Lösungen

1

a

D[5 x^5+4 x^4 +3 x^3+2 x^2+x+1+c+x^(-1),x]

b

Sgn[x]=const

D[Tan[x]+E^x+Log[x Sgn[x]],x]

c

D[Sin[x]+Log[pi x] x^2,x]

d

D[E^x/x-Cos[x] Log[x],x]

e

D[Sin[3 E^x]+2 E^(-x^3),x]

2

f[x_]:=(x-3)(x-1)(x+1)//Expand; f[x]

Plot[f[x],{x,-1.2,3.3},AspectRatio->Automatic]

a

Solve[f[x]==0,{x}]

b

Factor[f[x]]

c

winkelGrad[x_]:=N[ArcTan[D[f[u],u]/.u->x]/Degree]
winkelGrad[0]

winkelGrad[2]

plot1=Plot[{f[x],(x-0) Evaluate[(D[f[u],u]/.u->0)]+f[0],(x-2) Evaluate[(D[f[u],u]/.u->2)]+f[2]},{x,-1.2,3.3},AspectRatio->Automatic,PlotRange->{-3.2,3.2}]

d  Minimum bei x1, Minimum bei x2

f'[x]

solv=Solve[Evaluate[f'[x]==0],{x}]//Flatten

N[%]

x1=x/.solv[[1]];x2=x/.solv[[2]]; Print["x1 = ",x1//N," // ","x2 = ",x2//N]

e Monotoniebereiche

plot2=Plot[Evaluate[f'[u]/.u->x],{x,-1,3},AspectRatio->Automatic,PlotRange->{-4.1,4}]

Monoton wachsend bis x1 ungef. -0.154701 (Ableitung positiv), dann fallend bis x2 ungef 2.1547 (Ableitungnegativ), dann wieder wachsend (Ableitung positiv).

f  Wendepunkt

f''[x]

solv1=Solve[Evaluate[f''[x]==0],{x}] // Flatten;  x3=x/.solv1

Print["x3 = ",x3]

{x2, f[x3]}

N[%]

e  Graphen

{plot1,plot2}

Show[Graphics[{PointSize[0.03],Point[{x3,f[x3]}],Point[{x1,f[x1]}],Point[{x2,f[x2]}],Point[{0,f[0]}],Point[{2,f[2]}],Line[{{x1,0},{x1,f[x1]}}],Line[{{x2,0},{x2,f[x2]}}],Line[{{-1.2,0},{3.2,0}}],Line[{{2,0},{2,f[2]}}],Line[{{0,f[0]},{0,0}}]}],plot1,plot2,PlotRange->{-4.1,4}]

3

Remove["Global`*"]

a

f[x_]:=x^4-1; f[x]

x0=1; 
fLin[x_]:=f[x0]+Evaluate[f'[u]/.u->x0](x-x0);  fLin[x]// Expand

Plot[{f[x],fLin[x]},{x,0,1.5}]

b

FehlerProzent = f[1.1]-fLin[1.1]

4

Remove["Global`*"]

a

f[x_]:=0.1 x + Cos[x]; f[x]

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

FindRoot[f[x]==0,{x,1.5}]

b

x[1]=1.5;
x[n_]:=x[n-1]-f[x[n-1]]/Evaluate[D[f[x],x]/.x->x[n-1]];

x[3]

1.7457848640344924`

Table[{"x",n," = ",x[n]},{n,1,5}] // MatrixForm

(* Ein Klick auf den Output ergibt weitere Ziffern: *)

(({{"x", 1, " = ", 1.5`}, {"x", 2, " = ", 1.7459481166607174`}, {"x", 3, " = ", 1.7463292679341147`}, {"x", 4, " = ", 1.7463292822528527`}, {"x", 5, " = ", 1.7463292822528529`}}))