Bepaal $f(x)$ vanuit eerste beginsels as $f(x) = -x^2 + 4$ - NSC Mathematics - Question 8 - 2016 - Paper 1
Question 8
Bepaal $f(x)$ vanuit eerste beginsels as $f(x) = -x^2 + 4$.
Bepaal die afgelei van:
8.2.1 $y = 3x^2 + 10x$
8.2.2 $f(x) = (x - 3)^2$
Gegee: $f(x) = 2x^3 - 23x^2 +... show full transcript
Worked Solution & Example Answer:Bepaal $f(x)$ vanuit eerste beginsels as $f(x) = -x^2 + 4$ - NSC Mathematics - Question 8 - 2016 - Paper 1
Step 1
Bepaal $f(x)$ vanuit eerste beginsels as $f(x) = -x^2 + 4$
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Answer
Om f(x) vanuit die eerste beginsels te bepaal, gebruik ons die limietvolgorde:
f'(x) = rac{f(x+h) - f(x)}{h}
Die funksie f(x)=−x2+4 impliseer dat ons f(x+h)=−(x+h)2+4 kan skryf! Wanneer ons die limiet neem:
Bereken f(x+h):
f(x+h)=−((x+h)2)+4=−x2−2xh−h2+4
Substitueer in die limiet:
f'(x) = rac{-x^2 - 2xh - h^2 + 4 + x^2 - 4}{h}
Vereenvoudig:
f'(x) = rac{-2xh - h^2}{h} = -2x - h
Neem die limiet as h na 0 gaan:
f′(x)=−2x
Step 2
Bepaal die afgelei van 8.2.1 y = 3x^2 + 10x
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Om die afgeleide te bereken:
y′=6x+10.
Step 3
Bepaal die afgelei van 8.2.2 f(x) = (x - 3)^2
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Die afgeleide van f(x) is gegee deur:
f′(x)=2(x−3)(1)=2(x−3).
Step 4
Bepaal die $x$-koordinaat van die draaipunte van f$
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Om die draaipunte te vind, stel f′(x)=0:
ightarrow x^2 - 19x + 42 = 0$$
Die oplossings is $x = 8/3$ en $x = 5$.
Step 5
Skets die grafiek van f en benom ALLE draaipunte en snypunte met die as duidelik.
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Die grafiek van die funksie f(x) moet die intercepts en draaipunte duidelik weergee. Die snypunte met die x-as en y-as moet duidelik gemerk wees, sowel as die draaipunte op x=8/3 en x=5.
Step 6
Bepaal die koördinate van die $y$-afsnit van die raakylyn aan $f$ wat 'n helling van 40 het en wat raak by 'n punt waar die $x$-koordinaat 'n heelgetal is.
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Answer
Die helling m=40 impliseer die volgende:
y=mx+cy=40x+c
By die punt waar dit raak, moet f(x)=−25 wees:
