Lösungen / Solutions

1

a

Remove["Global`*"]

f[x_] := x^2/(x^4 + Cos[x]^2);
Plot[f[x], {x, -6, 6}];

b

Solve[f[x] == 80/100, {x}]

s = FindRoot[f[x] == 160/200, {x, 2}]

f[x] /. s

(f[x] /. s)  ==  0.8

f[0.9]

f[0.942]

Plot[f[x], {x, 0.9, 1.0}];

c

f[x_] := 2  Sin[Floor[x]] - x;
Plot[f[x], {x, -6, 6}];

d

f[x_] := Sin[Abs[Log[x]]];
Plot[f[x], {x, -2.5, 2.5}];

Plot[f[x], {x, -0.1, 0.1}];

Plot[f[x], {x, -0.01, 0.01}];

Plot[f[x], {x, -0.001, 0.001}];

Plot[f[x], {x, -0.0001, 0.0001}];

<< Graphics`Graphics`

LogPlot[f[x - 1], {x, 1.0435, 5}];

2

Exploration

1 + 
        1/(1 + 
              2/(1 + 1/(1 + 2/(1 + 1/(1 + 2/(1)))))) // 
      N

Exakt

solv = 
      Solve[x == 1 + 1/(1 + 2/(x)), {x}] // Flatten

N[%]

3

Remove["Global`*"]

f[x_] := Sinh[x];
p[x_] := Cosh[x^2];
p[f[x]] // Simplify

p[f[x]]

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

Evaluate[f[p[x]]]

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

f[p[x]^2] // Simplify

Plot[Evaluate[f[p[x]^2] - 1], {x, -1, 1}];

Plot[
        Evaluate[{p[f[x]], f[p[x]], f[p[x]^2] - 1}], {x, -1.25, 
          1.25}];

4

f[x_] := (2 - x) (4 - x^2)
p[x_] := (2 - x)^2 (4 - x);
f[x] p[x]

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

Plot[f[x] p[x], {x, -1. , 5}];

Plot[f[x] p[x], {x, -2.2, 5}];

f[x] /p[x]

f[x] /p[x] // Simplify

Plot[f[x] /p[x], {x, 0, 5}];

Plot[f[x] /p[x], {x, -4, 9}, 
        PlotRange -> {-3, 3}];

5

f[x_] := Log[1 + Sin[x]^2];
g[x_] := 1/2 x;

Plot[{f[x], g[x], 0.7} , {x, -6, 6}];

Plot[{f[x], 0.7} , {x, 1.4, 1.8}];

FindRoot[f[x] == g[x], {x, 0}]

FindRoot[f[x] == g[x], {x, -1}]

FindRoot[f[x] == g[x], {x, 0.5}]

FindRoot[f[x] == g[x], {x, 1.5}]

6

Plot[
        2^(6  x) - 6^x*E^x * 2^(-x) , {x, -7, 
          1}];

Plot[2^(6  x) - 6^x*E^x * 2^(-x) , {x, -14, 0}, 
        PlotRange -> {-0.00000001, 0}];

FindRoot[
      2^(6  x) - 6^x*E^x * 2^(-x) == 0, {x, 
        0}]

?FindRoot

s = 
      FindRoot[2^(6  x) - 6^x*E^x * 2^(-x) == 
          0, {x, -1}, WorkingPrecision -> 200]

2^(6  x) - 6^x*E^x * 2^(-x) 7 /. s

Solve[2^(6  x) == 6^x*E^x * 2^(-x), {x}]

NSolve[2^(6  x) == 6^x*E^x * 2^(-x), {x}]

7

Remove["Global`*"]

Sum[E/Pi^k, {k, 1, Infinity}] - 1

% // Simplify

% // Together

N[%]

E/(1 - 1/Pi) - E/Pi^0 - 1 // Simplify

8

Je nach Stellung der Terme wird die Lösung nicht gefunden!

Remove["Global`*"]

Limit[
      Evaluate[(n^3 (1) n^2 - Log[1 - 1/n] + n^4)/(n^6 - 2  n^2  + 
              1)], n -> Infinity]

Limit[
      Evaluate[(n^3 (Sin[n]) n^2 - Log[1 - 1/n] + n^4)/(n^6 - 
              2  n^2  + Cos[n])], n -> Infinity]

Limit[
      Evaluate[(n^5  Sin[n] - Log[1 - 1/n] + n^4)/(n^6 - 2  n^2  + 
              Cos[n])], n -> Infinity]

Limit[
      Abs[(n^5  Sin[n] - Log[1 - 1/n] + n^4)/(n^6 - 2  n^2  + 
              Cos[n])], n -> Infinity]

Limit[(  Sin[n] - Log[1 - 1/n]/n^5 + 1/n)/(n - 2/n^3  + 
            Cos[n]/n^5), n -> Infinity]

Limit[
      Evaluate[(n^3 (Sin[n]) n^2 - Log[1 - 1/n] + n^4)/(n^6 - 
              2  n^2  + Cos[n])], n -> Infinity]

Limit[(n^2 (Sin[n]) n^2 - Log[1 - 1/n] + n^4)/(n^5 - 
            2  n^2  + Cos[n]), n -> Infinity]

9

s1 = Sum[1/(6^k), {k, 0, Infinity}]

s2 = Sum[1/(4^k), {k, 0, Infinity}]

-s1 + s2 - 3/45

N[%]

10

Limit[((n^2 + 2  Log[n^2])/(3  n^2 - 4  Log[n^2] + 
                5  Tan[1/n^2])) + Sqrt[3], n -> Infinity]

N[%]

11

Remove[f1, f2, g]

f1[x_] :=  x^2 - x - 4;
f2[x_] := -x^2 + 2  x + 8;
g[x_] := 4  x + 10;
Plot[{f1[x], f2[x], g[x]}, {x, -3, 8}];

Solve[f1[x] == f2[x], {x}]

N[%]

Solve[f1[x] == g[x], {x}]

N[%]

f1 ist grösser als die andern Funktionen links und rechts aussen.