Lösungen/Solutions/I1S1Ue/Ex10

L=Lektion/Leçon-- Sprechstunde/heures de consultation
W=Work/Arbeit/Travail
A  =Anhang/  Annexe / Appendice

1a

Plot[{x^2 + Sin[x], x^2}, {x, -2, 5}];

Plot[{x^2 + Sin[x], x^2}, {x, -4, 10}];

1b

Plot[Floor[x^2] + 1, {x, -2, 4}];

1c

Plot[{Sin[Floor[x + 1]] - 1, Sin[x + 1] - 1}, {x, -7, 
          14}];

2a

Solve[x == 4 + 4/x, {x}]

% // N

x  > 4   ⇒   x ≈ 4.828....,  x ∉ Q

2b

Solve[x - 2 == 3 + 3/(x - 2), {x}]

% // N

x  > 5   ⇒   x ≈ 5.791....,  x ∉ Q

3a

f[x_] :=  ArcCos[x];  
    p[y_] :=  y^2 - 2  y + 2;
h[x_] := p[f[x]];       h[x]

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

Plot[h[x], {x, -1, 1}];

3b

{h[-1],  h[-1] // N}

h[1]

(Subscript[D, h]) = [-1,1],  (Subscript[W, h]) = [0, h(-1)]

3c

{h[0],  h[0] // N}

h[1]

4a

Remove[f, g, h]

f[x_] := x^2 + x + 1/4;  g[x_] := x^2 - x;  
    h[x_] := f[x] g[x]

h[x]

Plot[h[x], {x, -1, 2}];

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

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

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

4b

u[x_] := 1/(f[x] - g[x]);  u[x]

Plot[u[x], {x, -1, 2}];

Polstelle/ Place du pôle

Solve[1/u[x] == 0, {x}]

5a

Remove[h]

h[x_] := f[x]/g[x] + x;  h[x] // Simplify

Apart[h[x]]

Plot[h[x], {x, -1, 2}];

5b

Show[Plot[h[x], {x, -100, -5}], 
        Plot[h[x], {x, 5, 100}]];

Plot[x + 1, {x, -100, 100}];

6a

Remove[f, g]

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

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

6b

Plot[{f[x], g[x]}, {x, 0.650, 0.655}];

6c

m = 0; M = 1;

Plot[{f[x], m, M}, {x, -4, 4}];

6d

{f[0], f[Log[E]], f[1]}

% // N

6d

{f[0], f[Log[E]], f[1]}

% // N

7

expr =  
      Simplify[3^x == 2^(2  x)  E^3 3^(-2 x)]

Solve[expr, {x}]

% // N

expr1 =  
      Simplify[3^x (4/9)^(-x) ==  E^3]

Solve[x  Log[27/4] == 3, {x}]

% // N

Lösungen/ Solutions

Ueb/Ex 1      W

Limit[Cos[2  Pi + 1/5  n^3]/n^2, n -> Infinity]

Ueb/Ex 2      W

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

Ueb/Ex 3      W

Limit[Log[n]/n^1.5, n -> Infinity]

Ueb/Ex 4      W

Limit[(2 + 3/n + 4/n^2)/(5 + (6 + 7  n)/(8  n)), 
      n -> Infinity]

Ueb/Ex 5      W

Limit[5 E^(Tan[Pi + 2/n^3]), n -> Infinity]

Ueb/Ex 6      W

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

Ueb/Ex 7      W

Limit[(3 n + 1)/(n (2  n - 1)/(n + 2)), 
      n -> Infinity]

Ueb/Ex 8      W

l = 2 (1/2^0+1/2^1 +1/2^2  +1/2^3+  ...)

l = 2  Sum[1/2^n, {n, 0, Infinity}]

l = 2  (1/(1 - 1/2))

A = 2 * 1/2 + 2/2 * 1/2/2 + 2/2^2*1/2/2^2+....
    =  2 * 1/2 *(1+1/2*1/2+1/2^2*1/2^2+1/2^3*1/2^3+...)
    =  2 * 1/2 *(1+1/4+1/4^2+1/4^3+...)

A = 2 * 1/2   Sum[1/4^n, {n, 0, Infinity}]

A = 2 * 1/2   1/(1 - 1/4)

Ueb/Ex 9      W

Limit[(n Cos[2  n] - Sin[n^2] + 8  n^2 - 4  n + 5)/(2  n^2 + 
            4  n - 5 Sin[n]), n -> Infinity]

Ueb/Ex 10      W

x1 = -2;  x3 = 0;  x5 = 2;  x6 = 4;
f[x_] := a (x - x1) (x - x3) (x - x5);  f[x]

f[x6] == 48

solv3 = Solve[f[x6] == 48, {a}] // Flatten

f[x_] := a (x - x1) (x - x3) (x - x5) /. solv3 // Expand; 
    f[x]

Plot[f[x], {x, -2.5, 2.5}];

Annäherung / Approximation:

Plot[f[x] - 3.05, {x, -2.5, 2.5}];

Solve[f[x] - 3.05 == 0, {x}] //  Flatten

Solve[f[x] - 3.075 == 0, {x}] //  Flatten

Ueb/Ex 11      W

3+3/2^2+3/2^4+...=3((1/2^2)^0+(1/2^2)^1+(1/2^2)^2)+...

Limit[3 (Sum[(1/2^2)^k, {k, 0, n}]), 
      n -> Infinity]

3 (1/(1 - 1/2^2))

5/2+5/2^3+5/2^5+...=5/2 ((1/2^2)^0+(1/2^2)^1+(1/2^2)^2)+...

Limit[(5/2) (Sum[(1/2^2)^k, {k, 0, n}]), 
      n -> Infinity]

(5/2) Limit[(Sum[(1/2^2)^k, {k, 0, n}]), 
        n -> Infinity]

(5/2 ) (1/(1 - 1/2^2))

3 (1/(1 - 1/2^2)) * (5/
          2 ) (1/(1 - 1/2^2))

N[%]

Ueb/Ex 12      W

2^0 = 3^0 = 1   ===> k=0....Infinity

(1/(1 - 1/2)) - (1/(1 - 
              1/3))