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Summary Graad 12 wiskunde notas
Graad 12 wiskunde notas
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Graad 12Hierdie is die oorspronklike werk van die
Platinum Graad 12 Wiskunde handboek
wat slegs deur skool_notas opgesom is.
Dit mag nie herverkoop/versprei of
onder n nuwe lisensie verkoop word nie.
, Skool Instagram : @skool_notas
Epos : skoolnotas@icloud.com
Notas
Webtuiste : https://skoolnotas.company.site/
Whatsapp : 081 369 9131
Hierdie notas mag nie herverkoop, gekopieer, versprei of onder ‘n nuwe lisensie verkoop word nie.
Patrone rye en reekse ...................................................................................................................................................... 3
Rekeningkundige rye........................................................................................................................................................... 3
Meetkundige rye ..................................................................................................................................................................... 4
Die som van rekeningkundige reekse ................................................................................................................. 4
Die som van meetkundige reekse ........................................................................................................................... 5
Sigma-notasie ........................................................................................................................................................................... 6
Kwadratiese patrone en kombinasies van rekeningkundige en meetkundige rye .... 7
Funksies en inverse funksies ..................................................................................................................................... 8
Funksies .......................................................................................................................................................................................... 8
Die reguitlyn ................................................................................................................................................................................ 8
Die parabool............................................................................................................................................................................... 8
Die hiperbool.............................................................................................................................................................................. 9
Die eksonensiële grafiek .................................................................................................................................................10
Inverse funksies .......................................................................................................................................................................11
Eksponensiële en logaritmiese funksies..........................................................................................................12
Hersiening van eksponentwette en eksponensiële funksies .........................................................12
Logaritmes en logaritmiese funksies ..................................................................................................................13
Finansies groei en waardevermindering ....................................................................................................... 14
Hersiening Graad 11 : finansies, groei en waardevermindering ................................................. 14
Afleiding en gebruik van formules vir annuïteite .................................................................................... 14
Annuïteite : toepassing en probleemoplossing ........................................................................................ 16
Bereken tydperiodes met behulp van logaritmes .................................................................................. 18
Ontleed beleggings- en leningsopsies ............................................................................................................. 18
Trigonometrie : saamgestelde- en dubbel-identiteite ........................................................................ 20
Hersiening Graad 11 : trigonometrie ....................................................................................................................20
Afleiding van saamgestelde- en dubbel-identiteite ..............................................................................21
Los vergelykings op en bepaal die algemene oplossing................................................................ 23
Trigonometrie : probleemoplossing in twee en drie dimensies .................................................. 25
Probleme in twee dimensies ...................................................................................................................................... 25
Probleem in drie dimensies ........................................................................................................................................ 26
Veelterme (polinome)......................................................................................................................................................27
Faktoriseer derdegraadse veelterme ............................................................................................................... 27
Faktoriseer en los derdegraadse veelterme op deur die res- of faktorstelling te
gebruik .......................................................................................................................................................................................... 28
1
, Skool Instagram : @skool_notas
Epos : skoolnotas@icloud.com
Notas
Webtuiste : https://skoolnotas.company.site/
Whatsapp : 081 369 9131
Hierdie notas mag nie herverkoop, gekopieer, versprei of onder ‘n nuwe lisensie verkoop word nie.
Differensiasie ........................................................................................................................................................................ 30
Limiete ............................................................................................................................................................................................30
Gebruik limiete om die afgeleide van ’n funksie f te definieer ................................................... 32
Difrensiasie van ’n funksie vanuit eerste beginsels ............................................................................. 32
Gebruik die spesifieke reëls van difrensiasie............................................................................................. 33
Bepaal die vergelyking van raaklyne aan grafieke .............................................................................. 34
Die tweede afgeleide ........................................................................................................................................................ 34
Skets grafieke van derdegraadse funksies ................................................................................................. 36
Optimering en veranderingstempo ................................................................................................................... 36
Analitiese meetkunde ....................................................................................................................................................37
Vergelyking van ‘n sirkel ............................................................................................................................................... 37
Vergelyking van ’n raaklyn aan ’n sirkel.......................................................................................................... 38
Euklidiese meetkunde ...................................................................................................................................................39
Hersiening Graad 11 : meetkunde.......................................................................................................................... 39
Gelykvormige veelhoeke ............................................................................................................................................... 40
Die eweredigheidstelling .............................................................................................................................................. 40
Gelykhoekige driehoeke en gelykvormigheid ............................................................................................ 42
Driehoeke met eweredige sye en gelykvormigheid .............................................................................. 42
Pythagoras se Stelling en gelykvormigheid ................................................................................................ 43
Statistiek ................................................................................................................................................................................. 44
Hersiening van skewe en simmetriese data ............................................................................................... 44
Tweeveranderlike (bivariate) data : spredingsdiagramme regressielyne en
korrelasie .................................................................................................................................................................................... 44
Die telbeginsel en waarskynlikheid.................................................................................................................... 47
Hersiening van reëls vir onafhanklike, onderling uitsluitende en komplementêre
gebeurtenisse ......................................................................................................................................................................... 47
Gebruik Venn-diagramme boomdiagramme en gebeurlikheidstabelle ........................... 48
Die fundamentele telbeginsel .................................................................................................................................. 49
Toepassings van die telbeginsel om waarskynlikheidsprobleme op te los ................... 50
2
, Skool Instagram : @skool_notas
Epos : skoolnotas@icloud.com
Notas
Webtuiste : https://skoolnotas.company.site/
Whatsapp : 081 369 9131
Hierdie notas mag nie herverkoop, gekopieer, versprei of onder ‘n nuwe lisensie verkoop word nie.
→ a; a + d; a + 2d; a + 3d; a + 4d; a + 5d; ... a + (n - 1) d
→ A is die waarde van die eerste term.
→ D is die algemene verskil tussen die terme, d = T2 – T1 = T3 – T2 = Tn – Tn-1
→ Tn is die waarde van die term in posisie n, dus Tn = a + (n - 1)d
→ N is die posisie van 'n term en kan slegs 'n positiewe heelgetal wees, ook bekend
as 'n natuurlike getal.
Byvoorbeeld
→ Oorweeg die rekenkundige reeks 3; 7; 11; 15; … 99
→ T1 = 3; T2 = 7; T3 = 11; T4 = 15
d1 = T2 - T1 d2 = T3 - T2 d3 = T4 - T3
=7-3 = 11 - 7 = 15 - 11
=4 =4 =4
→ Aangesien d1 = d2 = d3, het ons 'n algemene verskil van 4.
→ Die eerste term word gegee deur a = 3 en die algemene verskil word gegee deur d
= 4.
→ Ons bepaal die formule vir die nde term in die ry deur a = 3 en d = 4 te vervang in
Tn = a + (n - 1) d.
→ Dit gee ons Tn = 3 + (n - 1)(4)
→ = 3 + 4n - 4
→ = 4n - 1
→ Kontroleer die formule deur n = 1 te vervang om die waarde van T1 te kry, n = 2 om
die waarde van T2 te kry, ensovoorts.
→ As n = 1, dan is T1 = 4 (1) - 1 = 3
→ As n = 2, dan is T2 = 4 (2) - 1 = 7
→ As n = 3, dan is T3 = 4 (3) - 1 = 11
→ Die nde termformule, Tn = 4n - 1, kan gebruik word om die posisie van enige term in
die ry te bepaal as die waarde van die term gegee word.
→ Om te bepaal watter term 'n waarde van 99 het, vervang Tn = 99 in Tn = 4n - 1.
→ 99 = 4n - 1 ⇒ 4n = 100 en n = 25, dus T25 = 99, wat beteken dat die vyf -en -twintigste
term 'n waarde van 99 het.
3
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