MATHHX A

MATHHX A

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#1.#2.#3\LWRsiunitxEND {\LWRsiunitxprintdecimalsubtwo #1,,\LWRsiunitxENDTWO \ifblank {#2}{}{{\LWRsiunitxdecimal }\LWRsiunitxprintdecimalsubtwo #2,,\LWRsiunitxENDTWO }}\) \(\newcommand {\LWRsiunitxprintdecimal }[1]{\LWRsiunitxprintdecimalsub #1...\LWRsiunitxEND }\) \(\def \LWRsiunitxnumplus #1+#2+#3\LWRsiunitxEND {\ifblank {#2}{\LWRsiunitxprintdecimal {#1}}{\ifblank {#1}{\LWRsiunitxprintdecimal {#2}}{\LWRsiunitxprintdecimal {#1}\unicode {x02B}\LWRsiunitxprintdecimal {#2}}}\LWRsiunitxdistribunit }\) \(\def \LWRsiunitxnumminus #1-#2-#3\LWRsiunitxEND {\ifblank {#2}{\LWRsiunitxnumplus #1+++\LWRsiunitxEND }{\ifblank {#1}{}{\LWRsiunitxprintdecimal {#1}}\unicode {x02212}\LWRsiunitxprintdecimal {#2}\LWRsiunitxdistribunit }}\) \(\def \LWRsiunitxnumpmmacro #1\pm #2\pm #3\LWRsiunitxEND {\ifblank {#2}{\LWRsiunitxnumminus #1---\LWRsiunitxEND }{\LWRsiunitxprintdecimal {#1}\unicode {x0B1}\LWRsiunitxprintdecimal {#2}\LWRsiunitxdistribunit }}\) \(\def \LWRsiunitxnumpm #1+-#2+-#3\LWRsiunitxEND {\ifblank 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{\morecmidrules }{}\) \(\newcommand {\specialrule }[3]{\hline }\) \(\newcommand {\addlinespace }[1][]{}\) \(\def \LWRsiunitxrangephrase { \protect \mbox {to (numerical range)} }\) \(\def \LWRsiunitxdecimal {.}\)

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:

\[y\leq 6 \quad \textrm {,} \quad x+2y\leq 14\quad \textrm {,}\quad 2x+y \leq 16\textrm {,}\quad x\geq 0 \quad \textrm {og}\quad y\geq 0.\]

  • a) Tegn polygonområdet givet ved ulighederne (du må gerne bruge GeoGebra). Gem dit polygonområde. Du skal bruge det i næste øvelse.

Løsning 3.1.1

  • a)

    (-tikz- diagram)

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:

    (-tikz- diagram)

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 \(f\).

Løsning 3.1.2

  • a) \(x+4y=40\)

  • b)

    (-tikz- diagram)

  • c) Maksimumspunktet er \((2,6)\).

  • d) Maksimumsværdien er \(26\).

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.

    (-tikz- diagram)

    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) \(f(x,y)=x^2-8x+9y^2-54y+98\)

Løsning 3.1.3

  • a) Parabel:

    (-tikz- diagram)

  • b) Ellipse:

    (-tikz- diagram)

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.

  • a) \(f(x)=x^2+x+y^2-3y-234\)

  • b) \(f(x)=-3x^2+5x-y^2+123\)

  • c) \(f(x)=-x^2-3x+y\)

Løsning 3.1.4

  • a) Cirkel

  • b) Ellipse

  • c) Parabel