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<title>Oppgavesamling i R1</title>
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<meta name="author" content="Tarjei Bærland" />
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FOR A PARTICULAR PURPOSE. See the GNU GPL for more details.
As additional permission under GNU GPL version 3 section 7, you
may distribute non-source (e.g., minimized or compacted) forms of
that code without the copy of the GNU GPL normally required by
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<body>
<div id="content">
<h1 class="title">Oppgavesamling i R1</h1>
<div id="table-of-contents">
<h2>Innhold</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org4fffc41">CAS</a></li>
</ul>
</div>
</div>
<script type="text/javascript" language="JavaScript"> org_html_manager.set("BASE_URL_QUESTIONS", "https://github.com/tarjeiba/r1/issues/new/");</script>
<div id="outline-container-org4fffc41" class="outline-2">
<h2 id="org4fffc41">CAS</h2>
<div class="outline-text-2" id="text-org4fffc41">
<p>
Vi skal nå komme oss dit hvor vi kan løse følgende eksamensoppgave:
</p>
<div class="TASK">
<p>
Funksjonen \(f\) er gitt ved \[f(x) = 2 \ln (x^2 - 4) - \frac{1}{2} x\]
</p>
<dl class="org-dl">
<dt>a</dt><dd>Bruk graftegner til å tegne grafen til \(f\) når \(x \in \langle -4, 16 \rangle\).</dd>
<dt>b</dt><dd>Bestem eventuelle topp- og bunnpunkter på grafen til \(f\).</dd>
</dl>
<p>
Funksjonen \(g\) er gitt ved \[g(x) = 2\ln (x^2 + k) - \frac{1}{2}x,\quad k>0.\]
</p>
<dl class="org-dl">
<dt>c</dt><dd>Bruk CAS til å bestemme \(k\) slik at \(g\) har et ekstremalpunkt i \(x-1\).</dd>
<dt>d</dt><dd>Bruk blant annet CAS til å bestemme hvor mange ekstremalpunkt \(g\) har for ulike verdier av \(k\).</dd>
</dl>
</div>
<p>
Veien dit, leder via CAS som kalkulator og som likningsløser.
</p>
</div>
<div id="outline-container-orgc1db89d" class="outline-3">
<h3 id="orgc1db89d">Som kalkulator</h3>
<div class="outline-text-3" id="text-orgc1db89d">
<p>
Aller først, må vi føle oss trygge på at vi klarer å bruke CAS som kalkulator. Da er det i utgangspunktet følgende knapper vi er interessert i:
</p>
<p>
<img src="figurer/cas-knapper.png" alt="cas-knapper.png"/>
</p>
<p>
Her har vi «eksakt utregning»-knappen, «numerisk utregning»-knappen og «behold inntasting»-knappen. Test hva de ulike knappene gjør med følgende beregninger.
</p>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
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<thead>
<tr>
<th scope="col" class="org-left">Utregning</th>
<th scope="col" class="org-left">Inntasting</th>
<th scope="col" class="org-left">Kommentar</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">\(2\cdot 2\)</td>
<td class="org-left"><code>2 * 2</code></td>
<td class="org-left"> </td>
</tr>
<tr>
<td class="org-left">$\frac{1}{2}+3</td>
<td class="org-left"><code>1/2 "hp" + 3</code></td>
<td class="org-left"><code>"hp"</code> er «høyrepil» på tastaturet</td>
</tr>
<tr>
<td class="org-left">\(2 + a + 2a\)</td>
<td class="org-left"><code>2 + a + 2 a</code></td>
<td class="org-left">Legg merke til mellomrommet i <code>2 a</code></td>
</tr>
<tr>
<td class="org-left">\(2 + \lg a\)</td>
<td class="org-left"><code>2 + lg(a)</code></td>
<td class="org-left">Legg merke til CAS' håndtering av logaritmer</td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="outline-container-org20e1868" class="outline-3">
<h3 id="org20e1868">Som likningsløser</h3>
<div class="outline-text-3" id="text-org20e1868">
<p>
Videre kan vi bruke CAS til å løse likninger. La oss forsøke å løse likninga \[2x^2 - 5x + 3 = 0.\] Om vi her gyver løs med abc-formelen, får vi \[x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4\cdot 2 \cdot 3}{2 \cdot 2} = 1 \vee \frac{3}{2}.\]
</p>
<p>
Dette kunne vi sjølsagt også ha løst med CAS som kalkulator, med følgende linjer
</p>
<p>
<img src="figurer/cas-uttrykk-abc.png" alt="cas-uttrykk-abc.png"/>
</p>
<p>
Enda enklere blir det om vi skriver likninga direkte inn i CAS, sjekker inntastinga, og velger en av likningsløsningsknappene.
</p>
<p>
<img src="figurer/cas-likningsknapper.png" alt="cas-likningsknapper.png"/>
</p>
<p>
Under taster jeg først inn i likninga i rad en, som så kan refereres til via <code>$1</code>, som jeg så løser eksakt med <img src="figurer/cas-knapp-eksakt.png" alt="cas-knapp-eksakt.png"/>-knappen og så numerisk med <img src="figurer/cas-knapp-numerisk.png" alt="cas-knapp-numerisk.png"/>-knappen.
</p>
<p>
<img src="figurer/cas-likningssolve.png" alt="cas-likningssolve.png"/>
</p>
<p>
Forsøk å løse følgende likninger i CAS:
</p>
<p>
\[2x^2 - 5 x + c = 0\]
\[2x^2 - b x = c\]
\[\lg x + 10 = 0\]
\[\lg x = x\]
</p>
<p>
Tenk over hva svarene vil si for de ulike løsningene.
</p>
<p>
Under vises grafene til de to funksjonene \(f(x) = \lg x\) og \(g(x) = x\). Hva forteller denne oss om forsøket på løsning av den siste likninga over?
</p>
<p>
<img src="figurer/graf-ingen.png" alt="graf-ingen.png"/>
</p>
</div>
</div>
<div id="outline-container-org423f063" class="outline-3">
<h3 id="org423f063">Som derivator</h3>
<div class="outline-text-3" id="text-org423f063">
<p>
For å derivere funksjoner i CAS. Kan vi velge mellom å skrive inn et funksjonsnavn og trykke derivasjonsknappen, <img src="figurer/cas-knapp-derivasjon.png" alt="cas-knapp-derivasjon.png"/>, eller å skrive inn derivasjonstegnet direkte. Se linjene under hvor vi deriverer funksjonen \(h(x) = 3 \ln x\) på to måter:
</p>
<p>
<img src="figurer/cas-derivasjon-framgang.png" alt="cas-derivasjon-framgang.png"/>
</p>
</div>
</div>
<div id="outline-container-org5b8e4cc" class="outline-3">
<h3 id="org5b8e4cc">Løsning av vår oppgave</h3>
<div class="outline-text-3" id="text-org5b8e4cc">
<p>
Vi er nå et godt stykke på vei for å få løst oppgave vi konkret står overfor.
</p>
</div>
</div>
</div>
</div>
</body>
</html>