Lösungen Analysis 
E+M 10 / 11

Ueb 1

a

Limit[Log[x]/x,x->Infinity]

b

Limit[Sin[x]/Sqrt[x-Pi],x->Pi]

c

Limit[Tan[x]^2/x,x->0]

Ueb 2

f1[x_] := (x^x)^x;  D[f1[x], x] // Simplify

Ueb 3

f2[x_] := x^(x^x);  D[f2[x], x] // Simplify

Ueb 4

f3[x_] := 1/2 x  -   Cos[x] ;  
    D[f3[x], x]  // Simplify

D[f3[x], {x, 2}]

Solve[D[f3[x], x] == 0, {x}]

Solve[D[f3[x], {x, 2}] == 0, {x}]

Plot[f3[x], {x, -2 Pi, 2  Pi}];

Ueb 5

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

Solve[D[f4[x], x] == 0, {x}]

Plot[f4[x], {x, 0, 3}];

Ueb 6

f5[x_] := E^(-x^2); D[f5[x], x]

Plot[f5[x], {x, -3, 3}];

Solve[D[f5[x], x] == 0, {x}]

D[f5[x], {x, 2}]

Solve[D[f5[x], {x, 2}] == 0, {x}]

N[%]

Ueb 7

f6[x_] := 3  x^3 + 4  x^2 - 5  x - 5; D[f6[x], x]

Solve[D[f6[x], x] == 0, {x}]

N[%]

Solve[D[f6[x], {x, 2}] == 0, {x}]

N[%]

Plot[f6[x], {x, -3, 2}];

Ueb 8

f7[x_] := x^4 - x^2 - 1; D[f7[x], x]

Solve[D[f7[x], x] == 0, {x}]

N[%]

Solve[D[f7[x], {x, 2}] == 0, {x}]

N[%]

Plot[f7[x], {x, -3, 2}];

Ueb 9

f1[x_] := Sin[x + b];      f1[x]

f2[x_] := Sin[x] Cos[b] + Cos[x] Sin[b];      f2[x]

f1[x] == f2[x]

D[f1[x], x] == D[f2[x], x]

Additionstheoreme / Théorèmes d'addition

Ueb 10

a

f[x_] := x^5/5 - x^4/4 - x^3/3 + x^2/2;    f[x]

Plot[f[x], {x, -2, 2}];

D[f[x], x]

solv1a = Solve[D[f[x], x] == 0, {x}]

b

f[x_] := x^4/12 - x^3/9 - x^2 + 1;    f[x]

Plot[f[x], {x, -3.5, 4.5}];

D[f[x], x]

solv1a = Solve[D[f[x], x] == 0, {x}]

c

f[x_] := 1/50 x^2 (x - 5) (x - 9);    f[x]

Plot[f[x], {x, -3.5, 11}];

D[f[x], x]

solv1a = Solve[D[f[x], x] == 0, {x}]

Ueb 11

f[x_] := a x (x - 5);  f[x]

f[2.5]

solv4 = Solve[f[2.5] == 25, {a}] // Flatten

a = a /. solv4

f[x]  //  Expand

aA[x_] :=  ((5 - x) - x) f[x];  aA[x]

aA[x] // Expand

Plot[aA[x], {x, -1.5, 6.5}];

D[aA[x], x] // Expand

solv1a = Solve[D[aA[x], x] == 0, {x}]

solv1a = 
      Solve[100 - 120  x + 24 (x^2) == 
          0, {x}]

Ueb 12

V[x_] := x (16 - 2  x) (6 - 2  x) // Expand;  V[x]

Plot[V[x], {x, -2, 16}];

D[V[x], x]

solv1a = Solve[D[V[x], x] == 0, {x}]

solv1a // N

Ueb 13

f1a[x_] := x^2;  
    f1b[x_] := (x - 2)^2 + 4;
D[f1a[x1], x1]

D[f1b[x2], x2] // Expand

solv1a = 
      Solve[D[f1a[x1], x1] == D[f1b[x2], x2], {x2}] // Flatten

τ1[x_] := f1a[x1] + (x - x1) 2 x1 //  Expand; τ1[
      x]

τ2[x_] := f1b[x2] + (x - x2) 2 x1 //  Expand; τ2[
      x]

eq = (τ1[x] - τ2[x] == 0 /.  solv1a  // 
          Simplify)

solv1b = Solve[Evaluate[eq], {x1}] // Flatten

x1 = x1 /. solv1b

y1 = f1a[x1]

x2 = x2 /. solv1a

f1b[x2]

Plot[{f1a[x], f1b[x], τ1[x], τ2[x]}, {x, 0, 
          3}];

Einfachere Lösung (Der 2. Graph entsteht durch Verschiebung des 1.):

Remove["Global`*"]

f1a[x_] := x^2;  
    f1b[x_] := (x - 2)^2 + 4;
D[f1a[x1], x1]

D[f1b[x2], x2] // Expand

x2 = x1 + 2

a = (f1a[x2] - f1b[x1])/(x2 - x1)  // Simplify

Solve[a == 2  x1, {x1}] // Flatten

Ueb 14

f2[x_] := Sin[x];  A[x0] := f2[x0] (Pi - x0 - x0)

expr2 = D[A[x0], x0] // Expand // Simplify

Plot[Evaluate[D[A[x0], x0], {x0, 0, Pi/2}]];

expr2/Cos[x0] == 0/Cos[x0] // Simplify

Pi/2 - x0 == Tan[x0]

Plot[{Pi/2 - x0, Tan[x0]}, {x0, 0, Pi/2 - 0.2}];

Plot[{Pi/2 - x0, Tan[x0], x0, Cot[x0]}, {x0, 0.2, 
          Pi/2 - 0.2}];

FindRoot[Pi/2 - x0 == Tan[x0], {x0, 0.5}]

Ueb 15

f1[x_] := x^2;  f2[x_] := x^4;  f1[x] - f2[x]

D[ f1[x] - f2[x], x]

Solve[Evaluate[D[ f1[x] - f2[x], x]] == 0, {x}]

N[%]

Plot[f1[x] - f2[x], {x, 0, 1}];