Lösungen/Solutions/E1S2AnUe/Ex6

L=Lektion/Leçon-- Sprechstunde/heures de consultation
W=Work/Arbeit/Travail
A  =Anhang/  Annexe / Appendice

Hinweis: Zur Konvergenzentscheidung werden hier auch noch gleich die Berechnungen druchgeführt.

Ueb/Ex 1      W

f1[x_, y_] := Tan[Sin[x + y]] - Sin[Tan[x*y]]

Plot3D[
        f1[x, y], {x, -Sqrt[Pi/2], Sqrt[Pi/2]}, {y, -Sqrt[Pi/2], 
          Sqrt[Pi/2]}, PlotPoints -> 30];

Plot3D[f1[x, y], {x, -Pi/2, Pi/2}, {y, -Pi/2, Pi/2}, 
        PlotPoints -> 40];

Ueb/Ex 2      W

a

Plot3D[Tan[Sin[x + y]], {x, -Pi/2, Pi/2}, {y, -Pi/2, Pi/2}, 
        PlotPoints -> 25];

Term ''Tan[Sin[x+y]]'' problemlos  (-1<=Sin...≤+1)

Plot3D[Sin[Tan[x y]], {x, -Pi/2, Pi/2}, {y, -Pi/2, Pi/2}, 
        PlotPoints -> 40];

Term ''Sin[Tan[x y]]]'' problematisch für  x*y = Pi/2 +n Pi  (Tan[x y]=+/- Infinity)
-> y = (Pi/2 +n Pi)/ x -> unstetig

Plot[y = (Pi/2)/x, {x, -Pi/2, Pi/2}];

b

Remove[x, y]

f2[x_, y_] := (x*y - x)/(x^2 + y^2)

f1[x,y] stetig für  x*y ungleich Pi/2 +n Pi

Plot3D[f2[x, y], {x, - .5, 0.5}, {y, -0.5, 0.5}, 
        PlotPoints -> 30];

Limit[f2[x, 0], x -> 0, Direction -> -1]

Limit[f2[x, 0], x -> 0, Direction -> +1]

Limit[f2[0, y], y -> 0, Direction -> -1]

Limit[f2[0, y], y -> 0, Direction -> +1]

f1[x,y] unstetig für  x = y = 0

Ueb/Ex 3      W

f3[x_, y_] := E^(x + y)  -  Sin[x^2]  +  Log[x*y]

a   n=1

D[f3[x, y], x]

D[f3[x, y], {x, 1}]

D[f3[x, y], y]

b   n=2

D[f3[x, y], {x, 2}]

D[f3[x, y], x, x]

D[f3[x, y], x, y]

D[f3[x, y], y, x]

D[f3[x, y], y, y]

c   n=3

D[f3[x, y], x, x, x]

D[f3[x, y], x, x, y]

D[f3[x, y], x, y, y]

D[f3[x, y], y, y, y]

Ueb/Ex 3      W

f4[x_, y_] := E^(-x^2 - y^2)

p1 = 
        Plot3D[f4[x, y], {x, -3, 3}, {y, -3, 3}, 
          PlotPoints -> 25, PlotRange -> {0, 1}];

Remove[Φ]

Φ[x_, y_] = 
      f4[1, 2] +  
        Evaluate[
            D[f4[x, 2], x] /. x -> 1] * (x - 
              1)
 +  
        Evaluate[D[f4[1, y], y] /. y -> 2] * (y - 2)

p2 = 
        Plot3D[Φ[x, y], {x, -3, 3}, {y, -3, 
            3}];

Show[p1, p2];

z0 = Φ[0, 0]

N[%]