Lösungen

1

Remove["Global`*"]

m[1] = 1/2; 
s[1] = m[1];

Modell

Momentenbedingung oder Hebelgesetz:

gleichung = ( 
          Gewicht*(k - 1)*mDist[k - 1] + 
              Gewicht*1*(mDist[k - 1] + 1/2) == 
            Gewicht*k*mDist[k] );
Solve[gleichung, {mDist[k] }] // Flatten // Simplify // 
      TraditionalForm

Erstetze  mDist  durch  m .

(m[k - 1]*(k - 1) + (m[k - 1] + 1/2)*1)/k      // 
        Simplify // TraditionalForm

m[k_] := (m[k - 1]*(k - 1) + (m[k - 1] + 1/2)*1)/
          k;

    
s[k_] := m[k] - m[k - 1];

a

Table[s[k], {k, 1, 8}]

% // N

Table[m[k], {k, 1, 8}]

% // N

b  Summe = unendlich (harmonische Reihe)

summe  = 2 Sum[1/k, {k, 1, Infinity}]

summe  == Infinity

2

Remove["Global`*"]

f[x_] := x Sin[x] + x /Sin[x] ;  f[x]

a

D[x Sin[x] + x /Sin[x], x] // Simplify

x Cos[x] + Csc[x] - x Cot[x] Csc[x] + 
        Sin[x] /. {Sin[x] -> sin[x], Cos[x] -> cos[x], 
        Tan[x] -> sin[x]/cos[x], Cot[x] -> cos[x]/sin[x], 
        Csc[x] -> 1/sin[x]}

% // Simplify

%% // Together

(D[x Sin[x] + x /Sin[x], x] // Simplify) /. 
      x -> Pi/2

b

Remove["Global`*"]

ln[x_] := Log[x];

f[x_] := ln[x]/x ;  f[x]

Methoden: Substitution oder partielle Integration.

Integrate[f[x], {x, 1,  E}]

c

Remove["Global`*"]

f[x_] := x^(E);  f[x]

i

Integrate[f[x], {x, 1, t}, GenerateConditions -> False]

ii

D[Evaluate[
        Integrate[f[x], {x, 1, t}, 
          GenerateConditions -> False]], {t}]

d

Remove["Global`*"]

f[x_] := (x + 4)/((x + 2) (x + 6));  f[x]

i

Plot[f[x], {x, -1, 1}];

ii

Apart[f[x]]

iii

Apart[f'[x]]

Apart[f'[x]] /. x -> 0

iv

% // ArcTan // N

%/Degree

v

Integrate[Evaluate[Apart[f[x]]], x]

Integrate[Evaluate[Apart[f[x]]], {x, -1, w}, 
      GenerateConditions -> False]

3

Remove["Global`*"]

f[x_, a2_, a1_, a0_] := a2 x^2 + a1 x + a0

g1 = (f[1, a2, a1, a0] == 6)

g2 = (f[6, a2, a1, a0] == 9)

g3 = (Evaluate[D[f[x, a2, a1, a0], x] /. x -> 6] == 
          Tan[60 Degree])

solv = Solve[{g1, g2, g3}, {a0, a1, a2}] // Flatten

f[x_] := a2 x^2 + a1 x + a0 /. solv

f[x]

f[x] // N

solv1 = 
      Solve[Evaluate[D[f[x], x] == 0], {x}] // Flatten

x3 = x /. solv1

% // N

f[x3]

f[x3] // N

4

Remove["Global`*"]

a

f[x_] :=  
        c  Cosh[(x - x0)/(c)]  +  y0 /. {c -> -4, 
            x0 -> 10, y0 -> 30};
Plot[f[x], {x, 0, 20}];

f[x_] :=  
        c  Cosh[(x - x0)/(c)]  +  y0 /. {c -> -4, 
            x0 -> 10, y0 -> 30};
Plot[f[x], {x, 0, 20}, PlotRange ->  {0, 27}];

b

m[a_] := (20 - 2  a) f[a] + (20 - 2  a) (f[10] - f[a])/
            2;  m[a]

m'[a] == 0

fRoot = FindRoot[Evaluate[m'[a] == 0], {a, 2}]

aa = a /. fRoot

m[aa]

Plot[Evaluate[m[a]], {a, 0, 20}];

5

Remove["Global`*"]

a

Gegeben ist diese Graphik

P1 = P1 ( x1, y1);  P2 = P2 ( x2, y2);  etc.   Dann gilt (Pythagoras):

g = (Sqrt[(10 - y)^2 + x^2] + 
            Sqrt[(12 - y)^2 + (8 - x)^2] == 11)

solv = Solve[g, {y}] // Flatten

y1[x_] := y /. solv[[1]];
y2[x_] := y /. solv[[2]];
Evaluate[y1[x]] /. x -> u

y[u_] := Evaluate[y1[x]] /. x -> u

y[u]

Plot[Evaluate[y[u]], {u, 0, 8}];

Plot[Evaluate[y2[x]], {x, 0, 8}];

y1  hat die erwarteten Masse.

b

solv1 = 
      Solve[Evaluate[y'[u] == 0] /. u -> x, {x}] // 
        Flatten

x0 = x /. solv1

N[%]

y[u]

y0 = y[u] /. u -> x0

N[%]

c

Tan[(10 - y0)/x0] // N

Tan[(12 - y0)/(8 - x0)] // N

(Tan[(10 - y0)/x0] // 
          N) == (Tan[(12 - y0)/(8 - x0)] // N)

6

Remove["Global`*"]

a

f[x_] := Sin[x]^2 + Cos[10 x]/4;

Plot[f[x], {x, 0, 2  Pi}, 
        PlotRange -> {-1, 1.3}];

Hinweis: Dei y-Koord. und damit die Radien r (x) = y (x)  können negativ sein. 
Daher muss bei der Integration allenfalls mit dem Betrag von y(x) gerechnet werden!!!!!!

b

Laenge  =  
      NIntegrate[
        Evaluate[Sqrt[1 + (f'[x])^2]], {x, 0, 2 Pi}]

c

Inhalt  =  Pi Integrate[f[x]^2, {x, 0, 2 Pi}]

Inhalt  =  Pi Integrate[f[x]^2, {x, 0, 2 Pi}] // N

d

OberflaecheFalsch  =  
      2 Pi NIntegrate[
            Evaluate[f[x]  Sqrt[1 + (f'[x])^2]], {x, 0, 2 Pi}] // 
        N

Betrag von f[x] verwenden!

OberflaecheRichtig  =  
      2 Pi NIntegrate[
            Evaluate[Abs[f[x]]  Sqrt[1 + (f'[x])^2]], {x, 0, 
              2 Pi}] // N

7

Remove["Global`*"]

f[x_, y_] := 
        x^2 + y^2 + (x - 2)^2 + (y + 
                1)^2;
Plot3D[f[x, y], {x, -4, 4}, {y, -4, 4}];

f[x, y] // Expand

a

solv = 
      Solve[Evaluate[{D[f[x, y], x] == 0,  
              D[f[x, y], y] == 0}], {x, y}] // Flatten

{xx, yy} = {x, y} /. solv;
z = f[xx, yy]

% // N

b

grad[h_] := {D[h[x, y], x], D[h[x, y], y]};   grad[h]

grad[f]

g[x_, y_] := 
        x + 10 + (y - 6)^2 - 25;
grad[g]

λ grad[g][[1]] == grad[f][[1]]

λ grad[g][[2]] == grad[f][[2]]

{λ grad[g][[1]] == grad[
            f][[1]], λ grad[g][[2]] == grad[
            f][[2]], g[x, y] == 0}

Solve[
          Evaluate[{λ grad[g][[1]] == grad[
                    f][[1]], λ grad[g][[2]] == grad[
                    f][[2]], g[x, y] == 0}], {x, y, λ}] // 
        N // Chop

x1 = 2.416171911191869`;  
    y1 = 9.547369178533316`;
x2 = 14.94153866199081`;  
    y2 = 6.241787795409923`;
x3 = 0.6422894268173227` ; y3 = 
        2.2108430260567613` ;

FindRoot[
          Evaluate[{λ grad[g][[1]] == grad[
                    f][[1]], λ grad[g][[2]] == grad[
                    f][[2]], 
              g[x, y] == 0}], 
{x, 1.86}, {y, 
            9.6}, {λ, 3.4}] // N // Chop

f[x1, y1]

f[x2, y2]

f[x3, y3]

8

DSolve[y'[x] == (x^3)/y[x], y, x]

Solve[-((Sqrt[(1^4) + 4 C[1]])/(Sqrt[2])) == 1, {C[1]}]

Solve[((Sqrt[(1^4) + 4 C[1]])/(Sqrt[2])) == 1, {C[1]}]

((Sqrt[(x^4) + 4  (1/4)])/(Sqrt[2]))

DSolve[{y'[x] == (x^3)/y[x], y[1] == 1}, y, 
      x]

((Sqrt[1 + (x^4)])/(Sqrt[2]))

9

Series[Cos[x], {x, 0, 4}]

Series[Cos[x], {x, 0, 4}] // Normal

a0 = 1;  a1 = 0;  a2 = 2 (-1/2) // Simplify;  a3 = 0;  
    a4 = 4! 1/24 // Simplify;
{a0, a1, a2, a3, a4}

a0 = 1;  a1 = 0;  a2 = 2 (-1/2);  a3 = 0;  
    a4 = 4! 1/24;
p[x_] := 
        a0 + a1 x + a2/2 x^2  + a3/3! x^3 + a4/4! x^4  //  
          Simplify;
Print[p[x] // Expand];
m = 2.5;
Plot[{Cos[x], p[x]}, {x, -m, m}];

m = 3;
Plot[{Cos[x], p[x]}, {x, -m, m}];

u = 2.73;  
  
Plot[{Cos[x], p[x], Cos[u], Cos[u] + 0.5}, {x, 1.5, 
          u}];