7.2. Differenzenquotientenfunktionen


Wir setzen nun die im einführenden Beispiel gewählte Strategie in ein allgemeines Verfahren um. Dabei war die Kenntnis der Sekantensteigungen von zentraler Bedeutung. In einem ersten Schritt werden wir daher jeder Funktion  f, bei Auswahl eines festen Punktes a, einen Überblick über alle Sekantensteigungen zuweisen.

Definition:  Es sei aA MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgIGiolaadgeacqGHckcZcqWIDesOaaa@3C85@ und  f:A MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacQdacaWGbbGaeyOKH4QaeSyhHekaaa@3BB5@ eine beliebige Funktion. Die Funktion
 
m a ff(a) Xa MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBaaaleaacaWGHbaabeaakiabg2da9maalaaabaGaamOzaiabgkHiTiaadAgacaGGOaGaamyyaiaacMcaaeaacaWGybGaeyOeI0Iaamyyaaaaaaa@40BF@
[7.2.1]

heißt die zu  f gehörige Differenzenquotientenfunktion bzgl. a.

Beachte:

  • m a :A\{a} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBaaaleaacaWGHbaabeaakiaacQdacaWGbbGaaiixaiaacUhacaWGHbGaaiyFaiabgkziUkabl2riHcaa@409E@   und  m a (x)= f(x)f(a) xa MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBaaaleaacaWGHbaabeaakiaacIcacaWG4bGaaiykaiabg2da9maalaaabaGaamOzaiaacIcacaWG4bGaaiykaiabgkHiTiaadAgacaGGOaGaamyyaiaacMcaaeaacaWG4bGaeyOeI0Iaamyyaaaaaaa@458B@ .
    Gemäß Konstruktion gehört der vorgewählte Punkt a nicht zum Definitionsbereich einer Differenzquotientenfunktion.
     
  • Gelegentlich schreiben wir m f,a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBaaaleaacaWGMbGaaiilaiaadggaaeqaaaaa@3988@ statt  m a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBaaaleaacaWGHbaabeaaaaa@37ED@ um die Zugehörigkeit zu  f deutlicher hervor zu heben.
     
  • Die Funktionswerte der Differenzenquotientenfunktion sind die Sekantensteigungszahlen.

    Oft sieht man in dem Wert m a (x) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBaaaleaacaWGHbaabeaakiaacIcacaWG4bGaaiykaaaa@3A4D@ aber auch ein Maß für das Änderungsverhalten der Funktion und nennt ihn dann die Änderungsrate (auch: mittlere Änderungsrate) von  f zwischen a und x.

  • Spezielle Notationen werden in der Physik benutzt: Bei der Bewegung eines Punktes etwa bezeichnet man den in t Zeiteinheiten zurückgelegten Weg mit s(t) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CaiaacIcacaWG0bGaaiykaaaa@3933@ . Die zur Funktion ts(t) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiablAAiHjaadohacaGGOaGaamiDaiaacMcaaaa@3BE5@ gehörenden Änderungsraten s( t 2 )s( t 1 ) t 2 t 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGZbGaaiikaiaadshadaWgaaWcbaGaaGOmaaqabaGccaGGPaGaeyOeI0Iaam4CaiaacIcacaWG0bWaaSbaaSqaaiaaigdaaeqaaOGaaiykaaqaaiaadshadaWgaaWcbaGaaGOmaaqabaGccqGHsislcaWG0bWaaSbaaSqaaiaaigdaaeqaaaaaaaa@4415@ , also die Quotienten

    zurückgelegter Weg verbrauchte Zeit MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaqG6bGaaeyDaiaabkhacaqG8dGaae4yaiaabUgacaqGNbGaaeyzaiaabYgacaqGLbGaae4zaiaabshacaqGLbGaaeOCaiaabccacaqGxbGaaeyzaiaabEgaaeaacaqG2bGaaeyzaiaabkhacaqGIbGaaeOCaiaabggacaqG1bGaae4yaiaabIgacaqG0bGaaeyzaiaabccacaqGAbGaaeyzaiaabMgacaqG0baaaaaa@5573@ ,

    notiert man meist in der Form Δv= Δs Δt MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaamODaiabg2da9maalaaabaGaeuiLdqKaam4Caaqaaiabfs5aejaadshaaaaaaa@3E1D@

     i

    Die Wahl des Buchstabens v erklärt sich aus dem Wort velocitas, lateinisch für Geschwindigkeit (engl. velocity).

    . Man spricht dann von der mittleren Geschwindigkeit (auch: Durchschnittsgeschwindigkeit) des Punktes zwischen den Zeitpunkten t 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDamaaBaaaleaacaaIXaaabeaaaaa@37C9@ und t 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDamaaBaaaleaacaaIYaaabeaaaaa@37CA@ , bzw. zwischen den Wegpunkten s( t 1 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CaiaacIcacaWG0bWaaSbaaSqaaiaaigdaaeqaaOGaaiykaaaa@3A24@ und s( t 2 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CaiaacIcacaWG0bWaaSbaaSqaaiaaikdaaeqaaOGaaiykaaaa@3A25@ .
     

Im folgenden Beispiel notieren wir Differenzenquotientenfunktionen zu einigen Standardfunktionen. Die über die bloße Aufstellung hinaus gehenden Umformungen benötigen wir erst im nächsten Abschnitt.

Beispiel:  Wir berechnen die Differenzenquotientenfunktion bzgl. a
  • zu einer linearen Funktion  mX+b MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiaadIfacqGHRaWkcaWGIbaaaa@3981@   für ein beliebiges a:
     
    m a = mX+b(ma+b) Xa = m(Xa) Xa =m Xa Xa MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBaaaleaacaWGHbaabeaakiabg2da9maalaaabaGaamyBaiaadIfacqGHRaWkcaWGIbGaeyOeI0Iaaiikaiaad2gacaWGHbGaey4kaSIaamOyaiaacMcaaeaacaWGybGaeyOeI0IaamyyaaaacqGH9aqpdaWcaaqaaiaad2gacaGGOaGaamiwaiabgkHiTiaadggacaGGPaaabaGaamiwaiabgkHiTiaadggaaaGaeyypa0JaamyBamaalaaabaGaamiwaiabgkHiTiaadggaaeaacaWGybGaeyOeI0Iaamyyaaaaaaa@5565@  .
    [7.2.2]

     
  • zur Potenzfunktion  X n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaCaaaleqabaGaamOBaaaaaaa@37E6@   für beliebiges a und n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgIGiolablwriLcaa@39CC@ :
     
    m a = X n a n Xa = () (Xa)( X n1 +a X n2 ++ a n2 X+ a n1 ) Xa = (Xa) i=0 n1 a i X ni1 Xa  . MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabmGaaaqaaiaad2gadaWgaaWcbaGaamyyaaqabaaakeaacqGH9aqpdaWcaaqaaiaadIfadaahaaWcbeqaaiaad6gaaaGccqGHsislcaWGHbWaaWbaaSqabeaacaWGUbaaaaGcbaGaamiwaiabgkHiTiaadggaaaaabaaabaGaeyypa0ZaaSaaaeaacaGGOaGaamiwaiabgkHiTiaadggacaGGPaGaaiikaiaadIfadaahaaWcbeqaaiaad6gacqGHsislcaaIXaaaaOGaey4kaSIaamyyaiaadIfadaahaaWcbeqaaiaad6gacqGHsislcaaIYaaaaOGaey4kaSIaeS47IWKaey4kaSIaamyyamaaCaaaleqabaGaamOBaiabgkHiTiaaikdaaaGccaWGybGaey4kaSIaamyyamaaCaaaleqabaGaamOBaiabgkHiTiaaigdaaaGccaGGPaaabaGaamiwaiabgkHiTiaadggaaaaabaaabaGaeyypa0ZaaSaaaeaacaGGOaGaamiwaiabgkHiTiaadggacaGGPaWaaabCaeaacaWGHbWaaWbaaSqabeaacaWGPbaaaOGaamiwamaaCaaaleqabaGaamOBaiabgkHiTiaadMgacqGHsislcaaIXaaaaaqaaiaadMgacqGH9aqpcaaIWaaabaGaamOBaiabgkHiTiaaigdaa0GaeyyeIuoaaOqaaiaadIfacqGHsislcaWGHbaaaaaaaaa@76E8@
    [7.2.3]

    Die Gleichheit (*) zeigen wir in einem Induktionsbeweis.
     
  • zur Kehrwertfunktion  1 X MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaamiwaaaaaaa@3791@ für a0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgcMi5kaaicdaaaa@3950@ :
     
    m a = 1 X 1 a Xa = aX aX(Xa) = Xa aX(Xa) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBaaaleaacaWGHbaabeaakiabg2da9maalaaabaWaaSaaaeaacaaIXaaabaGaamiwaaaacqGHsisldaWcaaqaaiaaigdaaeaacaWGHbaaaaqaaiaadIfacqGHsislcaWGHbaaaiabg2da9maalaaabaGaamyyaiabgkHiTiaadIfaaeaacaWGHbGaamiwaiaacIcacaWGybGaeyOeI0IaamyyaiaacMcaaaGaeyypa0JaeyOeI0YaaSaaaeaacaWGybGaeyOeI0IaamyyaaqaaiaadggacaWGybGaaiikaiaadIfacqGHsislcaWGHbGaaiykaaaaaaa@5414@  .
    [7.2.4]

     
  • zur Wurzelfunktion  X MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaWGybaaleqaaaaa@36E1@ für a0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgwMiZkaaicdaaaa@394F@ :
     
    m a = X a Xa = Xa ( X + a )(Xa) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBaaaleaacaWGHbaabeaakiabg2da9maalaaabaWaaOaaaeaacaWGybaaleqaaOGaeyOeI0YaaOaaaeaacaWGHbaaleqaaaGcbaGaamiwaiabgkHiTiaadggaaaGaeyypa0ZaaSaaaeaacaWGybGaeyOeI0IaamyyaaqaaiaacIcadaGcaaqaaiaadIfaaSqabaGccqGHRaWkdaGcaaqaaiaadggaaSqabaGccaGGPaGaaiikaiaadIfacqGHsislcaWGHbGaaiykaaaaaaa@4ACE@  .
    [7.2.5]

     
  • zur Betragsfunktion  |X| MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWGybaacaGLhWUaayjcSdaaaa@39E8@   für a=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabg2da9iaaicdaaaa@388F@  :
     
    m 0 = | X || 0 | X0 = | X | X MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBaaaleaacaaIWaaabeaakiabg2da9maalaaabaWaaqWaaeaacaWGybaacaGLhWUaayjcSdGaeyOeI0YaaqWaaeaacaaIWaaacaGLhWUaayjcSdaabaGaamiwaiabgkHiTiaaicdaaaGaeyypa0ZaaSaaaeaadaabdaqaaiaadIfaaiaawEa7caGLiWoaaeaacaWGybaaaaaa@4A1F@  .
    [7.2.6]

     

Es lohnt sich immer zu überprüfen, ob neu eingeführte Begriffe mit den Grundrechenarten verträglich sind, denn oft ergeben sich daraus leistungsfähige Rechentechniken. Die folgende Bemerkung belegt, dass die Zuweisung  f m f,a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiablAAiHjaad2gadaWgaaWcbaGaamOzaiaacYcacaWGHbaabeaaaaa@3C2C@   die vier Grundrechenarten respektiert.

Bemerkung:  Es sei  A,B MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiaacYcacaWGcbGaeyOGIWSaeSyhHekaaa@3B92@ und aAB MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgIGiolaadgeacqGHPiYXcaWGcbaaaa@3B7E@ . Dann gilt für  f:A MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacQdacaWGbbGaeyOKH4QaeSyhHekaaa@3BB5@ und g:B MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiaacQdacaWGcbGaeyOKH4QaeSyhHekaaa@3BB7@ :
 
1. m f+g,a = m f,a + m g,a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBaaaleaacaWGMbGaey4kaSIaam4zaiaacYcacaWGHbaabeaakiabg2da9iaad2gadaWgaaWcbaGaamOzaiaacYcacaWGHbaabeaakiabgUcaRiaad2gadaWgaaWcbaGaam4zaiaacYcacaWGHbaabeaaaaa@4491@ [7.2.7]
2. m f-g,a = m f,a - m g,a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBaaaleaacaWGMbGaeyOeI0Iaam4zaiaacYcacaWGHbaabeaakiabg2da9iaad2gadaWgaaWcbaGaamOzaiaacYcacaWGHbaabeaakiabgkHiTiaad2gadaWgaaWcbaGaam4zaiaacYcacaWGHbaabeaaaaa@44A7@ [7.2.8]
3. m fg,a = m f,a g+f(a) m g,a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBaaaleaacaWGMbGaeyyXICTaam4zaiaacYcacaWGHbaabeaakiabg2da9iaad2gadaWgaaWcbaGaamOzaiaacYcacaWGHbaabeaakiabgwSixlaadEgacqGHRaWkcaWGMbGaaiikaiaadggacaGGPaGaeyyXICTaamyBamaaBaaaleaacaWGNbGaaiilaiaadggaaeqaaaaa@4EA3@ [7.2.9]
4. m f g ,a = m f,a gf m g,a g(a)g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBaaaleaadaWcaaqaaiaadAgaaeaacaWGNbaaaiaacYcacaWGHbaabeaakiabg2da9maalaaabaGaamyBamaaBaaaleaacaWGMbGaaiilaiaadggaaeqaaOGaeyyXICTaam4zaiabgkHiTiaadAgacqGHflY1caWGTbWaaSbaaSqaaiaadEgacaGGSaGaamyyaaqabaaakeaacaWGNbGaaiikaiaadggacaGGPaGaeyyXICTaam4zaaaaaaa@50B0@ [7.2.10]

Beweis:  
1. ►   m f+g,a = (f+g)(f+g)(a) Xa = ff(a) Xa + gg(a) Xa = m f,a + m g,a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6515@  .

2. ►  Die Rechnung verläuft genauso wie in 1.

3. ►  Hier kommen wir mit dem Standardtrick "Addition der Null", hier 0=f(a)g+f(a)g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabg2da9iabgkHiTiaadAgacaGGOaGaamyyaiaacMcacaWGNbGaey4kaSIaamOzaiaacIcacaWGHbGaaiykaiaadEgaaaa@41A4@ , zum Ziel:
 

m fg,a = fgfg(a) Xa = fgf(a)g+f(a)gf(a)g(a) Xa = (ff(a))g Xa + f(a)(gg(a)) Xa = m f,a g+f(a) m g,a  . MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8868@

4. ►  Auch jetzt addieren wir die Null, und zwar in der Form 0=fgfg MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabg2da9iaadAgacaWGNbGaeyOeI0IaamOzaiaadEgaaaa@3C44@ .
 

m f g ,a = f g f g (a) Xa = g(a)ff(a)g (Xa)g(a)g = fgf(a)gfg+g(a)f (Xa)g(a)g = (ff(a))g Xa f(gg(a)) Xa g(a)g = m f,a gf m g,a g(a)g  . MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@A460@


7.1. 7.3.