Lösungen Modulprüfung Mathematik B 07

1B

a

D[Sum[1/k x^k,{k,1,100}],x]

D[Sum[1/k x^k,{k,1,1000}],x]/.x->1

b Diff., Indentität

(Sin[x/2]+Cos[x/2])^2 // Expand // TrigReduce

D[(Sin[x/2]+Cos[x/2])^2,x]==Cos[x]

(D[(Sin[x/2]+Cos[x/2])^2,x]// Expand // TrigReduce ) ==Cos[x]

c Partielle Integration

Integrate[(x-k+1)E^x,x]

d Diff, Indentität

D[Log[(x Sqrt[x])]-Log[(x Sqrt[x-k])],{x,1}]//Simplify

Limit[1/Evaluate[D[Log[(x Sqrt[x])]-Log[(x Sqrt[x-k])],{x,1}]//Simplify], {x ->k}]

Limit[Evaluate[1/D[Log[(x Sqrt[x])]-Log[(x Sqrt[x-k])],{x,1}]//Simplify], {x ->k^2}]

e

esin[x_]:=E^x Sin[x]; ecos[x_]:=E^x Cos[x];

Integrate[Integrate[esin[x],x],x]

Integrate[Integrate[ecos[x],x],x]

Integrate[Integrate[esin[x],x],x] == -ecos[x]/2

Integrate[Integrate[ecos[x],x],x] == esin[x]/2

{Table[D[E^x Sin[x],{x,n}],{n,1,10}]}//Transpose//MatrixForm

2B

Remove["Global`*"]

a

pA = {2,0}; pB = {7,1}; pC = {5,4};
pS = (pA+pB)/2+ 1/3 (pC-(pA+pB)/2)

pS = (pA+pB+pC)/3

N[%]

pB-pS

J[p_]:={-p[[2]],p[[1]]};
J[pB-pS]

d = (pB.J[pB-pS])/Norm[J[pB-pS]]

N[%]

b

v3[p_]:={p[[1]],p[[2]],0};

Cross[v3[pB-pA],v3[pC-pA]]/2

Norm[Cross[v3[pB-pA],v3[pC-pA]]/2] // N

c

mD[α_]:={{Cos[α],-Sin[α]},{Sin[α],Cos[α]}};
mV[p_]:=Transpose[{p}];

mD[30 Degree] // MatrixForm

mD[30 Degree] // N // MatrixForm

pS1 = mD[30 Degree].mV[pS];

pS1 // MatrixForm

pS1 // N // MatrixForm

flat[v_]:= Flatten[Transpose[v]];
pS1 = flat[pS1]

d

mSp[α_]:={{Cos[2α],Sin[2α]},{Sin[2α],-Cos[2α]}};

α = ArcCos[(pB-pA).{1,0}/Norm[pB-pA]]

α // N

α / Degree // N

mSp[α].mV[pS-pA]+ mV[pA] // N  // MatrixForm

mSp[α].mV[pC-pA]+mV[pA] // N  // MatrixForm

flat[v_]:= Flatten[Transpose[v]];
pS2 = flat[mSp[α].mV[pS-pA]+mV[pA]] // N

pC2 = flat[mSp[α].mV[pC-pA]+mV[pA]] // N

e

str[v_]:={2v[[1]],v[[2]]};
pS3 = str[pS2]

pS4 = mSp[α].mV[pS3-pA]+ mV[pA]  // MatrixForm

pS4 = flat[mSp[α].mV[pS3-pA]+mV[pA]] // N

f

points = Prepend[Map[Point,{{0,0},pA,pB,pC,pS,pS1,pS2,pC2,pS3,pS4}//N],PointSize[0.03]];
lin = {Line[{pA,pB,pC,pA,{7,0},{0,0}}]};
allP = Join[points,lin]

Show[Graphics[allP],AspectRatio->Automatic];

3B

a

r = 4;
h[a_,b_]:= Sqrt[r^2-(a/2)^2-(b/2)^2];
h[a,b]

b

V[a_,b_]:= a b h[a,b] + a b (r-h[a,b])/3;
V[a,b] // Simplify

c

Solve[Evaluate[{D[V[a,b],a]==0, D[V[a,b],b]==0}],{a,b}]

N[%]

d

Remove["Global`*"]

r = 4; V = 20;

Solve[V==a b h+ a b (r-h)/3,{h}]

h[a_, b_] := -((2 (-15 + a b))/(a b))

h[a,b]// Apart

e

Ob[a_,b_]:= 2 a h[a,b] +2 b h[a,b]+ 2 a Sqrt[(r-h[a,b])^2+(b/2)^2]/2+ 2 b Sqrt[(r-h[a,b])^2+(a/2)^2]/2;
Ob[a,b] //Simplify

Ob1[a_,b_]:= Ob[a,b]+ a b;
Ob1[a,b] //Simplify

Plot3D[Ob[a,b],{a,0.1,4},{b,0.1,4}];

Plot3D[Ob1[a,b],{a,0.1,4},{b,0.1,4}];

Plot[Ob[a,a],{a,0.1,4}];

Plot[Ob1[a,a],{a,0.1,4}];

Plot[Ob[a,4-a],{a,0.1,3.9}];

Plot[Ob1[a,4-a],{a,0.1,3.9}];

Plot3D[Ob[a,b],{a,0.5,4},{b,0.5,4},ViewPoint->{-2.056, -2.031, -0.0}];

(* Ableitungen zu kompliziert: Graphisch Schnitte studieren *)

D[Ob[a,b],a] // Simplify

D[Ob[a,b],b] // Simplify

NSolve[Evaluate[{D[Ob[a,b],a]==0,D[Ob[a,b],b]==0}],{a,b}]

(*   {a->3.3916,b->3.3916}   *)

NSolve[Evaluate[{D[Ob1[a,b],a]==0,D[Ob1[a,b],b]==0}],{a,b}]

(*   {a->2.66569,b->2.66569}   *)

(*  Achtung Zeitaufwand!!!  Solve[Evaluate[{D[Ob[a,b],a]==0,D[Ob[a,b],b]==0}],{a,b}] *)

4B

a

Remove["Global`*"]

f[x_,y_]:= 50-x^2-y^2

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

b

Integrate[f[x,y],{x,-5,5},{y,-5,5}]

N[%]

c

f[x,5]

len = Integrate[Evaluate[Sqrt[1+D[f[x,5],x]^2]],{x,-5,5}]

N[%]

d

Integrate[f[x,5],{x,-5,5}]

N[%]

e

v[x_,y_]:={x,y,f[x,y]};
Norm[Cross[D[v[x,y],x],D[v[x,y],y]]]

Sqrt[Cross[D[v[x,y],x],D[v[x,y],y]].Cross[D[v[x,y],x],D[v[x,y],y]]]

Integrate[Evaluate[Sqrt[Cross[D[v[x,y],x],D[v[x,y],y]].Cross[D[v[x,y],x],D[v[x,y],y]]]], {x,-5,5}]

Integrate[Integrate[Evaluate[Sqrt[Cross[D[v[x,y],x],D[v[x,y],y]].Cross[D[v[x,y],x],D[v[x,y],y]]]], {x,-5,5}], {y,-5,5}]

N[%]

NIntegrate[Evaluate[Sqrt[Cross[D[v[x,y],x],D[v[x,y],y]].Cross[D[v[x,y],x],D[v[x,y],y]]]], {x,-5,5}, {y,-5,5}]

B5

Remove["Global`*"]

a

I

(* b in Abh. von a *)

f[a_,x_,n_]:= a x^n;
h[b_,x_]:= 2-b x^2;
Solve[f[a,2,n]==h[b,2],{b}]

h[a_, x_, n_] :=  2 - (1/4) (2 - (2^n) a) x^2;

II

res[a_,n_]:=Integrate[f[a,x,n]-h[a,x,n],{x,0,2},GenerateConditions->False]

res[a,n]

res[a,n]//Apart

(* Ex. Max oder Min , in Abh. von a? - Unabh. vom Rand *)

D[res[a,n],a]

III

(* ist immer 0 für n = 2, sonst nie, d.h. monoton *)

IV

(* Tangenten senkrecht *)

D[f[a,x,n],{x}]

-1/D[h[a,x,n],{x}]

Solve[(Evaluate[D[f[a,x,n],{x}]==-1/D[h[a,x,n],{x}]]/.x->2),{a}]

Solve[(Evaluate[D[f[a,x,n],{x}]==-1/D[h[a,x,n],{x}]]/.x->2),{a}]/.n->2

Solve[(Evaluate[D[f[a,x,n],{x}]==-1/D[h[a,x,n],{x}]]/.x->2),{a}]/.n->3

Solve[(Evaluate[D[f[a,x,n],{x}]==-1/D[h[a,x,n],{x}]]/.x->2),{a}]/.n->4

b

Remove["Global`*"]

I

A[a_]:={{a,-1},{1,2}};
A[a]//N//Eigensystem//Simplify

A[a] // MatrixForm

Eigenvalues[A[a]]

(* Bedingung für 2 EW: a gröser 4 *)

II

A[5]// Eigensystem

(A[5]//N//Eigensystem)

Eigenvalues[A[5]]

Eigenvectors[A[5]]

A[5].(t {{1/2 (3-Sqrt[5])},{1}})//MatrixForm

%//Simplify//MatrixForm

N[%]

(* Geraden Richtung Eigenvektoren sind Fixgeraden *)

III

(Eigenvectors[A[5]])[[1]]

(Eigenvalues[A[5]])[[2]]

(A[5].(Eigenvectors[A[5]])//Flatten)//Simplify

(A[5].(Eigenvectors[A[5]])//Flatten)[[1]]//Simplify

(* 2.Koordinate = EW *)

(A[5].({{1/2 (3-Sqrt[5])},{1}})//Flatten)[[2]] //Simplify