Lösungen/Solutions/
E+M Analysis 09

Uebung 1

tanα = 80/360 2 Pi

f1[x_] := x^3;  D[f1[x], x]

solv1 = 
      Solve[3 (x^2) == Tan[tanα],  {x}] // Flatten

N[%]

x0 = x /. solv1[[2]]

N[%]

Uebung 2

f2[x_] := x^4;   D[f2[x], x]

tanβ =  D[f2[x], x] /. x -> 1

ArcTan[tanβ]

N[%]

Uebung 3

f3[x_] := 3 x^7  +  9 x^4  +  2; D[f3[x], x]

Uebung 4

Remove[a, b, c]

f41[x_]:=x^2; f42[x_]:=a x^2 + b x + c; 
tab1= Solve[Evaluate[{f41[1]== f42[1],0== f42[3],0== D[f42[x],x]/.x->3}],{a,b,c}]//Flatten

f42[x_] := 
        a x^2  +  b x  +  c /. tab1 ;  f42[
      x]

Plot[{f41[x], f42[x]}, {x, -1, 6}];

solv41 = Solve[f41[x] == f42[x], {x}] //  Flatten

x1 = x /. solv41[[2]]

{D[f41[x], x], D[f42[x], x]}

{D[f41[x], x], D[f42[x], x]} /. x -> x1

α1 = ArcTan[2] // N

α2 = ArcTan[-1] // N

Plot[{f41[x], f42[x],  
          2 (x - x1) + f41[x1],  -1 (x - x1) + 
            f41[x1]}, {x, -1, 4}, 
        AspectRatio -> Automatic];

{α1, α2}

{α1,α2}/Degree

(Abs[α1] + Abs[α2])/Degree

Uebung 5

Remove[x0, x1]

f5[x_] = x^2;
b = f5[x] /. x -> x0; 
a = f5'[x] /. x -> x0;
{a, b}

g[x_] :=  a (x - x0)  +  b;  g[x]

solv5 = Solve[g[x] == 0, {x}] // Flatten

x1 = x /.  solv5

A1 = Abs[(x0 - x1) f5[x0]/2]

A2 = Abs[(x1 - 0) g[0]/2]

Uebung 6

f1[x_] := x^2 Cos[x];  D[f1[x], x]

Uebung 7

f2[x_] := Sin[x]^3;   D[f2[x], x]

Uebung 8

f3[x_] := 3 x  -  x/Cos[x] ;  D[f3[x], x]  // Simplify

Uebung 9

f4[x_] := x  Sqrt[x]; D[f4[x], x]

Uebung 10

f5[x_] := 1/x^2 +  (x^2)/(x^2 + 2); D[f5[x], x]

% // Simplify

Uebung 11

f6[x_] := Sin[2  x]; D[f6[x], x]

Uebung 12

f7[x_] := 1/x  Cos[x]/Sin[x]; D[f7[x], x]

% // Simplify

Uebung 13

f8[x_] := 1/(x^3 - 2  x + Cos[x]); D[f8[x], x]

% // Simplify

Uebung 14

f9[x_] := Cos[Sin[Cos[x^4]]]; D[f9[x], x]

% // Simplify