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 | ||
