Lösungen

1

Remove["Global`*"]

a

f[x_] := (((a4 x + a3) x + a2) x + a1) x + a0  // 
          Expand;
Print[f[x]];
f'''[x]

b

Solve[6 a3 + 24 a4 1 == 0, {a3}]

Solve[6 a3 + 24 a4 1 == 0, {a4}]

c

Limit[(1 + 1/n)^(n x),  n -> Infinity]

Limit[(1 + 1/n)^(n x),  n -> Infinity] /. 
      x -> Pi

2

Remove["Global`*"]

f[x_] := (x/4 - Sin[x^2])^2 + 
          E^(-x^2);
h[x_] := x E^(-x^2) + 1/2 Sin[x] Cos[x];

a

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

b

f'[x]

f'[x] // Simplify

% // ExpandAll

c

f'[x] /. x -> 0

d

FindRoot[Evaluate[f'[x] == 0], {x, -0.8}]

f'[x] /. x -> -0.45906450412214833`

e

Limit[1/(1 + x)  h[x], {x  -> -Infinity}]

f

Integrate[h[x], x] + C[1]

g

Integrate[h[x], {x, 0, 1}]

Integrate[h[x], {x, 0, 1}] // N

h

Series[f[x], {x, 0, 3}]

Series[f[x], {x, 0, 3}] // N

i

FindRoot[Evaluate[f''[x] == 0], {x, -0.8}]

FindRoot[Evaluate[f''[x] == 0], {x, -0.1}]

3

Remove["Global`*"]

a

f[x_] := 
        4 - 2 ((E^x + E^(-x))/
                2);
f1[x_] := 4 - 2 Cosh[x];
Plot[{f[x], f1[x]}, {x, -2, 2}];

b

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

Solve[f[x] == 0, {x}] // N

x1 = -1.3169578969248164`;
x2 = 1.3169578969248164`;

c

gew = Integrate[f[x], {x, x1, x2}]

S = 1/gew Integrate[f[x]^2 /2, {x, x1, x2}]

d

rotV = Integrate[f[x]^2  Pi, {x, x1, x2}]

e

kurvL = 
      NIntegrate[
        Evaluate[Sqrt[1 + (f'[y])^2] /. y -> x], {x, x1, 
          x2}]

f

rotFl = 
      NIntegrate[ 
        Evaluate[
          f[y] 2 Pi Sqrt[1 + (f'[y])^2] /. y -> x], {x, x1, 
          x2}]

4

Remove["Global`*"]

f[x_, 
          y_] := (4 - x) x + (6 - 
                y) y^2;
h[x_] := f[x, 6];
g[x_, a_] :=  a x;
Plot3D[f[x, y], {x, 0, 4}, {y, 0, 6}];

Plot[h[x], {x, 0, 4}];

a

Solve[h[x] == g[x, a], {x}]

z[a_] := 4 - a

A1[a_] := 
        Integrate[h[x] - g[x, a], {x, 0, z[a]}];
A1[a]

A2[a_] := 
        z[a]  g[z[a], a]/2 + 
          Integrate[h[x], {x, z[a], 4}];
A2[a]

Solve[A1[a] == A2[a], {a}] // N // Chop

a1 = 0.8251978960636008`

Plot[{h[x], g[x, 0.8251978960636008`]}, {x, 0, 4}];

b

inh[a_] := z[a] g[z[a], a]/2;
inh[a]

solv = 
      Solve[Evaluate[inh'[u] == 0] /. u -> a, {a}] // 
        Flatten

a1 = a /. solv[[1]]

Plot[{h[x], g[x, a1]}, {x, 0, 4}];

u[x_] := f[x, 0];  u[x]

u[x] == g[x, a1]

solv1 = Solve[u[x] == g[x, a1], {x}] // Flatten

x1 = x /. solv1[[2]]

y = u[x1]

y == g[x1, a1]

flaeche = x1 y/2

% // N

c

D[f[x, y], x] == 0

D[f[x, y], y] == 0

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

Punkte (2, 4) sowie Eckpunkte.