Lösungen/Solutions/E1S2AnUe/Ex10

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

Remove["Global`"]

Aufgabe 1

f[x_, y_] := Cos[x y] - x;  
    f1[q_] := Normal[Series[Cos[q], {q, 0, 17}]]
 g[x_, y_] := E^y - x + y - 3; 
    g1[y_] := Normal[Series[E^y, {y, 0, 30}]]
{f[x, y], g[x, y],  f1[q], g1[y]}

Evaluate[f1[q] /. q -> (x y)] == x

Evaluate[g1[y]] == x - y + 3

<< Graphics`ImplicitPlot`

p1 = 
        ImplicitPlot[
          Evaluate[f1[q] /. q -> (x y)] == x, {x, -10, 
            10}, PlotRange -> {{-8, 5}, {-10, 10}}];

Plot3D[f [x, y], {x, -5, 5}, {y, -5, 5}];

Plot3D[f [x, y], {x, 0.5, 5}, {y, 0.5, 5}];

p2 = 
        ImplicitPlot[
          Evaluate[g1[y]] == x - y + 3, {x, -10, 10}];

Plot3D[g [x, y], {x, -15, 15}, {y, -15, 15}];

Show[p1, p2];

FindRoot[{f[x, y] == 0, g[x, y] == 0}, {x, 1}, {y, 
        1.5}]

FindRoot[{f[x, y] == 0, g[x, y] == 0}, {x, 4}, {y, 
        2}]

FindRoot[{f[x, y] == 0, g[x, y] == 0}, {x, 5}, {y, 
        3}]

Anderer Weg: Reduktion auf eine Variable

x1[y_] := E^y + y - 3

f1[y_] := f[x1[y], y]; f1[y]

Plot[f1[y], {y, -10, 10}];

r1 = FindRoot[f1[y], {y, 1}]

x1[y] /. r1[[1]]