Lösungen

1

a

f[x_,y_]:= Sqrt[x^2+y^2] E^(-Sqrt[x^2+y^2]/2);
Plot3D[f[x,y],{x,-2,2},{y,-2,2}];

Plot3D[f[x,y],{x,-2,2},{y,-2,2},ViewPoint->{3.667, 2.785, -1.120},PlotRange->{0,1}];

Plot3D[f[x,y],{x,-20,20},{y,-20,20},ViewPoint->{3.667, 2.785, -1.120},PlotRange->{0,1}];

h[r_]:= r E^(-r/2);
ParametricPlot3D[{r Cos[t],r Sin[t], h[r]}, {r,0,3}, {t,0,2Pi}, ViewPoint->{3.667, 2.785, -1.120}, AspectRatio->1];

ParametricPlot3D[{r Cos[t],r Sin[t], h[r]}, {r,0,2}, {t,0,2Pi}, AspectRatio->1];

b

D[f[x,y],x]/.x->u

Plot[Evaluate[D[f[x,0],x]/.x->u],{u,-3,3}];

Limit[Evaluate[D[f[x,0],x]/.x->u],u->0, Direction -> 1]

Limit[Evaluate[D[f[x,0],x]/.x->u],u->0, Direction -> -1]

c

Plot[h[r],{r,0,3}];

D[h[r],r]/.r->2

hMax=h[2]

hMax=h[2]//N

d

<<Calculus`VectorAnalysis`

Drop[Grad[f[x,y], Cartesian[x, y, z]],{3}]

Drop[Grad[f[x,y], Cartesian[x, y, z]],{3}] /. {x->1,y->1}

N[%]

e

Oberfl= Integrate[Evaluate[Sqrt[D[f[x,y],x]^2+D[f[x,y],x]^2+1]/.{x->x1,y->y1}],{x1,1,2},{y1,1,2}]

NIntegrate[Evaluate[Sqrt[D[f[x,y],x]^2+D[f[x,y],x]^2+1]/.{x->x1,y->y1}],{x1,1,2},{y1,1,2}]

NIntegrate[Evaluate[Sqrt[D[f[x,y],x]^2+D[f[x,y],x]^2+1]/.{x->x1,y->y1}],{x1,0,2},{y1,1,2}]

2

a

w=3-I; solv=Solve[(z-w)^5==6-9 I,{z}]//ComplexExpand

solv//N

s1=z/.solv//N; Map[Abs,s1]

1.9074+0.182468 i

solv=Solve[Reduce[(z)^5==6-9 I,{z}]/.z->z-w,{z}]

solv=Solve[Reduce[(z)^5==6-9 I,{z}]/.z->z-w,{z}];
s1=z/.solv//N; Map[Abs,s1]

(((-1))^(4/5)) (((6 - 9 i))^(1/5))

(-3)^(4/5)//ComplexExpand

b

w0=w;w1=1/Abs[w] w; w2=Conjugate[w];w3=1/w; w4=Conjugate[1/Abs[w] w];
mW={w0,w1,w2,w3,w4}

N[%]

(w0-w2)/Abs[w0-w2]

N[%]

(w1-w3)/Abs[w1-w3]

N[%]

c

toPoint[w_]:=Point[{Re[w],Im[w]}];
Map[toPoint,mW]

Show[Graphics[Join[{Line[{{-1.1,0},{2,0}}],Line[{{0,-1.1},{0,1.1}}],
Line[{{0,0},{3,-1}}],Line[{{0,0},{3,1}}],
Line[{toPoint[w0][[1]],toPoint[w4][[1]]}],
Line[{toPoint[w1][[1]],toPoint[w3][[1]]}]
,PointSize[0.02],Circle[{0,0},1]}, Map[toPoint,mW]]],AspectRatio->Automatic];

d

Winkel gespielgelt, Radius invertiert ==> r:1 = 1: (1/r)

3

Remove["Global`*"]

a

DSolve[{y''[x]  +  2  y[x]  == Cos[x]}, y[x], x ] // 
      Simplify

N[%]

b

DSolve[{y''[x]-y'[x] + 2 y[x] ==Cos[x]},y[x],x ]//Simplify

N[%]

c

DSolve[{y''[x] - y'[x] + 2  y[x] == Cos[x], 
          y[0] == 0, y'[0] == 0}, y[x], x ] // 
      Simplify

N[%]

solv=DSolve[{y''[x]-y'[x]+2y[x]==Cos[x],y[0]==0,y'[0]==0},y,x ]//Simplify//Flatten

N[%]

y=y/.solv;
y[z]

N[%]

d

pl=Plot[y[z],{z,0,12}];

e

FindRoot[y[x],{x,3}]

y[2.75558]

4

a

s1=Normal[Series[Sin[x],{x,0,10}]]

Plot[{s1,Sin[x]},{x,-10,10}];

s2=Normal[Series[Cos[x],{x,0,10}]]

b

Plot[s1-Sin[x],{x,-4,4},PlotRange->{-0.01,0.01}]

c

D[s1,x]

d

s1+s2

Sqrt[2] Sin[x+Pi/4]//TrigExpand

s3=Normal[Series[Sqrt[2] Sin[x+Pi/4],{x,0,10}]]

e

Plot[{Sin[x]/x,s1/x},{x,-1,1}]

f

Integrate[s1/x,x]

NIntegrate[Simplify[s1/x],{x,-1,1}]

NIntegrate[Sin[x]/x,{x,-1,1}]

5

a

M=Transpose[{{1,1,2},{4,-1,4},{-1,1,-2}}]; M//MatrixForm

b

Eigensystem[M]

Eigensystem[M]//N

Remove[u]

u[x_]:=x/Norm[x]

u /@ Eigenvectors[M]

%//N

c

Apply[Times,Eigenvalues[M]]

Apply[Times,Eigenvalues[M]]//Expand

Det[M]

d

Det[M-λ IdentityMatrix[3]]

Apply[Plus,Eigenvalues[M]]

Apply[Plus,Eigenvalues[M]]//Expand

e

Sum[Eigensystem[M][[2]][[k]]/Norm[Eigensystem[M][[2]][[k]]], {k,1,3}]

Sum[Eigensystem[M][[2]][[k]]/Norm[Eigensystem[M][[2]][[k]]], {k,1,3}]//N

Sum[1/Eigensystem[M][[1]][[k]] Eigensystem[M][[2]][[k]]/Norm[Eigensystem[M][[2]][[k]]], {k,1,3}]//Simplify

Sum[1/Eigensystem[M][[1]][[k]] Eigensystem[M][[2]][[k]]/Norm[Eigensystem[M][[2]][[k]]], {k,1,3}]//N

M.Sum[1/Eigensystem[M][[1]][[k]] Eigensystem[M][[2]][[k]]/Norm[Eigensystem[M][[2]][[k]]], {k,1,3}]//Simplify

M.Sum[1/Eigensystem[M][[1]][[k]] Eigensystem[M][[2]][[k]]/Norm[Eigensystem[M][[2]][[k]]] ,{k,1,3}]//N

6

a

p1={0,0,4}; p2={0,6,0}; p3={3,0,0}; q1={2,8,0};

θ[λ_, μ_] := 
        p1  +  λ (p2 - p1)  +  μ (p3 - 
                p1);
n = 
        Cross[(p2 - p1) , (p3 - p1)];
s[t_] :=  q1 + t n;
{θ[λ, μ], n, s[t]}

solvθs = Solve[θ[λ,μ]==s[t],{λ,μ,t}]//Flatten

θs = s[t] /. solvθs

N[%]

q2 = q1 + 2 (θs-q1)

N[%]

m=(q1+q2)/2

N[%]

b

V = 1/6 Det[{p1-q1,p2-q1,p3-q1}]

c

Flae = 1/2 Norm[Cross[p2-p1,p3-p1]]

N[%]

d

ArcCos[(q2-p1).(q1-p1)/(Norm[(q2-p1)] Norm[(q1-p1)])]

N[%]

ArcCos[(179/203)] /(2  Pi) 360  // N

ArcCos[(179/203)] / Degree  // N

e

d = 2 Norm[Cross[q1,q2]/2] / Norm[q1-q2]

N[%]

7

a

tr[x_]:= (Pi+ (Pi-2 x))/2 Sin[x]; tr[x]//Simplify

Plot[Evaluate[D[tr[x],x]/.x->u],{u,0,Pi}];

FindRoot[(Evaluate[D[tr[x],x]]==0) /.x->u,{u,1}]

(Evaluate[D[tr[x],x]]) /.x->1.11283

tr[x] /.x->1.11283

Integrate[Sin[x],{x,0,Pi}]

Integrate[Sin[x],{x,0,Pi}] - (tr[x] /.x->1.11283)

b

h[α_] := 2.5 + 4 Tan[α];
g[α_] := 1/Tan[α] h[α];
laenge[α_] := Sqrt[h[α]^2+g[α]^2]

Plot[laenge[α],{α,0,Pi/2}];

D[laenge[α],α]

fr=FindRoot[(Evaluate[D[laenge[α],α]==0] /.α->u),{u,0.75}]

laenge[u] /. fr