Lösungen E+M Analysis 5 - 6

Uebung 1

?Limit

a

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

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

b

Limit[2 + 1/Sqrt[n], n -> Infinity]

c

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

Apart[2 n^2/(1 + 3  n^2)]

d

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

Apart[(4  n^3 - 3  n + 1)/(n^4 - 2  n^2)]

e

Limit[Sin[n]/n, n -> Infinity]

f

Limit[E^(1/n), n -> Infinity]

Uebung 2

Gerade/ Paire

Plot[
        E^(x^2 - Cos[2  x]), {x, -1.2, 1.2}];

Uebung 3

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

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

p1 = Plot[{f[x], x}, {x, 0, 4}];

p2 = ParametricPlot[{f[x], x}, {x, 0, 4}];

Show[p1, p2, AspectRatio -> 1];

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

Uebung 4

Log[x^2] + Log[1/x] - Log[x]

log[x_^2] := 2 log[x];  log[1/x_] := -log[x];

log[x^2] + log[1/x] - log[x]

Uebung 5

Sin[x + Pi/3]

sin[x_ + y_] := Sin[x] Cos[y]  +  Cos[x] Sin[y]

sin[x + Pi/3]

cos[x_ + y_] :=  Cos[x]^2  +   Sin[y]^2

tan[x_ + y_] := sin[x + y]/cos[x + y]

tan[x + Pi/3]

tan[x + Pi/3] // TrigReduce

Uebung 6

a

ParametricPlot[{Cos[x] (1 + Cos[x/2]), 
          Sin[x] (1 + Cos[x/2])}, {x, 0, 2  Pi}];

ParametricPlot[{Cos[x] (1 + Cos[x/2]), 
          Sin[x] (1 + Cos[x/2])}, {x, 0, 4  Pi}];

b

<< Graphics`Graphics`

PolarPlot[1 + Cos[t/2], {t, 0, 4 Pi}];

Uebung 7

Solve[2 3^x  == 5^x, {x}]

N[%]

Uebung 8

Solve[999  x == 367, x]

1000  a  - a  == 367  //  Solve

Uebung 9

Limit[Sin[3 Pi  +  4/5 n^2]/n^2, n -> Infinity]

Uebung 10

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

Uebung 11

Limit[Log[n]/n^2, n -> Infinity]

Uebung 12

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

Uebung 13

Limit[E^(Sin[Pi + 1/n]), n -> Infinity]

Uebung 14

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

Uebung 15

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

Uebung 16

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

Uebung 17

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

Uebung 18

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

Uebung 19

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

Uebung 20

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

Uebung 21

a

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

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

b

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

c

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

a

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

% // N

x  > 4   ⇒   x ≈ 4.828....,  x ∉ Q

Uebung 22

b

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

% // N

x  > 5   ⇒   x ≈ 5.791....,  x ∉ Q

Uebung 23

a

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}];

b

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

h[1]

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

c

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

h[1]

Uebung 24

a

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}]

b

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}]

Uebung 25

a

Remove[h]

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

Apart[h[x]]

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

b

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

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

Uebung 26

a

Remove[f, g]

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

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

b

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

c

m = 0; M = 1;

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

d

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

% // N

Uebung 27

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

Uebung 28

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)

Uebung 29

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

Uebung 30

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

Uebung 31

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[%]

Uebung 32

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

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

Uebung 33

Nest[Cos,x,5]

Plot[Evaluate[Table[Nest[Cos,x,n],{n,1,10}]],{x,0,2Pi}];

Plot[Evaluate[Table[Nest[Cos,x,n],{n,1,10}]],{x,0,2Pi},PlotRange->{-1,1}];

Plot[Evaluate[Table[Nest[Sin,x,n],{n,1,10}]],{x,0,2Pi}];

Plot[Evaluate[Table[Nest[Sin,x,n],{n,1,100}]],{x,0,2Pi}];

Vermutung:   Limit[Nest[Sin,x,n],n->Infinity] = 0