3.4 Övningar
Sommarmatte 1
(Skillnad mellan versioner)
Versionen från 29 april 2007 kl. 14.02 (redigera) Ossiang (Diskussion | bidrag) (→Övning 3.4:2) ← Gå till föregående ändring |
Nuvarande version (31 maj 2007 kl. 07.03) (redigera) (ogör) KTH.SE:u1rp004j (Diskussion | bidrag) |
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(8 mellanliggande versioner visas inte.) | |||
Rad 1: | Rad 1: | ||
+ | |||
==Övning 3.4:1== | ==Övning 3.4:1== | ||
Rad 4: | Rad 5: | ||
Lös ekvationerna | Lös ekvationerna | ||
<table width="100%" cellspacing="10px"> | <table width="100%" cellspacing="10px"> | ||
- | <tr height="5px"><td\></tr> | ||
<tr align="left"> | <tr align="left"> | ||
- | <td class="ntext>a)</td> | + | <td class="ntext">a)</td> |
- | <td class="ntext>$e^x=13$</td> | + | <td class="ntext" width="33%">$e^x=13$</td> |
- | <td class="ntext>b)</td> | + | <td class="ntext">b)</td> |
- | <td class="ntext>$13e^x=2\cdot3^{-x}$</td> | + | <td class="ntext" width="33%">$13e^x=2\cdot3^{-x}$</td> |
- | <td class="ntext>c)</td> | + | <td class="ntext">c)</td> |
- | <td class="ntext>$3e^x=7\cdot2^x$</td> | + | <td class="ntext" width="33%">$3e^x=7\cdot2^x$</td> |
+ | <tr><td height="5px"/></tr> | ||
</tr> | </tr> | ||
</table> | </table> | ||
Rad 19: | Rad 20: | ||
<div class=NavHead>Facit </div> | <div class=NavHead>Facit </div> | ||
<div class=NavContent> | <div class=NavContent> | ||
- | Facit till alla delfrågorna<br \> | + | Facit till alla delfrågor |
<table width="100%" cellspacing="10px"> | <table width="100%" cellspacing="10px"> | ||
- | <tr height="5px"><td\></tr> | ||
<tr align="left"> | <tr align="left"> | ||
- | <td class="ntext>a)</td> | + | <td class="ntext">a)</td> |
- | <td class="ntext>$x=\ln 13$</td> | + | <td class="ntext" width="33%">$x=\ln 13$</td> |
- | <td class="ntext>b)</td> | + | <td class="ntext">b)</td> |
- | <td class="ntext>$x=\displaystyle\frac{\ln 2 - \ln 13}{1+\ln 3}$</td> | + | <td class="ntext" width="33%">$x=\displaystyle\frac{\ln 2 - \ln 13}{1+\ln 3}$</td> |
- | <td class="ntext>c)</td> | + | <td class="ntext">c)</td> |
- | <td class="ntext>$x=\displaystyle\frac{\ln 7 - \ln 3}{1-\ln 2}$</td> | + | <td class="ntext" width="33%">$x=\displaystyle\frac{\ln 7 - \ln 3}{1-\ln 2}$</td> |
+ | <tr><td height="5px"/></tr> | ||
</tr> | </tr> | ||
</table> | </table> | ||
Rad 35: | Rad 36: | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning a </div> | + | <div class=NavHead>Lösning a) </div> |
<div class=NavContent> | <div class=NavContent> | ||
- | Lösning till delfråga a<br \>[[Bild:3_4_1a-1(2).gif]]<br \>[[Bild:3_4_1a-2(2).gif]] | + | Lösning till delfråga a) |
+ | <table width="100%"> | ||
+ | <tr> | ||
+ | <td align="center"> | ||
+ | [[Bild:3_4_1a-1(2).gif]] | ||
+ | </td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td align="center"> | ||
+ | [[Bild:3_4_1a-2(2).gif]] | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
</div> | </div> | ||
</div> | </div> | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning b </div> | + | <div class=NavHead>Lösning b) </div> |
<div class=NavContent> | <div class=NavContent> | ||
- | Lösning till delfråga b<br \>[[Bild:3_4_1b-1(2).gif]]<br \>[[Bild:3_4_1b-2(2).gif]] | + | Lösning till delfråga b) |
+ | <table width="100%"> | ||
+ | <tr> | ||
+ | <td align="center"> | ||
+ | [[Bild:3_4_1b-1(2).gif]] | ||
+ | </td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td align="center"> | ||
+ | [[Bild:3_4_1b-2(2).gif]] | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
</div> | </div> | ||
</div> | </div> | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning </div> | + | <div class=NavHead>Lösning c) </div> |
<div class=NavContent> | <div class=NavContent> | ||
- | Lösning till delfråga <br \>[[Bild:3_4_1c.gif]] | + | Lösning till delfråga c) |
+ | <table width="100%"> | ||
+ | <tr> | ||
+ | <td align="center"> | ||
+ | [[Bild:3_4_1c.gif]] | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
</div> | </div> | ||
</div> | </div> | ||
Rad 62: | Rad 94: | ||
<tr align="left"> | <tr align="left"> | ||
<td class="ntext">a)</td> | <td class="ntext">a)</td> | ||
- | <td class="ntext">$2^{\scriptstyle x^2-2}=1$</td> | + | <td class="ntext" width="33%">$2^{\scriptstyle x^2-2}=1$</td> |
<td class="ntext">b)</td> | <td class="ntext">b)</td> | ||
- | <td class="ntext">$e^{2x}+e^x=4$</td> | + | <td class="ntext" width="33%">$e^{2x}+e^x=4$</td> |
<td class="ntext">c)</td> | <td class="ntext">c)</td> | ||
- | <td class="ntext">$3e^{x^2}=2^x$</td> | + | <td class="ntext" width="33%">$3e^{x^2}=2^x$</td> |
</tr> | </tr> | ||
<tr><td height="5px"/></tr> | <tr><td height="5px"/></tr> | ||
Rad 75: | Rad 107: | ||
<div class=NavHead>Facit </div> | <div class=NavHead>Facit </div> | ||
<div class=NavContent> | <div class=NavContent> | ||
- | Facit till alla delfrågorna<br \> | + | Facit till alla delfrågor |
<table width="100%" cellspacing="10px"> | <table width="100%" cellspacing="10px"> | ||
<tr align="left"> | <tr align="left"> | ||
<td class="ntext">a)</td> | <td class="ntext">a)</td> | ||
- | <td class="ntext">$x=$</td> | + | <td class="ntext" width="33%">$ \left\{ \eqalign{ x_1&=\sqrt2 \cr x_2&=-\sqrt2 } \right. $</td> |
<td class="ntext">b)</td> | <td class="ntext">b)</td> | ||
- | <td class="ntext">$x=$</td> | + | <td class="ntext" width="33%">$x=\ln \left(\displaystyle\frac{\sqrt17}{2}-\frac{1}{2}\right)$</td> |
<td class="ntext">c)</td> | <td class="ntext">c)</td> | ||
- | <td class="ntext">$ \left\{ \matrix{ x_1=\sqrt2 \cr x_2=-\sqrt2 } \right $</td> | + | <td class="ntext" width="33%">Saknar lösning</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%"> | ||
+ | <tr> | ||
+ | <td align="center"> | ||
+ | [[Bild:3_4_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%"> | ||
+ | <tr> | ||
+ | <td align="center"> | ||
+ | [[Bild:3_4_2b-1(2).gif]] | ||
+ | </td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td align="center"> | ||
+ | [[Bild:3_4_2b-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%"> | ||
+ | <tr> | ||
+ | <td align="center"> | ||
+ | [[Bild:3_4_2c.gif]] | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | </div> | ||
+ | </div> | ||
+ | |||
+ | ==Övning 3.4:3== | ||
+ | |||
+ | <div class="ovning"> | ||
+ | Lös ekvationerna | ||
+ | <table width="100%" cellspacing="10px"> | ||
+ | <tr align="left"> | ||
+ | <td class="ntext">a)</td> | ||
+ | <td class="ntext" width="50%">$2^{-x^2}=2e^{2x}$</td> | ||
+ | <td class="ntext">b)</td> | ||
+ | <td class="ntext" width="50%">$\ln{(x^2+3x)}=\ln{(3x^2-2x)}$</td> | ||
+ | </tr> | ||
+ | <tr align="left"> | ||
+ | <td class="ntext">c)</td> | ||
+ | <td class="ntext" width="50%">$\ln{x}+\ln{(x+4)}=\ln{(2x+3)}$</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="50%">$x=-\,\displaystyle\frac{1}{\ln{2}}\pm\sqrt{\left(\displaystyle\frac{1}{\ln{2}}\right)^2-1}$</td> | ||
+ | <td class="ntext">b)</td> | ||
+ | <td class="ntext" width="50%">$x=\displaystyle \frac{5}{2}$</td> | ||
+ | </tr> | ||
+ | <tr align="left"> | ||
+ | <td class="ntext">c)</td> | ||
+ | <td class="ntext" width="50%">$x=1$</td> | ||
</tr> | </tr> | ||
<tr><td height="5px"/></tr> | <tr><td height="5px"/></tr> | ||
Rad 92: | Rad 209: | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning </div> | + | <div class=NavHead>Lösning a) </div> |
<div class=NavContent> | <div class=NavContent> | ||
- | Lösning till delfråga <br \>[[Bild:]] | + | Lösning till delfråga a) |
+ | <table width="100%"> | ||
+ | <tr> | ||
+ | <td align="center"> | ||
+ | [[Bild:3_4_3a-1(2).gif]] | ||
+ | </td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td align="center"> | ||
+ | [[Bild:3_4_3a-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%"> | ||
+ | <tr> | ||
+ | <td align="center"> | ||
+ | [[Bild:3_4_3b-1(2).gif]] | ||
+ | </td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td align="center"> | ||
+ | [[Bild:3_4_3b-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%"> | ||
+ | <tr> | ||
+ | <td align="center"> | ||
+ | [[Bild:3_4_3c-1(2).gif]] | ||
+ | </td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td align="center"> | ||
+ | [[Bild:3_4_3c-2(2).gif]] | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
</div> | </div> | ||
</div> | </div> |
Nuvarande version
[redigera] Övning 3.4:1
Lös ekvationerna
a) | $e^x=13$ | b) | $13e^x=2\cdot3^{-x}$ | c) | $3e^x=7\cdot2^x$ |
Facit
Facit till alla delfrågor
a) | $x=\ln 13$ | b) | $x=\displaystyle\frac{\ln 2 - \ln 13}{1+\ln 3}$ | c) | $x=\displaystyle\frac{\ln 7 - \ln 3}{1-\ln 2}$ |
[redigera] Övning 3.4:2
Lös ekvationerna
a) | $2^{\scriptstyle x^2-2}=1$ | b) | $e^{2x}+e^x=4$ | c) | $3e^{x^2}=2^x$ |
Facit
Facit till alla delfrågor
a) | $ \left\{ \eqalign{ x_1&=\sqrt2 \cr x_2&=-\sqrt2 } \right. $ | b) | $x=\ln \left(\displaystyle\frac{\sqrt17}{2}-\frac{1}{2}\right)$ | c) | Saknar lösning |
[redigera] Övning 3.4:3
Lös ekvationerna
a) | $2^{-x^2}=2e^{2x}$ | b) | $\ln{(x^2+3x)}=\ln{(3x^2-2x)}$ |
c) | $\ln{x}+\ln{(x+4)}=\ln{(2x+3)}$ | ||
Facit
Facit till alla delfrågor
a) | $x=-\,\displaystyle\frac{1}{\ln{2}}\pm\sqrt{\left(\displaystyle\frac{1}{\ln{2}}\right)^2-1}$ | b) | $x=\displaystyle \frac{5}{2}$ |
c) | $x=1$ | ||