Lösungen

1

a

f[x_]:=(t+1)x^101+t x^22+(t-1)x^5+t(x-1);
f[x]

f'[x]

b

f'[x]/. {x->1,t->1}

c

f[x_]:=(a-1)x^(-a)+a x^a;
f[x]

f'[x]

f'[x]/. {x->a}

d

f[x_]:=Sin[x] Log[x];
f[x]

f'[x]

e

f'[x]/. x->Pi/4

N[%]

f

f[x_]:=Sin[x]/x^3+E^(Sin[x]);
f[x]

f'[x]

g

f'[x]/. x->Pi

N[%]

h

f[x_]:=E^(3 x^3-2x+1);
f[x]

f'[x]

u = f'[x]/. x->1

ArcTan[u]//N

ArcTan[u]/Degree//N

2

a

f[x_]:= (x-3) (x-1)/(x-1);
Plot[f[x],{x,-5,7}];

D[f[x],x]//Together

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

N[%]

D[f[x],{x,2}]//Together

Solve[Evaluate[D[f[x],{x,2}]==0],{x}]

Keine Extreme uns WP

b

f[x_]:=(Sin[x](x-3)/(x (x-4)^2)+1);
Plot[f[x],{x,-5,7}];

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

Numerator[D[f[x],x]]//Together

Plot[Evaluate[Numerator[D[f[x],x]]],{x,-7,12},PlotRange->{-2,3}];

FindRoot[Evaluate[D[f[x],x]==0],{x,3}]

FindRoot[Evaluate[D[f[x],x]==0],{x,8}]

FindRoot[Evaluate[D[f[x],x]==0],{x,1}]

Nur 1. pos Extremum 0.5055...

c

f[x_]:=E^(-(x+3)^2);

Plot[E^(-(x+3)^2),{x,-6,1}];

D[f[x],x]//Together

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

N[%]

D[f[x],{x,2}]//Together

Solve[Evaluate[D[f[x],{x,2}]==0],{x}]

N[%]

d

f[x_]:=Sin[Sqrt[x]];

Plot[Sin[Sqrt[x]],{x,0,4Pi^2}];

D[f[x],x]//Together

Plot[Evaluate[D[f[x],x]],{x,0,100}];

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

N[%]

D[f[x],{x,2}]//Together

FindRoot[Evaluate[D[f[x],{x,1}]==0],{x,20}]

FindRoot[Evaluate[D[f[x],{x,1}]==0],{x,50}]

FindRoot[Evaluate[D[f[x],{x,1}]==0],{x,100}]

Plot[Evaluate[D[f[x],{x,2}]],{x,0,100}];

FindRoot[Evaluate[D[f[x],{x,2}]==0],{x,5}]

FindRoot[Evaluate[D[f[x],{x,2}]==0],{x,40}]

FindRoot[Evaluate[D[f[x],{x,2}]==0],{x,90}]

3

a

f[x_]:=2 (x-3) (x-2)/(2x-4)^2;
{Limit[f[x],{x->0}],Limit[f[x],{x->Infinity}]}

b

f[x_]:=Sin[x] (x-3) /(x (x-4)^2) +1;
{Limit[f[x],{x->0}],Limit[f[x],{x->Infinity}]}

c

f[x_]:=E^(-x^2+2 x) x+(Cos[x]+1)/x;
{Limit[f[x],{x->0}],Limit[f[x],{x->Infinity}]}

d

f[x_]:=Sin[E^(-x)+1/x];
{Limit[f[x],{x->0}],Limit[f[x],{x->Infinity}]}

4

a

I

Remove["Global`*"]

Plot[Sin[x],{x,0,Pi/2},Epilog->{Line[{{Pi/2,0},{Pi/2,1}}],Line[{{0.6,0},{0.6,Sin[0.6]}}],Line[{{0.6,Sin[0.6]},{Pi/2,Sin[0.6]}}]}];

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

D[V[x],x]//Simplify

Plot[Evaluate[D[V[x],x]],{x,0,Pi/2}];

FindRoot[Evaluate[D[V[x],x]==0],{x,0.7}]

II

Remove["Global`*"]

f[x_]:=Sin[x]; g[x_,x0_]:= Evaluate[f[x0]+(D[f[u],u]/.u->x0)(x-x0)]; g[x,x0]

g[u, v]

PyO[x0_]:=g[Pi/2,x0];
PyO[x0]

PyO[2]

xU=x/.(Solve[g[x,x0]==0,{x}]//Flatten)

x1U[x0_] := x /. (Solve[g[x, x0] == 0, {x}] // Flatten); 
    x1U[z]

x0=0.6;
Plot[Sin[x],{x,xU,Pi/2},Epilog->{Line[{{xU,0},{Pi/2,PyO[x0]}}],
Line[{{Pi/2,0},{Pi/2,PyO[x0]}}]},PlotRange->{0,PyO[x0]}];

A[x1_]:=PyO[x1] (Pi/2- x1U[x1])/2; A[x2]

Plot[Evaluate[A[u]], {u, 0, Pi/2}];

Plot[Evaluate[A[u]], {u, 0, 1.05}];

D[Evaluate[A[u]], {u}]

FindRoot[D[Evaluate[A[u]], {u}], {u, 0.6}]

b

f[x_,n_]:= -x^(2 n)+2; f[x,n]

Plot[
        Evaluate[Table[f[x, n], {n, 0, 5}]], {x, -1.6, 
          1.6}];

2 (x  + f[x, n]) // Expand

rechteck[x_, n_] := 2 (x  + f[x, n])
rechteck[x, n] // Expand

D[rechteck[x, n], {x}] // Expand

Evaluate[Table[rechteck[x, n], {n, 0, 5}]]

Evaluate[Table[D[rechteck[x, n], {x}] // Expand, {n, 0, 5}]]

Evaluate[
      Table[Expand[D[rechteck[x, n], {x}]] == 0, {n, 0, 5}]]

Evaluate[
      Table[Solve[Expand[D[rechteck[x, n], {x}]] == 0, {x}], {n, 0, 
          5}]]

Evaluate[
          Table[Solve[Expand[D[rechteck[x, n], {x}]] == 0, {x}], {n, 0, 
              5}]] // N // MatrixForm

tab1 = 
      ToRadicals[
        Flatten[Evaluate[
            Table[Solve[Expand[D[rechteck[x, n], {x}]] == 0, {x}], {n, 
                0, 5}]]]]

tab2 = 
      Table[ToRadicals[
              Flatten[Evaluate[
                  Table[Solve[
                      Expand[D[rechteck[x, n], {x}]] == 0, {x}], {n, 0, 
                      5}]]]][[n]][[2]], {n, 1, 
          Length[tab1]}]

tab2[[5]]

If[Element[tab2[[1]], Reals], tab2[[1]], {}]

Table[
        If[Element[tab2[[n]], Reals], tab2[[n]], {}], {n, 1, 
          Length [tab2]}] // Flatten

Table[
          If[Element[tab2[[n]], Reals], tab2[[n]], {}], {n, 1, 
            Length [tab2]}] // Flatten // N