S1 eksempeloppgave 2015 vår LØSNING

Oppgave 1

a)

$f(x)=3x^2-4x+2 \\ f ´(x)= 6x-4$

b)

$g(x)= 3x^3-3 \\ g ´(x)= 9x^2 \\ g ´(2) = 9 \cdot 4 = 36$

Oppgave 2

a)

$\frac{2^{-1}\cdot a \cdot b^{-1}}{4^{-1} \cdot a^{-2} \cdot b^2} = \frac{4 a^3}{2b^3} = 2 (\frac ab)^3$

b)

$lg(a^2b)+lg(ab^2)+lb(\frac{a}{b^3}) = 2lga+ lgb + lga + 2lgb + lga - 3lgb = 4lga$

c)

$ \frac{3a^2-75}{6a+30} = \frac{3(a+5)(a-5)}{6(a+5)}= \frac{a-5}{2}$

Oppgave 3

a)

$\frac{61^2-39^2}{51^2-49^2} = \\ \frac{(61+39)(61-39)}{(51+49)(51-49)} =\\ \frac{100 \cdot 22}{100 \cdot 2} =11 $

b)

$1997 \cdot 2003 - 1993 \cdot 2007 = \\ (2000 - 3)(2000 + 3) - ( 2000 - 7)( 2000+7) \\ 2000^2 -3^2 - 2000^2 + 7^2 \\ -9 + 49 = 40$

Oppgave 4

$f(x) = g(x) \\ x^2-x-2 = x+1 \\ x^2-2x-3 = 0 \\ x= \frac{2 \pm \sqrt{4+12}}{2} \\ x=-1 \vee x= 3$