Lösungen

1

Lösung: Siehe Skript Analysis von R. Wirz 
Ausgabe d/f Seite 263
Ausgabe d   Seite 234

2

Remove["Global`*"]

f[x_,y_]:=1-x^2-2(y+1)^2;
f[x,y]

f[{x_,y_}]:=f[x,y];
p={0,2};

a

grad[f_,x_,y_]:={D[f,x],D[f,y]};
grad[f[x,y],x,y]

Solve[Evaluate[grad[f[x,y],x,y]==0],{x,y}]

b

e30={Cos[30 Degree],Sin[30 Degree]}

grad[f[x,y],x,y].e30

grad[f[x,y],x,y].e30 /. {x->0,y->2}

c

ArcTan[-6] // N

ArcTan[-6]/Degree // N

d

Norm[grad[f[x,y],x,y]]

Norm[grad[f[x,y],x,y]] /. {x->0,y->2}

ArcTan[12] // N

ArcTan[12]/Degree // N

e

gradP = grad[f[x,y],x,y] /. {x->0,y->2}

Φ[x_, y_] := 
        f[p] + gradP[[1]] (x - p[[1]]) + 
          gradP[[2]] (y - p[[2]]);
Φ[x, y]

Solve[{Φ[x,y]==0,y==0},{x}]

%// N

f

pl1=Plot3D[f[x,y],{x,-6,6},{y,-6,6}];

pl1=Plot3D[f[x,y],{x,-6,6},{y,-6,6},Shading -> False];

v[t_]:= {Expand[t(t-1)(t+2)-2],t};  v[t]

pl2=ParametricPlot3D[{v[t][[1]],v[t][[2]],0},{t,-3,2}];

pl3=ParametricPlot3D[{v[t][[1]],v[t][[2]],f[v[t]]},{t,-2.7,1.7}];

Show[pl1,pl3,pl2];

Show[pl1,pl3,pl2,ViewPoint->{0.424, -0.630, 1.583}];

Show[pl1,pl3,pl2,ViewPoint->{0.210, 2.200, 0.760}];

g

v[t_]:= {Expand[t(t-1)(t+2)-2],t};  v[t]

g[x_,y_]:= Evaluate[v[t][[1]] - x  /.t->y];
g[x,y] ==0

grad[f[x,y],x,y]== λ grad[g[x,y],x,y]

NSolve[{grad[f[x,y],x,y]== λ grad[g[x,y],x,y], g[x,y]==0}, {λ,x,y}]

f[v[t]]

solv1 = Solve[Evaluate[D[f[v[t]],t]==0],{t}] //N // Flatten

v[t] /. solv1[[1]]

f[v[t]] /. solv1[[1]]

v[t] /. solv1[[4]]

f[v[t]] /. solv1[[4]]

v[t] /. solv1[[5]]

f[v[t]] /. solv1[[5]]

==> Punkt {0,-1,1}

h

D[f[v[t]],t]

D[f[v[t]],t] // Expand

3

Remove["Global`*"]

f[x_,y_]:=10/(3 (x-1)^2+5(y+1)^2+10)+ 14/(2 (x+1)^2+5(y-2)^2+10)

a

pl1=Plot3D[f[x,y],{x,-6,6},{y,-6,6}];

pl1=Plot3D[f[x,y],{x,-6,6},{y,-6,6},Shading -> False];

ContourPlot[f[x,y],{x,-6,6},{y,-6,6}];

b

grad[f_,x_,y_]:={D[f,x],D[f,y]};
grad[f[x,y],x,y]

grad[f[x,y],x,y] // Simplify

<<Graphics`PlotField`

PlotGradientField[f[x,y],
             {x, -3, 3}, {y, -3, 3}];

NSolve[Evaluate[grad[f[x,y],x,y]==0],{x,y}]

c

e30={Cos[30 Degree],Sin[30 Degree]}

grad[f[x,y],x,y].e30

%//N

grad[f[x,y],x,y].e30 /. {x->0,y->2}

%//N

d

i

f[{x_,y_}]:=f[x,y];
p={0,2};

gradP = grad[f[x,y],x,y] /. {x->0,y->2}

(gradP = grad[f[x,y],x,y] /. {x->0,y->2} ) //N

Φ[x_, y_] := 
        f[p] + gradP[[1]] (x - p[[1]]) + 
          gradP[[2]] (y - p[[2]]);
Φ[x, y]

%//N //Simplify

Solve[{Φ[x,y]==z,x==0,y==0},{z}]

%// N

ii

Norm[grad[f[x,y],x,y]]

Norm[grad[f[x,y],x,y]] /. {x->0,y->2}

%//N

ArcTan[%] // N

ArcTan[%%]/Degree // N

iii

g[x_,y_]:= y-x^2

solv2= NSolve[{grad[f[x,y],x,y]== λ grad[g[x,y],x,y], g[x,y]==0}, {λ,x,y}]

Table[{k,solv2[[k]]},{k,1,Length[solv2]}]//MatrixForm

{solv2[[3]],solv2[[8]],solv2[[11]]}

f[x,y]/.solv2[[3]]

f[x,y]/.solv2[[8]]

f[x,y]/.solv2[[11]]

Punkt { -1.37488, 1.89028, 1.49939}

4

a

Series[E^(-x^2),{x,0,8}]

N[%]

b

Evaluate[Normal[Series[E^(-x^2),{x,0,8}]]]

N[%]

Integrate[Evaluate[Normal[Series[E^(-x^2),{x,0,8}]]],{x,-2,2}]

%//N

Integrate[Evaluate[Normal[Series[E^(-x^2),{x,0,100}]]],{x,-2,2}]//N

Integrate[E^(-x^2),{x,-2,2}]//N

c

Evaluate[Normal[Series[Cos[x^2]+E^(x^2),{x,0,8}]]]

N[%]

Integrate[Evaluate[Normal[Series[E^(-x^2),{x,0,8}]]],{x,-2,2}]

%//N

Integrate[E^(-x^2),{x,-2,2}]//N

Integrate[Evaluate[Normal[Series[E^(-x^2),{x,0,100}]]],{x,-2,2}]//N

d

Evaluate[Normal[Series[Sqrt[x],{x,1,6}]]]

N[%]

e

r=1

d

Evaluate[Normal[Series[Sqrt[x],{x,1,6}]]]

N[%]

e

r=1

f

Evaluate[Normal[Series[Log[x]-Sin[x],{x,1,6}]]]

N[%]

g

1-1/(1-1/2)+1/(1-1/3)

N[%]

h

Sum[1/k,{k,1,n}]

Limit[Sum[1/k,{k,1,n}],{n->Infinity}]

Limit[Sum[1/k,{k,1,n}]-Log[n],{n->Infinity}]

%//N