Lösungen

Aufgabe 1

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

Aufgabe 2

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}]

Aufgabe 3

f3[x_] := a x^3  +  b x^2  +  c x  +  d

D[f3[x], x]

D[f3[x], x] == 0 /. x -> 1

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

solv3 = 
      Solve[{
f3[-6] == 
              0, 
D[f3[x], x] == 0 /. 
              x -> 1, 
D[f3[x], x] == 0 /. 
              x -> 3, 
D[f3[x], {x, 2}] == 0 /. 
              x -> 2, 
f3[2] == 
              4
}, {a, b, c, d}] // Flatten

f3new[x_] := a x^3  +  b x^2  +  c x  +  d /. solv3;   f3new[
      x]

Plot[ f3new[x], {x, -6, 6}];

Plot[ f3new[x], {x, -6.1, 6.1}, 
        PlotRange -> {-1, 5}];

Aufgabe 4

f4[x_] := Cosh[x]

l = Integrate[Sqrt[1 + Evaluate[D[f4[x], x]]^2], x]

FullSimplify[%] /.  Sqrt[x_^2] -> x

lNum = 
      Integrate[Sqrt[1 + Evaluate[D[f4[x], x]]^2], {x, -1, 1}]

FullSimplify[%] /.  (Sqrt[1 + Sinh[x_]^2]) -> Cosh[x]

N[lNum]

Ob = 
      2 Pi Integrate[f4[x] Sqrt[1 + Evaluate[D[f4[x], x]]^2], 
          x]

FullSimplify[%] /.  (Sqrt[1 + Sinh[x_]^2]) -> 
          Cosh[x]
 /.  (Sqrt[Cosh])[x_]^2 -> 
        Cosh[x]

ObNum = 
      2 Pi Integrate[
          f4[x] Sqrt[1 + Evaluate[D[f4[x], x]]^2], {x, -1, 
            1}]

FullSimplify[%] /.  (Sqrt[1 + Sinh[x_]^2]) -> 
          Cosh[x]
 /.  (Sqrt[Cosh])[x_]^2 -> 
        Cosh[x]

N[ObNum]

Vol = Pi Integrate[f4[x]^2, x]

VolNum = Pi Integrate[f4[x]^2, {x, -1, 1}]

N[VolNum]

Aufgabe 5

a

Limit[(1/x)^Sin[x] ,x -> 0,Direction -> -1]

Limit[Log[(1/x)] Sin[x] ,x -> 0,Direction -> -1]

b

Apart[1/(2  x^2 - 1)]

Factor[(2  x^2 - 1)]

Factor[(2  x^2 - 1), Extension -> (Sqrt[2])]

1/Factor[(2  x^2 - 1), Extension -> (Sqrt[2])]

Apart[1/Factor[(2  x^2 - 1), Extension -> (Sqrt[2])], 
      x]

Integrate[1/(2  x^2 - 1), x]

Integrate[1/(2  x^2 - 1), {x, 5, 7}]

% // N // Chop

Integrate[x Sinh[x], x]

Integrate[x Sinh[x], {x, 5, 7}]

% // N

Integrate[E^(-5 x + 16), x]

Integrate[E^(-5 x + 16), {x, 5, 7}]

% // N

Integrate[1/(2  x^2 - 1) + x Sinh[x] + E^(-5 x + 16), 
      x]

Integrate[
      1/(2  x^2 - 1) + x Sinh[x] + E^(-5 x + 16), {x, 5, 
        7}]

% // N // Chop