Mathe Q4 - Lösungen zum Pflichtteil

13. März 2012


(%i1) "Lösungen zur Übungsklausur"$
(%i2) "Pflichtteil"$

(%i3) "Nr. 1"$

(%i4) f(x) := x^2 * %e^(-2*x);
                                      2   (- 2) x
(%o4)                        f(x) := x  %e

(%i5) diff(x^2 * %e^(-2*x),x);
                                - 2 x      2   - 2 x
(%o5)                     2 x %e      - 2 x  %e

(%i6) ratsimp(%);
                                  2          - 2 x
(%o6)                       - (2 x  - 2 x) %e

(%i7) g(x) := 8 + 16/x^2;
                                            16
(%o7)                           g(x) := 8 + --
                                             2
                                            x

(%i8) integrate(8 + 16/x^2,x);
                                         16
(%o8)                              8 x - --
                                         x

(%i9) "Nr. 2"$

(%i10) "a)"$

(%i11) solve(x^3 + 2*x^2 - 4*x - 5 = 0,x);
                       sqrt(21) + 1      sqrt(21) - 1
(%o11)          [x = - ------------, x = ------------, x = - 1]
                            2                 2

(%i12) ev(%,numer);
(%o12)      [x = - 2.79128784747792, x = 1.79128784747792, x = - 1]

Die Aufgabe soll vermutlich per Polynomdivision gelöst werden:
(%i68) divide((x^3 + 2*x^2 - 4*x - 5),(x + 1));
                                  2
(%o68)                          [x  + x - 5, 0]

(%i69) solve(x^2 + x  - 5 = 0,x);
                           sqrt(21) + 1      sqrt(21) - 1
(%o69)              [x = - ------------, x = ------------]
                                2                 2

(%i70) ev(%,numer);
(%o70)          [x = - 2.79128784747792, x = 1.79128784747792]

(%i13) "b)"$

(%i14) solve(%e^x + %e^(0.5*x) - 2 = 0,x);

rat: replaced 0.5 by 1/2 = 0.5
                                  x         x/2
(%o14)                         [%e  = 2 - %e   ]

(%i15) ev(%,numer);
                                 x         0.5 x
(%o15)                        [%e  = 2 - %e     ]

(%i16) find_root( %e^x + %e^(0.5*x) - 2 = 0,x,-10,10);
(%o16)                                0.0

(%i17) "Nr. 3 => Steckbrief-Funktion: f(x) := a*x^3 + b*x^2 + c*x + d"$

(%i18) "f'(x) = 3*a*x^2 + 2*b*x + c"$

(%i19) "f''(x) = 6*a*x + 2*b"$

(%i20) "f'''(x) = 6*a "$

(%i21) linsolve([8.0*x+4.0*y+2.0*z+1.0*u=3.0,12.0*x+4.0*y+1.0*z+0.0*u=-1.0,12.0*x+2.0*y+0.0*z+0.0*u=0.0],[x,y,z,u]);

                 %r3 - 5      3 %r3 - 15        3 %r3 - 13
(%o21)    [x = - -------, y = ----------, z = - ----------, u = %r3]
                    8             4                 2

(%i22) "Loesung: u <> 5 => f'''(2) <> 0 "$

(%i23) "Nr. 4"$

(%i24) "a)"$

(%i25) "besondere Punkte: Extremstelle x ca. 2"$

(%i26) "besondere Bereiche: steigend: 0 - 2, fallend 2 - 8"$

(%i27) "b)"$

(%i28) "notwendig: f'(x_E) = 0, hinreichend: f''(x_E) <> 0"$

(%i29) "c)"$

(%i30) f(x) := 3 * %e^(-x^2);
                                              2
                                           - x
(%o30)                         f(x) := 3 %e

(%i31) diff(3 * %e^(-x^2),x);
                                            2
                                         - x
(%o31)                           - 6 x %e

(%i32) "=> x_E = 0, da e-Funktion <> 0"$


(%i33) diff(3 * %e^(-x^2),x,2);
                                       2          2
                                2   - x        - x
(%o33)                      12 x  %e     - 6 %e

(%i34) "=> f''(x_e) = f''(0) = -6 <> 0"$


(%i35) "Nr. 5"$


(%i36) linsolve([1.0*x+1.0*y+-1.0*z=0.0,2.0*x+-1.0*y+-5.0*z=3.0,1.0*x+3.0*y+1.0*z=-2.0],[x,y,z]);

solve: dependent equations eliminated: (3)
(%o36)              [x = 2 %r4 + 1, y = - %r4 - 1, z = %r4]

(%i37) "=> unterbestimmtes Gleichungssystem mit einem Parameter!"$


(%i38) "Nr. 6"$


(%i39) "Ebene F senkrecht zu Ebene E und enthaelt Gerade g"$


(%i40) "=> Stuetzvektor u + Richtungsvektor v von g"$


(%i41) "Normalenvektor von E = 2. Richtungsvektor  von F"$


(%i42) "Vektor v1"$


(%i43) v1: transpose(matrix ([2.0,2.0,0.0]));
                                    [ 2.0 ]
                                    [     ]
(%o43)                              [ 2.0 ]
                                    [     ]
                                    [ 0.0 ]

(%i44) "Vektor v2"$


(%i45) v2: transpose(matrix ([2.0,4.0,-1.0]));
                                   [  2.0  ]
                                   [       ]
(%o45)                             [  4.0  ]
                                   [       ]
                                   [ - 1.0 ]

(%i46) "Kreuzprodukt zwischen den Vektoren: v1 x v2 "$


(%i47) v1_v2 : transpose(matrix ([-2.0,2.0,4.0]));
                                   [ - 2.0 ]
                                   [       ]
(%o47)                             [  2.0  ]
                                   [       ]
                                   [  4.0  ]

(%i48) "=> n^T = ([-2.0,2.0,4.0])"$


(%i49) "Aufpunkt stuetzF"$


(%i50) stuetzF : transpose(matrix ([3.0,1.0,2.0]));
                                    [ 3.0 ]
                                    [     ]
(%o50)                              [ 1.0 ]
                                    [     ]
                                    [ 2.0 ]

(%i51) "Richtungsvektoren uvec und vvec"$


(%i52) uvecF : transpose(matrix ([2.0,0.0,-1.0]));
                                   [  2.0  ]
                                   [       ]
(%o52)                             [  0.0  ]
                                   [       ]
                                   [ - 1.0 ]

(%i53) vvecF : transpose(matrix ([-2.0,2.0,4.0]));
                                   [ - 2.0 ]
                                   [       ]
(%o53)                             [  2.0  ]
                                   [       ]
                                   [  4.0  ]

(%i54) "Ebenengleichung"$


(%i55) display(stuetzF +  r * uvecF +  s * vvecF)$
        [ 3.0 ]   [  2.0 r  ]   [ - 2.0 s ]   [ - 2.0 s + 2.0 r + 3.0 ]
        [     ]   [         ]   [         ]   [                       ]
        [ 1.0 ] + [   0.0   ] + [  2.0 s  ] = [      2.0 s + 1.0      ]
        [     ]   [         ]   [         ]   [                       ]
        [ 2.0 ]   [ - 1.0 r ]   [  4.0 s  ]   [  4.0 s - 1.0 r + 2.0  ]


(%i56) "Aufpunkt stuetzE1"$


(%i57) stuetzE1 : transpose(matrix ([2.0,0.0,0.0]));
                                    [ 2.0 ]
                                    [     ]
(%o57)                              [ 0.0 ]
                                    [     ]
                                    [ 0.0 ]

(%i58) "Richtungsvektoren uvec und vvec"$


(%i59) uvecE1 : transpose(matrix ([2.0,2.0,0.0]));
                                    [ 2.0 ]
                                    [     ]
(%o59)                              [ 2.0 ]
                                    [     ]
                                    [ 0.0 ]

(%i60) vvecE1 : transpose(matrix ([2.0,4.0,-1.0]));
                                   [  2.0  ]
                                   [       ]
(%o60)                             [  4.0  ]
                                   [       ]
                                   [ - 1.0 ]

(%i61) "Ebenengleichung"$


(%i62) display(stuetzE1 +  r * uvecE1 +  s * vvecE1)$
          [ 2.0 ]   [ 2.0 r ]   [  2.0 s  ]   [ 2.0 s + 2.0 r + 2.0 ]
          [     ]   [       ]   [         ]   [                     ]
          [ 0.0 ] + [ 2.0 r ] + [  4.0 s  ] = [    4.0 s + 2.0 r    ]
          [     ]   [       ]   [         ]   [                     ]
          [ 0.0 ]   [  0.0  ]   [ - 1.0 s ]   [       - 1.0 s       ]


(%i63) "Schnittgerade der Ebenen E1 und F"$


(%i64) display(transpose(matrix([0.0, -1.50, -0.25])) +  r * transpose(matrix([4.0,2.0,1.0])))$
                   [  0.0   ]   [ 4.0 r ]   [    4.0 r     ]
                   [        ]   [       ]   [              ]
                   [ - 1.5  ] + [ 2.0 r ] = [ 2.0 r - 1.5  ]
                   [        ]   [       ]   [              ]
                   [ - 0.25 ]   [ 1.0 r ]   [ 1.0 r - 0.25 ]


(%i65) "Schnittwinkel alpha =  90.00"$


(%i66) "=>  Ebene F ist senkrecht zur Ebene E"$