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Övningar 2.1 - Versionshistorik
2026-06-21T11:10:45Z
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KTH.SE:u1zpa8nw den 18 juli 2007 kl. 08.34
2007-07-18T08:34:16Z
<p></p>
<table border='0' width='98%' cellpadding='0' cellspacing='4' style="background-color: white;">
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<td colspan='2' width='50%' align='center' style="background-color: white;">← Äldre version</td>
<td colspan='2' width='50%' align='center' style="background-color: white;">Versionen från 18 juli 2007 kl. 08.34</td>
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<tr><td colspan="2" align="left"><strong>Rad 96:</strong></td>
<td colspan="2" align="left"><strong>Rad 96:</strong></td></tr>
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<tr><td colspan="2"> </td><td>+</td><td style="background: #cfc; font-size: smaller;"><br><br><br></td></tr>
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KTH.SE:u1zpa8nw den 18 juli 2007 kl. 08.33
2007-07-18T08:33:39Z
<p></p>
<table border='0' width='98%' cellpadding='0' cellspacing='4' style="background-color: white;">
<tr>
<td colspan='2' width='50%' align='center' style="background-color: white;">← Äldre version</td>
<td colspan='2' width='50%' align='center' style="background-color: white;">Versionen från 18 juli 2007 kl. 08.33</td>
</tr>
<tr><td colspan="2" align="left"><strong>Rad 91:</strong></td>
<td colspan="2" align="left"><strong>Rad 91:</strong></td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;"><tr align="left" valign="top"></td><td> </td><td style="background: #eee; font-size: smaller;"><tr align="left" valign="top"></td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;"><td class="ntext">b)</td></td><td> </td><td style="background: #eee; font-size: smaller;"><td class="ntext">b)</td></td></tr>
<tr><td>-</td><td style="background: #ffa; font-size: smaller;"><td class="ntext" width="100%">$\displaystyle \int \sin^2 x\quad$ (Ledning: skriv om integranden <span style="color: red; font-weight: bold;">med en </span>trigonometrisk formel)</td></td><td>+</td><td style="background: #cfc; font-size: smaller;"><td class="ntext" width="100%">$\displaystyle \int \sin^2 x\quad$ (Ledning: skriv om integranden <span style="color: red; font-weight: bold;">m.h.a. </span>trigonometrisk formel)</td></td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;"></tr></td><td> </td><td style="background: #eee; font-size: smaller;"></tr></td></tr>
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KTH.SE:u1zpa8nw den 18 juli 2007 kl. 08.32
2007-07-18T08:32:05Z
<p></p>
<table border='0' width='98%' cellpadding='0' cellspacing='4' style="background-color: white;">
<tr>
<td colspan='2' width='50%' align='center' style="background-color: white;">← Äldre version</td>
<td colspan='2' width='50%' align='center' style="background-color: white;">Versionen från 18 juli 2007 kl. 08.32</td>
</tr>
<tr><td colspan="2" align="left"><strong>Rad 16:</strong></td>
<td colspan="2" align="left"><strong>Rad 16:</strong></td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;"></tr></td><td> </td><td style="background: #eee; font-size: smaller;"></tr></td></tr>
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<tr><td>-</td><td style="background: #ffa; font-size: smaller;">Ledning d) För $a < b < 0$ gäller $\displaystyle\int_{a}^{b}|x|\, dx=\int_{a}^{b} -x\,dx$</td><td colspan="2"> </td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;"></div></td><td> </td><td style="background: #eee; font-size: smaller;"></div></td></tr>
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<tr><td colspan="2" align="left"><strong>Rad 35:</strong></td>
<td colspan="2" align="left"><strong>Rad 34:</strong></td></tr>
<tr><td> </td><td style="background: #eee; font-size: smaller;"></tr></td><td> </td><td style="background: #eee; font-size: smaller;"></tr></td></tr>
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<tr><td>-</td><td style="background: #ffa; font-size: smaller;">Ledning c,d) skriv om $\sqrt x=x^{1/2}$, och använd eventuellt potenslagarna.</td><td colspan="2"> </td></tr>
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<tr><td colspan="2" align="left"><strong>Rad 54:</strong></td>
<td colspan="2" align="left"><strong>Rad 52:</strong></td></tr>
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<tr><td>-</td><td style="background: #ffa; font-size: smaller;">Ledning b) Använd att $\sin2v=2\sin v\cos v$<br\></td><td colspan="2"> </td></tr>
<tr><td>-</td><td style="background: #ffa; font-size: smaller;">Ledning d) $\displaystyle\int\frac{x^2+1}{x}\, dx=\int\frac{x^2}{x}\, dx+\int\frac{1}{x}\, dx$</td><td colspan="2"> </td></tr>
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KTH.SE:u1zpa8nw: Ny sida: __NOTOC__ '''Övning 2.1:1''' <div class="ovning" style="margin-top:-20px; margin-bottom:-18px;">Tolka integralerna som areor och bestäm deras värde <table width="100%" cellspacing="10px"...
2007-07-18T08:29:44Z
<p>Ny sida: __NOTOC__ '''Övning 2.1:1''' <div class="ovning" style="margin-top:-20px; margin-bottom:-18px;">Tolka integralerna som areor och bestäm deras värde <table width="100%" cellspacing="10px"...</p>
<p><b>Ny sida</b></p><div>__NOTOC__<br />
'''Övning 2.1:1'''<br />
<div class="ovning" style="margin-top:-20px; margin-bottom:-18px;">Tolka integralerna som areor och bestäm deras värde<br />
<table width="100%" cellspacing="10px"><br />
<tr align="left"><br />
<td class="ntext">a)</td><br />
<td class="ntext">$\displaystyle\int_{-1}^{2} 2\, dx$</td><br />
<td class="ntext">b)</td><br />
<td class="ntext">$\displaystyle\int_{0}^{1} (2x+1)\, dx$</td><br />
</tr><br />
<tr align="left"><br />
<td class="ntext">c)</td><br />
<td class="ntext">$\displaystyle \int_{0}^{2} (3-2x)\, dx$</td><br />
<td class="ntext">d)</td><br />
<td class="ntext">$\displaystyle\int_{-1}^{2}|x| \, dx$</td><br />
</tr><br />
</table><br />
Ledning d) För $a < b < 0$ gäller $\displaystyle\int_{a}^{b}|x|\, dx=\int_{a}^{b} -x\,dx$<br />
</div><br />
<br />
'''Övning 2.1:2'''<br />
<div class="ovning" style="margin-top:-20px; margin-bottom:-18px;">Beräkna integralerna<br />
<table width="100%" cellspacing="10px"><br />
<tr align="left"><br />
<td class="ntext">a)</td><br />
<td class="ntext" width="50%">$\displaystyle\int_{0}^{2} (x^2+3x^3)\, dx$</td><br />
<td class="ntext">b)</td><br />
<td class="ntext" width="50%">$\displaystyle\int_{-1}^{2} (x-2)(x+1)\, dx$</td><br />
</tr><br />
<tr align="left"><br />
<td class="ntext">c)</td><br />
<td class="ntext" width="50%">$ \displaystyle\int_{4}^{9} \left(\sqrt{x} - \displaystyle\frac{1}{\sqrt{x}}\right)\, dx$</td><br />
<td class="ntext">d)</td><br />
<td class="ntext" width="50%">$\displaystyle\int_{1}^{4} \displaystyle\frac{\sqrt{x}}{x^2}\, dx$</td><br />
</tr><br />
</table><br />
Ledning c,d) skriv om $\sqrt x=x^{1/2}$, och använd eventuellt potenslagarna.<br />
</div><br />
<br />
'''Övning 2.1:3'''<br />
<div class="ovning" style="margin-top:-20px; margin-bottom:-18px;">Beräkna integralerna<br />
<table width="100%" cellspacing="10px"><br />
<tr align="left"><br />
<td class="ntext">a)</td><br />
<td class="ntext" width="50%">$\displaystyle\int \sin x\, dx$</td><br />
<td class="ntext">b)</td><br />
<td class="ntext" width="50%">$\displaystyle\int 2\sin x \cos x\, dx$</td><br />
</tr><br />
<tr align="left"><br />
<td class="ntext">c)</td><br />
<td class="ntext" width="50%">$ \displaystyle\int e^{2x}(e^x+1)\, dx$</td><br />
<td class="ntext">d)</td><br />
<td class="ntext" width="50%">$\displaystyle\int \displaystyle\frac{x^2+1}{x}\, dx$</td><br />
</tr><br />
</table><br />
Ledning b) Använd att $\sin2v=2\sin v\cos v$<br\><br />
Ledning d) $\displaystyle\int\frac{x^2+1}{x}\, dx=\int\frac{x^2}{x}\, dx+\int\frac{1}{x}\, dx$<br />
</div><br />
<br />
'''Övning 2.1:4'''<br />
<div class="ovning" style="margin-top:-20px; margin-bottom:-18px;"><br />
<table cellspacing="10px"><br />
<tr align="left" valign="top"><br />
<td class="ntext">a)</td><br />
<td class="ntext" width="100%">Beräkna arean mellan kurvan $y=\sin x$ och $x$-axeln när $0\le x \le \frac{5\pi}{4}$</td><br />
</tr><br />
<tr align="left" valign="top"><br />
<td class="ntext">b)</td><br />
<td class="ntext" width="100%">Beräkna arean av det område under kurvan $y=-x^2+2x+2$ och ovanför $x$-axeln</td><br />
</tr><br />
<tr align="left" valign="top"><br />
<td class="ntext">c)</td><br />
<td class="ntext" width="100%">Beräkna arean av det ändliga området mellan kurvorna $y=\frac{1}{4}x^2+2$ och $y=8-\frac{1}{8}x^2$ (studentexamen 1965).</td><br />
</tr><br />
<tr align="left" valign="top"><br />
<td class="ntext">d)</td><br />
<td class="ntext" width="100%">Beräkna arean av det ändliga området som kurvorna $y=x+2, y=1$ och $y=\frac{1}{x}$ innesluter.</td><br />
</tr><br />
<tr align="left" valign="top"><br />
<td class="ntext">e)</td><br />
<td class="ntext" width="100%">Beräkna arean av området som ges av olikheterna $x^2\le y\le x+2$.</td><br />
</tr><br />
<tr><td height="5px"/></tr><br />
</table><br />
</div><br />
<br />
'''Övning 2.1:5'''<br />
<div class="ovning" style="margin-top:-20px; margin-bottom:-18px;"><br />
Beräkna integralerna<br />
<table width="100%" cellspacing="10px"><br />
<tr align="left" valign="top"><br />
<td class="ntext">a)</td><br />
<td class="ntext" width="100%">$\displaystyle \int \displaystyle\frac{dx}{\sqrt{x+9}-\sqrt{x}}\quad$ (Ledning: förläng med nämnarens konjugat)</td><br />
</tr><br />
<tr align="left" valign="top"><br />
<td class="ntext">b)</td><br />
<td class="ntext" width="100%">$\displaystyle \int \sin^2 x\quad$ (Ledning: skriv om integranden med en trigonometrisk formel)</td><br />
</tr><br />
<tr><td height="5px"/></tr><br />
</table><br />
</div></div>
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