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Lösning 1.3:1c - Versionshistorik
2026-05-25T01:28:31Z
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Tek den 29 juni 2010 kl. 11.05
2010-06-29T11:05:58Z
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<td colspan='2' style="background-color: white; color:black;">← Äldre version</td>
<td colspan='2' style="background-color: white; color:black;">Versionen från 29 juni 2010 kl. 11.05</td>
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<tr><td colspan="2" class="diff-lineno">Rad 1:</td>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>{{<del style="color: red; font-weight: bold; text-decoration: none;">NAVCONTENT_START}</del>}</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Funktionen har derivata lika med noll i tre punkter <math>x=-2</math>, <math>x=-1</math> och <math>x=\tfrac</ins>{<ins style="color: red; font-weight: bold; text-decoration: none;">1}</ins>{<ins style="color: red; font-weight: bold; text-decoration: none;">2</ins>}<ins style="color: red; font-weight: bold; text-decoration: none;"></math> (se figuren nedan) som därmed är de kritiska punkterna till funktionen.</ins></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><center> <del style="color: red; font-weight: bold; text-decoration: none;">[[Bild</del>:<del style="color: red; font-weight: bold; text-decoration: none;">1_3_1c</del>-1<del style="color: red; font-weight: bold; text-decoration: none;">(</del>3<del style="color: red; font-weight: bold; text-decoration: none;">)</del>.<del style="color: red; font-weight: bold; text-decoration: none;">gif]] </del></center></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>{{<del style="color: red; font-weight: bold; text-decoration: none;">NAVCONTENT_STOP}</del>}</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><center><ins style="color: red; font-weight: bold; text-decoration: none;">{{</ins>:<ins style="color: red; font-weight: bold; text-decoration: none;">1.3 </ins>- <ins style="color: red; font-weight: bold; text-decoration: none;">Figur - Grafen till övning </ins>1<ins style="color: red; font-weight: bold; text-decoration: none;">.</ins>3<ins style="color: red; font-weight: bold; text-decoration: none;">:1c med horisontella tangenter}}</center></ins></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>{{<del style="color: red; font-weight: bold; text-decoration: none;">NAVCONTENT_START</del>}}</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><center><small>Grafen har horisontella tagenter i ''x''&nbsp;=&nbsp;-2, ''x''&nbsp;=&nbsp;-1 och ''x''&nbsp;=&nbsp;½</ins>.<ins style="color: red; font-weight: bold; text-decoration: none;"></small></ins></center></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><center> <del style="color: red; font-weight: bold; text-decoration: none;">[[Bild:1_3_1c</del>-2(<del style="color: red; font-weight: bold; text-decoration: none;">3</del>)<del style="color: red; font-weight: bold; text-decoration: none;">.gif]] </del></<del style="color: red; font-weight: bold; text-decoration: none;">center</del>></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>{{<del style="color: red; font-weight: bold; text-decoration: none;">NAVCONTENT_STOP</del>}}</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>{{<del style="color: red; font-weight: bold; text-decoration: none;">NAVCONTENT_START</del>}}</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Punkten <math>x=-1</math> är en terasspunkt eftersom derivatan är positiv i en omgivning både till vänster och höger om punkten.</ins></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><<del style="color: red; font-weight: bold; text-decoration: none;">center</del>> <del style="color: red; font-weight: bold; text-decoration: none;">[[Bild:1_3_1c-3(3).gif]] </del></<del style="color: red; font-weight: bold; text-decoration: none;">center</del>></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">{{NAVCONTENT_STOP}</del>}</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">I definitionsintervallets vänstra ändpunkt <math>x=-3</math> och i punkten <math>x=\tfrac</ins>{<ins style="color: red; font-weight: bold; text-decoration: none;">1}</ins>{<ins style="color: red; font-weight: bold; text-decoration: none;">2</ins>}<ins style="color: red; font-weight: bold; text-decoration: none;"></math> har funktionen lokala maximipunkter eftersom i alla punkter i närheten av respektive punkt antar funktionen mindre funktionsvärden. I punkten <math>x=-2</math> och i den högre ändpunkten <math>x=2</math> har funktionen lokala minimipunkter.</ins></div></td></tr>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Vi ser också att den vänstra ändpunkten är en global maximipunkt (funktionen antar sitt största värde där) och <math>x=-2</math> är en global minimipunkt.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><center></ins>{{<ins style="color: red; font-weight: bold; text-decoration: none;">:1.3 - Figur - Grafen till övning 1.3:1c med max och min</ins>}}<<ins style="color: red; font-weight: bold; text-decoration: none;">/</ins>center></div></td></tr>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Mellan den vänstra ändpunkten och <math>x=</ins>-2<ins style="color: red; font-weight: bold; text-decoration: none;"></math> liksom mellan <math>x=\tfrac{1}{2}</math> och den högra ändpunkten är funktionen strängt avtagande </ins>(<ins style="color: red; font-weight: bold; text-decoration: none;">ju större <math>x</math> desto mindre är <math>f(x</ins>)</<ins style="color: red; font-weight: bold; text-decoration: none;">math</ins>><ins style="color: red; font-weight: bold; text-decoration: none;">) medan funktionen är strängt växande mellan <math>x=-2</math> och <math>x=\tfrac{1}{2}</math> (grafen planar ut i <math>x=-1</math> men den är inte konstant där).</ins></div></td></tr>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>{<ins style="color: red; font-weight: bold; text-decoration: none;">| align="center"</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">|-</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">||</ins>{<ins style="color: red; font-weight: bold; text-decoration: none;">{:1.3 - Figur - Grafen till övning 1.3:1c med avtagande område</ins>}}</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">| width="10px" |</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">||</ins>{{<ins style="color: red; font-weight: bold; text-decoration: none;">:1.3 - Figur - Grafen till övning 1.3:1c med växande område</ins>}}</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">|-</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">||</ins><<ins style="color: red; font-weight: bold; text-decoration: none;">small</ins>><ins style="color: red; font-weight: bold; text-decoration: none;">Område där funktionen är strängt avtagande</ins></<ins style="color: red; font-weight: bold; text-decoration: none;">small</ins>></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">| width="10px" |</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">||<small>Område där funktionen är strängt växande</small></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">|</ins>}</div></td></tr>
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