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## Inleiding

In Graad 10 het ons die idee van gemiddelde gradiënt ondersoek and gesien dat die gradiënt van sommige funksies verskillend is by verskillende intervalle. In Graad 11 kyk ons verder na die idee van gemiddelde gradiënt, en stel die idee van 'n gradiënt van 'n kurwe by 'n punt bekend.

Ons het gesien dat die gemiddelde gradiënt tussen twee punte op 'n kurwe die gradiënt is van die reguitlyn wat deur twee punte gaan.

Wat gebeur met die gradiënt as ons die posisie van een punt vasstel en die tweede punt nader aan die vaste punt beweeg?

### Ondersoek: Gradiënt by 'n Enkele Punt op 'n Kurwe

Die kurwe gewys is gedefineër deur y=-2x2-5y=-2x2-5. Punt B is vasgestel by koördinate (0, -5). Die posisie van punt A wissel. Voltooi die tabel hieronder deur die yy-koördinate van punt A te bereken vir die gegewe xx-koördinate. Bereken dan die gemiddelde gradiënt tussen punte A en B.

 x A x A y A y A gemiddelde gradiënt -2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Wat gebeur met die gemiddelde gradiënt soos A in die rigting van B beweeg? Wat gebreur met die gemiddelde gradieënt soos A wegbeweeg van B? Wat is die gemiddelde gradiënt wanneer A oorvleuel met B?

In figuur 3 verander die gradiënt van die reguitlyn wat deur punte A en C gaan soos A nader beweeg aan C. By die punt wat A en C oorvleuel gaan die reguitlyn slegs deur een punt op die kurwe. So 'n lyn is bekend as 'n raaklyn aan die kurwe.

Ons stel dan die idee bekend van 'n gradiënt by 'n enkele punt op 'n kurwe. Die gradiënt by 'n punt op 'n kurwe is eenvoudig die gradiënt van die raaklyn aan die kurwe by die gegewe punt.

Kry die gemiddelde gradiënt tussen twee punte P(a;g(a))(a;g(a)) en Q(a+h;g(a+h))(a+h;g(a+h)) op die kurwe g(x)=x2g(x)=x2. Kry dan die gemiddelde gradiënt tussen P(2;g(2))(2;g(2)) en Q(4;g(4))(4;g(4)). Laastens, verduidelik wat gebeur met die gemiddelde gradiënt soos P nader beweeg aan Q.

#### Solution

1. Stap 1. Benoem die xx-koördinate:
x 1 = a x 1 = a
(1)
x 2 = a + h x 2 = a + h
(2)
2. Stap 2. Bepaal die yy -koördinate:

Deur die funksie g(x)=x2g(x)=x2 te gebruik, kry ons:

y 1 = g ( a ) = a 2 y 1 = g ( a ) = a 2
(3)
y 2 = g ( a + h ) = ( a + h ) 2 = a 2 + 2 a h + h 2 y 2 = g ( a + h ) = ( a + h ) 2 = a 2 + 2 a h + h 2
(4)
3. Stap 3. Bereken die gemiddelde gradiënt:
y 2 - y 1 x 2 - x 1 = ( a 2 + 2 a h + h 2 ) - ( a 2 ) ( a + h ) - ( a ) = a 2 + 2 a h + h 2 - a 2 a + h - a = 2 a h + h 2 h = h ( 2 a + h ) h = 2 a + h y 2 - y 1 x 2 - x 1 = ( a 2 + 2 a h + h 2 ) - ( a 2 ) ( a + h ) - ( a ) = a 2 + 2 a h + h 2 - a 2 a + h - a = 2 a h + h 2 h = h ( 2 a + h ) h = 2 a + h
(5)

Die gemiddelde gradiënt tussen P(a;g(a))(a;g(a)) en Q(a+h;g(a+h))(a+h;g(a+h)) op die kurwe g(x)=x2g(x)=x2 is 2a+h2a+h.

4. Stap 4. Bereken die gemiddelde gradiënt tussen P(2;g(2))(2;g(2)) en Q(4;g(4))(4;g(4)) :

Ons kan die resultaat in vergelyking 5 gebruik, maar ons sal moet bepaal wat aa en hh is. Ons doen dit deur te kyk na die definisies van P en Q. Die xx-koërdinaat van P is aa en die xx-koördinaat van Q is a+ha+h. Daarom as ons aanneem dat a=2a=2 en a+h=4a+h=4, dan is h=2h=2.

2 a + h = 2 ( 2 ) + ( 2 ) = 6 2 a + h = 2 ( 2 ) + ( 2 ) = 6
(6)
5. Stap 5. Soos P nader beweeg aan Q... :

soos punt P nader beweeg aan punt Q, word hh kleiner. Dit beteken dat die gemiddelde gradiënt ook kleiner word. As die punt Q oorvleuel met die punt P, is h=0h=0 en die gemiddelde gradiënt word gegee deur 2a2a.

Ons sien nou dat ons die vergelyking kan skryf om die gemiddelde gradiënt op 'n effens anderse manier te bereken. As ons 'n kurwe het gedefineer deur f(x)f(x), dan vir twee punte P en Q met P(a;f(a))(a;f(a)) en Q(a+h;f(a+h))(a+h;f(a+h)), word die gemiddelde gradiënt tussen P en Q gegee deur f(x)f(x):

y 2 - y 1 x 2 - x 1 = f ( a + h ) - f ( a ) ( a + h ) - ( a ) = f ( a + h ) - f ( a ) h y 2 - y 1 x 2 - x 1 = f ( a + h ) - f ( a ) ( a + h ) - ( a ) = f ( a + h ) - f ( a ) h
(7)

Hierdie resultaat is belangrik om die gradiënt te bereken by 'n punt op 'n kurwe en dit sal in groter detail ondersoek word in Graad 12.

## Hoofstukoefeninge

1. Bepaal die gemiddelde gradiënt van die kurwe f(x)=x(x+3)f(x)=x(x+3) tussen x=Â5x=Â5 en x=Â3x=Â3.
2. Sê nou wat jy kan aflei oor die funksie ff tussen x=Â5x=Â5 en x=Â3x=Â3.
1. A(1;3) is 'n punt op f(x)=3x2f(x)=3x2.
1. Bepaal die gradiënt van die kurwe by punt A.
2. Bepaal nou die vergelyking van die raaklyn by A.
2. Given: f(x)=2x2f(x)=2x2.
1. Bepaal die gemiddelde gradiënt tussen x=-2x=-2 en x=1x=1.
2. Bepaal die gradiënt van die kurwe van ff waar x=2x=2.

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