Lösungen

1

Remove["Global`*"]

f1[x_] := (x - 1) (x + 1) (x^2 + 1) + 4  x^3 + 3  x^2 + 
          2  x;
f1[x] // Expand

f1'[x]

f1'[x] // Expand

2

Remove["Global`*"]

f2[x_] := Sin[x] E^x Cosh[x];
f2'[x]

% /. {Cosh[x] -> (E^x + E^-x)/2, 
        Sinh[x] -> (E^x - E^-x)/2}

% // Simplify

3

Remove["Global`*"]

f3[x_] := (x - 1) (x + 2)/(x^2 - 1);  
    f3[x] // Simplify

D[%, x]

% // Together

4

Remove["Global`*"]

f4[x_] := Cos[Cos[x]];  f4'[x]

5

Remove["Global`*"]

f5[x_] := Cos[Cos[Cos[x]]];  f5'[x]

6

Remove["Global`*"]

f6[x_] := (2  x)^(3  x);  f6'[
      x]

% // Simplify

7

Remove["Global`*"]

Sign[x]

f7[x_] := x^3 Log[Abs[x^3]]; f7'[x]/.Abs'[x]->0

f7[x_] := x^3 Log[x^3 Sign[x^3]]; f7'[x]/. Sign'[x]->0

% // Simplify

8

Remove["Global`*"]

f8[x_] := 2  Sin[2 x];  f8'[x]

Solve[4 Cos[2 x] == Tan[30 Degree], {x}]

% // N

9

Remove["Global`*"]

f9[x_] := E^-x - 2  x^3 + 4;  f9'[
      x]

f91[x_] := Evaluate[ f9'[x]]

w[0] = 1;
w[n_] := w[n - 1] - f9[w[n - 1]]/f91[w[n - 1]] // N

Table[{n, w[n]}, {n, 0, 9}] // TableForm

Plot[f9[x], {x, 0, 2}];

10

Remove["Global`*"]

f10[x_] := Sqrt[4 - x^2];
 Dreieck10[x_] := x f10[x]/2;

Plot[f10[x], {x, 0, 2}, AspectRatio -> Automatic];

Dreieck10'[x]

Solve[Evaluate[Dreieck10'[x] == 0], {x}]

% // N

1.4142135623730951/2

11

Remove["Global`*"]

f11[x_] := Sqrt[4 - (x/2)^2];
 Dreieck11[x_] := x f11[x]/2;

Plot[f11[x], {x, 0, 4}, AspectRatio -> Automatic];

Dreieck11'[x]

Solve[Evaluate[Dreieck11'[x] == 0], {x}]

% // N

2.8284271247461903/4

12

Remove["Global`*"]

f12[x_] := x^2;
t12[x_, x1_] := 
        Evaluate[
          f12[y] + f12'[y] (x - y) /. y -> x1];

a = 1;
Plot[{f12[x], t12[x, a]}, {x, 0, 1.4}, AspectRatio -> Automatic, 
        Epilog -> {Line[{{a, 0}, {a, f12[a]}}]}];

13

Remove["Global`*"]

f13[x_] := Sin[x];
h13[x_] := (Pi - x - x) f13[x]

h13'[x]

Solve[Evaluate[h13'[x] == 0], {x}]

fr = FindRoot[Evaluate[h13'[x] == 0], {x, 0.5}]

x1 = x /. fr

Plot[f13[x], {x, 0, Pi}, AspectRatio -> Automatic, 
        Epilog -> {Line[{{x1, 0}, {x1, f13[x1]}, {Pi - x1, 
                  f13[x1]}, {Pi - x1, 0}}]}];

14

Remove["Global`*"]

f14[
          x_] := (x - 1) (x - 2) (x + 3) (x + 
              5);
f14[x] // Expand

f14''[x] // Simplify

Solve[Evaluate[f14'[x] == 0], {x}] // N // 
      Chop

sv = Solve[Evaluate[f14''[x] == 0], {x}] // N // 
        Chop

{x1 = x /. sv[[1]], x2 = x /. sv[[2]]}

Plot[f14[x], {x, -6, 3}, 
        Epilog -> {PointSize[0.03], Point[{x1, f14[x1]}], 
            Point[{x2, f14[x2]}]}];

15

Remove["Global`*"]

x  : 2.3 =(x+3) : y;   L^2=(x+3)^2+y^2 = (x^2+2.3^2)/x  *(x+3)

f15[x_] := Sqrt[x^2 + (2.3)^2] (x + 3)/x;  f15[x]

Plot[f15[x],{x,0,10}];

f15'[x]//Simplify

so1 = Solve[Evaluate[f15'[x]==0],{x}]

xMax = x /. so1[[3]]

Laenge = Sqrt[(xMax^2 + 2.3^2)]/xMax*(xMax + 3)

WinkelInRad = ArcTan[2.3/xMax]

WinkelInGrad = ArcTan[2.3/xMax]/Degree

Das ist fast 45 Grad, jedoch nicht exakt 45 Grad!