Övn 2.3
Sommarmatte 1
(Skillnad mellan versioner)
| Versionen från 21 juni 2007 kl. 14.56 (redigera) KTH.SE:u1xsetv1 (Diskussion | bidrag) (Ny sida: ==Övning 2.3:1== <div class="ovning"> Kvadratkomplettera följande uttryck <table width="100%" cellspacing="10px"> <tr align="left"> <td class="ntext">a)</td> <td class="ntext" wi...) ← Gå till föregående ändring |
Nuvarande version (16 juli 2007 kl. 07.59) (redigera) (ogör) KTH.SE:u1zpa8nw (Diskussion | bidrag) (Tar bort sidans innehåll) |
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| (3 mellanliggande versioner visas inte.) | |||
| Rad 1: | Rad 1: | ||
| - | ==Övning 2.3:1== | ||
| - | <div class="ovning"> | ||
| - | Kvadratkomplettera följande uttryck | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr align="left"> | ||
| - | <td class="ntext">a)</td> | ||
| - | <td class="ntext" width="25%">$x^2-2x$</td> | ||
| - | <td class="ntext">b)</td> | ||
| - | <td class="ntext" width="25%">$x^2+2x-1$</td> | ||
| - | <td class="ntext">c)</td> | ||
| - | <td class="ntext" width="25%">$5+2x-x^2$</td> | ||
| - | <td class="ntext">d)</td> | ||
| - | <td class="ntext" width="25%">$x^2+5x+3$</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="25%">$(x-1)^2-1$</td> | ||
| - | <td class="ntext">b)</td> | ||
| - | <td class="ntext" width="25%">$(x+1)^2-2$</td> | ||
| - | <td class="ntext">c)</td> | ||
| - | <td class="ntext" width="25%">$-(x-1)^2+6$</td> | ||
| - | <td class="ntext">d)</td> | ||
| - | <td class="ntext" width="25%">$\bigl(x+\frac{5}{2}\bigr)^2-\frac{13}{4}$</td> | ||
| - | </tr> | ||
| - | <tr><td height="5px"/></tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning a) </div> | ||
| - | <div class=NavContent> | ||
| - | Lösning till delfråga a) | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_1a.gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning b) </div> | ||
| - | <div class=NavContent> | ||
| - | Lösning till delfråga b) | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_1b.gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning c) </div> | ||
| - | <div class=NavContent> | ||
| - | Lösning till delfråga c) | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_1c.gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning d) </div> | ||
| - | <div class=NavContent> | ||
| - | Lösning till delfråga d) | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_1d.gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | ==Övning 2.3:2== | ||
| - | |||
| - | <div class="ovning"> | ||
| - | Lös följande andragradsekvationer med kvadratkomplettering | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr align="left"> | ||
| - | <td class="ntext">a)</td> | ||
| - | <td class="ntext" width="33%">$x^2-4x+3=0$</td> | ||
| - | <td class="ntext">b)</td> | ||
| - | <td class="ntext" width="33%">$y^2+2y-15=0$</td> | ||
| - | <td class="ntext">c)</td> | ||
| - | <td class="ntext" width="33%">$y^2+3y+4=0$</td> | ||
| - | </tr> | ||
| - | <tr align="left"> | ||
| - | <td class="ntext">d)</td> | ||
| - | <td class="ntext" width="33%">$4x^2-28x+13=0$</td> | ||
| - | <td class="ntext">e)</td> | ||
| - | <td class="ntext" width="33%">$5x^2+2x-3=0$</td> | ||
| - | <td class="ntext">f)</td> | ||
| - | <td class="ntext" width="33%">$3x^2-10x+8=0$</td> | ||
| - | </tr> | ||
| - | <tr><td height="5px"/></tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Facit </div> | ||
| - | <div class=NavContent> | ||
| - | Facit till alla delfrågor<br \> | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr><td height="5px"/></tr> | ||
| - | <tr align="left"> | ||
| - | <td class="ntext">a)</td> | ||
| - | <td class="ntext" width="33%">$\left\{ \eqalign{ x_1 &= 1 \cr x_2 &= 3\cr }\right.$</td> | ||
| - | <td class="ntext">b)</td> | ||
| - | <td class="ntext" width="33%">$\left\{ \eqalign{ y_1 &= -5 \cr y_2 &= 3\cr }\right.$</td> | ||
| - | <td class="ntext">c)</td> | ||
| - | <td class="ntext" width="33%"> saknar (reella) lösning</td> | ||
| - | </tr> | ||
| - | <tr align="left"> | ||
| - | <td class="ntext">d)</td> | ||
| - | <td class="ntext" width="33%">$ \left\{ \eqalign{ x_1 &= \textstyle\frac{1}{2}\cr x_2 &= \textstyle\frac{13}{2}\cr }\right.$</td> | ||
| - | <td class="ntext">e)</td> | ||
| - | <td class="ntext" width="33%">$\left\{ \eqalign{ x_1 &= -1 \cr x_2 &= \textstyle\frac{3}{5}\cr }\right.$</td> | ||
| - | <td class="ntext">f)</td> | ||
| - | <td class="ntext" width="33%">$ \left\{ \eqalign{ x_1 &= \textstyle\frac{4}{3}\cr x_2 &= 2\cr }\right.$</td> | ||
| - | </tr> | ||
| - | <tr><td height="5px"/></tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning a) </div> | ||
| - | <div class=NavContent> | ||
| - | Lösning till delfråga a) | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_2a.gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | |||
| - | </div> | ||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning b) </div> | ||
| - | <div class=NavContent> | ||
| - | Lösning till delfråga b) | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_2b.gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | |||
| - | </div> | ||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning c) </div> | ||
| - | <div class=NavContent> | ||
| - | Lösning till delfråga c) | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_2c.gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning d) </div> | ||
| - | <div class=NavContent> | ||
| - | Lösning till delfråga d) | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_2d.gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning e) </div> | ||
| - | <div class=NavContent> | ||
| - | Lösning till delfråga e) | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_2e.gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning f) </div> | ||
| - | <div class=NavContent> | ||
| - | Lösning till delfråga f) | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_2f.gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | ==Övning 2.3:3== | ||
| - | |||
| - | <div class="ovning"> | ||
| - | Lös följande ekvationer direkt | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr align="left"> | ||
| - | <td class="ntext">a)</td> | ||
| - | <td class="ntext" width="50%">$x(x+3)=0$</td> | ||
| - | <td class="ntext">b)</td> | ||
| - | <td class="ntext" width="50%">$(x-3)(x+5)=0$</td> | ||
| - | </tr> | ||
| - | <tr align="left"> | ||
| - | <td class="ntext">c)</td> | ||
| - | <td class="ntext" width="50%">$5(3x-2)(x+8)=0$</td> | ||
| - | <td class="ntext">d)</td> | ||
| - | <td class="ntext" width="50%">$x(x+3)-x(2x-9)=0$</td> | ||
| - | </tr> | ||
| - | <tr align="left"> | ||
| - | <td class="ntext">e)</td> | ||
| - | <td class="ntext" width="50%">$(x+3)(x-1)-(x+3)(2x-9)=0$</td> | ||
| - | <td class="ntext">f)</td> | ||
| - | <td class="ntext" width="50%">$x(x^2-2x)+x(2-x)=0$</td> | ||
| - | </tr> | ||
| - | <tr><td height="5px"/></tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Facit </div> | ||
| - | <div class=NavContent> | ||
| - | Facit till alla delfrågor<br \> | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr align="left"> | ||
| - | <td class="ntext">a)</td> | ||
| - | <td class="ntext" width="50%">$\left\{ \eqalign{ x_1 &= 0 \cr x_2 & = -3\cr }\right.$</td> | ||
| - | <td class="ntext">b)</td> | ||
| - | <td class="ntext" width="50%">$\left\{ \eqalign{ x_1 &= 3 \cr x_2 & = -5\cr }\right. $</td> | ||
| - | </tr> | ||
| - | <tr align="left"> | ||
| - | <td class="ntext">c)</td> | ||
| - | <td class="ntext" width="50%">$\left\{ \eqalign{ x_1 & = \textstyle\frac{2}{3} \cr x_2 & = -8\cr }\right. $</td> | ||
| - | <td class="ntext">d)</td> | ||
| - | <td class="ntext" width="50%">$\left\{ \eqalign{ x_1 & = 0\cr x_2 & = 12\cr }\right. $</td> | ||
| - | </tr> | ||
| - | <tr align="left"> | ||
| - | <td class="ntext">e)</td> | ||
| - | <td class="ntext" width="50%">$\left\{ \eqalign{ x_1 & = -3 \cr x_2 & = 8\cr }\right. $</td> | ||
| - | <td class="ntext">f)</td> | ||
| - | <td class="ntext" width="50%">$\left\{ \eqalign{ x_1 & = 0 \cr x_2 & = 1 \cr x_3 & = 2 }\right. $</td> | ||
| - | </tr> | ||
| - | <tr><td height="5px"/></tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning a) </div> | ||
| - | <div class=NavContent> | ||
| - | Lösning till delfråga a) | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_3a.gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning b) </div> | ||
| - | <div class=NavContent> | ||
| - | Lösning till delfråga b) | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_3b.gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning c) </div> | ||
| - | <div class=NavContent> | ||
| - | Lösning till delfråga c) | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_3c.gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning d) </div> | ||
| - | <div class=NavContent> | ||
| - | Lösning till delfråga d) | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_3d.gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning e) </div> | ||
| - | <div class=NavContent> | ||
| - | Lösning till delfråga e) | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_3e.gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning f) </div> | ||
| - | <div class=NavContent> | ||
| - | Lösning till delfråga f) | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_3f.gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | ==Övning 2.3:4== | ||
| - | |||
| - | <div class="ovning"> | ||
| - | Bestäm en andragradsekvation som har rötterna | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr align="left"> | ||
| - | <td class="ntext">a)</td> | ||
| - | <td class="ntext" width="100%">$-1\ $ och $\ 2$</td> | ||
| - | </tr> | ||
| - | <tr align="left"> | ||
| - | <td class="ntext">b)</td> | ||
| - | <td class="ntext" width="100%">$1+\sqrt{3}\ $ och $\ 1-\sqrt{3}$</td> | ||
| - | </tr> | ||
| - | <tr align="left"> | ||
| - | <td class="ntext">c)</td> | ||
| - | <td class="ntext" width="100%">$3\ $ och $\ \sqrt{3}$</td> | ||
| - | </tr> | ||
| - | <tr><td height="5px"/></tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Facit </div> | ||
| - | <div class=NavContent> | ||
| - | Facit till alla delfrågor | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr align="left"> | ||
| - | <td class="ntext">a)</td> | ||
| - | <td class="ntext" width="100%">$ax^2-ax-2a=0\,$, där $\,a\ne 0\,$ är en konstant.</td> | ||
| - | </tr> | ||
| - | <tr align="left"> | ||
| - | <td class="ntext">b)</td> | ||
| - | <td class="ntext" width="100%">$ax^2-2ax-2a=0\,$, där $\,a\ne 0\,$ är en konstant.</td> | ||
| - | </tr> | ||
| - | <tr align="left"> | ||
| - | <td class="ntext">c)</td> | ||
| - | <td class="ntext" width="100%">$ax^2-(3+\sqrt{3}\,)ax+3\sqrt{3}\,a=0\,$, där $\,a\ne 0\,$ är en konstant.</td> | ||
| - | </tr> | ||
| - | <tr><td height="5px"/></tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning a) </div> | ||
| - | <div class=NavContent> | ||
| - | Lösning till delfråga a) | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_4a-1(2).gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_4a-2(2).gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning b) </div> | ||
| - | <div class=NavContent> | ||
| - | Lösning till delfråga b) | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_4b-1(2).gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_4b-2(2).gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning c) </div> | ||
| - | <div class=NavContent> | ||
| - | Lösning till delfråga c) | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_4c.gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | ==Övning 2.3:5== | ||
| - | |||
| - | <div class="ovning"> | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr align="left"> | ||
| - | <td class="ntext">a)</td> | ||
| - | <td class="ntext" width="100%">Bestäm en andragradsekvation som bara har $\,-7\,$ som rot.</td> | ||
| - | </tr> | ||
| - | <tr> | ||
| - | <td class="ntext">b)</td> | ||
| - | <td class="ntext" width="100%">Bestäm ett värde på $\,x\,$ som gör att uttrycket $\,4x^2-28x+48\,$ är negativt.</td> | ||
| - | </tr> | ||
| - | <tr> | ||
| - | <td class="ntext">c)</td> | ||
| - | <td class="ntext" width="100%">Ekvationen $\,x^2+4x+b=0\,$ har en rot $\,x=1\,$. Bestäm värdet på konstanten $\,b\,$.</td> | ||
| - | </tr> | ||
| - | <tr><td height="5px"/></tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Facit </div> | ||
| - | <div class=NavContent> | ||
| - | Facit till alla delfrågor | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr align="left"> | ||
| - | <td class="ntext">a)</td> | ||
| - | <td class="ntext" width="100%">Exempelvis $\ x^2+14x+49=0\,$.</td> | ||
| - | </tr> | ||
| - | <tr align="left"> | ||
| - | <td class="ntext">b)</td> | ||
| - | <td class="ntext" width="100%">$3< x<4$</td> | ||
| - | </tr> | ||
| - | <tr align="left"> | ||
| - | <td class="ntext">c)</td> | ||
| - | <td class="ntext" width="100%">$b=-5$</td> | ||
| - | </tr> | ||
| - | <tr><td height="5px"/></tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning a </div> | ||
| - | <div class=NavContent> | ||
| - | Lösning till delfråga a) | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_5a.gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning b) </div> | ||
| - | <div class=NavContent> | ||
| - | Lösning till delfråga b) | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_5b.gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning c) </div> | ||
| - | <div class=NavContent> | ||
| - | Lösning till delfråga c) | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_5c.gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | ==Övning 2.3:6== | ||
| - | |||
| - | <div class="ovning"> | ||
| - | Bestäm det minsta värde som följande polynom antar | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr align="left"> | ||
| - | <td class="ntext">a)</td> | ||
| - | <td class="ntext" width="33%">$x^2-2x+1$</td> | ||
| - | <td class="ntext">b)</td> | ||
| - | <td class="ntext" width="33%">$x^2-4x+2$</td> | ||
| - | <td class="ntext">c)</td> | ||
| - | <td class="ntext" width="33%">$x^2-5x+7$</td> | ||
| - | </tr> | ||
| - | <tr><td height="5px"/></tr> | ||
| - | </table> | ||
| - | |||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Facit </div> | ||
| - | <div class=NavContent> | ||
| - | Facit till alla delfrågor | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr align="left"> | ||
| - | <td class="ntext">a)</td> | ||
| - | <td class="ntext" width="33%">$0$</td> | ||
| - | <td class="ntext">b)</td> | ||
| - | <td class="ntext" width="33%">$-2$</td> | ||
| - | <td class="ntext">c)</td> | ||
| - | <td class="ntext" width="33%">$\displaystyle \frac{3}{4}$</td> | ||
| - | </tr> | ||
| - | <tr><td height="5px"/></tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning a) </div> | ||
| - | <div class=NavContent> | ||
| - | Lösning till delfråga a) | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_6a.gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning b) </div> | ||
| - | <div class=NavContent> | ||
| - | Lösning till delfråga b) | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_6b.gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning c) </div> | ||
| - | <div class=NavContent> | ||
| - | Lösning till delfråga c) | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_6c.gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | ==Övning 2.3:7== | ||
| - | |||
| - | <div class="ovning"> | ||
| - | Bestäm det största värde som följande polynom antar | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr align="left"> | ||
| - | <td class="ntext">a)</td> | ||
| - | <td class="ntext" width="33%">$1-x^2$</td> | ||
| - | <td class="ntext">b)</td> | ||
| - | <td class="ntext" width="33%">$-x^2+3x-4$</td> | ||
| - | <td class="ntext">c)</td> | ||
| - | <td class="ntext" width="33%">$x^2+x+1$</td> | ||
| - | </tr> | ||
| - | <tr><td height="5px"/></tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Facit </div> | ||
| - | <div class=NavContent> | ||
| - | Facit till alla delfrågor<br \> | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr align="left"> | ||
| - | <td class="ntext">a)</td> | ||
| - | <td class="ntext" width="33%">$1$</td> | ||
| - | <td class="ntext">b)</td> | ||
| - | <td class="ntext" width="33%">$\displaystyle -\frac{7}{4}$</td> | ||
| - | <td class="ntext">c)</td> | ||
| - | <td class="ntext" width="33%">saknar max</td> | ||
| - | </tr> | ||
| - | <tr><td height="5px"/></tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning a) </div> | ||
| - | <div class=NavContent> | ||
| - | Lösning till delfråga a) | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_7a.gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning b) </div> | ||
| - | <div class=NavContent> | ||
| - | Lösning till delfråga b) | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_7b.gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning c) </div> | ||
| - | <div class=NavContent> | ||
| - | Lösning till delfråga c) | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_7c.gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | ==Övning 2.3:8== | ||
| - | |||
| - | <div class="ovning"> | ||
| - | Skissera grafen till följande funktioner | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr align="left"> | ||
| - | <td class="ntext">a)</td> | ||
| - | <td class="ntext" width="33%">$f(x)=x^2+1$</td> | ||
| - | <td class="ntext">b)</td> | ||
| - | <td class="ntext" width="33%">$f(x)=(x-1)^2+2$</td> | ||
| - | <td class="ntext">c)</td> | ||
| - | <td class="ntext" width="33%">$f(x)=x^2-6x+11$</td> | ||
| - | </tr> | ||
| - | <tr><td height="5px"/></tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Facit </div> | ||
| - | <div class=NavContent> | ||
| - | Facit till alla delfrågor<br \> | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr align="left"> | ||
| - | <td>Se lösningarna.</td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning a) </div> | ||
| - | <div class=NavContent> | ||
| - | Lösning till delfråga a) | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_8a.gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning b) </div> | ||
| - | <div class=NavContent> | ||
| - | Lösning till delfråga b) | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_8b.gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning c) </div> | ||
| - | <div class=NavContent> | ||
| - | Lösning till delfråga c) | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr> | ||
| - | <td align="center"> | ||
| - | [[Bild:2_3_8c.gif]] | ||
| - | </td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | ==Övning 2.3:9== | ||
| - | |||
| - | <div class="ovning"> | ||
| - | Hitta alla skärningspunkter mellan x-axeln och kurvan | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr align="left"> | ||
| - | <td class="ntext">a)</td> | ||
| - | <td class="ntext" width="33%">$y=x^2-1$</td> | ||
| - | <td class="ntext">b)</td> | ||
| - | <td class="ntext" width="33%">$y=x^2-5x+6$</td> | ||
| - | <td class="ntext">c)</td> | ||
| - | <td class="ntext" width="33%">$y=3x^2-12x+9$</td> | ||
| - | </tr> | ||
| - | <tr><td height="5px"/></tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Facit </div> | ||
| - | <div class=NavContent> | ||
| - | Facit till alla delfrågor | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr align="left"> | ||
| - | <td class="ntext">a)</td> | ||
| - | <td class="ntext" width="33%">$(-1,0)\ $ och $\ (1,0)$</td> | ||
| - | <td class="ntext">b)</td> | ||
| - | <td class="ntext" width="33%">$(2,0)\ $ och $\ (3,0)$</td> | ||
| - | <td class="ntext">c)</td> | ||
| - | <td class="ntext" width="33%">$(1,0)\ $ och $\ (3,0)$</td> | ||
| - | </tr> | ||
| - | <tr><td height="5px"/></tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
| - | |||
| - | |||
| - | |||
| - | ==Övning 2.3:10== | ||
| - | |||
| - | <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="50%">$y \geq x^2\ $ och $\ y \leq 1 $</td> | ||
| - | <td class="ntext">b)</td> | ||
| - | <td class="ntext" width="50%">$y \leq 1-x^2\ $ och $\ x \geq 2y-3 $</td> | ||
| - | </tr> | ||
| - | <tr align="left"> | ||
| - | <td class="ntext">c)</td> | ||
| - | <td class="ntext" width="50%">$1 \geq x \geq y^2 $</td> | ||
| - | <td class="ntext">d)</td> | ||
| - | <td class="ntext" width="50%">$x^2 \leq y \leq x $</td> | ||
| - | </tr> | ||
| - | <tr><td height="5px"/></tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | |||
| - | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Facit </div> | ||
| - | <div class=NavContent> | ||
| - | Facit till alla delfrågor | ||
| - | <table width="100%" cellspacing="10px"> | ||
| - | <tr align="left"> | ||
| - | <td>Se lösningarna.</td> | ||
| - | </tr> | ||
| - | </table> | ||
| - | </div> | ||
| - | </div> | ||
