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Rad 1: Rad 1:
__NOTOC__ __NOTOC__
-[[övn 2.3]]+[[övn 2]]
==Övning 1.1:1== ==Övning 1.1:1==
Rad 695: Rad 695:
<td class="ntext">f)</td> <td class="ntext">f)</td>
<td class="ntext" width="33%">$3^{40}>2^{56}$</td> <td class="ntext" width="33%">$3^{40}>2^{56}$</td>
-</tr> 
-<tr><td height="5px"/></tr> 
-</table> 
-</div> 
- 
-==Övning 2.1:1== 
-<div class="ovning"> 
-Utveckla 
-<table width="100%" cellspacing="10px"> 
-<tr align="left"> 
-<td class="ntext">a)</td> 
-<td class="ntext" width="33%">$3x(x-1)$</td> 
-<td class="ntext">b)</td> 
-<td class="ntext" width="33%">$(1+x-x^2)xy$</td> 
-<td class="ntext">c)</td> 
-<td class="ntext" width="33%">$-x^2(4-y^2)$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">d)</td> 
-<td class="ntext" width="33%">$x^3y^2\left(\displaystyle \frac{1}{y} - \frac{1}{xy}+1\right)$</td> 
-<td class="ntext">e) </td> 
-<td class="ntext" width="33%">$(x-7)^2$</td> 
-<td class="ntext">f)</td> 
-<td class="ntext" width="33%">$(5+4y)^2$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">g)</td> 
-<td class="ntext" width="33%">$(y^2-3x^3)^2$</td> 
-<td class="ntext">h)</td> 
-<td class="ntext" width="33%">$(5x^3+3x^5)^2$</td> 
-</tr> 
-<tr><td height="5px"/></tr> 
-</table> 
-</div> 
- 
-<div class="svar"> 
-<table width="100%" cellspacing="10px"> 
-<tr align="left"> 
-<td class="ntext">a)</td> 
-<td class="ntext" width="33%">$3x^2-3x$</td> 
-<td class="ntext">b)</td> 
-<td class="ntext" width="33%">$xy+x^2y-x^3y$</td> 
-<td class="ntext">c)</td> 
-<td class="ntext" width="33%">$-4x^2+x^2y^2$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">d)</td> 
-<td class="ntext" width="33%">$x^3y-x^2y+x^3y^2$</td> 
-<td class="ntext">e)</td> 
-<td class="ntext" width="33%">$x^2-14x+49$</td> 
-<td class="ntext">f)</td> 
-<td class="ntext" width="33%">$16y^2+40y+25$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">g)</td> 
-<td class="ntext" width="33%">$9x^6-6x^3y^2+y^4$</td> 
-<td class="ntext">h)</td> 
-<td class="ntext" width="33%">$9x^{10}+30x^8+25x^6$</td> 
-</tr> 
-<tr><td height="5px"/></tr> 
-</table> 
-</div> 
- 
-==Övning 2.1:2== 
-<div class="ovning"> 
-Utveckla och förenkla så långt som möjligt 
-<table width="100%" cellspacing="10px"> 
-<tr align="left"> 
-<td class="ntext">a)</td> 
-<td class="ntext" width="50%">$(x-4)(x-5)-3x(2x-3)$</td> 
-<td class="ntext">b)</td> 
-<td class="ntext" width="50%">$(1-5x)(1+15x)-3(2-5x)(2+5x)$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">c)</td> 
-<td class="ntext" width="50%">$(3x+4)^2-(3x-2)(3x-8)$</td> 
-<td class="ntext">d)</td> 
-<td class="ntext" width="50%">$(3x^2+2)(3x^2-2)(9x^4+4)$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">e)</td> 
-<td class="ntext" width="50%">$(a+b)^2+(a-b)^2$</td> 
-</tr> 
-<tr><td height="5px"/></tr> 
-</table> 
-</div> 
- 
-<div class="svar"> 
-<table width="100%" cellspacing="10px"> 
-<tr align="left"> 
-<td class="ntext">a)</td> 
-<td class="ntext" width="50%">$-5x^2+20$</td> 
-<td class="ntext">b)</td> 
-<td class="ntext" width="50%">$10x-11$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">c)</td> 
-<td class="ntext" width="50%">$54x$</td> 
-<td class="ntext">d)</td> 
-<td class="ntext" width="50%">$81x^8-16$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">e)</td> 
-<td class="ntext" width="50%">$2a^2+2b^2$</td> 
-</tr> 
-<tr><td height="5px"/></tr> 
-</table> 
-</div> 
- 
-==Övning 2.1:3== 
- 
-<div class="ovning"> 
-Faktorisera s&aring; l&aring;ngt som m&ouml;jligt 
-<table width="100%" cellspacing="10px"> 
-<tr align="left"> 
-<td class="ntext">a)</td> 
-<td class="ntext" width="33%">$x^2-36$</td> 
-<td class="ntext">b)</td> 
-<td class="ntext" width="33%">$5x^2-20$</td> 
-<td class="ntext">c)</td> 
-<td class="ntext" width="33%">$x^2+6x+9$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">d)</td> 
-<td class="ntext" width="33%">$x^2-10x+25$</td> 
-<td class="ntext">e)</td> 
-<td class="ntext" width="33%">$18x-2x^3$</td> 
-<td class="ntext">f)</td> 
-<td class="ntext" width="33%">$16x^2+8x+1$</td> 
-</tr> 
-<tr><td height="5px"/></tr> 
-</table> 
-</div> 
- 
-<div class="svar"> 
-<table width="100%" cellspacing="10px"> 
-<tr align="left"> 
-<td class="ntext">a)</td> 
-<td class="ntext" width="33%">$(x+6)(x-6)$</td> 
-<td class="ntext">b)</td> 
-<td class="ntext" width="33%">$5(x+2)(x-2)$</td> 
-<td class="ntext">c)</td> 
-<td class="ntext" width="33%">$(x+3)^2$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">d)</td> 
-<td class="ntext" width="33%">$(x-5)^2$</td> 
-<td class="ntext">e)</td> 
-<td class="ntext" width="33%">$-2x(x+3)(x-3)$</td> 
-<td class="ntext">f)</td> 
-<td class="ntext" width="33%">$(4x+1)^2$</td> 
-</tr> 
-<tr><td height="5px"/></tr> 
-</table> 
-</div> 
- 
-==Övning 2.1:4== 
-<div class="ovning"> 
-Bestäm koefficienterna framför $\,x\,$ och $\,x^2\,$ när följande uttryck utvecklas 
-<table width="100%" cellspacing="10px"> 
-<tr align="left"> 
-<td class="ntext">a)</td> 
-<td class="ntext" width="100%">$(x+2)(3x^2-x+5)$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">b)</td> 
-<td class="ntext" width="100%">$(1+x+x^2+x^3)(2-x+x^2+x^4)$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">c)</td> 
-<td class="ntext" width="100%">$(x-x^3+x^5)(1+3x+5x^2)(2-7x^2-x^4)$</td> 
-</tr> 
-<tr><td height="5px"/></tr> 
-</table> 
-</div> 
- 
-<div class="svar"> 
-<table width="100%" cellspacing="10px"> 
-<tr align="left"> 
-<td class="ntext">a)</td> 
-<td class="ntext" width="100%">$5\,$ framf&ouml;r $\,x^2\,$, $\,3\,$ framf&ouml;r $\,x$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">b)</td> 
-<td class="ntext" width="100%">$2\,$ framf&ouml;r $\,x^2\,$, $\,1\,$ framf&ouml;r $\,x$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">$\textrm{c) }$</td> 
-<td class="ntext" width="100%">$6\,$ framf&ouml;r $\,x^2\,$, $\,2\,$ framf&ouml;r $\,x$</td> 
-</tr> 
-<tr><td height="5px"/></tr> 
-</table> 
-</div> 
- 
-==Övning 2.1:5== 
-<div class="ovning"> 
-Förenkla så långt som möjligt 
-<table width="100%" cellspacing="10px"> 
-<tr align="left"> 
-<td class="ntext">a)</td> 
-<td class="ntext" width="50%">$\displaystyle \frac{1}{x-x^2}-\displaystyle \frac{1}{x}$</td> 
-<td class="ntext">b)</td> 
-<td class="ntext" width="50%">$\displaystyle \frac{1}{y^2-2y}-\displaystyle \frac{2}{y^2-4}$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">c)</td> 
-<td class="ntext" width="50%">$\displaystyle \frac{(3x^2-12)(x^2-1)}{(x+1)(x+2)}$</td> 
-<td class="ntext">d)</td> 
-<td class="ntext" width="50%">$\displaystyle \frac{(y^2+4y+4)(2y-4)}{(y^2+4)(y^2-4)}$</td> 
-</tr> 
-</table> 
-</div> 
- 
-<div class="svar"> 
-<table width="100%" cellspacing="10px"> 
-<tr align="left"> 
-<td class="ntext">a)</td> 
-<td class="ntext" width="50%">$\displaystyle \frac{1}{1-x}$</td> 
-<td class="ntext">b)</td> 
-<td class="ntext" width="50%">$-\displaystyle \frac{1}{y(y+2)}$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">c)</td> 
-<td class="ntext" width="50%">$3(x-2)(x-1)$</td> 
-<td class="ntext">d)</td> 
-<td class="ntext" width="50%">$\displaystyle \frac{2(y+2)}{y^2+4}$</td> 
-</tr> 
-</table> 
-</div> 
- 
-==Övning 2.1:6== 
-<div class="ovning"> 
-Förenkla så långt som möjligt 
-<table width="100%" cellspacing="10px"> 
-<tr align="left"> 
-<td class="ntext">a)</td> 
-<td class="ntext" width="50%">$\left(x-y+\displaystyle\frac{x^2}{y-x}\right)$ $\left(\displaystyle\frac{y}{2x-y}-1\right)$</td> 
-<td class="ntext">b)</td> 
-<td class="ntext" width="50%">$\displaystyle \frac{x}{x-2}+\displaystyle \frac{x}{x+3}-2$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">c)</td> 
-<td class="ntext" width="50%">$\displaystyle \frac{2a+b}{a^2-ab}-\frac{2}{a-b}$</td> 
-<td class="ntext">d)</td> 
-<td class="ntext" width="50%">$\displaystyle\frac{a-b+\displaystyle\frac{b^2}{a+b}}{1-\left(\displaystyle\frac{a-b}{a+b}\right)^2}$</td> 
-</tr> 
-<tr><td height="5px"/></tr> 
-</table> 
-</div> 
- 
-<div class="svar"> 
-<table width="100%" cellspacing="10px"> 
-<tr align="left"> 
-<td class="ntext">a)</td> 
-<td class="ntext" width="50%">$2y$</td> 
-<td class="ntext">b)</td> 
-<td class="ntext" width="50%">$\displaystyle\frac{-x+12}{(x-2)(x+3)}$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">c)</td> 
-<td class="ntext" width="50%">$\displaystyle\frac{b}{a(a-b)}$</td> 
-<td class="ntext">d)</td> 
-<td class="ntext" width="50%">$\displaystyle\frac{a(a+b)}{4b}$</td> 
-</tr> 
-</table> 
-</div> 
- 
-==Övning 2.1:7== 
-<div class="ovning"> 
-Förenkla följande bråkuttryck genom att skriva på gemensamt bråkstreck 
-<table width="100%" cellspacing="10px"> 
-<tr align="left"> 
-<td class="ntext">a)</td> 
-<td class="ntext" width="33%">$\displaystyle \frac{2}{x+3}-\frac{2}{x+5}$</td> 
-<td class="ntext">b)</td> 
-<td class="ntext" width="33%">$x+\displaystyle \frac{1}{x-1}+\displaystyle \frac{1}{x^2}$</td> 
-<td class="ntext">c)</td> 
-<td class="ntext" width="33%">$\displaystyle \frac{ax}{a+1}-\displaystyle \frac{ax^2}{(a+1)^2}$</td> 
-</tr> 
-</table> 
-</div> 
- 
-<div class="svar"> 
-<table width="100%" cellspacing="10px"> 
-<tr align="left"> 
-<td class="ntext">a)</td> 
-<td class="ntext" width="33%">$\displaystyle \frac{4}{(x+3)(x+5)}$</td> 
-<td class="ntext">b)</td> 
-<td class="ntext" width="33%">$\displaystyle \frac{x^4-x^3+x^2+x-1}{x^2(x-1)}$</td> 
-<td class="ntext">c)</td> 
-<td class="ntext" width="33%">$\displaystyle \frac{ax(a+1-x)}{(a+1)^2}$</td> 
-</tr> 
-</table> 
-</div> 
- 
-==Övning 2.1:8== 
- 
-<div class="ovning"> 
-Förenkla följande bråkuttryck genom att skriva på gemensamt bråkstreck 
-<table width="100%" cellspacing="10px"> 
-<tr> 
-<td class="ntext">a)</td> 
-<td class="ntext" width="33%">$\displaystyle \frac{\displaystyle\ \frac{x}{x+1}\ }{\ 3+x\ }$</td> 
-<td class="ntext">b)</td> 
-<td class="ntext" width="33%">$\displaystyle \frac{\displaystyle \frac{3}{x}-\displaystyle \frac{1}{x}}{\displaystyle \frac{1}{x-3}}$</td> 
-<td class="ntext">c)</td> 
-<td class="ntext" width="33%">$\displaystyle \frac{1}{1+\displaystyle \frac{1}{1+\displaystyle \frac{1}{1+x}}}$</td> 
-</tr> 
-<tr><td height="5px"/></tr> 
-</table> 
-</div> 
- 
-<div class="svar"> 
-<table width="100%" cellspacing="10px"> 
-<tr align="left"> 
-<td class="ntext">a)</td> 
-<td class="ntext" width="33%">$\displaystyle \frac{x}{(x+3)(x+1)}$</td> 
-<td class="ntext">b)</td> 
-<td class="ntext" width="33%">$\displaystyle \frac{2(x-3)}{x}$</td> 
-<td class="ntext">c)</td> 
-<td class="ntext" width="33%">$\displaystyle \frac{x+2}{2x+3}$</td> 
-</tr> 
-<tr><td height="5px"/></tr> 
-</table> 
-</div> 
- 
-==Övning 2.2:1== 
-<div class="ovning"> 
-L&ouml;s ekvationerna 
-<table width="100%" cellspacing="10px"> 
-<tr align="left"> 
-<td class="ntext">a)</td> 
-<td class="ntext" width="50%">$x-2=-1$</td> 
-<td class="ntext">b)</td> 
-<td class="ntext" width="50%">$2x+1=13$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">c)</td> 
-<td class="ntext" width="50%">$\displaystyle\frac{1}{3}x-1=x$</td> 
-<td class="ntext">d)</td> 
-<td class="ntext" width="50%">$5x+7=2x-6$</td> 
-</tr> 
-<tr><td height="5px"/></tr> 
-</table> 
-</div> 
- 
-<div class="svar"> 
-<table width="100%" cellspacing="10px"> 
-<tr align="left"> 
-<td class="ntext">a)</td> 
-<td class="ntext" width="50%">$x=1$</td> 
-<td class="ntext">b)</td> 
-<td class="ntext" width="50%">$x=6$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">c) </td> 
-<td class="ntext" width="50%">$x=-\displaystyle\frac{3}{2}$</td> 
-<td class="ntext">d)</td> 
-<td class="ntext" width="50%">$x=-\displaystyle\frac{13}{3}$</td> 
-</tr> 
-<tr><td height="5px"/></tr> 
-</table> 
-</div> 
- 
-==Övning 2.2:2== 
-<div class="ovning"> 
-L&ouml;s ekvationerna 
-<table width="100%" cellspacing="10px"> 
-<tr align="left"> 
-<td class="ntext">a)</td> 
-<td class="ntext" width="50%">$\displaystyle\frac{5x}{6}-\displaystyle\frac{x+2}{9}=\displaystyle\frac{1}{2}$</td> 
-<td class="ntext">b)</td> 
-<td class="ntext" width="50%">$\displaystyle\frac{8x+3}{7}-\displaystyle\frac{5x-7}{4}=2$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">c)</td> 
-<td class="ntext" width="50%">$(x+3)^2-(x-5)^2=6x+4$</td> 
-<td class="ntext">d)</td> 
-<td class="ntext" width="50%">$(x^2+4x+1)^2+3x^4-2x^2=(2x^2+2x+3)^2$</td> 
-</tr> 
-<tr><td height="5px"/></tr> 
-</table> 
-</div> 
- 
-<div class="svar"> 
-<table width="100%" cellspacing="10px"> 
-<tr align="left"> 
-<td class="ntext">a)</td> 
-<td class="ntext" width="50%">$x=1$</td> 
-<td class="ntext">b)</td> 
-<td class="ntext" width="50%">$x=\displaystyle\frac{5}{3}$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">c)</td> 
-<td class="ntext" width="50%">$x=2$</td> 
-<td class="ntext">d)</td> 
-<td class="ntext" width="50%">$x=-2$</td> 
-</tr> 
-<tr><td height="5px"/></tr> 
-</table> 
-</div> 
- 
-==Övning 2.2:3== 
-<div class="ovning"> 
-L&ouml;s ekvationerna 
-<table width="100%" cellspacing="10px"> 
-<tr align="left"> 
-<td class="ntext">a)</td> 
-<td class="ntext" width="100%">$\displaystyle\frac{x+3}{x-3}-\displaystyle\frac{x+5}{x-2}=0$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">b)</td> 
-<td class="ntext" width="100%">$\displaystyle\frac{4x}{4x-7}-\displaystyle\frac{1}{2x-3}=1$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">c)</td> 
-<td class="ntext" width="100%">$\left(\displaystyle\frac{1}{x-1}-\frac{1}{x+1}\right)\left(x^2+\frac{1}{2}\right)=\displaystyle\frac{6x-1}{3x-3}$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">d) </td> 
-<td class="ntext" width="100%"> $\left(\displaystyle\frac{2}{x}-3\right)\left(\displaystyle\frac{1}{4x}+\frac{1}{2}\right)-\left(\displaystyle\frac{1}{2x}-\frac{2}{3}\right)^2-\left(\displaystyle\frac{1}{2x}+\frac{1}{3}\right)\left(\displaystyle\frac{1}{2x}-\frac{1}{3}\right)=0$</td> 
-</tr> 
-<tr><td height="5px"/></tr> 
-</table> 
-</div> 
- 
-<div class="svar"> 
-<table width="100%" cellspacing="10px"> 
-<tr align="left"> 
-<td class="ntext">a)</td> 
-<td class="ntext" width="100%">$x=9$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">b)</td> 
-<td class="ntext" width="100%">$x=\displaystyle\frac{7}{5}$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">c) </td> 
-<td class="ntext" width="100%">$x=\displaystyle\frac{4}{5}$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">d) </td> 
-<td class="ntext" width="100%">$x=\displaystyle\frac{1}{2}$</td> 
-</tr> 
-<tr><td height="5px"/></tr> 
-</table> 
-</div> 
- 
-==&Ouml;vning 2.2:4== 
- 
-<div class="ovning"> 
-<table width="100%" cellspacing="10px"> 
-<tr align="left"> 
-<td>a)</td> 
-<td width="100%">Skriv ekvationen f&ouml;r linjen $\,y=2x+3\,$ på formen $\,ax+by=c\,$.</td> 
-</tr> 
-<tr align="left"> 
-<td>b)</td> 
-<td width="100%">Skriv ekvationen f&ouml;r linjen $\,3x+4y-5=0\,$ på formen $\,y=kx+m\,$.</td> 
-</tr> 
-<tr><td height="5px"\></tr> 
-</table> 
-</div> 
- 
-<div class="svar"> 
-<table width="100%" cellspacing="10px"> 
-<tr align="left"> 
-<td>a)</td> 
-<td width="100%">$-2x+y=3$</td> 
-</tr> 
-<tr align="left"> 
-<td>b)</td> 
-<td width="100%">$y=-\displaystyle\frac{3}{4}x+\frac{5}{4}$</td> 
-</tr> 
-<tr><td height="5px"\></tr> 
-</table> 
-</div> 
- 
-==&Ouml;vning 2.2:5== 
- 
-<div class="ovning"> 
-<table width="100%" cellspacing="10px"> 
-<tr align="left" valign="top"> 
-<td class="ntext">a)</td> 
-<td class="ntext" width="100%">Best&auml;m ekvationen f&ouml;r den r&auml;ta linje som g&aring;r genom punkterna $\,(2,3)\,$ och $\,(3,0)\,$.</td> 
-</tr> 
-<tr align="left" valign="top"> 
-<td class="ntext">b)</td> 
-<td class="ntext" width="100%">Best&auml;m ekvationen f&ouml;r den r&auml;ta linje som har riktningskoefficient $\,-3\,$ och g&aring;r genom punkten $\,(1,-2)\,$.</td> 
-</tr> 
-<tr align="left" valign="top"> 
-<td class="ntext">c)</td> 
-<td class="ntext" width="100%">Best&auml;m ekvationen f&ouml;r den r&auml;ta linje som g&aring;r genom punkten $\,(-1,2)\,$ och &auml;r parallell med linjen $\,y=3x+1\,$.</td> 
-</tr> 
-<tr align="left" valign="top"> 
-<td class="ntext">d)</td> 
-<td class="ntext" width="100%">Best&auml;m ekvationen f&ouml;r den r&auml;ta linje som g&aring;r genom punkten $\,(2,4)\,$ och &auml;r vinkelr&auml;t mot linjen $\,y=2x+5\,$.</td> 
-</tr> 
-<tr align="left" valign="top"> 
-<td class="ntext">e)</td> 
-<td class="ntext" width="100%">Best&auml;m riktningskoefficienten, $\,k\,$, f&ouml;r den r&auml;ta linje som sk&auml;r ''x''-axeln i punkten $\,(5,0)\,$ och ''y''-axeln i punkten $\,(0,-8)\,$.</td> 
-</tr> 
-<tr><td height="5px"/></tr> 
-</table> 
-</div> 
- 
-<div class="svar"> 
-<table width="100%" cellspacing="10px"> 
-<tr align="left"> 
-<td class="ntext">a)</td> 
-<td class="ntext" width="100%">$y=-3x+9$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">b)</td> 
-<td class="ntext" width="100%">$y=-3x+1$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">c)</td> 
-<td class="ntext" width="100%">$y=3x+5$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">d)</td> 
-<td class="ntext" width="100%">$y=-\displaystyle \frac{1}{2}x+5$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">e)</td> 
-<td class="ntext" width="100%">$k = \displaystyle\frac{8}{5}$</td> 
-</tr> 
-<tr><td height="5px"/></tr> 
-</table> 
-</div> 
- 
-==&Ouml;vning 2.2:6== 
- 
-<div class="ovning"> 
-Finn sk&auml;rningspunkten mellan f&ouml;ljande linjer 
-<table width="100%" cellspacing="10px"> 
-<tr align="left"> 
-<td class="ntext">a)</td> 
-<td class="ntext" width="50%">$y=3x+5\ $ och ''x''-axeln</td> 
-<td class="ntext">b)</td> 
-<td class="ntext" width="50%">$y=-x+5\ $ och ''y''-axeln</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">c)</td> 
-<td class="ntext" width="50%">$4x+5y+6=0\ $ och ''y''-axeln</td> 
-<td class="ntext">d)</td> 
-<td class="ntext" width="50%">$x+y+1=0\ $ och $\ x=12$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">e)</td> 
-<td class="ntext" width="50%">$2x+y-1=0\ $ och $\ y-2x-2=0$</td> 
-</tr> 
-<tr><td height="5px"/></tr> 
-</table> 
-</div> 
- 
-<div class="svar"> 
-<table width="100%" cellspacing="10px"> 
-<tr align="left"> 
-<td class="ntext">a)</td> 
-<td class="ntext" width="50%">$\bigl(-\frac{5}{3},0\bigr)$</td> 
-<td class="ntext">b)</td> 
-<td class="ntext" width="50%">$(0,5)$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">c)</td> 
-<td class="ntext" width="50%">$\bigl(0,-\frac{6}{5}\bigr)$</td> 
-<td class="ntext">d)</td> 
-<td class="ntext" width="50%">$(12,-13)$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">e)</td> 
-<td class="ntext" width="50%">$\bigl(-\frac{1}{4},\frac{3}{2}\bigr)$</td> 
-</tr> 
-<tr><td height="5px"/></tr> 
-</table> 
-</div> 
- 
-==&Ouml;vning 2.2:7== 
- 
-<div class="ovning"> 
-Skissera grafen till f&ouml;ljande funktioner 
-<table width="100%" cellspacing="10px"> 
-<tr align="left"> 
-<td class="ntext">a)</td> 
-<td class="ntext" width="33%">$f(x)=3x-2$</td> 
-<td class="ntext">b)</td> 
-<td class="ntext" width="33%">$f(x)=2-x$</td> 
-<td class="ntext">c)</td> 
-<td class="ntext" width="33%">$f(x)=2$</td> 
-</tr> 
-<tr><td height="5px"/></tr> 
-</table> 
-</div> 
- 
-<div class="svar"> 
-<table width="100%"> 
-<tr align="left"> 
-<td class="ntext">a)</td> 
-<td class="ntext" width="33%">[[Bild:Svar_o2_2_7a.gif‎]] </td> 
-<td class="ntext">b)</td> 
-<td class="ntext" width="33%">[[Bild:Svar_o2_2_7b.gif‎]] </td> 
-</tr><tr> 
-<td class="ntext">c)</td> 
-<td class="ntext" width="33%">[[Bild:Svar_o2_2_7c.gif‎]] </td> 
-</tr> 
-</table> 
-</div> 
- 
-==&Ouml;vning 2.2:8== 
- 
-<div class="ovning"> 
-Rita in i ett ''xy''-plan alla punkter vars koordinater $\,(x,y)\,$ uppfyller 
-<table width="100%" cellspacing="10px"> 
-<tr align="left"> 
-<td class="ntext">a)</td> 
-<td class="ntext" width="33%">$y \geq x $</td> 
-<td class="ntext">b)</td> 
-<td class="ntext" width="33%">$y &lt; 3x -4 $</td> 
-<td class="ntext">c)</td> 
-<td class="ntext" width="33%">$2x+3y \leq 6 $</td> 
-</tr> 
-<tr><td height="5px"/></tr> 
-</table> 
-</div> 
- 
-<div class="svar"> 
-<table width="100%"> 
-<tr align="left"> 
-<td class="ntext">a)</td> 
-<td class="ntext" width="33%">[[Bild:Svar_o2_2_8a.gif‎]]</td> 
-<td class="ntext">b)</td> 
-<td class="ntext" width="33%">[[Bild:Svar_o2_2_8b.gif‎]]</td> 
-</tr><tr> 
-<td class="ntext">c)</td> 
-<td class="ntext" width="33%">[[Bild:Svar_o2_2_8c.gif‎]]</td> 
-</tr> 
-</table> 
-</div> 
- 
-==&Ouml;vning 2.2:9== 
- 
-<div class="ovning"> 
-Ber&auml;kna arean av den triangel som  
-<table width="100%" cellspacing="10px"> 
-<tr align="left"> 
-<td class="ntext">a)</td> 
-<td class="ntext" width="100%">har h&ouml;rn i punkterna $\,(1,4)\,$, $\,(3,3)\,$ och $\,(1,0)\,$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">b)</td> 
-<td class="ntext" width="100%">begr&auml;nsas av linjerna $\ x=2y\,$, $\ y=4\ $ och $\ y=10-2x\,$</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">c)</td> 
-<td class="ntext" width="100%">beskrivs av olikheterna $\ x+y \geq -2\,$, $\ 2x-y \leq 2\ $ och $\ 2y-x \leq 2\,$</td> 
-</tr> 
-<tr><td height="5px"/></tr> 
-</table> 
-</div> 
- 
-<div class="svar"> 
-<table width="100%" cellspacing="10px"> 
-<tr align="left"> 
-<td class="ntext">a)</td> 
-<td class="ntext" width="100%">$4\,$ a.e.</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">b)</td> 
-<td class="ntext" width="100%">$5\,$ a.e.</td> 
-</tr> 
-<tr align="left"> 
-<td class="ntext">c)</td> 
-<td class="ntext" width="100%">$6\,$ a.e.</td> 
</tr> </tr>
<tr><td height="5px"/></tr> <tr><td height="5px"/></tr>
</table> </table>
</div> </div>

Versionen från 21 juni 2007 kl. 15.07


övn 2

Övning 1.1:1

Beräkna (utan hjälp av räknedosa)

a) $3-7-4+6-5$ b) $3-(7-4)+(6-5)$
c) $3-(7-(4+6)-5)$ d) $3-(7-(4+6))-5$
a) $-7$ b) $1$
c) $11$ d) $1$

Övning 1.1:2

Beräkna (utan hjälp av räknedosa)

a) $(3-(7-4))(6-5)$ b) $3-(((7-4)+6)-5)$
c) $3\cdot(-7)-4\cdot(6-5)$ d) $3\cdot(-7)-(4+6)/(-5)$
a) $0$ b) $-1$
c) $-25$ d) $-19$

Övning 1.1:3

Vilka av följande tal tillhör de naturliga talen? heltalen? rationella talen? irrationella talen? Förenkla först!

a) $8$ b) $-4$ c) $8-4$
d) $4-8$ e) $8(-4)$ f) $(-8)(-4)$
g) $\displaystyle \frac{4}{-8}$ h) $\displaystyle \frac{-8}{-4}$ i) $\displaystyle \frac{\sqrt{2}}{3}$
j) $\displaystyle \Bigl(\frac{4}{\sqrt{2}}\Bigr)^2$ k) $-\pi$ l) $\pi+1$
a) naturliga talen, heltalen, rationella talen b) heltalen, rationella talen c) naturliga talen, heltalen, rationella talen
d) heltalen, rationella talen e) heltalen, rationella talen f) naturliga talen, heltalen, rationella talen
g) rationella talen h) naturliga talen, heltalen, rationella talen i) irrationella talen
j) naturliga talen, heltalen, rationella talen k) irrationella talen l) irrationella talen

Övning 1.1:4

Ordna följande tal i storleksordning

a) $\displaystyle 2,\ \frac{3}{5},\ \frac{5}{3}\ $ och $\ \displaystyle \frac{7}{3}$
b) $\displaystyle -\frac{1}{2},\ -\frac{1}{5},\ -\frac{3}{10}\ $ och $\ \displaystyle -\frac{1}{3}$
c) $\displaystyle \frac{1}{2},\ \frac{2}{3},\ \frac{3}{5},\ \frac{5}{8}\ $ och $\ \displaystyle \frac{21}{34}$
a) $\displaystyle \frac{3}{5}<\frac{5}{3}<2<\frac{7}{3}$
b) $\displaystyle -\frac{1}{2}<-\frac{1}{3}<-\frac{3}{10}<-\frac{1}{5}$
c) $\displaystyle \frac{1}{2}<\frac{3}{5}<\frac{21}{34}<\frac{5}{8}<\frac{2}{3}$

Övning 1.1:5

Ange decimalutvecklingen med tre korrekta decimaler till

a) $\displaystyle \frac{7}{6}$ b) $\displaystyle \frac{9}{4}$ c) $\displaystyle \frac{2}{7}$ d) $\sqrt{2}$
a) $1{,}167$ b) $2{,}250$ c) $0{,}286$ d) $1{,}414$

Övning 1.1:6

Vilka av följande tal är rationella? Ange dem som en kvot mellan heltal.

a) $3,14$
b) $3{,}1416\,1416\,1416\,\dots$
c) $0{,}2\,001\,001\,001\,\dots\,$ (därefter är var tredje decimal en 1:a och övriga 0)
d) $0{,}10\,100\,1000\,10000\,1\dots\, $ (en 1:a, en 0:a, en 1:a, två 0:or, en 1:a, tre 0:or osv.)
a) Talet är rationellt och lika med $\,314/100 = 157/50\,$.
b) Talet är rationellt och är lika med $\,31413/9999 = 10471/3333\,$.
c) Talet är rationellt och lika med $\,1999/9990\,$.
d) Talet är irrationellt.


Övning 1.2:1

Skriv på gemensamt bråkstreck

a) $\displaystyle \frac{7}{4}+\frac{11}{7}$ b) $\displaystyle \frac{2}{7}-\frac{1}{5}$ c) $\displaystyle \frac{1}{6}-\frac{2}{5}$
d) $\displaystyle \frac{1}{3}+\frac{1}{4}+\frac{1}{5}$ e) $\displaystyle \frac{8}{7}+\frac{3}{4}-\frac{4}{3}$
a) $\displaystyle \frac{93}{28}$ b) $\displaystyle \frac{3}{35}$ c) $\displaystyle -\frac{7}{30}$
d) $\displaystyle \frac{47}{60}$ e) $\displaystyle \frac{47}{84}$

Övning 1.2:2

Bestäm minsta gemensamma nämnare

a) $\displaystyle \frac{1}{6}+\frac{1}{10}$ b) $\displaystyle \frac{1}{4}-\frac{1}{8}$
c) $\displaystyle \frac{1}{12}-\frac{1}{14}$ d) $\displaystyle \frac{2}{45}+\frac{1}{75}$
a) $\displaystyle {30}$ b) $\displaystyle {8}$
c) $\displaystyle {84}$ d) $\displaystyle {225}$

Övning 1.2:3

Beräkna följande uttryck genom att använda minsta gemensamma nämnare:

a) $\displaystyle \frac{3}{20}+\frac{7}{50}-\frac{1}{10}$ b) $\displaystyle \frac{1}{24}+\frac{1}{40}-\frac{1}{16}$
a) $\displaystyle \frac{19}{100}$ b) $\displaystyle \frac{1}{240}$

Övning 1.2:4

Förenkla följande uttryck genom att skriva på gemensamt bråkstreck. Bråket ska vara färdigförkortat.

a) $\displaystyle\frac{\displaystyle\frac{3}{5}}{\displaystyle\frac{7}{10}}$ b) $\displaystyle\frac{\displaystyle\frac{2}{7}}{\displaystyle\frac{3}{8}}$ c) $\displaystyle\frac{\displaystyle\frac{1}{4}-\frac{1}{5}}{\displaystyle\frac{3}{10}}$
a) $\displaystyle \frac{6}{7}$ b) $\displaystyle \frac{16}{21}$ c) $\displaystyle \frac{1}{6}$

Övning 1.2:5

Förenkla följande uttryck genom att skriva på gemensamt bråkstreck. Bråket ska vara färdigförkortat.

a) $\displaystyle \frac{2}{\displaystyle \frac{1}{7}\displaystyle -\frac{1}{15}}$ b) $\displaystyle\frac{\displaystyle\frac{1}{2}\displaystyle+\frac{1}{3}}{\displaystyle\frac{1}{3}\displaystyle-\frac{1}{2}}$ c) $\displaystyle\frac{\displaystyle\frac{3}{10}\displaystyle-\frac{1}{5}}{\displaystyle\frac{7}{8}\displaystyle-\frac{3}{16}}$
a) $\displaystyle \frac{105}{4}$ b) $-5$ c) $\displaystyle \frac{8}{55}$

Övning 1.2:6

Förenkla $\ \,\displaystyle \frac{\displaystyle \frac{2}{\displaystyle 3+\frac{1}{2}}\displaystyle + \frac{\displaystyle \frac{1}{2}}{\displaystyle \frac{1}{4}\displaystyle -\frac{1}{3}}}{\displaystyle \frac{1}{2}\displaystyle - \frac{3}{\displaystyle 2-\frac{2}{7}}}$

$\displaystyle \frac{152}{35}$


Övning 1.3:1

Beräkna

a) $2^3\cdot3^2$ b) $3^5\cdot9^{-2}$ c) $(-5)^3$ d) $\Bigl(\displaystyle \frac{2}{3}\Bigr)^{-3}$
a) $72$ b) $3$ c) $-125$ d) $\displaystyle \frac{27}{8}$

Övning 1.3:2

Skriv som en potens av 2

a) $2\cdot4\cdot8$ b) $0{,}25$ c) $1$
a) $2^6$ b) $2^{-2}$ c) $2^0$

Övning 1.3:3

Skriv som en potens av 3

a) $\displaystyle \frac{1}{3}$ b) $243$ c) $9^2$ d) $\displaystyle \frac{1}{27}$ e) $\displaystyle \frac{3}{9^2}$
a) $3^{-1}$ b) $3^5$ c) $3^4$ d) $3^{-3}$ e) $3^{-3}$

Övning 1.3:4

Beräkna

a) $2^{9}\cdot2^{-7}$ b) $3^{13}\cdot9^{-3}\cdot27^{\,-2}$ c) $\displaystyle \frac{5^{12}}{5^{-4}}\cdot(5^{2})^{-6}$
d) $2^{2^{\scriptstyle3}}\cdot(-2)^{\scriptstyle-4}$ e) $625\cdot(5^{8}+5^{9})^{-1}$
a) $4$ b) $3$ c) $625$
d) $16$ e) $\displaystyle \frac{1}{3750}$

Övning 1.3:5

Beräkna

a) $4^{1/2}$ b) $4^{-1/2}$ c) $9^{3/2}$
d) $\left(47^{2/3} \right) ^{3}$ e) $3^{1{,}4}\cdot3^{0{,}6}$ f) $\bigl( 125 ^{1/3} \bigr)^2\cdot \bigl( 27^{1/3} \bigr)^{-2}\cdot9^{1/2}$
a) $2$ b) $\displaystyle \frac{1}{2}$ c) $27$
d) $2209$ e) $9$ f) $\displaystyle \frac{25}{3}$


Övning 1.3:6

Avgör vilket tal som är störst av

a) $256^{1/3}\ $ och $\ 200^{1/3}$ b) $0{,}5^{-3}\ $ och $\ 0{,}4^{-3}$ c) $0{,}2^5\ $ och $\ 0{,}2^{7}$
d) $400^{1/3}\ $ och $\ \bigl(5^{1/3}\bigr)^{4}$ e) $125^{1/2}\ $ och $\ 625^{1/3}$ f) $2^{56}\ $ och $\ 3^{40}$
a) $256^{1/3}>200^{1/3}$ b) $0{,}4^{-3}>0{,}5^{-3}$ c) $0{,}2^{5}>0{,}2^{7}$
d) $\bigl(5^{1/3}\bigr)^{4}>400^{1/3}$ e) $125^{1/2}>625^{1/3}$ f) $3^{40}>2^{56}$
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