(%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"$