Lösungen / Solutions

Remove["Global`*"]

1

D[5 x^4 - 2 x^3 + 8 x^2 + 4 x - 3 , x]

D[5 x^4 - 2 x^3 + 8 x^2 + 4 x - 3 , {x, 2}]

D[5 x^4 Cos[10 x] - x^(2 x), x]

D[E^(Sin[3  x + 7]) - Log[Tan[x^2]], x]

2

f[x_] := 4  x^3 - 6  x^2 + 5  x - 8

Plot[f[x], {x, -1, 3}];

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

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

f[1/2]

s[t_] := t + 1/2

g[t_] := f[s[t]] + (13/2)

g[t]

f[s[t]] + 13/2  // Expand

Ungerade, punktsymmetrisch zu O

Plot[g[t], {t, -2, 2}];

f1[x_] := (x - 1) (x - 2) (x + 3) (x + 5)

Plot[f1[t], {t, -6, 6}];

3

f[x_] := x (x - 1) (x - 3) - 3 Log[x]

D[f[x], x] == 2.5

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

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

Solve[D[f[x], x] == 2.5, {x}]

Plot[Evaluate[D[f[x], x]], {x, 1, 3}];

Plot[Evaluate[{2.5, D[f[x], x]}], {x, 1, 3}];

Plot[
        Evaluate[D[Sin[x (x - 1) (x - 3)] - 3 Log[x], x]], {x, 1, 
          5}];

FindRoot[
      Evaluate[D[Sin[x (x - 1) (x - 3)] - 3 Log[x], x]], {x, 
        2.5}]

FindRoot[
      Evaluate[D[Sin[x (x - 1) (x - 3)] - 3 Log[x], x] - 2.5], {x, 
        2.5}]

Plot[
        Evaluate[D[Cos[x (x - 1) (x - 3)] - 3 Log[x], x]], {x, 1, 
          5}];

FindRoot[
      Evaluate[D[Cos[x (x - 1) (x - 3)] - 3 Log[x], x]], {x, 
        2.5}]

FindRoot[
      Evaluate[D[Cos[x (x - 1) (x - 3)] - 3 Log[x], x] - 2.5], {x, 
        3}]

FindRoot[
      Evaluate[D[Cos[x (x - 1) (x - 3)] - 3 Log[x], x] - 2.5], {x, 
        3.5}]

4

Remove["Global`*"]

y[1] =   0;
f[x_] :=   Pi Cos[x] - 1 E^x;
g[x_] := Evaluate[D[f[s], s]] /. s -> x;

Plot[f[x], {x, 0.25, Pi/2}];

Plot[g[s], {s, 0.25, Pi/2}];

g[s]

y[n_] := 
      s - Evaluate[f[s]/g[s]] /. s -> y[n - 1]; ?y

y[2]

Table[y[k], {k, 1, 5}] // N

y[1] =   1;
Table[y[k], {k, 1, 5}] // N

x[1] = 0;
x[n_] := f[x[n - 1]];
Table[x[n], {n, 1, 5}] // N

Remove["Global`*"]

y[1, a_] := 0;
f[x_, a_] :=   Pi Cos[x] - a E^x;
g[s_, a_] := D[f[s, a], s];
y[n_, a_] := 
        s - Evaluate[f[s, a]/g[s, a]] /. s -> y[n - 1, a] // 
          N;

y[5, 1]

y[5, 1]

Plot[y[5, x], {x, 0.1, 5}];

Plot[y[5, x], {x, 0.1, 10}];

Plot[y[10, x], {x, 0.1, 10}];

5

Programm CM18.nb

tab1 = {{1, 2}, {3, 5}, {5, 1}, {7, 3}, {9, 4}};

f1Abl[
        n_] := (tab1[[n + 
                    2]][[2]] - tab1[[n]][[2]])/(tab1[[n + 
                    2]][[1]] - tab1[[n]][[1]])

tab2 = Table[f1Abl[n], {n, 1, Length[tab1] - 2}]

f2Abl[
        n_] := (tab2[[n + 
                    2]][[2]] - tab2[[n]][[2]])/(tab2[[n + 
                    2]][[1]] - tab2[[n]][[1]])

tab2 = Table[f1Abl[n], {n, 1, Length[tab2] - 2}]

New1

Remove["Global`*"]

y[1, a_] := 0.1;
f[x_, a_] :=   (Pi + 1) Cot[x] - 
          a E^x;
g[s_, a_] := D[f[s, a], s];
y[n_, a_] := 
        s - Evaluate[f[s, a]/g[s, a]] /. s -> y[n - 1, a] // 
          N;

y[5, 1]

y[5, 1]

Plot[y[5, x], {x, 0.1, 5}];

Plot[y[5, x], {x, 0.1, 10}];

Plot[y[10, x], {x, 0.1, 10}];

New2

Remove["Global`*"]

y[1, a_] := 0.1;
f[x_, a_] :=   (E + 1) Cot[x] - 
          a E^x;
g[s_, a_] := D[f[s, a], s];
y[n_, a_] := 
        s - Evaluate[f[s, a]/g[s, a]] /. s -> y[n - 1, a] // 
          N;

y[5, 1]

y[5, 1]

Plot[y[5, x], {x, 0.1, 5}];

Plot[y[5, x], {x, 0.1, 10}];

Plot[y[10, x], {x, 0.1, 10}];