Lösungen / Solutions

1

a

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

FindRoot[x^2 - Cos[x] == 0, {x, 1}]

===>    Definitionslücke / Ne pas difini pour  x ~ 0.824132

b

f[x_] := Floor[Cos[x]] + 1;
Plot[f[x], {x, -2 Pi, 2  Pi}];

f[0]

f[2  Pi]

f[-2 Pi]

c

f[x_] := E^Log[x^2 + 1]; 
    Plot[{f[x], x^2 + 1}, {x, -Pi, Pi}];

d

f[x_] := Cos[E^Log[x^2 + 1]];
Plot[f[x], {x, -Pi, Pi}];

2

x^(3  Log x^2) == x

Plot[{x^(3  Log[ x^2]), x}, {x, 1, 1.5}];

Log[x] 3  Log[ x^2]  == Log[x] /. x -> 1

{Log[x] 3  Log[ x^2], Log[x]} /. x -> 1

Solve[3  Log[ x^2]  == 1, {x}]

N[%]

3

p[x_] := (x - 2) (x + 1) (x^2 - 
            16)
h[x_] := (x^2 - 2  x - 3);
Solve[p[x] h[x] == 0, {x}]

Plot[p[x] h[x], {x, -5, 5}];

Plot[p[x]/ h[x], {x, -5, 9}];

4

Limit[(3  n^2 - 3  n + n^2 Sin[n]) n/(+2 n^2  + 4  n^4), 
      n -> Infinity]

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

N[%]

s1 =  Sum[1/(5^k), {k, 0, Infinity}]

N[%]

s2 =  Sum[4^k/(5^k), {k, 0, Infinity}]

s2/s1

N[%]

5

s =  Sum[1/(q^k), {k, 0, Infinity}]

Solve[s == val, {q}]

Solve[s == 10, {q}]

N[%]

Solve[s == 1, {q}]