Övn 3
Sommarmatte 1
(Skillnad mellan versioner)
| Versionen från 25 juni 2007 kl. 07.34 (redigera) KTH.SE:u1xsetv1 (Diskussion | bidrag) (→Övning 3.3:5) ← Gå till föregående ändring |
Nuvarande version (25 juni 2007 kl. 07.47) (redigera) (ogör) KTH.SE:u1xsetv1 (Diskussion | bidrag) |
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| (En mellanliggande version visas inte.) | |||
| Rad 722: | Rad 722: | ||
| </div> | </div> | ||
| - | <div class=NavFrame style="CLEAR: both"> | + | <div class="svar"> |
| - | <div class=NavHead>Facit </div> | + | |
| - | <div class=NavContent> | + | |
| - | Facit till alla delfrågor | + | |
| <table width="100%" cellspacing="10px"> | <table width="100%" cellspacing="10px"> | ||
| <tr align="left"><td height="5px"/></tr> | <tr align="left"><td height="5px"/></tr> | ||
| Rad 743: | Rad 740: | ||
| </table> | </table> | ||
| </div> | </div> | ||
| + | |||
| + | ==Övning 3.4:1== | ||
| + | |||
| + | <div class="ovning"> | ||
| + | Lös ekvationerna | ||
| + | <table width="100%" cellspacing="10px"> | ||
| + | <tr align="left"> | ||
| + | <td class="ntext">a)</td> | ||
| + | <td class="ntext" width="33%">$e^x=13$</td> | ||
| + | <td class="ntext">b)</td> | ||
| + | <td class="ntext" width="33%">$13e^x=2\cdot3^{-x}$</td> | ||
| + | <td class="ntext">c)</td> | ||
| + | <td class="ntext" width="33%">$3e^x=7\cdot2^x$</td> | ||
| + | <tr><td height="5px"/></tr> | ||
| + | </tr> | ||
| + | </table> | ||
| </div> | </div> | ||
| - | <div class=NavFrame style="CLEAR: both"> | + | <div class="svar"> |
| - | <div class=NavHead>Lösning a) </div> | + | <table width="100%" cellspacing="10px"> |
| - | <div class=NavContent> | + | <tr align="left"> |
| - | Lösning till delfråga a) | + | <td class="ntext">a)</td> |
| - | <table width="100%"> | + | <td class="ntext" width="33%">$x=\ln 13$</td> |
| - | <tr> | + | <td class="ntext">b)</td> |
| - | <td align="center"> | + | <td class="ntext" width="33%">$x=\displaystyle\frac{\ln 2 - \ln 13}{1+\ln 3}$</td> |
| - | [[Bild:3_3_6a-1(2).gif]] | + | <td class="ntext">c)</td> |
| - | </td> | + | <td class="ntext" width="33%">$x=\displaystyle\frac{\ln 7 - \ln 3}{1-\ln 2}$</td> |
| + | <tr><td height="5px"/></tr> | ||
| </tr> | </tr> | ||
| - | <tr> | + | </table> |
| - | <td align="center"> | + | </div> |
| - | [[Bild:3_3_6a-2(2).gif]] | + | |
| - | </td> | + | |
| + | ==Övning 3.4:2== | ||
| + | |||
| + | <div class="ovning"> | ||
| + | Lös ekvationerna | ||
| + | <table width="100%" cellspacing="10px"> | ||
| + | <tr align="left"> | ||
| + | <td class="ntext">a)</td> | ||
| + | <td class="ntext" width="33%">$2^{\scriptstyle x^2-2}=1$</td> | ||
| + | <td class="ntext">b)</td> | ||
| + | <td class="ntext" width="33%">$e^{2x}+e^x=4$</td> | ||
| + | <td class="ntext">c)</td> | ||
| + | <td class="ntext" width="33%">$3e^{x^2}=2^x$</td> | ||
| </tr> | </tr> | ||
| + | <tr><td height="5px"/></tr> | ||
| </table> | </table> | ||
| - | </div> | ||
| </div> | </div> | ||
| - | <div class=NavFrame style="CLEAR: both"> | + | <div class="svar"> |
| - | <div class=NavHead>Lösning b) </div> | + | <table width="100%" cellspacing="10px"> |
| - | <div class=NavContent> | + | <tr align="left"> |
| - | Lösning till delfråga b) | + | <td class="ntext">a)</td> |
| - | <table width="100%"> | + | <td class="ntext" width="33%">$ \left\{ \eqalign{ x_1&=\sqrt2 \cr x_2&=-\sqrt2 } \right. $</td> |
| - | <tr> | + | <td class="ntext">b)</td> |
| - | <td align="center"> | + | <td class="ntext" width="33%">$x=\ln \left(\displaystyle\frac{\sqrt17}{2}-\frac{1}{2}\right)$</td> |
| - | [[Bild:3_3_6b.gif]] | + | <td class="ntext">c)</td> |
| - | </td> | + | <td class="ntext" width="33%">Saknar lösning</td> |
| </tr> | </tr> | ||
| + | <tr><td height="5px"/></tr> | ||
| </table> | </table> | ||
| - | </div> | ||
| </div> | </div> | ||
| - | <div class=NavFrame style="CLEAR: both"> | + | ==Övning 3.4:3== |
| - | <div class=NavHead>Lösning c) </div> | + | |
| - | <div class=NavContent> | + | <div class="ovning"> |
| - | Lösning till delfråga c) | + | Lös ekvationerna |
| - | <table width="100%"> | + | <table width="100%" cellspacing="10px"> |
| - | <tr> | + | <tr align="left"> |
| - | <td align="center"> | + | <td class="ntext">a)</td> |
| - | [[Bild:3_3_6c-1(2).gif]] | + | <td class="ntext" width="50%">$2^{-x^2}=2e^{2x}$</td> |
| - | </td> | + | <td class="ntext">b)</td> |
| + | <td class="ntext" width="50%">$\ln{(x^2+3x)}=\ln{(3x^2-2x)}$</td> | ||
| </tr> | </tr> | ||
| - | <tr> | + | <tr align="left"> |
| - | <td align="center"> | + | <td class="ntext">c)</td> |
| - | [[Bild:3_3_6c-2(2).gif]] | + | <td class="ntext" width="50%">$\ln{x}+\ln{(x+4)}=\ln{(2x+3)}$</td> |
| - | </td> | + | |
| </tr> | </tr> | ||
| + | <tr><td height="5px"/></tr> | ||
| </table> | </table> | ||
| </div> | </div> | ||
| + | |||
| + | <div class="svar"> | ||
| + | <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><td height="5px"/></tr> | ||
| + | </table> | ||
| </div> | </div> | ||
Nuvarande version
[redigera] Övning 3.1:1
Skriv i potensform
| a) | $\sqrt{2}$ | b) | $\sqrt{7^5}$ | c) | $\bigl(\sqrt[\scriptstyle3]{3}\,\bigr)^4$ | d) | $\sqrt{\sqrt{3}}$ |
| a) | $2^{1/2}$ | b) | $7^{5/2}$ | c) | $3^{4/3}$ | d) | $3^{1/4}$ |
[redigera] Övning 3.1:2
Förenkla så långt som möjligt
| a) | $\sqrt{3^2}$ | b) | $\sqrt{\left(-3\right)^2}$ | c) | $\sqrt{-3^2}$ | d) | $\sqrt{5}\cdot\sqrt[\scriptstyle3]{5}\cdot5$ |
| e) | $\sqrt{18}\cdot\sqrt{8}$ | f) | $\sqrt[\scriptstyle3]{8}$ | g) | $\sqrt[\scriptstyle3]{-125}$ | ||
| a) | $3$ | b) | $3$ | c) | ej definierad | d) | $5^{11/6}$ |
| e) | $12$ | f) | $2$ | g) | $-5$ | ||
[redigera] Övning 3.1:3
Förenkla så långt som möjligt
| a) | $\bigl(\sqrt{5}-\sqrt{2}\,\bigr)\bigl(\sqrt{5}+\sqrt{2}\,\bigr)$ | b) | $\displaystyle \frac{\sqrt{96}}{\sqrt{18}}$ |
| c) | $\sqrt{16+\sqrt{16}}$ | d) | $\sqrt{\displaystyle \frac{2}{3}}\bigl(\sqrt{6}-\sqrt{3}\,\bigr)$ |
| a) | $3$ | b) | $\displaystyle \frac{4\sqrt{3}}{3}$ |
| c) | $2\sqrt{5}$ | d) | $2-\sqrt{2}$ |
[redigera] Övning 3.1:4
Förenkla så långt som möjligt
| a) | $\sqrt{0{,}16}$ | b) | $\sqrt[\scriptstyle3]{0{,}027}$ | |
| c) | $\sqrt{50}+4\sqrt{20}-3\sqrt{18}-2\sqrt{80}$ | d) | $\sqrt{48}+ \sqrt{12} +\sqrt{3} -\sqrt{75}$ | |
| a) | $0{,}4$ | b) | $0{,}3$ |
| c) | $-4\sqrt{2}$ | d) | $2\sqrt{3}$ |
[redigera] Övning 3.1:5
Skriv som ett uttryck utan rottecken i nämnaren.
| a) | $\displaystyle \frac{2}{\sqrt{12}}$ | b) | $\displaystyle \frac{1}{\sqrt[\scriptstyle3]{7}}$ | c) | $\displaystyle \frac{2}{3+\sqrt{7}}$ | d) | $\displaystyle \frac{1}{\sqrt{17}-\sqrt{13}}$ |
| a) | $\displaystyle \frac{\sqrt{3}}{3}$ | b) | $\displaystyle \frac{7^{2/3}}{7}$ | c) | $3-\sqrt{7}$ | d) | $\displaystyle \frac{\sqrt{17}+\sqrt{13}}{4}$ |
[redigera] Övning 3.1:6
Skriv som ett uttryck utan rottecken i nämnaren.
| a) | $\displaystyle \frac{\sqrt{2}+3}{\sqrt{5}-2}$ | b) | $\displaystyle \frac{1}{\left(\sqrt{3}-2\right)^2-2}$ |
| c) | $\displaystyle \frac{\displaystyle \frac{1}{\sqrt{3}}-\displaystyle \frac{1}{\sqrt{5}}}{\displaystyle \frac{1}{\sqrt{2}}-\displaystyle \frac{1}{2}}$ | d) | $\displaystyle \frac{1}{\sqrt{2}+\sqrt{3}+\sqrt{6}}$ |
| a) | $6+2\sqrt{2}+3\sqrt{5}+\sqrt{10}$ | b) | $-\displaystyle \frac{5+4\sqrt{3}}{23}$ |
| c) | $\displaystyle \frac{2}{3}\sqrt{6}+\displaystyle \frac{2}{3}\sqrt{3}-\displaystyle \frac{2}{5}\sqrt{10}-\displaystyle \frac{2}{5}\sqrt{5}$ | d) | $\displaystyle \frac{5\sqrt{3}+7\sqrt{2}-\sqrt{6}-12}{23}$ |
[redigera] Övning 3.1:7
Förenkla så långt som möjligt
| a) | $\displaystyle \frac{1}{\sqrt{6}-\sqrt{5}} - \displaystyle \frac{1}{\sqrt{7}-\sqrt{6}}$ | b) | $\displaystyle \frac{5\sqrt{7}-7\sqrt{5}}{\sqrt{7}-\sqrt{5}}$ | c) | $\displaystyle \sqrt{153}-\sqrt{68}$ |
| a) | $\sqrt{5}-\sqrt{7}$ | b) | $-\sqrt{35}$ | c) | $\sqrt{17}$ |
[redigera] Övning 3.1:8
Avgör vilket tal som är störst av
| a) | $\sqrt[\scriptstyle3]5\ $ och $\ \sqrt[\scriptstyle3]6$ | b) | $\sqrt7\ $ och $\ 7$ |
| c) | $\sqrt7\ $ och $\ 2{,}5$ | d) | $\sqrt2\bigl(\sqrt[\scriptstyle4]3\,\bigr)^3\ $ och $\ \sqrt[\scriptstyle3]2\cdot3$ |
| a) | $\sqrt[\scriptstyle3]6 > \sqrt[\scriptstyle3]5$ | b) | $7 > \sqrt7$ |
| c) | $\sqrt7 > 2{,}5$ | d) | $\sqrt[\scriptstyle3]2\cdot3 > \sqrt2\bigl(\sqrt[\scriptstyle4]3\,\bigr)^3$ |
[redigera] Övning 3.2:1
| Lös ekvationen $\ \sqrt{x-4}=6-x\,$. |
| $x=5$ |
[redigera] Övning 3.2:2
| Lös ekvationen $\ \sqrt{2x+7}=x+2\,$. |
| $x=1$ |
[redigera] Övning 3.2:3
| Lös ekvationen $\ \sqrt{3x-8}+2=x\,$. |
| $\left \{ \eqalign{ x_1 & = 3 \cr x_2 & = 4\cr } \right.$ |
[redigera] Övning 3.2:4
| Lös ekvationen $\ \sqrt{1-x}=2-x\,$. |
| Saknar lösning. |
[redigera] Övning 3.2:5
| Lös ekvationen $\ \sqrt{3x-2}=2-x\,$. |
| $x=1$ |
[redigera] Övning 3.2:6
| Lös ekvationen $\ \sqrt{x+1}+\sqrt{x+5}=4\,$. |
| $x=\displaystyle\frac{5}{4}$ |
[redigera] Övning 3.3:1
Bestäm $\,x\,$ om
| a) | $10^x=1\,000$ | b) | $10^x=0{,}1$ |
| c) | $\displaystyle \frac{1}{10^x}=100$ | d) | $\displaystyle \frac{1}{10^x}=0{,}000\,1$ |
| a) | $x=3$ | b) | $x=-1$ |
| c) | $x=-2$ | d) | $x=4$ |
[redigera] Övning 3.3:2
Beräkna
| a) | $\lg{ 0{,}1}$ | b) | $\lg{ 10\,000}$ | c) | $\lg {0{,}001}$ | d) | $\lg {1}$ |
| e) | $10^{\lg{2}}$ | f) | $\lg{10^3}$ | g) | $10^{-\lg{0{,}1}}$ | h) | $\lg{\displaystyle \frac{1}{10^2}}$ |
| a) | $-1$ | b) | $4$ | c) | $-3$ | d) | $0$ |
| e) | $2$ | f) | $3$ | g) | $10$ | h) | $-2$ |
[redigera] Övning 3.3:3
Beräkna
| a) | $\log_2{8}$ | b) | $\log_9{\displaystyle \frac{1}{3}}$ | c) | $\log_2{0{,}125}$ |
| d) | $\log_3{\left(9\cdot3^{1/3}\right)}$ | e) | $2^{\log_{\scriptstyle2}{4}}$ | f) | $\log_2{4}+\log_2{\displaystyle \frac{1}{16}}$ |
| g) | $\log_3{12}-\log_3{4}$ | h) | $\log_a{\bigl(a^2\sqrt{a}\,\bigr)}$ | ||
| a) | $3$ | b) | $-\displaystyle \frac{1}{2}$ | c) | $-3$ |
| d) | $\displaystyle \frac{7}{3}$ | e) | $4$ | f) | $-2$ |
| g) | $1$ | h) | $\displaystyle \frac{5}{2}$ | ||
[redigera] Övning 3.3:4
Förenkla
| a) | $\lg{50}-\lg{5}$ | b) | $\lg{23}+\lg{\displaystyle \frac{1}{23}}$ | c) | $\lg{27^{1/3}}+\displaystyle \frac{\lg{3}}{2}+\lg{\displaystyle \frac{1}{9}}$ |
| a) | $1$ | b) | $0$ | c) | $-\displaystyle \frac{1}{2}\lg{3}$ |
[redigera] Övning 3.3:5
Förenkla
| a) | $\ln{e^3}+\ln{e^2}$ | b) | $\ln{8}-\ln{4}-\ln{2}$ | c) | $(\ln{1})\cdot e^2$ |
| d) | $\ln{e}-1$ | e) | $\ln{\displaystyle \frac{1}{e^2}}$ | f) | $\left(e^{\ln{e}}\right)^2$ |
| a) | $5$ | b) | $0$ | c) | $0$ |
| d) | $0$ | e) | $-2$ | f) | $e^2$ |
[redigera] Övning 3.3:6
Använd miniräknaren till höger för att beräkna med tre decimaler (Knappen LN betecknar den naturliga logaritmen i basen e):
| a) | $\log_3{4}$ |
| b) | $\lg{46}$ |
| c) | $\log_3{\log_2{(3^{118})}}$ |
| a) | $1{,}262$ |
| b) | $1{,}663$ |
| c) | $4{,}762$ |
[redigera] Övning 3.4:1
Lös ekvationerna
| a) | $e^x=13$ | b) | $13e^x=2\cdot3^{-x}$ | c) | $3e^x=7\cdot2^x$ |
| 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) | $3e^{x^2}=2^x$ |
| 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) | $\ln{x}+\ln{(x+4)}=\ln{(2x+3)}$ | ||
| 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) | $x=1$ | ||
