R2 2012 vår LØSNING

Oppgave 1

a)

1)

<tex> f(x) = 3sin(2x)\\ u=2x, \quad u' = 2 \\ f'(x) = 2 \cdot 3 cos(2x) \\ f'(x) = 6cos(2x)</tex>

2)

<tex>g(x) = x^2sinx \\ u= x^2, \quad v = sinx \\ g'(x) = 2xsinx + x^2cosx =x(2sinx+xcosx)</tex>

3)

<tex>k(x) = 5cos(\frac{\pi}{12}x-2)+7 \\ k'(x) = - \frac{5 \pi}{12} sin( \frac{\pi}{13}x-2)</tex>

b)

<tex>\int xe^{2x}dx = \frac 12 x e^{2x} - \int \frac12 e^{2x}dx \\ = \frac 12 x e^{2x} - \frac 14 e^{2x} +C \\ = \frac 14 e^{2x}(2x-1) + C</tex>

c)

<tex>\int^7_3 \frac{2x}{x^2-4}dx \\ \frac{2x}{x^2+4} = \frac{A}{x-2}+ \frac{B}{x-2} \\ 2x= A(x+2) + B(x-2) \\ x=2 \Rightarrow A = 1 \\ x= -2 \Rightarrow B=1 \\ \int^7_3 \frac{2x}{x^2-4}dx =\int^7_3 \frac{1}{x-2}dx + \int^7_3 \frac {1}{x+2}dx \\ = [ln|x-2|]^7_3 + [ln|x+2|]^7_3 \\ = ln5-ln1+ln9-ln5 = ln3^2 = 2ln3</tex>

d)

<tex> y' -2y = 3 \\ y' \cdot e^{-2x}-2ye^{-2x} = 3e^{-2x} \\ (ye^{-2x})' =3e^{-2x} \\ ye^{-2x} = - \frac 32 e^{-2x} + C \\ y = - \frac 32 +Ce^{2x} \\y(o) = 8 \Rightarrow 8 = - \frac 32 + C \Rightarrow C = \frac{19}{2} \\ y = - \frac 32 + \frac{19}{2}e^{2x}</tex>

e)

<tex>1+e^{-x} + e^{-2x}+ .... \quad x > 0</tex>

1)

<tex>k= \frac{e^{-x}}{1} = \frac{e^{-2x}}{e^{-x}} = e^{-x}</tex>

<tex> -1 < e^{-x}<1 </tex>

Dvs: rekken konvergerer.

2)

<tex>S = \frac{a_1}{1-k} = \frac{1}{1-e^{-x}} = \frac {e^x}{e^x -1}</tex>

Oppgave 3

<tex>f(x) = x \cdot e^x</tex>

a)

<tex>f'(x) = </tex>