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The Death Of Tommy Grimes Essay

4949 words - 20 pages

1 Ligninger
Du skal redegøre for løsningsmetoder for ligninger. Herunder også andengradsligninger.
Ligninger med én ubekendt:
En ligning med én ubekendt er et udtryk med en variabel (oftest x) og et lighedstegn. En løsning til ligningen er en værdi for den variable (x), der giver samme resultat på begge sider af lighedstegnet. Der findes mange forskellige ligninger, båge enkle og vanskelige ligninger. En af de enkle ligninger er:
x-4=5
Løsningen er det x der gør at venstresiden (3x+4) er lig højresiden (7). I praksis løser man ligninger ved at isolere x på den ene side af lighedstegnet.

Regler for omformning af ligninger:
Det er tilladt at:
* Lægge samme tal eller ...view middle of the document...

Sætning for rødder i andelgradspolynomiet:
Hvis d≥0, udregnes rod eller rødder ved formlen:
x=-b±d2a
Hvor diskriminanten (d) som er defineret ved d=b2-4ac.
Når der står ± foran kvartatrodstegnet betyder det at man til den ene løsning skal regne med +, til den anden regne med -. Man kan derfor opskrive formlen som 2 løsninger:
x1=-b+d2a
x2=-b-d2a
Hvis d = 0 ser vi at x1 = x2, eller med andre ord, at der netop er en rod.
Se eksempler i bogen side 66 – 69.

Bevis for nulpunkter i andengradspolynomium:
Ideen i beviset er at omskrive ligningen ax2+bx+c=0 ved hjælp af kvadratrodsætningen:
(a+b)2=a+b∙a+b=a2+b2+2ab.
Derved får man omskrevet ligningen til en form hvor det ses at beviset må opdeles i tre tilfælde: d<0, d=0 eller d>0.
Jeg starter med at gange ligningen ax2+bx+c=0 med 4a på begge sider af lighedstegnet (vi husker at vi antog at a≠0, så jeg har lov til at gange med 4a). Da fås:
4aax2+4abx+4ac=0
Jeg lægger b2-4ac til på begge sider af lighedstegnet og får:
4aax2+4abx+4ac+b2-4ac=b2-4ac
Som kan reduceres til:
4aax2+4abx+b2=b2-4ac
For nemmere at kunne anvende kvadratrodssætningen omskrives venstresiden:
(2ax)2+b2+2∙2axb=b2-4ac
Her genkender jeg venstresiden som ”kvadratet på en toleddet størrelse” Derfor kan jeg omskrive venstresiden til:
2ax+b∙2ax+b=b2-4ac
(2ax+b)2=b2-4ac
Højresiden kan jeg genkende som diskriminanten (d), så nu har jeg:
(2ax+b)2=d
Jeg har omskrevet ax2+bx+c=0 til ligningen (2ax+b)2=d, og vi kunne gå den anden vej og komme tilbage til ax2+bx+c=0.

Jeg må nu opdele løsningen af ligningen i tre tilfælde:
Hvis d<0:
Venstre siden er et kvadrat (2ax+b)2, så det kan aldrig være et negativt tal. Så hvis diskriminanten d<0 er der ingen løsning på ligningen.
Hvis d=0: Jeg har da ligningen:
(2ax+b)2=0, Da må gælde at:
2ax+b=0, da 0=02, nu kan jeg bare løse ligningen:
2ax=-b
x=-b2a
Hvis d>0: Jeg har altså ligningen:
(2ax+b)2=d, da d>0 er det tilladt at skrive:
(d)2=d Da både negative og positive tal i anden kan give det samme må altså gælde:
2ax+b=-d, eller 2ax+b=d
Hvis jeg kalder den ene løsning for x1og den anden for x2 , kan jeg altså løse ligningerne:
2ax1=-b-d
x1=-b-d2a
Og: 2ax2=-b+d
x2=-b+d2a
Man skal være opmærksom på at en andengradsligning, netop fordi det er en ligning, godt kan tage sig anderledes ud end grundformen ax2 + bx + c = 0.

2. vækstmodeller
Du skal redegøre for karakteristiske egenskaber ved den lineære funktion f(x) = ax + b, samt lineær regression:
Lineære funktioner er karakteriseret ved:
1. At grafen er en ret linje (eller en del af en ret linje)’
2. At regneforskriften er f(x) = ax + b
3. At de absolutte tilvækster for x og for y er proportionale: ∆y=a∙∆x
At de absolutte tilvækster for x og y er proportionale detyder følgende:
Ser man på en absolut tilvækst for x kaldet ∆x og den tilsvarende absolutte tilvækst for y (∆y), så vil ∆y∆x = a.
Hvis a = 1 vil den absolutte tilvækst i x (∆x) være lige så stor som den absolutte tilvækst i y...

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