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Gemiddelde gradient: Reguit lyn en parabool

Module by: Free High School Science Texts Project. E-mail the author

Inleiding

Die gradiënt van 'n reguitlyngrafiek word bereken met:

y 2 - y 1 x 2 - x 1 y 2 - y 1 x 2 - x 1
(1)

vir 2 punte (x1;y1)(x1;y1) en (x2;y2)(x2;y2) op die grafiek.

Ons kan nou die gemiddelde gradiënt tussen 2 punte (x1;y1)(x1;y1) en (x2;y2)(x2;y2) bepaal, selfs al word hulle gedefinieer deur 'n funksie wat nie 'n reguitlyn is nie, met:

y 2 - y 1 x 2 - x 1 y 2 - y 1 x 2 - x 1
(2)

Dit is dieselfde as Equation 1.

Reguitlynfunksies

Ondersoek: Gemiddelde Gradiënt - Reguitlynfunksie

Voltooi die tabel deur die gemiddelde gradiënt oor die aangeduide intervalle te bereken vir die funksie f(x)=2x-2f(x)=2x-2. Let daarop dat (x1x1;y1y1) die koördinate is van die eerste punt en dat (x2x2;y2y2) die koördinate is van die tweede punt. So, vir AB is (x1x1;y1y1) die koördinate van punt A en (x2x2;y2y2) is die koördinate van punt B.

Table 1
  x 1 x 1 x 2 x 2 y 1 y 1 y 2 y 2 y 2 - y 1 x 2 - x 1 y 2 - y 1 x 2 - x 1
A-B          
A-C          
B-C          
Figure 1
Figure 1 (MG10C12_001.png)

Wat let jy op van die gradiënte oor elke interval?

Die gemiddelde gradiënt van 'n reguitlynfunksie is dieselfde oor enige twee intervalle in die funksie.

Paraboliese Funksie

Ondersoek : Gemiddelde Gradiënt - Paraboliese Funksie

Vul die tabel in deur die gemiddelde gradiënt oor die aangeduide intervalle te bereken vir die funksie f(x)=2x-2f(x)=2x-2:

Table 2
  x 1 x 1 x 2 x 2 y 1 y 1 y 2 y 2 y 2 - y 1 x 2 - x 1 y 2 - y 1 x 2 - x 1
A-B          
B-C          
C-D          
D-E          
E-F          
F-G          

Wat let jy op van die gemiddelde gradiënt oor elke interval? Wat kan jy sê oor die gemiddelde gradiënte tussen A en D in vergelyking met die gemiddelde gradiënte tussen D en G?

Figure 2
Figure 2 (MG10C12_002.png)

Die gemiddelde gradiënt van 'n paraboliese funksie hang af van die interval en is die gradiënt van 'n reguitlyn wat deur die betrokke punte op daardie interval loop.

Byvoorbeeld, in Figure 3 is die verskeie punte verbind deur reguitlyne. Die gemiddelde gradiënte tussen die betrokke punte is dan die gradiënte van die reguitlyne wat deur daardie punte loop.

Figure 3: Die gemiddelde gradiënt tussen twee punte op 'n kurwe is die gradiënt van die reguitlyn wat deur die punte loop.
Figure 3 (MG10C12_003.png)

Metode: Gemiddelde Gradiënt

Gegee, die vergelyking van 'n kromme en twee punte (x1x1; x2x2):

  1. Skryf die vergelyking van die kromme in die vorm y=...y=....
  2. Bereken y1y1 deur x1x1 in die vergelyking vir die kromme in te stel.
  3. Bereken y2y2 deur x2x2 in die vergelyking vir die kromme in te stel.
  4. Bereken die gemiddelde gradiënt deur gebruik te maak van:
    y2-y1x2-x1y2-y1x2-x1
    (3)

Exercise 1: Gemiddelde Gradiënt

Vind die gemiddelde gradiënt van die kromme y=5x2-4y=5x2-4 tussen die punte x=-3x=-3 en x=3x=3.

Solution
  1. Step 1. Merk (benoem) die punte :

    Merk die punte as volg:

    x 1 = - 3 x 1 = - 3
    (4)
    x 2 = 3 x 2 = 3
    (5)

    om dit makliker te maak om die gradiënt te bereken.

  2. Step 2. Bereken die yy koördinate :

    Ons gebruik die vergelyking van die kromme om die yy-waarde van die kromme by x1x1 en x2x2 te vind.

    y 1 = 5 x 1 2 - 4 = 5 ( - 3 ) 2 - 4 = 5 ( 9 ) - 4 = 41 y 1 = 5 x 1 2 - 4 = 5 ( - 3 ) 2 - 4 = 5 ( 9 ) - 4 = 41
    (6)
    y 2 = 5 x 2 2 - 4 = 5 ( 3 ) 2 - 4 = 5 ( 9 ) - 4 = 41 y 2 = 5 x 2 2 - 4 = 5 ( 3 ) 2 - 4 = 5 ( 9 ) - 4 = 41
    (7)
  3. Step 3. Bereken die gemiddelde gradiënt :
    y 2 - y 1 x 2 - x 1 = 41 - 41 3 - ( - 3 ) = 0 3 + 3 = 0 6 = 0 y 2 - y 1 x 2 - x 1 = 41 - 41 3 - ( - 3 ) = 0 3 + 3 = 0 6 = 0
    (8)
  4. Step 4. Skryf die finale antwoord neer :

    Die gemiddelde gradiënt tussen x=-3x=-3 en x=3x=3 op die kromme y=5x2-4y=5x2-4 is 0.

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