Mathematik HTL 2, Schulbuch

302 Wichtige Formeln auf einen Blick Quadratische Gleichungen Lösung von x 2 + px + q = 0 (sofern Diskriminante º 0) – p _ 2 + 9 ____ 2 p _ 2 3 2 – q und – p _ 2 – 9 ____ 2 p _ 2 3 2 – q Lösung von ax 2 + bx + c = 0 (sofern Diskriminante º 0) –b + 9 _____ b 2 – 4ac __ 2a und –b – 9 _____ b 2 – 4ac __ 2a Trigonometrie Sinussatz sin( α ) _ a = sin( β ) _ b = sin( γ ) _ c Cosinussatz a 2 = b 2 + c 2 – 2·b·c·cos( α ) b 2 = a 2 + c 2 – 2·a·c·cos( β ) c 2 = a 2 + b 2 – 2·a·b·cos( γ ) Trigonometrische Flächenformel 1 _ 2 c·b·sin( α ) = 1 _ 2 a·c·sin( β ) = 1 _ 2 a·b·sin( γ ) Summensätze cos( α + β ) = cos( α )·cos( β ) – sin( α )·sin( β ) sin( α + β ) = sin( α )·cos( β ) + cos( α )·sin( β ) Komplexe Zahlen z = a + bj j 2 = –1 konjugiert komplexe Zahl: z* = a – bj Betrag: † z † = 9 ____ a 2 + b 2 Polarform: z = † z † ·e α j Addition: a + bj + c + dj = a + c + (b + d)j Subtraktion: a + bj – c + dj = a – c + (b – d)j Multiplikation: (a + bj)·(c + dj) = ac – bd + (ad + bc)j Division: c + dj _ a + bj = (c + dj)· a – bj _ a 2 + b 2 Statistik f: G ¥ R quantitatives Merkmal; G = {g 1 , g 2 , …, g n }; f(g 1 ) = x 1 , f(g 2 ) = x 2 , …, f(g n ) = x n Arithmetisches Mittel _ x = 1 _ n ; i = 1 n x i = x 1 + x 2 + … + x n __ n Geometrisches Mittel n 9 _______ x 1 ·x 2 ·…·x n Varianz Var(f) = 1 _ n ; i = 1 n (x i – _ x) 2 = (x 1 – _ x) 2 + (x 2 – _ x) 2 + … + (x n – _ x) 2 _____ n Standardabweichung 9 ___ Var(f) Variationskoeffizient 9 ___ Var(f) _ _ x Exponential- und Logarithmusfunktionen exp a : R ¥ R , x ¦ a x exp a (0) = a 0 = 1 exp a (1) = a 1 = a exp a (–1) = a –1 = 1 _ a exp a (x + y) = exp a (x)·exp a (y), also a x + y = a x ·a y exp a (x – y) = exp a (x) _ exp a (y) , also a x – y = a x _ a y exp a (x·y) = exp a (x) y = exp a (y) x , also a x·y = (a x ) y = (a y ) x log a : R + ¥ R , a x ¦ x log a (1) = 0 log a (a) = 1 log a 2 1 _ a 3 = –1 log a (x·y) = log a (x) + log a (y) log a 2 x _ y 3 = log a (x) – log a (y) log a (x y ) = y·log a (x) Skalarprodukt Abstand zwischen Punkten in der Ebene: A = (a 1 1 a 2 ), B = (b 1 1 b 2 ) u A – B u = 9 ___________ (b 1 – a 1 ) 2 + (b 2 – a 2 ) 2 im Raum: A = (a 1 1 a 2 1 a 3 ), B = (b 1 1 b 2 1 b 3 ) u A – B u = 9 _________________ (b 1 – a 1 ) 2 + (b 2 – a 2 ) 2 + (b 3 – a 3 ) 2 Skalarprodukt in der Ebene: P = (p 1 1 p 2 ), Q = (q 1 1 q 2 ) P·Q = p 1 q 1 + p 2 q 2 im Raum: P = (p 1 1 p 2 1 p 3 ), Q = (q 1 1 q 2 1 q 3 ) P·Q = p 1 q 1 + p 2 q 2 + p 3 q 3 Normalvektorform der Geraden durch P {(x 1 y) ‡ N·(x 1 y) = N·P} Winkel cos( α ) = P·Q __ u P u · u Q u ô õ ó b a c Nur zu Prüfzwecken – Eigentum des Verlags öbv