{VERSION 3 0 "IBM INTEL LINUX" "3.0" } {USTYLETAB {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE " " -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 258 " Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 259 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 0 "" 0 "" {TEXT 23 26 "LEHRSTUHL A F\334R MATHEMATIK" }} {PARA 0 "" 0 "" {TEXT 23 20 "Prof. Dr. E. G\366rlich" }}{PARA 0 "" 0 " " {TEXT 23 75 " W intersemester 99/00 " }}{PARA 0 "" 0 "" {TEXT 23 70 " \+ 7. 2. 2000" }}{PARA 0 "" 0 "" {TEXT 23 74 "______________________________________________________ ____________________" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 " " {TEXT 23 15 "6) Zu Uebung 15" }}{PARA 0 "" 0 "" {TEXT 23 13 "======= ======" }}{PARA 0 "" 0 "" {TEXT 23 10 "Aufgabe 2:" }}{PARA 0 "" 0 "" {TEXT 23 75 "Fuer welche 0 < a,0 < b; ist die Funktion f(x); im Punkt \+ 0 differenzierbar," }}{PARA 0 "" 0 "" {TEXT 23 23 "die definiert ist d urch" }}{PARA 0 "" 0 "" {TEXT 23 54 "> f(x) = abs(x)^a*sin(1/(abs(x)^b )); , falls x <> 0; ," }}{PARA 0 "" 0 "" {TEXT 23 49 "> f(x) = 0; , \+ falls x = 0; ." }}{PARA 0 "" 0 "" {TEXT 23 61 "Fuer welche dieser a,b ist die Ableitung stetig im Nullpunkt?" }}{PARA 0 " " 0 "" {TEXT 23 10 "> restart:" }}{PARA 0 "" 0 "" {TEXT 23 39 "> f := \+ x -> abs(x)^a*sin(1/(abs(x)^b));" }}{PARA 0 "" 0 "" {TEXT 23 20 "Diffe renzenquotient:" }}{PARA 0 "" 0 "" {TEXT 23 17 "> (f(h)-f(0))/h; " }} {PARA 0 "" 0 "" {TEXT 23 23 "Ableitung im Nullpunkt:" }}{PARA 0 "" 0 " " {TEXT 23 41 "> NullAbleitung:=limit ((f(h)-0)/h, h=0);" }}{PARA 0 " " 0 "" {TEXT 23 66 "Offensichtlich existiert der Grenzwert genau dann, wenn 1 < a; ist" }}{PARA 0 "" 0 "" {TEXT 23 27 "und hat dann den Wert Null." }}{PARA 0 "" 0 "" {TEXT 23 50 "=============================== ===================" }}{PARA 0 "" 0 "" {TEXT 23 15 "Zur Stetigkeit:" } }{PARA 0 "" 0 "" {TEXT 23 66 "Ableitung fuer allgemeine x, (x <> 0; st illschweigend anzunehmen):" }}{PARA 0 "" 0 "" {TEXT 23 26 "> diff(x^a* sin(x^(-b)),x);" }}{PARA 0 "" 0 "" {TEXT 23 12 "> normal(%);" }}{PARA 0 "" 0 "" {TEXT 23 20 "> simplify(%,power);" }}{PARA 0 "" 0 "" {TEXT 23 11 "> ?simplify" }}{PARA 0 "" 0 "" {TEXT 23 73 "simplify benoetigt \+ oft die Information, dass alle involvierten Parameter " }}{PARA 0 "" 0 "" {TEXT 23 68 "reell sein sollen. Die Zusammenfassung der Exponente n erfordert dann" }}{PARA 0 "" 0 "" {TEXT 23 69 "immer noch ein wenig \+ Fantasie. Hier wende simplify auf die einzelnen " }}{PARA 0 "" 0 "" {TEXT 260 13 "Summanden an!" }}{PARA 0 "" 0 "" {TEXT 23 26 "> simplify (%,assume=real);" }}{PARA 0 "" 0 "" {TEXT 23 32 "> diff(x^a*sin(x^(-b) ),x):op(%);" }}{PARA 0 "" 0 "" {TEXT 23 68 "> A:=op(1,diff(x^a*sin(x^( -b)),x)):B:=op(2,diff(x^a*sin(x^(-b)),x)):" }}{PARA 0 "" 0 "" {TEXT 23 26 "> simplify(A,assume=real);" }}{PARA 0 "" 0 "" {TEXT 23 26 "> si mplify(B,assume=real);" }}{PARA 0 "" 0 "" {TEXT 23 71 "Die Ableitung i st stetig im Nullpunkt, wenn der Grenzwert fuer x gegen " }}{PARA 0 " " 0 "" {TEXT 261 19 "Null des folgenden " }}{PARA 0 "" 0 "" {TEXT 256 40 "Ausdrucks existiert und gleich Null ist:" }}{PARA 0 "" 0 "" {TEXT 23 7 "> %%+%;" }}{PARA 0 "" 0 "" {TEXT 23 37 "Dazu ist offensichtlich \+ erforderlich:" }}{PARA 0 "" 0 "" {TEXT 23 8 "> 1 < a;" }}{PARA 0 "" 0 "" {TEXT 23 10 "> 1+b < a;" }}{PARA 0 "" 0 "" {TEXT 23 48 "----------- -------------------------------------" }}{PARA 0 "" 0 "" {TEXT 23 44 " Einige Tests mit festen Werten fuer a und b:" }}{PARA 0 "" 0 "" {TEXT 23 56 "> Grenzwert:=limit(normal(diff(x^a*sin(x^(-b)),x)),x=0);" }} {PARA 0 "" 0 "" {TEXT 23 40 "> a:=1/2;b:=1/2;NullAbleitung;Grenzwert; " }}{PARA 0 "" 0 "" {TEXT 23 38 "> a:=1;b:=1/2;NullAbleitung;Grenzwert ;" }}{PARA 0 "" 0 "" {TEXT 23 74 "Die Angabe eines Intervalls als Erge bnis heisst nicht, dass der Grenzwert " }}{PARA 0 "" 0 "" {TEXT 262 10 "existiert." }}{PARA 0 "" 0 "" {TEXT 23 16 "> ?limit[return]" }} {PARA 0 "" 0 "" {TEXT 23 77 "If limit returns a numeric range it means that the value of the limiting " }}{PARA 0 "" 0 "" {TEXT 23 76 "e xpression is known to lie in that range for arguments restricted to so me " }}{PARA 0 "" 0 "" {TEXT 257 74 "neighborhood of the limit point . It does not necessarily imply that the " }}{PARA 0 "" 0 "" {TEXT 258 79 "limiting expression is known to achieve every value infinitely often in this " }}{PARA 0 "" 0 "" {TEXT 259 7 "range. " }}{PARA 0 " " 0 "" {TEXT 23 36 "> a:=1;b:=1;NullAbleitung;Grenzwert;" }}{PARA 0 " " 0 "" {TEXT 23 40 "> a:=11/10;b:=1;NullAbleitung;Grenzwert;" }}{PARA 0 "" 0 "" {TEXT 23 42 "> a:=11/10;b:=100;NullAbleitung;Grenzwert;" }} {PARA 0 "" 0 "" {TEXT 23 42 "> a:=11/10;b:=.01;NullAbleitung;Grenzwert ;" }}{PARA 0 "" 0 "" {TEXT 23 41 "> a:=101; b:=0.9;NullAbleitung;Grenz wert;" }}{PARA 0 "" 0 "" {TEXT 23 36 "> a:=2;b:=1;NullAbleitung;Grenzw ert;" }}{PARA 0 "" 0 "" {TEXT 23 2 "> " }}{PARA 0 "" 0 "" {TEXT 23 9 " Ergebnis:" }}{PARA 0 "" 0 "" {TEXT 23 58 "Ableitung im Nullpunkt exist iert genau dann, wenn 1 < a; ." }}{PARA 0 "" 0 "" {TEXT 23 23 "Ihr Wer t ist dann Null." }}{PARA 0 "" 0 "" {TEXT 23 62 "Ableitung ist stetig \+ im Nullpunkt genau dann, wenn zusaetzlich" }}{PARA 0 "" 0 "" {TEXT 23 15 "gilt 1+b < a; ." }}}{MARK "51 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }