(%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"$