (x, y) options operator, arrow; x*y/(x^2+y^2) end proc;
f4 := (x, y)  xy
x2+y2
> z:=subs(x=r*cos(phi), y=r*sin(phi), f4(x,y)):
> plot3d([r*cos(phi), r*sin(phi), simplify(z)], r=0..1, phi=-Pi..Pi, grid=[15,45],
orientation=[15,45]);
-1
-0,5
0
0,5
-1
-0,4
-0,5
0
0,5 1
1
-0,2
0
0,2
0,4
> f5:=proc(x,y) if x=0 and y=0 then 0 else x^2*y/(x^4+y^2) fi end;
f5 := proc(xy) if x = 0 and y = 0 then 0 else x2
 y  (x4
y2
)-1
end if; end proc;
> plot3d(f5, -2..2, -2..2, grid=[100,100], style=patchcontour, orientation=[-46,35],
contours=12);
2
-0,4
1
-0,2
0
-1
0
-2
-2
-1
0,2
0
0,4
1
2
> f6:=(x,y)->1/(x^2+y^2);
f6 := (x, y)  x2
+ y2 -1
> plot3d(f6(x,y), x=-1..1, y=-1..1, view=0..6);
-1
-0,5 y
0
0,5
1
-1
-0,5
0
x0,5
1
> f7:=(x,y)->2/(x^2+y^2-9);
f7 := (x, y)  2 x2
+ y2
- 9
-1
> s1:=plot3d([u*cos(v), u*sin(v), subs(x=u*cos(v), y=u*sin(v), f7(x,y))], v=0..2*Pi,
u=0..2.999):
> s2:=plot3d([u*cos(v), u*sin(v), subs(x=u*cos(v), y=u*sin(v), f7(x,y))], v=0..2*Pi,
u=3.001..6):
> display3d(s1,s2, view=-8..8);
-6
-4
-2
0
2
4
6
-6
-4
-2
0
2
4
6
> f10:=(x,y)->2/(x^2-9);
f10 := (x, y)  2 x2
- 9
-1
> d1:=plot3d(f10(x,y), x=2..2.99, y=0..1):
> d2:=plot3d(f10(x,y), x=3.01..4, y=0..1):
> display3d(d1,d2);
0
0,2
y0,4
0,6
0,8
1
2
2,5
3
x
3,5
4
> f8:=(x,y)->(exp(x*y)-1)/x;
f8 := (x, y)  exy
-1
x
> plot3d(f8,-.5..0.5, 3..5, grid=[8,8]);
3
3,5
4
4,5
5
-0,4
-0,2
0
0,2
0,4
> f9:=(x,y)->sin(x+y)/(Pi-x-y);
f9 := (x, y)  sin(x+y)
-x-y
> plot3d(f9(x,y), x=0..Pi, y=0..Pi, grid=[30,60], style=patchcontour);
0
0,5
1 y
1,5
2
2,5
3
0
0,5
1
1,5
2 x
2,5
3