R1 2011 vår LØSNING

a)

<tex>O(x)= \frac{500}{x} + 8x^2 \\ O(x) = 500x^{-1} + 8x^2 \\ O'(x) = -500x^{-2}+ 16x = \frac{-500}{x^2} + 16x = \frac{-500 +16x^3}{x^2}</tex>

b)

1)

<tex>f(x)= 3ln(2x) \\ f'(x) = 3 \cdot \frac{1}{(2x)}\cdot 2 = \frac {6}{2x} = \frac 3x</tex>

2)

<tex>g(x) = 3x \cdot e^{x^2} \\ g'(x) = 3e^{x^2}+3x \cdot 2x \cdot e^{x^2} = (3+6x^2)e^{x^2}</tex>

c)

1)

<tex>f(x)= x^3-3x^2-13x+15 \\ f(1)= 1-3-13+15 = 0 \\ \quad(x^3-3x^2-13x+15):(x-1)= x^2-2x-15 \\-(x^3-x^2) \\ \quad \quad\quad \quad \quad-2x^2-13x \\\quad \quad\quad -(-2x^2+2x)\\\quad \quad\quad \quad\quad \quad\quad \quad \quad \quad-15x+15 \\ \quad \quad \quad\quad \quad \quad\quad\quad -(-15x+15) \\\quad \quad\quad \quad \quad\quad \quad\quad \quad \quad\quad \quad\quad \quad \quad\quad \quad\quad \quad \quad 0 </tex>

Faktoriserer svaret fra divisjonen:

<tex>x= \frac{2 \pm \sqrt{4+60}}{2}= \frac{2 \pm 8}{2}\\ x=-3 \vee x= 5</tex>

<tex>f(x) = (x-1)(x+3)(x-5)</tex>

2)

<tex>f(x)\leq 0</tex>

<tex>x \in < \leftarrow, -3] \cup [1,5]</tex>

e)

<tex>lg(a^2b)+lg(ab^2)+lg(\frac{a}{b^3})= \\ lga^2 + lgb + lga + lgb^2 + lga - lgb^3 = \\ 2lga + lgb + lga + 2lgb + lga - 3lgb = 4 lga</tex>