3.2 Övningar
Sommarmatte 2
(Skillnad mellan versioner)
| Versionen från 28 juni 2007 kl. 13.42 (redigera) Mikael (Diskussion | bidrag) (→Övning 3.2:2) ← Gå till föregående ändring |
Versionen från 3 juli 2007 kl. 17.28 (redigera) (ogör) KTH.SE:u1tyze7e (Diskussion | bidrag) (Korrekturläst) Gå till nästa ändring → |
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| Rad 1: | Rad 1: | ||
| ==Övning 3.2:1== | ==Övning 3.2:1== | ||
| <div class="ovning"> | <div class="ovning"> | ||
| - | Givet de komplexa talen $z=2+i , \, w=2+3i$ och $u=-1-2i$. Markera följande tal i det komplexa talplanet | + | Givet de komplexa talen $\,z=2+i\,$, $\,w=2+3i\,$ och $\,u=-1-2i\,$. Markera följande tal i det komplexa talplanet |
| <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" width="50%">$z$ och $w$</td> | + | <td class="ntext" width="50%">$z\,$ och $\,w$</td> |
| <td class="ntext">b)</td> | <td class="ntext">b)</td> | ||
| - | <td class="ntext" width="50%">$z+u$ och $z-w$</td> | + | <td class="ntext" width="50%">$z+u\,$ och $\,z-u$</td> |
| </tr> | </tr> | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">c)</td> | <td class="ntext">c)</td> | ||
| - | <td class="ntext" width="50%">$ 2z+w$</td> | + | <td class="ntext" width="50%">$2z+w$</td> |
| <td class="ntext">d)</td> | <td class="ntext">d)</td> | ||
| - | <td class="ntext" width="50%">$z-\overline{w} +u$</td> | + | <td class="ntext" width="50%">$z-\overline{w}+u$</td> |
| </tr> | </tr> | ||
| </table> | </table> | ||
| Rad 21: | Rad 21: | ||
| <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<br \> |
| <table width="100%" cellspacing="10px"> | <table width="100%" cellspacing="10px"> | ||
| <tr align="left"> | <tr align="left"> | ||
| Rad 29: | Rad 29: | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">c)<br\>[[Bild:f_3_2_1c.gif]]</td> | <td class="ntext">c)<br\>[[Bild:f_3_2_1c.gif]]</td> | ||
| - | <td class="ntext">d)<br\>[[Bild:f_3_2_1d.gif]]></td> | + | <td class="ntext">d)<br\>[[Bild:f_3_2_1d.gif]]</td> |
| </tr> | </tr> | ||
| </table> | </table> | ||
| Rad 63: | Rad 63: | ||
| <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<br \> |
| <table width="100%" cellspacing="10px"> | <table width="100%" cellspacing="10px"> | ||
| <tr align="left"> | <tr align="left"> | ||
| Rad 83: | Rad 83: | ||
| ==Övning 3.2:3== | ==Övning 3.2:3== | ||
| <div class="ovning"> | <div class="ovning"> | ||
| - | De komplexa talen $1+i, 3+2i$ och $3i$ bildar i det komplexa talplanet tre hörn i en kvadrat. Bestäm kvadratens fjärde hörn. | + | De komplexa talen $\,1+i\,$, $\,3+2i\,$ och $\,3i\,$ bildar i det komplexa talplanet tre hörn i en kvadrat. Bestäm kvadratens fjärde hörn. |
| </div> | </div> | ||
| Rad 113: | Rad 113: | ||
| <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<br \> |
| <table width="100%" cellspacing="10px"> | <table width="100%" cellspacing="10px"> | ||
| <tr align="left"> | <tr align="left"> | ||
| Rad 145: | Rad 145: | ||
| <td class="ntext" width="50%">$ (\sqrt{3} +i)(1-i)$</td> | <td class="ntext" width="50%">$ (\sqrt{3} +i)(1-i)$</td> | ||
| <td class="ntext">d)</td> | <td class="ntext">d)</td> | ||
| - | <td class="ntext" width="50%">$\displaystyle\frac{1}{1+i}$</td> | + | <td class="ntext" width="50%">$\displaystyle\frac{i}{1+i}$</td> |
| </tr> | </tr> | ||
| </table> | </table> | ||
| Rad 153: | Rad 153: | ||
| <div class=NavHead>Facit </div> | <div class=NavHead>Facit </div> | ||
| <div class=NavContent> | <div class=NavContent> | ||
| - | Facit till alla delfrågorna<br \> | + | Facit till alla delfrågor<br \> |
| <table width="100%" cellspacing="10px"> | <table width="100%" cellspacing="10px"> | ||
| <tr align="left"> | <tr align="left"> | ||
| Rad 163: | Rad 163: | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">c)</td> | <td class="ntext">c)</td> | ||
| - | <td class="ntext" width="50%">$-\displaystyle\frac{\pi}{12}$</td> | + | <td class="ntext" width="50%">$-\displaystyle\frac{\pi}{12}\,$ eller $\,\displaystyle\frac{23}{12}\pi$</td> |
| <td class="ntext">d)</td> | <td class="ntext">d)</td> | ||
| <td class="ntext" width="50%">$\displaystyle\frac{\pi}{4}$</td> | <td class="ntext" width="50%">$\displaystyle\frac{\pi}{4}$</td> | ||
| Rad 185: | Rad 185: | ||
| <td class="ntext" width="50%">$ -4-4i$</td> | <td class="ntext" width="50%">$ -4-4i$</td> | ||
| <td class="ntext">d)</td> | <td class="ntext">d)</td> | ||
| - | <td class="ntext" width="50%">$\sqrt{10} + \sqrt{30}i$</td> | + | <td class="ntext" width="50%">$\sqrt{10} + \sqrt{30}\,i$</td> |
| </tr> | </tr> | ||
| <tr align="left"> | <tr align="left"> | ||
| Rad 191: | Rad 191: | ||
| <td class="ntext" width="50%">$ \displaystyle\frac{1+i\sqrt{3}}{1+i}$</td> | <td class="ntext" width="50%">$ \displaystyle\frac{1+i\sqrt{3}}{1+i}$</td> | ||
| <td class="ntext">f)</td> | <td class="ntext">f)</td> | ||
| - | <td class="ntext" width="50%">$\displaystyle\frac{(2+2i)(1+i\sqrt{3})}{3i(\sqrt{12} -2i)}$</td> | + | <td class="ntext" width="50%">$\displaystyle\frac{(2+2i)(1+i\sqrt{3}\,)}{3i(\sqrt{12} -2i)}$</td> |
| </tr> | </tr> | ||
| </table> | </table> | ||
| Rad 199: | Rad 199: | ||
| <div class=NavHead>Facit </div> | <div class=NavHead>Facit </div> | ||
| <div class=NavContent> | <div class=NavContent> | ||
| - | Facit till alla delfrågorna<br \> | + | Facit till alla delfrågor |
| + | <br \> | ||
| <table width="100%" cellspacing="10px"> | <table width="100%" cellspacing="10px"> | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">a)</td> | <td class="ntext">a)</td> | ||
| - | <td class="ntext" width="50%">$\displaystyle3(\cos 0 + i \sin 0)$</td> | + | <td class="ntext" width="50%">$\displaystyle3(\cos 0 + i\,\sin 0)$</td> |
| <td class="ntext">b)</td> | <td class="ntext">b)</td> | ||
| - | <td class="ntext" width="50%">$\displaystyle11\left(\cos \frac{3\pi}{2} + i\sin\frac{3\pi}{2}\right)$</td> | + | <td class="ntext" width="50%">$\displaystyle11\left(\cos \frac{3\pi}{2} + i\,\sin\frac{3\pi}{2}\right)$</td> |
| </tr> | </tr> | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">c)</td> | <td class="ntext">c)</td> | ||
| - | <td class="ntext" width="50%">$\displaystyle4\sqrt2\left(\cos \frac{5\pi}{4} + i\sin\frac{5\pi}{4}\right)$</td> | + | <td class="ntext" width="50%">$\displaystyle4\sqrt2\left(\cos \frac{5\pi}{4} + i\,\sin\frac{5\pi}{4}\right)$</td> |
| <td class="ntext">d)</td> | <td class="ntext">d)</td> | ||
| - | <td class="ntext" width="50%">$\displaystyle2\sqrt{10}\left(\cos \frac{\pi}{3} + i\sin\frac{\pi}{3}\right)$</td> | + | <td class="ntext" width="50%">$\displaystyle2\sqrt{10}\left(\cos \frac{\pi}{3} + i\,\sin\frac{\pi}{3}\right)$</td> |
| </tr> | </tr> | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">e)</td> | <td class="ntext">e)</td> | ||
| - | <td class="ntext" width="50%">$\displaystyle2\sqrt2\left(\cos \frac{\pi}{12} + i\sin\frac{\pi}{12}\right)$</td> | + | <td class="ntext" width="50%">$\displaystyle\sqrt2\left(\cos \frac{\pi}{12} + i\,\sin\frac{\pi}{12}\right)$</td> |
| <td class="ntext">f)</td> | <td class="ntext">f)</td> | ||
| - | <td class="ntext" width="50%">$\displaystyle\frac{\sqrt2}{3}\left(\cos \frac{\pi}{4} + i\sin\frac{\pi}{4}\right)$</td> | + | <td class="ntext" width="50%">$\displaystyle\frac{\sqrt2}{3}\left(\cos \frac{\pi}{4} + i\,\sin\frac{\pi}{4}\right)$</td> |
| </tr> | </tr> | ||
| </table> | </table> | ||
| </div> | </div> | ||
| </div> | </div> | ||
Versionen från 3 juli 2007 kl. 17.28
Innehåll |
Övning 3.2:1
Givet de komplexa talen $\,z=2+i\,$, $\,w=2+3i\,$ och $\,u=-1-2i\,$. Markera följande tal i det komplexa talplanet
| a) | $z\,$ och $\,w$ | b) | $z+u\,$ och $\,z-u$ |
| c) | $2z+w$ | d) | $z-\overline{w}+u$ |
Facit
Facit till alla delfrågor
a) |
b) |
c) |
d) |
Övning 3.2:2
Rita in följande mängder i det komplexa talplanet
| a) | $0\le \mbox{Im}\, z \le 3$ | b) | $0 \le \mbox{Re} \, z \le \mbox{Im}\, z \le 1$ |
| c) | $ |z|=2$ | d) | $|z-1-i|=3$ |
| e) | $ \mbox{Re}\, z = i + \bar z$ | f) | $2<|z-i|\le3$ |
Facit
Facit till alla delfrågor
a) |
b) |
c) |
d) |
e) |
f) |
Övning 3.2:3
De komplexa talen $\,1+i\,$, $\,3+2i\,$ och $\,3i\,$ bildar i det komplexa talplanet tre hörn i en kvadrat. Bestäm kvadratens fjärde hörn.
Facit
$2+4i$
Övning 3.2:4
Bestäm beloppet av
| a) | $3+4i$ | b) | $(2-i) + (5+3i)$ |
| c) | $(3-4i)(3+2i)$ | d) | $\displaystyle\frac{3-4i}{3+2i}$ |
Facit
Facit till alla delfrågor
| a) | $5$ | b) | $\sqrt{53}$ |
| c) | $5\sqrt{13}$ | d) | $\displaystyle\frac{5}{\sqrt{13}}$ |
Övning 3.2:5
Bestäm argumentet av
| a) | $-10$ | b) | $-2+2i$ |
| c) | $ (\sqrt{3} +i)(1-i)$ | d) | $\displaystyle\frac{i}{1+i}$ |
Facit
Facit till alla delfrågor
| a) | $\pi$ | b) | $\displaystyle\frac{3\pi}{4}$ |
| c) | $-\displaystyle\frac{\pi}{12}\,$ eller $\,\displaystyle\frac{23}{12}\pi$ | d) | $\displaystyle\frac{\pi}{4}$ |
Övning 3.2:6
Skriv följande tal i polär form
| a) | $3$ | b) | $-11i$ |
| c) | $ -4-4i$ | d) | $\sqrt{10} + \sqrt{30}\,i$ |
| e) | $ \displaystyle\frac{1+i\sqrt{3}}{1+i}$ | f) | $\displaystyle\frac{(2+2i)(1+i\sqrt{3}\,)}{3i(\sqrt{12} -2i)}$ |
Facit
Facit till alla delfrågor
| a) | $\displaystyle3(\cos 0 + i\,\sin 0)$ | b) | $\displaystyle11\left(\cos \frac{3\pi}{2} + i\,\sin\frac{3\pi}{2}\right)$ |
| c) | $\displaystyle4\sqrt2\left(\cos \frac{5\pi}{4} + i\,\sin\frac{5\pi}{4}\right)$ | d) | $\displaystyle2\sqrt{10}\left(\cos \frac{\pi}{3} + i\,\sin\frac{\pi}{3}\right)$ |
| e) | $\displaystyle\sqrt2\left(\cos \frac{\pi}{12} + i\,\sin\frac{\pi}{12}\right)$ | f) | $\displaystyle\frac{\sqrt2}{3}\left(\cos \frac{\pi}{4} + i\,\sin\frac{\pi}{4}\right)$ |
