MATHHX A
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3.1 Introduktion til kvadratisk programmering
I kvadratisk programmering kigger vi på funktioner af typen:
\[ f(x,y)=ax^2+bx+cy^2+dy+e\quad , \quad \textrm {hvor } a \textrm { og } c \textrm { ikke begge er nul}.\]
Den slags funktioner kaldes kvadratiske funktioner i to variable fordi de indeholder to variable, hvoraf en eller begge indgår i anden potens.
Kvadratisk programmering er lidt som lineære programmering – bare mere besværligt, så lad os da lige starte med at repetere noget lineær programmering.
Repetition af lineær programmering
Øvelse 3.1.1
Betragt ulighederne:
\[y\leq 6 \quad \textrm {,} \quad x+2y\leq 14\quad \textrm {,}\quad 2x+y \leq 16\textrm {,}\quad x\geq 0 \quad \textrm {og}\quad y\geq 0.\]
Vi husker at man regner niveaulinjen \(N(t)\) ved at opstille ligningen:
\[f(x,y)=t.\]
Den fremkomne ligning kan så tastes i GeoGebra.
-
Eksempel 3.1.1
Betragt forskriften for en lineære funktion i to variable:
\[f(x,y)=x+4y.\]
Vi vil tegne niveaulinjen \(N(30)\) ind i polygonrådet fra ovenstående øvelse. Vi skal altså opstille ligningen:
\[f(x,y)=30.\]
Vi indsætter vores forskrift i ligningen:
\[x+4y=30.\]
Denne ligning kan vi taste ind i GeoGebra og man får noget i stil med:
Øvelse 3.1.2
Med udgangspunkt i eksemplet ovenover:
-
a) Opstil ligningen for niveaulinjen \(N(40)\)
-
b) Indtegn \(N(30)\) og \(N(40)\) i dit polygonområde fra øvelse 3.1.1.
-
c) Bestem ud fra niveaulinjerne maksimumspunktet for \(f\).
-
d) Bestem maksimumsværdien for \(f\).
Niveaukurver for kvadratiske funktioner i to variable
I forbindelse med kvadratiske funktioner i to variable, kan vi også opstille ligningen:
\[f(x,y)=t.\]
Dette vil dog ikke give os en linje (se nedenstående øvelse). Derfor taler vi om niveaukurver i stedet for niveaulinjer.
-
Eksempel 3.1.2
Lad
\[f(x, y)=x^2 - 10x + y^2 - 8y + 41.\]
Vi vil nu bestemme en ligning for \(N(4)\):
\[f(x,y)=4\]
Vi indsætter forskriften og får den ønskede ligning.
\[x^2 - 10x + y^2 - 8y + 41=4\]
Hvad mon den ligning udtrykker? Lad os prøve at taste den ind i en graftegner.
Det gav en cirkel. Interessant.
Man kunne nu tro at niveaukurver for kvadratiske funktioner altid er cirkler, men som vi skal se i næste øvelse, er der flere muligheder.
Øvelse 3.1.3
Tegn \(N(10)\) i GeoGebra for følgende kvadratiske funktioner, og beskriv hvilken figur niveaukurven danner.
-
a) \(f(x,y)=2x^2 - 24x - y + 85\)
-
b) \(f(x,y)=x^2-8x+9y^2-54y+98\)
Det viser sig at formen på niveaukurven afhænger af koefficienterne \(a\) og \(c\) i forskriften for den kvadratiske funktion:
\[ f(x,y)=ax^2+bx+cy^2+dy+e.\]
Der gælder følgende resultat som vi vil bevise senere:
. |
Værdi af \(a\) og \(c\) |
Niveaukurverne er |
\(c=0\) |
Parabler |
\(a=c\) |
Cirkler |
\(a\) og \(c\) har samme fortegn |
Ellipser |
|
|
Der er selvfølgelig også andre muligheder for \(a\) og \(c\), men mulighederne i tabellen udgør dem vi møder her på HHX. Vi bemærker at en cirkel bare er en særligt pæn ellipse, så vi møder kun to grundlæggende situationer her:
Enten er \(c=0\) og ellers har \(a\) og \(c\) samme fortegn.
Øvelse 3.1.4
Bestem ud fra forskriften (ikke noget GeoGebra), hvilken type niveaukurver funktionen har.
Løsning 3.1.4
-
a) Cirkel
-
b) Ellipse
-
c) Parabel