1T 2011 høst LØSNING

DEL EN

a)

<tex> \frac{x^2-25}{x^2+10x+25} = \frac{(x+5)(x-5)}{(x+5)(x+5)} = \frac{x-5}{x+5}</tex>

b)

<tex> 3^{2x-1} = 1 \\ 3^{2x-1} = 3^0 \\ 2x-1 = 0 \\ x= \frac 12</tex>

c)

<tex> \frac{a^{\frac 14} \sqrt a}{(a^{\frac 34})^3 \cdot a^{-2}}= a^{\frac 14 + \frac 24 - \frac 94 + \frac 84} = a^{\frac 12} = \sqrt a </tex>

d)

<tex> A= \frac {gh}{2} \\6= \frac {5h}{2} \\ h = \frac {12}{5} </tex>

e)

Ser fra figuren at:

<tex> f(x) \leq 0 \quad \quad \quad x \in <\leftarrow,1] \cup [3, \rightarrow> \\ f(x) > g(x) \quad \quad x \in <0,5></tex>

f)

<tex>tanC =2 \\ 2= \frac{AB}{AC} \\ AC = 1,5 </tex>

g)

3 Blå, 2 røde, 1 grønn. Totalen er 6.

1) <tex> \frac 56 \cdot \frac 45 = \frac 23 </tex>

2) <tex> \frac 36 \cdot \frac 25 + \frac 26 \cdot \frac 35 = \frac 25 </tex>

h)

<tex> f(x)=x^2+1 \\ \lim_{\Delta x\to\0}\quad\frac{f(x+ \Delta x) - f(x)}{\Delta x} \\ \lim_{\Delta x\to\0}\quad\frac{(x+ \Delta x)^2 +1 - (x^2+1)}{\Delta x} \\ \lim_{\Delta x\to\0}\quad\frac{x^2+2x \Delta x +( \Delta x)^2+1-x^2-1 }{\Delta x} \\ \lim_{\Delta x\to\0}\quad\frac{2x \Delta x +( \Delta x)^2}{\Delta x} \\ \lim_{\Delta x\to\0}\quad\frac{\Delta x(2x + \Delta x}{\Delta x} \\ \lim_{\Delta x\to\0} \quad 2x + \Delta x = 2x</tex>

a)

<tex>f(x) = -x^2+2x-2</tex>

Desom ingen nullpunkter må

<tex>b^2-4ac <0 \\ 2^2-4 \cdot(-1) \cdot (-2) =-4</tex>

Dvs. ingen nullpunkter

b)

<tex>f'(x) = -2x+2 \\ f'(x)=0 \\ -2x+2 =0 ++ x=1</tex>

b)

DEL TO