Lösungen

1

f[x_] := 5  x^3 - 2 x^2 + x - 6

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

D[f[x], x]

res = D[f[x], x] /. x -> 0

ArcTan[res]

ArcTan[res]/Degree  // N

Solve[Evaluate[D[f[x], x] == 1], {x}]

N[%]

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

Solve[Evaluate[D[f[x], {x, 2}] == 0], {x}]

N[%]

2

Integrate[f[x], x]

3

f[x_] := E^(-x^2)

D[f[x], x]

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

Solve[Evaluate[D[f[x], {x, 1}] == 0], {x}]

Solve[Evaluate[D[f[x], {x, 2}] == 0], {x}]

N[%]

4

Remove[g, a, b, x, x0]

f[x_] := x^2;

g[x_, b_] := 2  x0 x  +  b;
Solve[{ g[x0, b] == x0^2},  {b}]

g[x_, x0_] := 2  x0 x  - x0^2;

n[x_, x0_, b_] := -1/(2  x0) x + 
          b;
Solve[{ n[x0, x0, b] == x0^2},  {b}]

n[x_, 
          x0_] := -1/(2  x0) x + (1 + 2  x0^2)/2;

Plot[{x^2, g[x, 1], n[x, 1]}, {x, -3, 3}, 
        Epilog -> {PointSize[0.03], Point[{1, g[1, 1]}], 
            Line[{{1, 0}, {1, 1}}]}, AspectRatio -> Automatic, 
        PlotRange -> {-5, 5}];

Solve[g[x, x0] == 0, {x}]

A1 = x0/2 f[x0]/2

A2 = x0/2 x0^2/2

B = x0 ((1 + 2  x0^2)/2  +  x0^2)/2  // Simplify

5

D[Cos[x] Sin[2  x],  x]

D[Cos[x]  - 2  Sin[x],  x]

D[(E^x + 1)/Sin[x],  
        x]  //Simplify

D[Sin[Cos[Tan[x]]],  
        x]  //Simplify

D[(x^4 + 1) (x^2 - 1) + 1,  
        x]  //Simplify

D[(x^4 + 1) (x^2 - 1) + 1,  x]   /.  x -> 1

6 Schlechter Ansatz

Remove[f]

f[x_, a0_, a2_, a4_, a6_] := 
      a6 x^6 + a4 x^4 + a2 x^2 +  a0

Evaluate[{
f[0, a0, a2, a4, a6] == 
          10, 
f[5, a0, a2, a4, a6] == 
          0, 
(D[f[t, a0, a2, a4, a6], t] == 
              Tan[45 Degree] /. 
            t -> 
              3), 
(D[f[t, a0, a2, a4, a6], t] == 
              Tan[110 Degree] /. t -> 5)}]

Solve[
      Evaluate[{
f[0, a0, a2, a4, a6] == 
            10, 
f[5, a0, a2, a4, a6] == 
            0, 
(D[f[t, a0, a2, a4, a6], t] == 
                Tan[45 Degree] /. 
              t -> 
                3), 
(D[f[t, a0, a2, a4, a6], 
                  t] == Tan[110 Degree] /. t -> 5)}], {a0, a2, 
        a4, a6}]

solv = Solve[
            Evaluate[{
f[0, a0, a2, a4, a6] == 
                  10, 
f[5, a0, a2, a4, a6] == 
                  0, 
(D[f[t, a0, a2, a4, a6], 
                        t] == Tan[45 Degree] /. 
                    t -> 
                      3), 
(D[f[t, a0, a2, a4, a6], 
                        t] == Tan[110  Degree] /. 
                    t -> 5)}], {a0, a2, a4, a6}] // N  // 
        Flatten

f[x_] := a6 x^6 + a4 x^4 + a2 x^2 +  a0 /. solv

Plot[f[x], {x, -5, 5}];

Plot[f[x], {x, 4.9, 5}]

f'[x] /. x -> 5

Resultat: Geht nicht mit n = 4

6 Versuch mit n = 8

Remove[f]

a8 = 0.000265;
f[x_, a0_, a2_, a4_, a6_, a8_] := 
        a8 x^8 + a6 x^6 + a4 x^4 + a2 x^2 +  
          a0;
solv = Solve[
              Evaluate[{
f[0, a0, a2, a4, a, a86] == 
                    10, 
f[5, a0, a2, a4, a6, a8] == 
                    0, 
(D[f[t, a0, a2, a4, a6, a8], 
                          t] == Tan[45 Degree] /. 
                      t -> 
                        3), 
(D[
                          f[t, a0, a2, a4, a6, a8], t] == 
                        Tan[110  Degree] /. t -> 5)}], {a0, a2, a4, 
                a6}] // N  // Flatten;
f[x_] := 
        a8 x^8 + a6 x^6 + a4 x^4 + a2 x^2 +  a0 /. 
          solv;
Plot[f[x], {x, -5, 5}];
Plot[f[x], {x, 4.9, 5}];
f'[x] /. x -> 5

Resultat: Geht nicht mit n = 4

7

f[x_] := x  Sin[1/x]

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

Plot[f[x], {x, -20, 20}];

Limit[ x Sin[1/x], x -> Infinity]

8

Limit[E^n/(Log[n] + E^(3 n)) + n^2 /E^n, 
      n -> Infinity]

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

(E^(x-1)-1)Sin[x]/(x-1)