Lösungen

1

Remove["Global`*"]

a

f[x_]:=4x^4+5x^5-6x^6+x^0-2/x^2

f[x]

i

Integrate[f[x],x]+c

N[%]

ii

Integrate[f[x],{x,1,2}]

N[%]

iii

Integrate[f[x],{x,-1,1}]

iv

D[Evaluate[Integrate[f[x],{x,0,t}]],t]

b

f[x_]:=1/Sin[x/3-3]^2

f[x]

i

Integrate[f[x],x]+c

N[%]

ii

Integrate[f[x],{x,2,t}]

Assuming[t ∈ Reals, Integrate[f[x], {x, 2, t}]]

Integrate[f[x],{x,2,t},GenerateConditions->False]

D[Evaluate[Integrate[f[x],{x,2,t},GenerateConditions->False]],t]

Integrate[Evaluate[Evaluate[D[Evaluate[Integrate[f[x],{x,2,t},GenerateConditions->False]],t]]],{t,1,u},GenerateConditions->False]

c

f[x_]:=1/(x(x+2)(x-2))

f[x]

Apart[f[x]]

i

Integrate[f[x],x]+c

Integrate[f[x],{x,3,5}]

N[%]

ii

Integrate[f[x],{x,3,Infinity}]

N[%]

d

f[x_]:=7 Log[x]-(E^x-E^(-x))/E^(3x)

f[x]

i

Integrate[f[x],{x,1,E}]

N[%]

e

f[x_]:=x Log[x]^2+x

f[x]

i

Integrate[f[x],x]+c

N[%]

ii

Integrate[f[x],{x,t,2t}]

N[%]

2

Remove["Global`*"]

a

f[x_]:=Sin[x]; x0=2 Pi; n=8;

Series[f[x],{x,x0,n}]

f[x_]:=Sin[2x];Series[f[x],{x,x0,n}]

N[%]

b

f[x_]:=Cos[x^2]+E^(-x^2); x0=0; n=8;

Series[f[x],{x,x0,n}]

N[%]

c

Integrate[Normal[Series[f[x],{x,x0,n}]],{x,-2,2}]

N[%]

n=100;Integrate[Normal[Series[f[x],{x,x0,n}]],{x,-2,2}]

N[%]

NIntegrate[f[x],{x,-2,2}]

d

\(Konvergenzradius\ von\ e\^x\ und\ von\ cos \((x)\) 
      ist\ unendlich\ \((wegen\ den\ Fakultäten\ im\ Nenner, \ 
          folgt\ aus\ der\ Formel\ für\ den\ Konvergenzradius)\) . Wird\ auch\ für\ x\^2\ u . s . 
        w\ \(\(übernommen\)\(.\)\)\)

3

Remove["Global`*"]

f[x_,y_]:=Sin[x y]+Sin[x];
f[{x_,y_}]:=f[x,y];

a

Plot3D[f[x,y],{x,0,Pi},{y,-Pi,Pi}];

ContourPlot[f[x,y],{x,0,Pi},{y,-Pi,Pi}];

b

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

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

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

FindRoot[Evaluate[grad[f[x,y],x,y]=={0,0}],{x,Pi/2},{y,0}]

f[0,-1]

FindRoot[Evaluate[grad[f[x,y],x,y]=={0,0}],{x,Pi/2},{y,0.5}]

f[Pi/2,1]

FindRoot[Evaluate[grad[f[x,y],x,y]=={0,0}],{x,Pi/2},{y,3}]

f[Pi/2,3]

Rand

Evaluate[(D[f[x,y],y]==0)/.x->0]

Alles Lösungen

Evaluate[(D[f[x,y],y]==0)/.x->Pi]

Solve[Evaluate[(D[f[x,y],y]==0)/.x->Pi],{y}]

Evaluate[(D[f[x,y],x]==0)/.y->-Pi]

Solve[Evaluate[(D[f[x,y],x]==0)/.y->-Pi],{x}]

FindRoot[Evaluate[(D[f[x,y],x]==0)/.y->-Pi],{x,0}]

FindRoot[Evaluate[(D[f[x,y],x]==0)/.y->-Pi],{x,0.1}]

FindRoot[Evaluate[(D[f[x,y],x]==0)/.y->-Pi],{x,1}]

FindRoot[Evaluate[(D[f[x,y],x]==0)/.y->-Pi],{x,1.1}]

FindRoot[Evaluate[(D[f[x,y],x]==0)/.y->-Pi],{x,2}]

FindRoot[Evaluate[(D[f[x,y],x]==0)/.y->-Pi],{x,2.1}]

Evaluate[(D[f[x,y],x]==0)/.y->Pi]

FindRoot[Evaluate[(D[f[x,y],x]==0)/.y->Pi],{x,0}]

FindRoot[Evaluate[(D[f[x,y],x]==0)/.y->Pi],{x,0.05}]

FindRoot[Evaluate[(D[f[x,y],x]==0)/.y->Pi],{x,0.1}]

FindRoot[Evaluate[(D[f[x,y],x]==0)/.y->Pi],{x,1}]

FindRoot[Evaluate[(D[f[x,y],x]==0)/.y->Pi],{x,0.5}]

FindRoot[Evaluate[(D[f[x,y],x]==0)/.y->Pi],{x,2}]

FindRoot[Evaluate[(D[f[x,y],x]==0)/.y->Pi],{x,2.1}]

c

grad[f[x,y],x,y].{1,2} 1/Norm[{1,2}]/.{x->1,y->1}

N[%]

d

ArcTan[%]

%/Degree

e

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

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

Evaluate[{grad[f[x,y],x,y]== λ grad[g[x,y],x,y]}]/.x->y^2

Evaluate[grad[f[x,y],x,y][[2]]== λ grad[g[x,y],x,y][[2]]/. λ->-grad[f[x,y],x,y][[1]]] /.x->y^2

InputForm[%]

tabl1=Table[FindRoot[y^2*Cos[y^3] == 2*y*(-Cos[y^2] - y*Cos[y^3]),{y,k Pi}],{k,-1,1,0.0125}]//Flatten;tabl1=Union[tabl1//Chop];
tabl1//MatrixForm

tabl2=Union[Round[100000 Table[{u=y/.tabl1[[k]],u^2},{k,1,Length[tabl1]}]]/100000]//N;
tabl3=Map[Reverse,tabl2]  ;
tabl4 = Select[tabl3,#[[1]]<=Pi&]; tabl4//MatrixForm

Extrema

Funktionswerte in den Extrema

Map[f,tabl4]//MatrixForm

4

Remove["Global`*"]

k=1;
f[g_,h_,exp_]:=1/(1/g+1/h+k 10^(exp)/(g+h)); f[g,h,exp]

fehler[g_,h_,exp_,Δg_,Δh_]:= Abs[D[f[g,h,exp],g]] Δg + Abs[D[f[g,h,exp],h]] Δh;
fehler[g,h,exp,Δg,Δh]

f[g,h,exp]/. {g->14.28, h->25.62,exp->4}

fehler[g,h,exp,Δg,Δh]/. {g->14.28, h->25.62,exp->4,Δg->0.1,Δh->0.25}

f[g,h,exp]/. {g->14.28, h->25.62,exp->-4}

fehler[g,h,exp,Δg,Δh]/. {g->14.28, h->25.62,exp->-4,Δg->0.1,Δh->0.25}

k=0;
f[g_,h_,exp_]:=1/(1/g+1/h+k 10^(exp)/(g+h));

f[g,h,exp]/. {g->14.28, h->25.62,exp->-4}

fehler[g,h,exp,Δg,Δh]/. {g->14.28, h->25.62,exp->-4,Δg->0.1,Δh->0.25}

Anders

h[x_,y_]:=1/(1/x+1/y+10^(-4)/(x+y))

Fehler= Abs[D[h[x,y],x]] 0.1+Abs[D[h[x,y],y]] 0.25

Fehler= Abs[D[h[x,y],x]] 0.1+Abs[D[h[x,y],y]] 0.25 /.{x->14.28,y->25.62}

h[x,y]/.{x->14.28,y->25.62}

h[x_,y_]:=1/(1/x+1/y+10^(4)/(x+y))

Fehler= Abs[D[h[x,y],x]] 0.1+Abs[D[h[x,y],y]] 0.25

Fehler= Abs[D[h[x,y],x]] 0.1+Abs[D[h[x,y],y]] 0.25 /.{x->14.28,y->25.62}

h[x,y]/.{x->14.28,y->25.62}

5

Remove["Global`*"]

a

f[x_,a_,b_,c_,f_]:=(a x^2+b x+c)/(x^2+f)-2;
Solve[{f[x,a,b,c,f]==f[-x,a,b,c,f]},{b}]

Solve[{f[x,a,0,c,f]==f[-x,a,0,c,f],f[0,a,0,c,f]==1,f[1,a,0,c,f]==3/2},{a,c,f}]

f[x,7/2+f/2,0,3f,f]//Simplify

Limit[f[x,7/2+f/2,0,3f,f],x->Infinity]==2

f[x,7/2+1/2,0,3 1,1]

D[f[x,7/2+1/2,0,3 1,1],x]

(D[f[x,7/2+1/2,0,3 1,1],x]/.x->1)==1/2

b

Integrate[f[x,7/2+1/2,0,3 1,1],{x,-2,2}]

N[%]

c

Integrate[Pi (f[2,7/2+1/2,0,3 1,1]^2-f[x,7/2+1/2,0,3 1,1]^2),{x,-2,2}]

N[%]