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Rapport om differentialregning

Rapporten skal behandle følgende punkter:

1. En oversigt over de vigtigste begreber, sætninger og metoder i differentialregning:

• Hvad udtrykker en differenskvotient? Opskriv det generelle udtryk for differenskvotienten og illustrér den grafiske betydning.

• Hvad udtrykker en differentialkvotient? Opskriv det generelle udtryk for differentialkvotienten og illustrér den grafiske betydning.

2. Vælg tre af de sætninger inden for differentialregning, som er bevist herhenne, og hvor I synes, at beviserne er meget forskellige (I skal have jeres valg godkendt af læreren). For hver af de tre sætninger skal I gennemgå beviset, dvs.:

• Forklar hvad sætningen siger

• Forklar ideen/metoden i beviset

• Gør rede for de definitioner, som sætningen bygger på.

Gør desuden rede for de vigtigste forskelle og ligheder mellem de tre beviser.

Beviste sætninger: Sætn. 1 (s. 15), sætn. 2 (s. 16), sætn. 3 og 4 (s. 17), sætn. 5 (s. 18), sætn. 17 (s. 194), sætn. 18 (195) og sætn. 19 (s.196)

3. Redegør – ved hjælp af opgaven på næste side – for hvordan differentialregning kan benyttes til optimering.

Rapportens omfang forventes at være på ca. 4-5 sider.
Design af chokoladeæsker

Du er ansat i et rådgivningsfirma, som netop har fået en opgave fra en chokoladefabrikant – se vedlagte brev. Du bliver af din chef sat til at løse opgaven og målet er at chokoladefabrikanten bliver tilfreds. Din opgave er dels at udarbejde et brev til chokoladefabrikanten, hvor han får løsningen på sit problem og dels at udarbejde en rapport til din chef, hvori du redegør for problemet der skulle løses og hvordan du løste det.

1. Brevet til chokoladefabrikanten. Brevet skal indeholde et klart svar på, hvilke dimensioner chokoladeæsken skal have, for at have så stort rumfang som muligt. Husk at chokoladefabrikanten ikke kender meget til matematik og derfor ikke er interesseret i, hvordan opgaven bliver løst. Han er kun interesseret i selve løsningen formuleret i et sprog, han kan forstå – altså dagligdags sprog.

2. Rapporten til chefen. Chefen har ansvaret for, at rådgivningsfirmaet altid udarbejder de bedste løsninger. Derfor skal hver medarbejder lave en kortfattet rapport over de problemer, som de har arbejdet med. Denne rapport skal indeholde følgende punkter:

• Beskrivelse af problemet

• Valg af løsningsmetoden samt beskrivelse af denne

• Løsningen

Hint: En æske uden låg foldes af et stykke karton. Figuren nedenfor viser kartonets mål, hvor x (cm) angiver æskens højde.

1. Gør rede for, at æskens rumfang R(x), målt i cm3, er bestemt ved:

[pic]

og gør rede for, at [pic]

2. Bestem x, så æskens rumfang bliver størst muligt (optimeringsopgave).
Snask A/S
Slikvej 25
7654 Nørre Flødebo

Til Rådgivningsfirmaet – Godt-Råd-Igen A/S
Idévej 17
5500 Intelligensstrup

Til brug for fremstilling af æsker til fyldt chokolade har jeg indkøbt rigtigt mange stykker guldkarton. Hvert stykke måler 20 cm i bredden og er 28 cm langt. Disse skal foldes til en firkantet æske. Der skal ikke være låg til, da de skal pakkes ind i cellofan. Jeg er interesseret i at fremstile æsker til min ”Supersize Deluxe”-blanding og er derfor interesseret i at vide, hvordan dimensionerne af en æske skal være, så æskerne får så stort rumfang som muligt.

Jeg håber, at De kan løse mit problem hurtigt og godt.

De bedste hilsener fra

C. H. Okoladesen
Fabrikant.

----------------------- f (x0)

f (x)

(x

(f

x0

x

Differenskvotient og sekanthældning:

[pic]

Differentialkvotient og tangenthældning:

[pic]

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