Lösungen

1

1a

Integrate[x E^(-x), x]

Achtung: Konstante C anfügen!

1b

Integrate[(x + 1)/(x + 2)/x, x]

Integrate[(x + 1)/(x + 2)/x, {x, 1, 2}]

(% // Together) /. (c_ Log[a_] +  Log[b_]) -> 
        Log[a^c*b]

N[%]

2

2a

p1 = Normal[Series[Sin[x], {x, 0, 2}]]

p11 = Normal[Series[Sin[x], {x, 0, 5}]]

Normal[Series[E^z, {z, 0, 2}]]

p2 = Normal[Series[E^z, {z, 0, 2}]] /. z -> -x^2

1 + p1 p2 // Expand

1 + p11 p2 // Expand

Normal[Series[1 + Sin[x] E^(-x^2), {x, 0, 5}]]

(* Näherung max. Grad 2 *)       x

2b

r1 = NIntegrate[1 + x, {x, 0, 0.2}]

r2 = NIntegrate[1 + Sin[x] E^(-x^2), {x, 0, 0.2}]

AbweichungProzent  =  (r1 - r2)/r1 100

Brauchbar

3

Normal[Series[1/x, {x, 2, 20}]]

Plot[
        Evaluate[{Normal[Series[1/x, {x, 2, 20}]], 1/x}], {x, -1, 
          5}];

Plot[
        Evaluate[{Normal[Series[1/x, {x, 2, 50}]], 1/x}], {x, 0, 
          4}];

k [n_] := 1/2^(n + 1)

r = k[n]/k[n + 1]

Limit[k[n]/k[n + 1], n -> Infinity]

4

Remove["Global`*"]

v = {2, -1};   P = {2, 1};   f[x_, y_] := 
      Sin[x] + Cos[y];
grad[f_] := {D[f, x], D[f, y]};    grad[f[x, y]]

richtAbleit = 
      grad[f[x, y]] . v/Norm[v] /. {x -> P[[1]], 
          y -> P[[2]]}

N[%]

ArcTan[richtAbleit]

ArcTan[richtAbleit] // N

ArcTan[richtAbleit]/Degree // N

5

Remove["Global`*"]

DSolve[y'[x] == x^2/E^y[x], y, x]

DSolve[{y'[x] == x^2/E^y[x], y[1] == 1}, y, x]

% // N

6

Remove["Global`*"]

f[x] := 1/x;  a = 2; b = 6;  
    ParametricPlot3D[{Cos[s]/x, x, Sin[s]/x}, {x, 2, 6}, {s, 0, 
        2 π}];

6a

Integrate[
      Evaluate[Sqrt[(D[f[x], {x, 1}])^2 + 1]], {x, a, b}]

Mantellaenge = 
      NIntegrate[
        Evaluate[Sqrt[(D[f[x], {x, 1}])^2 + 1]], {x, a, b}]

Sqrt[4^2 + 0.5^2] // N

6b

2 Pi Integrate[
        Evaluate[f[x] Sqrt[(D[f[x], {x, 1}])^2 + 1]], {x, a, 
          b}]

Mantelflaeche = 
      2 Pi NIntegrate[
          Evaluate[f[x] Sqrt[(D[f[x], {x, 1}])^2 + 1]], {x, a, 
            b}]

6c

Pi Integrate[f[x] ^2, {x, a, b}]

Rotationsvolumen = Pi NIntegrate[f[x] ^2, {x, a, b}]