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samenvatting examencommissie 3ASO 2021 Basis : Rekenregels Machte...
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SAMENVATTING WISKUNDE
3ASO - EXAMENCOMMISSIE
ANNAV1 STUVIA | KNOOWY
,Inhoudsopgave
Reële functies................................................................................................................................................. 5
Herhaling............................................................................................................................................................ 5
Het begrip functie ......................................................................................................................................... 5
Definitie van een functie ............................................................................................................................... 5
Voorbeeld ................................................................................................................................................. 5
Notatie .......................................................................................................................................................... 6
Voorbeeld ................................................................................................................................................. 6
Absolute waarde....................................................................................................................................... 6
Kenmerken .................................................................................................................................................... 6
Domein en bereik ..................................................................................................................................... 6
Nulwaarden en nulpunten........................................................................................................................ 7
Voorbeeld ............................................................................................................................................ 7
Symmetrieën ............................................................................................................................................ 7
Symmetrie ten opzichte van de y-as .................................................................................................... 7
Symmetrie ten opzichte van de x-as .................................................................................................... 8
Symmetrie ten opzichte van de oorsprong.......................................................................................... 8
Symmetrie ten opzichte van de eerste bissectrice .............................................................................. 9
Stijgen, dalen of constant ......................................................................................................................... 9
Tekenverandering ................................................................................................................................... 10
Tekenschema wanneer de grafiek gegeven is ................................................................................... 11
Tekenschema zonder grafiek ............................................................................................................. 11
Overzicht ............................................................................................................................................ 11
Even en oneven functies ............................................................................................................................. 12
Voorbeeld ............................................................................................................................................... 12
Punt- en lijnsymmetrie ........................................................................................................................... 14
Voorbeeld lijnsymmetrie ................................................................................................................... 14
Voorbeeld puntsymmetrie................................................................................................................. 14
Verschillende soorten ................................................................................................................................. 15
Oefening ............................................................................................................................................ 15
Veeltermfuncties .............................................................................................................................................. 16
Definitie van een veelterm .......................................................................................................................... 16
Nulwaarden ................................................................................................................................................. 16
Ontbindingstechnieken ............................................................................................................................... 17
Regel van Horner .................................................................................................................................... 17
Voorbeeld A ....................................................................................................................................... 17
Stappenplan regel van Horner ........................................................................................................... 17
Voorbeeld B ....................................................................................................................................... 18
Discriminant ........................................................................................................................................... 19
Gemeenschappelijke factor .................................................................................................................... 19
Afgeleide van een veeltermfunctie ............................................................................................................. 20
Voorbeeld ............................................................................................................................................... 20
Rekenregels afgeleiden ............................................................................................................................... 20
Veelvoud ................................................................................................................................................. 20
Som ......................................................................................................................................................... 20
Verschil ................................................................................................................................................... 20
Product ................................................................................................................................................... 20
Quotiënt ................................................................................................................................................. 20
Uitbreiding .............................................................................................................................................. 20
Oefeningen .................................................................................................................................................. 21
Het differentiequotiënt ............................................................................................................................... 22
Voorbeeld ............................................................................................................................................... 22
De afgeleide van een veeltermfunctie en de helling van de grafiek in een punt ........................................ 23
Verloop van een veeltermfunctie................................................................................................................ 23
1
, Voorbeeld ............................................................................................................................................... 23
a) Domein van de functie ................................................................................................................... 23
b) Nulwaarden van de functie ........................................................................................................... 23
c) Nulwaarden van de afgeleide functie ............................................................................................ 24
d) Tekenschema van de afgeleide functie ......................................................................................... 24
e) Stijgen en dalen van de functie...................................................................................................... 24
Oefening ................................................................................................................................................. 25
a) Domein van de functie ................................................................................................................... 25
b) Nulwaarden van de functie ........................................................................................................... 25
c) Nulwaarden van de afgeleide functie ............................................................................................ 25
d) Tekenschema van de afgeleide functie ......................................................................................... 25
e) Stijgen en dalen van de functie...................................................................................................... 25
Toepassingen van het begrip afgeleide in domeinen buiten de wiskunde ................................................. 26
Plaatsfunctie ........................................................................................................................................... 26
Snelheidsfunctie ..................................................................................................................................... 26
Versnellingsfunctie ................................................................................................................................. 26
Voorbeeld A ....................................................................................................................................... 26
Voorbeeld B ....................................................................................................................................... 26
Extremawaarden en het stijgen en dalen ................................................................................................... 27
Vraagstukken .......................................................................................................................................... 27
Voorbeeld .......................................................................................................................................... 27
Werkschema ...................................................................................................................................... 28
a) Wiskundige formulering van het probleem .............................................................................. 28
b) Extremalisatie van f .................................................................................................................. 28
c) Antwoord en controle ............................................................................................................... 28
Exponentiële en logaritmische functies ............................................................................................................ 29
Machten met natuurlijke exponenten ........................................................................................................ 29
Definitie .................................................................................................................................................. 29
Rekenregels ............................................................................................................................................ 29
Voorbeeld ............................................................................................................................................... 29
Machten met gehele exponenten ............................................................................................................... 30
Definitie .................................................................................................................................................. 30
Rekenregels ............................................................................................................................................ 30
Machten met rationale exponenten ........................................................................................................... 30
Voorbeeld ............................................................................................................................................... 30
N-de machtswortels .................................................................................................................................... 30
Voorbeeld ............................................................................................................................................... 30
Rekenregels ............................................................................................................................................ 30
De exponentiële functie .............................................................................................................................. 31
Definitie .................................................................................................................................................. 31
Logaritmische functie .................................................................................................................................. 31
Voorbeeld ............................................................................................................................................... 31
Definitie .................................................................................................................................................. 32
Rekenregels ............................................................................................................................................ 32
Overgang naar andere grondtallen ........................................................................................................ 33
Voorbeeldoefeningen ........................................................................................................................ 33
Voorkennis .................................................................................................................................................. 34
Lineaire en exponentiële groei .................................................................................................................... 35
Exponentiële vergelijkingen ........................................................................................................................ 36
Verband tussen exponentiële en logaritmische functie .............................................................................. 37
Goniometrische functies................................................................................................................................... 38
Hoeken in graden en radialen ..................................................................................................................... 38
De grondbegrippen van hoeken ............................................................................................................. 38
Definitie van een hoek ....................................................................................................................... 38
De goniometrische cirkel ................................................................................................................... 38
De radiaal ........................................................................................................................................... 38
2
, Verwante hoeken en hun goniometrische getallen .................................................................................... 39
Sinus en cosinus...................................................................................................................................... 39
Tangens en cotangens ............................................................................................................................ 39
Tekens en bijzondere waarden van de goniometrische getallen ........................................................... 39
Bijzondere hoeken .................................................................................................................................. 39
Verwante hoeken ................................................................................................................................... 40
De goniometrische functies ........................................................................................................................ 40
Periodieke functies ................................................................................................................................. 40
Voorbeeld .......................................................................................................................................... 40
De sinusfunctie ....................................................................................................................................... 41
De algemene sinusfunctie ...................................................................................................................... 41
Definitie ............................................................................................................................................. 41
Goniometrische vergelijkingen.................................................................................................................... 43
Elementaire goniometrische vergelijkingen ........................................................................................... 43
Statistiek ...................................................................................................................................................... 44
Definitie, gemiddelde, mediaan en modus ....................................................................................................... 44
Definitie ....................................................................................................................................................... 44
Gemiddelde ................................................................................................................................................. 45
Mediaan ...................................................................................................................................................... 45
Modus ......................................................................................................................................................... 46
Soorten variabelen ........................................................................................................................................... 47
Kwalitatieve en kwantitatieve variabelen ................................................................................................... 47
Discrete en continue variabelen ................................................................................................................. 48
Meetniveaus: nominaal, ordinaal .................................................................................................................... 48
Nominaal niveau ......................................................................................................................................... 48
Ordinaal niveau ........................................................................................................................................... 49
Steekproef en populatie ................................................................................................................................... 49
Populatie ..................................................................................................................................................... 49
Steekproef ................................................................................................................................................... 49
Centrummaten en spreidingsmaten................................................................................................................. 50
Centrummaten ............................................................................................................................................ 50
Gemiddelde ............................................................................................................................................ 50
Mediaan.................................................................................................................................................. 50
Modus..................................................................................................................................................... 51
Spreidingsmaten ......................................................................................................................................... 51
Variantie ................................................................................................................................................. 51
Standaardafwijking ................................................................................................................................. 52
Normaalverdeling ............................................................................................................................................ 53
Normale dichtheidsfuncties en de standaardnormale verdeling ..................................................................... 54
De standaardnormale verdeling .................................................................................................................. 54
De normale dichtheidsfunctie ..................................................................................................................... 54
Voorbeeld ............................................................................................................................................... 55
Samenvatting .............................................................................................................................................. 55
Opmerking .............................................................................................................................................. 55
Normale verdelingen vergelijken ..................................................................................................................... 56
Voorbeeld .................................................................................................................................................... 56
Relatieve frequentie en kans ............................................................................................................................ 57
Uitdaging ..................................................................................................................................................... 57
Absolute, totale en relatieve frequentie ..................................................................................................... 57
Absolute frequentie................................................................................................................................ 57
Totale frequentie .................................................................................................................................... 57
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