Nogle bump p\303\245 vejen i ligningsl\303\270sningLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=IndledningI dette dokument vil vi kigge p\303\245 nogle eksempler p\303\245 ligninger, hvor der opst\303\245r problemer, hvis man fors\303\270ger at l\303\270se ligningerne p\303\245 den helt simple m\303\245de ved hj\303\246lp af en solve-kommando. Vi vil se, hvordan vi kan omg\303\245 disse forhindringer. Vi kigger kun p\303\245 ligninger med en variabel her. Eksakte l\303\270sninger kontra numeriske l\303\270sningerMaple vil normalt - hvis den ikke f\303\245r besked p\303\245 andet - fors\303\270ge at l\303\270se en ligning eksakt. Det kan i nogle tilf\303\246lde v\303\246re lige det man \303\270nsker. I de fleste anvendelser i gymnasiet \303\270nsker man imidlertid blot at f\303\245 l\303\270sningerne i kommatal. Lad os kigge p\303\245 nogle eksempler, hvor man f\303\245r et meget underligt svar fra Maple.Eksempel 1L\303\270s ligningen 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. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==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print(); # input placeholderJSFHDet er imponerende, at Maple kan give resultatet eksakt med dette vanvittige udtryk, men det er ikke det vi \303\270nsker: Vi \303\270nsker numeriske l\303\270sninger, dvs. svarene i kommatal. Derfor bruger vi kommafidusen og skriver et af tallene i ligningen som et kommatal. Vi kan for eksempel skrive tallet 12 i ligningen som 12 efterfulgt af et punktum for at indikere at det er tallet 12.0 fremfor bare 12. Det vil f\303\245 Maple til at angive resultaterne i kommatal. 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(); # input placeholderNu ser svaret straks meget mere brugbart ud. Der er dog stadig problemer, for vi f\303\245r tilsyneladende en r\303\246kke l\303\270sninger, hvori der figurere et m\303\246rkeligt symbol I. Vi kommer tilbage til det i n\303\246ste afsnit. Eksempel 2L\303\270s ligningen 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. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==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print(); # input placeholderHer f\303\245r et indirekte og ubrugeligt svar. Igen kan det l\303\270ses ved, at man bruger i kommafidusen, dvs. skriver et af tallene i ligningen som et kommatal. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==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print(); # input placeholderJSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Reelle og komplekse l\303\270sningerLad os kigge videre p\303\245 eksempel 1 fra forrige afsnit:Eksempel 3LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==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(); # input placeholderVi ser, at der er flere l\303\270sninger, der er adskilt af et komma. Nogle af dem indeholder et underligt stort I. Det er s\303\245kaldte komplekse l\303\270sninger. I gymnasiet kigger vi normalt kun p\303\245 de reelle l\303\270sninger, der svarer til m\303\246ngden af alle kommatal. M\303\246ngden af komplekse tal er en st\303\270rre m\303\246ngde. Den nemmeste l\303\270sning er at negligere dem og bare skrive til sidst i en konkluderende tekstlinje:Vi ser at l\303\270sningen er 11.79.afh\303\246ngig af hvor man mange decimaler, man synes man har brug for. Eksempel 4Denne gang vil vi kigge p\303\245 en ligning, hvori der indg\303\245r en potensfunktion: 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. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==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print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Igen genkender vi en masse komplekse l\303\270sninger. Der er to l\303\270sningsforslag: Enten kan man g\303\270re som i eksempel 3 og blot sortere de komplekse fra i hovedet. Det er helt fint. Er man dog usikker p\303\245, om man f\303\245r \303\270je p\303\245 alle i den lange liste, kan man dog alternativt anbringe et kald af pakken RealDomain, som g\303\270r at man kun f\303\245r de komplekse svar. Den anbringes mellem restart kommandoen og solve-kommandoen. Og husk at markere alt fra sidste restart og genberegne:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRK1JlYWxEb21haW5GJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC1JI21vR0YkNi1RIjpGJ0ZALyUmZmVuY2VHRj8vJSpzZXBhcmF0b3JHRj8vJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVkY9RkA=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print(); # input placeholderPakken vil v\303\246re tilg\303\246ngelig indtil n\303\246ste restart kommando udf\303\270res eller indtil man skriver kommandoen LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEndW53aXRoRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVErUmVhbERvbWFpbkYnRi9GMi9GM1Enbm9ybWFsRidGPUY9. KommentarTil den avancerede l\303\246ser kan det oplyses, at det man egentlig kommer til at l\303\270se i eksempel 4 er den s\303\245kaldte binome ligning LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjVGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIj1GJy9GNlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkgvJSlzdHJldGNoeUdGSC8lKnN5bW1ldHJpY0dGSC8lKGxhcmdlb3BHRkgvJS5tb3ZhYmxlbGltaXRzR0ZILyUnYWNjZW50R0ZILyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVy1GLzYlUSJrRidGMkY1RkQ=, der altid har n komplekse l\303\270sninger, der ligger p\303\245 en cirkel i den komplekse plan. Uheldig symbolbrug i MapleVi vil starte kigge p\303\245 et eksempel, hvor man kommer ind i problemer p\303\245 grund af de symboler man v\303\246lger i Maple. Heldigvis kan problemet nemt l\303\270ses ved at bruge andre symboler. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Eksempel 5Fru Hansen s\303\246tter 2500 kr. ind p\303\245 en bankkonto. Efter 23 \303\245r er bel\303\270bet vokset til 13670 kr. Hvor stor har den gennemsnitlige rente v\303\246ret? Vi ser, at det er oplagt at benytte den s\303\245kaldte renteformel, der eventuelt kan have f\303\270lgende udseende: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSI9RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZILyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRldGLi1GQTYtUScmc2RvdDtGJ0ZERkZGSUZLRk1GT0ZRRlMvRlZRJjAuMGVtRicvRllGaG4tSSVtc3VwR0YkNiUtSShtZmVuY2VkR0YkNiQtRiM2Ji1JI21uR0YkNiRRIjFGJ0ZELUZBNi1RIitGJ0ZERkZGSUZLRk1GT0ZRRlMvRlZRLDAuMjIyMjIyMmVtRicvRllGam8tRi82JVEickYnRjJGNUZERkRGOC8lMXN1cGVyc2NyaXB0c2hpZnRHRj9GRA==. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==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(); # input placeholderVi ser, at vi har f\303\245et et uhyrligt svar, endda selv om vi har brugt kommafidusen ved at skrive 2500 som 2500 med et punktum efter. Problemerne er dobbelte her: Dels har man benyttet en variabel K b\303\245de med og uden indeks, og s\303\245 har man endda tildelt indekset en v\303\246rdi ved at s\303\246tte LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkjbW5HRiQ2JFEjMjNGJ0Y5Rjk=. Der er to l\303\270sningsforslag: L\303\270sning 1De variable skrives helt uden indeks: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==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print(); # input placeholderHer kan man s\303\245 bruge teknikken fra eksempel 4 i forrige afsnit: Enten sorterer man selv de komplekse l\303\270sninger fra i hovedet, eller ogs\303\245 kalder man pakken RealDomain. Vi ser at l\303\270sningen er 0.07666, hvilket betyder at den gennemsnitlige rente har v\303\246ret 7,67%. L\303\270sning 2Man bruger et literal subscript i stedet for et s\303\246dvanligt subscript eller indeks, som det hedder p\303\245 dansk. Det ser ud pr\303\246cist som et s\303\246dvanligt indeks, men det indtastes via tastekombinationen Ctrl+Shift+- i stedet for det normale, som er Shift+- . Bruger man Expression paletten skal man bruge 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 i stedet for 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LSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIlZ3YhIiItJSlCT1VORFNfWUc2IyQiJD8iRiotJS1CT1VORFNfV0lEVEhHNiMkIiQ/JEYqLSUuQk9VTkRTX0hFSUdIVEc2IyQiJF0kRiotJSlDSElMRFJFTkc2Ig==. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==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(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=ForklaringForklaringen p\303\245 problemerne ligger i, at hvis man indf\303\270rer en variabel LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn frembragt med et almindeligt indeks (underscore), s\303\245 vil Maple opfatte det som en liste-variabel K, der har en indgang, indekseret ved n. Hvis man s\303\245 senere s\303\246tter K til noget ved kommandoen for eksempel LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkjbW5HRiQ2JFElMjUwMEYnRjlGOQ==, s\303\245 vil Maple glemme v\303\246rdien af LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn. Denne m\303\245de at g\303\270re det p\303\245 kan v\303\246re hensigtsm\303\246ssig i en del sammenh\303\246nge, hvis liste-variablen K indeholder en masse indgange og man \303\270nsker at g\303\270re noget ved dem alle p\303\245 \303\251n gang. Det har vi sj\303\246ldent brug for i gymnasieopgaver. Heldigvis er der s\303\245 et literal subscript. Ved brug at dette bliver LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJy9GNlEnbm9ybWFsRic= opfattet som et helt symbol og ikke den n'te indgang i en listevariabel K. Dermed er problemet l\303\270st! LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=S\303\246rligt genstridige ligningerDer findes ligninger, der er s\303\246rligt genstridige. Blandt dem er de s\303\245kaldte transcendente ligninger, hvor der optr\303\246der for eksempel et polynomium sammen med en transcendent funktion s\303\245som LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYyLUkjbWlHRiQ2JVEjbG5GJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEieEYnL0YwUSV0cnVlRicvRjNRJ2l0YWxpY0YnRjJGMi1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRkI2LVEifkYnRjJGRS9GSEYxRklGS0ZNRk9GUUZTL0ZXRlUtSSVtc3VwR0YkNiUtRiw2JVEiYUYnRj1GPy1GIzYlRjpGPUY/LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0ZBRlktRmluNiUtRkI2LVEvJkV4cG9uZW50aWFsRTtGJ0YyRkVGZm5GSUZLRk1GT0ZRRlMvRldRLDAuMTExMTExMWVtRictRiM2J0Y6Rj0vJStmb3JlZ3JvdW5kR1EsWzEyOCwwLDEyOF1GJy8lLHBsYWNlaG9sZGVyR0Y+Rj9GYG9GQUZZLUYsNiVRJGNvc0YnRi9GMkY1RkEtRiw2JVEkc2luRidGL0YyRjVGMg== etc. En brugbar metode til at l\303\270se s\303\245danne ligninger er at plotte grafen og finde l\303\270sningerne \303\251n ad gangen ved hj\303\246lp af kommandoen fsolve. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==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(); # input placeholderSelv om vi har benyttet kommafidusen, s\303\245 f\303\245r vi det irriterende svar, at Maple m\303\245ske ikke har f\303\245et alle l\303\270sninger med. I virkeligheden er der fire l\303\270sninger. Vi ser, at ingen af metoderne leverer det \303\270nskede. Hvis man i \303\270vrigt giver ligningen til for eksempel en Texas TI 89 Titanium grafregner, s\303\245 finder den tre af de fire l\303\270sninger, og det er jo heller ikke godt nok. Vi leder efter en metode, som med meget stor sandsynlighed finder alle l\303\270sninger. Jeg siger med meget stor sandsynlighed, for man kan faktisk aldrig v\303\246re fuldst\303\246ndig sikker p\303\245, at en l\303\270sning smutter igennem nettet - med mindre man ved noget s\303\246rligt om funktionen. Vi skal g\303\270re brug af et plot af grafen for den funktion, som h\303\270rer til ligningen og kigge p\303\245 nulpunkter for funktionen. Vi er n\303\270dt til at v\303\246lge et endeligt interval at tegne grafen i. Allerede her er faren for, at der kunne ligge l\303\270sninger udenfor dette interval. For at im\303\270deg\303\245 dette problem en smule, v\303\246lger vi et stort interval at tegne grafen i fra begyndelsen, fx fra -1000 til 1000. Da vi kun er interesseret i nulpunkter, s\303\245 er det hensigtsm\303\246ssigt kun at vise grafen for y-v\303\246rdier omkring 0, for eksempel mellem -1 og 1. Hvis du ikke foretager den begr\303\246nsning bliver grafen undertiden uoverskuelig.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%;Vi ser, at der kun sker noget t\303\246t p\303\245 0. Derfor indskr\303\246nker vi intervallet. N\303\245r du g\303\270r det i praksis, skal du blot lige \303\246ndre i x-gr\303\246nserne og trykke p\303\245 Enter, mens cursoren st\303\245r i plot kommandoen, s\303\245 vil grafen blive vist i et mindre interval. Ingen grund til at aflevere flere grafer. Jeg er dog n\303\270dt til at g\303\270re det i denne demonstration: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%;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Nu kan vi for alvor se nulpunkterne. Der er klart fire styk. Vi har stadig ikke fundet nulpunkterne, kun gjort os klart, at der er fire, og at de ligger t\303\246t p\303\245 v\303\246rdierne -1, 0, 2 og 6. Metode: For hvert nulpunkt bruges fsolve kommandoen med den omtrentlige v\303\246rdi for nulpunktet som startv\303\246rdiLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Vi vil finde nulpunktet t\303\246t p\303\245 -1 ved at udskifte x med LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSomdW1pbnVzMDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GUy1JI21uR0YkNiRRIjFGJ0Y5Rjk=: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(); # input placeholderSamme procedure gentages for de \303\270vrige tre nulpunkter ... og de giver henholdsvis 0, LUkjbW5HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JFEsMi4yMjY5ODMwMTBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= og LUkjbW5HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JFEsNi41OTI2MTkzODNGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=