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MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}\n div#h5p_6a6d75294fe0f  section.h5p_page p span.doxMathFormula .MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}\n div#h5p_6a6d75294fe0f  section.h5p_page p span.doxMathFormula .MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}\n div#h5p_6a6d75294fe0f  section.h5p_page p span.doxMathFormula .MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}\n div#h5p_6a6d75294fe0f  section.h5p_page p span.doxMathFormula .MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}\n div#h5p_6a6d75294fe0f  section.h5p_page p span.doxMathFormula .MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}\n div#h5p_6a6d75294fe0f  section.h5p_page p span.doxMathFormula .MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}\n div#h5p_6a6d75294fe0f  section.h5p_page p span.doxMathFormula .MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}\n div#h5p_6a6d75294fe0f  section.h5p_page p span.doxMathFormula .MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}\n div#h5p_6a6d75294fe0f  section.h5p_page p span.doxMathFormula .MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}\n div#h5p_6a6d75294fe0f  section.h5p_page p span.doxMathFormula .MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}\n div#h5p_6a6d75294fe0f  section.h5p_page p span.doxMathFormula .MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}\n div#h5p_6a6d75294fe0f  section.h5p_page p span.doxMathFormula .MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}\n div#h5p_6a6d75294fe0f  section.h5p_page p span.doxMathFormula .MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}\n div#h5p_6a6d75294fe0f  section.h5p_page p span.doxMathFormula .MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}\n div#h5p_6a6d75294fe0f  section.h5p_page p span.doxMathFormula .MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}\n div#h5p_6a6d75294fe0f  section.h5p_page p span.doxMathFormula .mjx-sub span.mjx-char {font-size: 70.7%}\n div#h5p_6a6d75294fe0f  section.h5p_page p span.doxMathFormula .mjx-sup span.mjx-char {font-size: 70.7%}\n div#h5p_6a6d75294fe0f  section.h5p_page p span.doxMathFormula .mjx-under span.mjx-char {font-size: 70.7%}\n div#h5p_6a6d75294fe0f  section.h5p_page p span.doxMathFormula .mjx-over span.mjx-char {font-size: 70.7%}\n div#h5p_6a6d75294fe0f  section.h5p_page p span.doxMathFormula .mjx-root span.mjx-char {font-size: 70.7%;}\n div#h5p_6a6d75294fe0f  section.h5p_page p span.doxMathFormula .mjx-root {font-size: auto !important ; width: 0.5em !important; vertical-align: 1.2em !important}\n<\/style><script src=\"https:\/\/cdn.mathjax.org\/mathjax\/2.6-latest\/MathJax.js?config=MML_CHTML\" type=\"text\/javascript\"> <\/script>\n<div id=\"h5p_6a6d75294fe0f\"><section xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"section_0\" class=\"h5p_layout h5p_page h5p_page_PL0\"><div class=\"PL0\"><p class=\"P1\">&nbsp;<\/p><div style=\"left: 0;height: 750.80016pt;width: 578.90016pt;\" class=\"a14\"><div style=\"left: 0;height: 750.80016pt;position: absolute;width: 578.90016pt;\" class=\"\"><div style=\"position: absolute; float: none !important;height: 10.42778in;width: 8.04028in;margin-left: 0.0125in;margin-top: 0in;\" class=\"a1\"><svg style=\"position: absolute; top: 0; left: 0;padding: 0 !important; border: none !important\" preserveAspectRatio=\"none\" fill-rule=\"evenodd\" class=\"a1\" height=\"10.42778in\" width=\"8.04028in\" viewBox=\"0 0 772 1001\"><g><path d=\"M 0 0 L 772 0 772 1001 0 1001 Z\"\/><\/g><\/svg><\/div><div style=\"position: absolute; float: none !important;height: 10.42778in;width: 5.88339in;margin-left: 2.16939in;margin-top: 0in;\" class=\"a2\"><svg style=\"position: absolute; top: 0; left: 0;padding: 0 !important; border: none !important\" preserveAspectRatio=\"none\" fill-rule=\"evenodd\" class=\"a2\" height=\"10.42778in\" width=\"5.88339in\" viewBox=\"0 0 565 1001\"><g><path d=\"M 0 0 L 565 0 565 1001 0 1001 Z\"\/><\/g><\/svg><p class=\"Ingenafstand\"><span class=\"T6\">Differentialligninger<\/span><\/p><p class=\"P7\">&nbsp;<\/p><p class=\"P8\">&nbsp;<\/p><p class=\"P9\">&nbsp;<\/p><p class=\"P10\">&nbsp;<\/p><\/div><div style=\"left: 0;height: 453.978pt;position: absolute;width: 78.12288pt;\" class=\"\"><div style=\"position: absolute; float: none !important;height: 1.05284in;width: 1.08504in;margin-left: 1.08504in;margin-top: 4.19957in;\" class=\"a3\"><svg style=\"position: absolute; top: 0; left: 0;padding: 0 !important; border: none !important\" preserveAspectRatio=\"none\" fill-rule=\"evenodd\" class=\"a3\" height=\"1.05284in\" width=\"1.08504in\" viewBox=\"0 0 104 101\"><g><path d=\"M 0 0 L 104 0 104 101 0 101 Z\"\/><\/g><\/svg><\/div><div style=\"position: absolute; float: none !important;height: 1.05284in;width: 1.08504in;margin-left: 1.08504in;margin-top: 3.14672in;\" class=\"a4\"><svg style=\"position: absolute; top: 0; left: 0;padding: 0 !important; border: none !important\" preserveAspectRatio=\"none\" fill-rule=\"evenodd\" class=\"a4\" height=\"1.05284in\" width=\"1.08504in\" viewBox=\"0 0 104 101\"><g><path d=\"M 0 0 L 104 0 104 101 0 101 Z\"\/><\/g><\/svg><\/div><div style=\"position: absolute; float: none !important;height: 1.05284in;width: 1.08504in;margin-left: 0in;margin-top: 3.14672in;\" class=\"a5\"><svg style=\"position: absolute; top: 0; left: 0;padding: 0 !important; border: none !important\" preserveAspectRatio=\"none\" fill-rule=\"evenodd\" class=\"a5\" height=\"1.05284in\" width=\"1.08504in\" viewBox=\"0 0 104 101\"><g><path d=\"M 0 0 L 104 0 104 101 0 101 Z\"\/><\/g><\/svg><\/div><div style=\"position: absolute; float: none !important;height: 1.05284in;width: 1.08504in;margin-left: 0in;margin-top: 2.09388in;\" class=\"a6\"><svg style=\"position: absolute; top: 0; left: 0;padding: 0 !important; border: none !important\" preserveAspectRatio=\"none\" fill-rule=\"evenodd\" class=\"a6\" height=\"1.05284in\" width=\"1.08504in\" viewBox=\"0 0 104 101\"><g><path d=\"M 0 0 L 104 0 104 101 0 101 Z\"\/><\/g><\/svg><\/div><div style=\"position: absolute; float: none !important;height: 1.05284in;width: 1.08504in;margin-left: 0in;margin-top: 4.19957in;\" class=\"a7\"><svg style=\"position: absolute; top: 0; left: 0;padding: 0 !important; border: none !important\" preserveAspectRatio=\"none\" fill-rule=\"evenodd\" class=\"a7\" height=\"1.05284in\" width=\"1.08504in\" viewBox=\"0 0 104 101\"><g><path d=\"M 0 0 L 104 0 104 101 0 101 Z\"\/><\/g><\/svg><\/div><div style=\"position: absolute; float: none !important;height: 1.05284in;width: 1.08504in;margin-left: 1.08504in;margin-top: 5.25241in;\" class=\"a8\"><svg style=\"position: absolute; top: 0; left: 0;padding: 0 !important; border: none !important\" preserveAspectRatio=\"none\" fill-rule=\"evenodd\" class=\"a8\" height=\"1.05284in\" width=\"1.08504in\" viewBox=\"0 0 104 101\"><g><path d=\"M 0 0 L 104 0 104 101 0 101 Z\"\/><\/g><\/svg><\/div><\/div><div style=\"position: absolute; float: none !important;height: 1.05354in;width: 1.08504in;margin-left: 1.64457in;margin-top: 0in;\" class=\"a9\"><svg style=\"position: absolute; top: 0; left: 0;padding: 0 !important; border: none !important\" preserveAspectRatio=\"none\" fill-rule=\"evenodd\" class=\"a9\" height=\"1.05354in\" width=\"1.08504in\" viewBox=\"0 0 104 101\"><g><path d=\"M 0 0 L 104 0 104 101 0 101 Z\"\/><\/g><\/svg><p class=\"P11\">&nbsp;<\/p><\/div><\/div><div style=\"left: 0;height: 736.11216pt;position: absolute;width: 354.88296pt;\" class=\"\"><div style=\"left: 0;height: 736.11144pt;position: absolute;width: 19.53pt;\" class=\"\"><div style=\"position: absolute; float: none !important;height: 0.26561in;width: 0.27125in;margin-left: 7.29592in;margin-top: 9.95816in;\" class=\"a10\"><svg style=\"position: absolute; top: 0; left: 0;padding: 0 !important; border: none !important\" preserveAspectRatio=\"none\" fill-rule=\"evenodd\" class=\"a10\" height=\"0.26561in\" width=\"0.27125in\" viewBox=\"0 0 26 25\"><g><path d=\"M 0 0 L 26 0 26 25 0 25 Z\"\/><\/g><\/svg><\/div><div style=\"position: absolute; float: none !important;height: 0.26561in;width: 0.27125in;margin-left: 7.29592in;margin-top: 9.69642in;\" class=\"a11\"><svg style=\"position: absolute; top: 0; left: 0;padding: 0 !important; border: none !important\" preserveAspectRatio=\"none\" fill-rule=\"evenodd\" class=\"a11\" height=\"0.26561in\" width=\"0.27125in\" viewBox=\"0 0 26 25\"><g><path d=\"M 0 0 L 26 0 26 25 0 25 Z\"\/><\/g><\/svg><\/div><div style=\"position: absolute; float: none !important;height: 0.26561in;width: 0.27125in;margin-left: 7.56717in;margin-top: 9.69642in;\" class=\"a12\"><svg style=\"position: absolute; top: 0; left: 0;padding: 0 !important; border: none !important\" preserveAspectRatio=\"none\" fill-rule=\"evenodd\" class=\"a12\" height=\"0.26561in\" width=\"0.27125in\" viewBox=\"0 0 26 25\"><g><path d=\"M 0 0 L 26 0 26 25 0 25 Z\"\/><\/g><\/svg><\/div><\/div><div style=\"position: absolute; float: none !important;height: 0.95896in;width: 4.92893in;margin-left: 2.17136in;margin-top: 9.26482in;\" class=\"a13\"><svg style=\"position: absolute; top: 0; left: 0;padding: 0 !important; border: none !important\" preserveAspectRatio=\"none\" fill-rule=\"evenodd\" class=\"a13\" height=\"0.95896in\" width=\"4.92893in\" viewBox=\"0 0 473 92\"><g><path d=\"M 0 0 L 473 0 473 92 0 92 Z\"\/><\/g><\/svg><p class=\"P12\">&nbsp;<\/p><p class=\"P13\">&nbsp;<\/p><p class=\"P14\">&nbsp;<\/p><\/div><\/div><\/div><p class=\"Normal\"\/><p class=\"P15\">&nbsp;<\/p><p class=\"Overskrift\"><span>Indholdsfortegnelse<\/span><\/p><div><p class=\"P16\"><a href=\"#_Toc449013653\" target=\"_top\" title=\"\"><span class=\"Hyperlink\">Indledning<\/span>&emsp;<span>2<\/span><\/a><\/p><p class=\"P17\"><a href=\"#_Toc449013654\" target=\"_top\" title=\"\"><span class=\"Hyperlink\">Differentialligninger<\/span>&emsp;<span>2<\/span><\/a><\/p><p class=\"P18\"><a href=\"#_Toc449013655\" target=\"_top\" title=\"\"><span class=\"Hyperlink\">L\u00f8sningsprincippet<\/span>&emsp;<span>2<\/span><\/a><\/p><p class=\"P19\"><a href=\"#_Toc449013656\" target=\"_top\" title=\"\"><span class=\"Hyperlink\">Separable differentialligninger<\/span>&emsp;<span>2<\/span><\/a><\/p><p class=\"P20\"><a href=\"#_Toc449013657\" target=\"_top\" title=\"\"><span class=\"Hyperlink\">Opgave<\/span>&emsp;<span>2<\/span><\/a><\/p><p class=\"P21\"><a href=\"#_Toc449013658\" target=\"_top\" title=\"\"><span class=\"Hyperlink\">Type I<\/span>&emsp;<span>2<\/span><\/a><\/p><p class=\"P22\"><a href=\"#_Toc449013659\" target=\"_top\" title=\"\"><span class=\"Hyperlink\">Type II<\/span>&emsp;<span>3<\/span><\/a><\/p><p class=\"P23\"><a href=\"#_Toc449013660\" target=\"_top\" title=\"\"><span class=\"Hyperlink\">Type III<\/span>&emsp;<span>4<\/span><\/a><\/p><p class=\"P24\"><a href=\"#_Toc449013661\" target=\"_top\" title=\"\"><span class=\"Hyperlink\">Type IV<\/span>&emsp;<span>5<\/span><\/a><\/p><p class=\"P25\"><a href=\"#_Toc449013662\" target=\"_top\" title=\"\"><span class=\"Hyperlink\">Type I<\/span>&emsp;<span>6<\/span><\/a><\/p><p class=\"P26\"><a href=\"#_Toc449013663\" target=\"_top\" title=\"\"><span class=\"Hyperlink\">Type II<\/span>&emsp;<span>7<\/span><\/a><\/p><p class=\"P27\"><a href=\"#_Toc449013664\" target=\"_top\" title=\"\"><span class=\"Hyperlink\">Type III<\/span>&emsp;<span>7<\/span><\/a><\/p><p class=\"P28\"><a href=\"#_Toc449013665\" target=\"_top\" title=\"\"><span class=\"Hyperlink\">Type IV<\/span>&emsp;<span>7<\/span><\/a><\/p><p class=\"P29\"><a href=\"#_Toc449013666\" target=\"_top\" title=\"\"><span class=\"Hyperlink\">Differentialligningen for logistisk v\u00e6kst og dens l\u00f8sninger<\/span>&emsp;<span>11<\/span><\/a><\/p><\/div><p class=\"Normal\">&nbsp;<\/p><p class=\"P30\">&nbsp;<\/p><h1 class=\"Overskrift1\"><span id=\"_Toc449013653\"\/><span>Indledning<\/span><\/h1><p class=\"P31\"><span>I denne emneopgave vil vi kort gennemg\u00e5 de mest basale elementer i emnet differentialligninger. Vi vil undervejs gennemg\u00e5 en r\u00e6kke teori for differentialligninger, samtidig med at vi undervejs vil l\u00f8se de i opgaveformuleringen givne opgaver.<\/span>&#x200b;&#x200b;&nbsp;<\/p><h1 class=\"Overskrift1\"><span id=\"_Toc449013654\"\/><span>Differentialligninger<\/span><\/h1><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P32\"><span>Ved en differentialligning forst\u00e5s en ligning hvor differentialkvotienten indg\u00e5r som den ubekendte. Ligningen<\/span>&#x200b;&#x200b;&nbsp;<span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfrac><mml:mrow><mml:mi>d<\/mml:mi><mml:mi>y<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>d<\/mml:mi><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfrac><mml:mo>=<\/mml:mo><mml:mi>f<\/mml:mi><mml:mo>(<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T33\">&#x200b;&#x200b;&nbsp;kaldes for en differentiall<\/span><span class=\"T34\">igning.<\/span><\/p><h2 class=\"Overskrift2\"><span id=\"_Toc449013655\"\/><span>L\u00f8sningsprincippet<\/span><\/h2><p class=\"P35\"><span class=\"T36\">Man kan altid tjekke om en given funktion er en l\u00f8sning til en differentialligning ved at s\u00e6tte l\u00f8sningen ind p\u00e5 begge sider, og derefter se om det giver det samme.&#x200b;&#x200b;&nbsp;<\/span><\/p><h1 class=\"Overskrift1\"><span id=\"_Toc449013656\"\/><span>Separable differentialligninger<\/span><\/h1><p class=\"P37\"><span>Ved separable differentialligninger forst\u00e5s ligninger hvor vi kan isolere x og y p\u00e5 henholdsvis den ene og den anden side af lighedstegnet. Der findes tre typer af disse:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span class=\"T38\">Type I<\/span><span>:<\/span>&#x200b;&#x200b;&nbsp;<span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfrac><mml:mrow><mml:mi>d<\/mml:mi><mml:mi>y<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>d<\/mml:mi><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfrac><mml:mo>=<\/mml:mo><mml:mi>g<\/mml:mi><mml:mo>(<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:math><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span class=\"T39\">Denne af afh\u00e6nger kun af x, og den generelle l\u00f8sning lyder&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u222b<\/mml:mo><mml:mi mathvariant=\"normal\">g<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi mathvariant=\"normal\">x<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mi mathvariant=\"normal\">d<\/mml:mi><mml:mi mathvariant=\"normal\">x<\/mml:mi><\/mml:math><\/span><\/span><span class=\"T40\">.<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span class=\"T41\">Type II<\/span><span class=\"T42\">:&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfrac><mml:mrow><mml:mi>d<\/mml:mi><mml:mi>y<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>d<\/mml:mi><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfrac><mml:mo>=<\/mml:mo><mml:mi>g<\/mml:mi><mml:mo>(<\/mml:mo><mml:mi>y<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:math><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span class=\"T43\">Denne afh\u00e6nger kun af y, og den g<\/span><span class=\"T44\">enerelle l\u00f8sning er&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u222b<\/mml:mo><mml:mfrac><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">g<\/mml:mi><mml:mo>(<\/mml:mo><mml:mi mathvariant=\"normal\">y<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:mfrac><mml:mi mathvariant=\"normal\">d<\/mml:mi><mml:mi mathvariant=\"normal\">y<\/mml:mi><mml:mo>=<\/mml:mo><mml:mi mathvariant=\"normal\">x<\/mml:mi><\/mml:math><\/span><\/span><span class=\"T45\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span class=\"T46\">Type III<\/span><span class=\"T47\">:&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfrac><mml:mrow><mml:mi>d<\/mml:mi><mml:mi>y<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>d<\/mml:mi><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfrac><mml:mo>=<\/mml:mo><mml:mi>h<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mi>*<\/mml:mi><mml:mi>g<\/mml:mi><mml:mo>(<\/mml:mo><mml:mi>y<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:math><\/span><\/span><\/p><p class=\"Normal\"><span>Her er der tale om, at det b\u00e5de er x og y der er den variable.<\/span><\/p><p class=\"Normal\">&nbsp;<\/p><h2 class=\"Overskrift2\"><span id=\"_Toc449013657\"\/><span>Opgave<\/span><\/h2><p class=\"P48\"><span>Find den fuldst\u00e6ndige l\u00f8sning til nedenst\u00e5ende differentialligninger uden brug af Maple og angiv for I, II og III hvilke typer funktioner, der er l\u00f8sninger.<\/span><\/p><h3 class=\"Overskrift3\"><span id=\"_Toc449013658\"\/><span>Type I<\/span><\/h3><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mfrac><mml:mrow><mml:mi>d<\/mml:mi><mml:mi>y<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>d<\/mml:mi><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfrac><mml:mo>=<\/mml:mo><mml:mi>a<\/mml:mi><\/mml:math><\/span><\/span><span class=\"T49\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"Normal\"><span>Vi ganger med dx p\u00e5 begge sider, og f\u00e5r s\u00e5ledes:<\/span><\/p><p class=\"Normal\"><span>dy = a*dx<\/span><\/p><p class=\"Normal\"><span>Og ved simpel integration f\u00e5s s\u00e5:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mo>\u222b<\/mml:mo><mml:mi>d<\/mml:mi><mml:mi>y<\/mml:mi><mml:mo>=<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:mo>\u222b<\/mml:mo><mml:mi>a<\/mml:mi><mml:mi>d<\/mml:mi><mml:mi>x<\/mml:mi><\/mml:math><\/span><\/span><span class=\"T50\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"P51\"><span>Vi f\u00e5r s\u00e5 den fuldst\u00e6ndige l\u00f8sning:<\/span><\/p><p class=\"P52\"><span>y = ax + c<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span class=\"T53\">Hvor meget vokser y med, n\u00e5r x vokser med&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u2206<\/mml:mo><mml:mi mathvariant=\"bold-italic\">x<\/mml:mi><mml:mo>?<\/mml:mo><\/mml:math><\/span><\/span><\/p><p class=\"Normal\"><span>Vi finder funktionstilv\u00e6ksten<\/span><\/p><p class=\"Normal\"><span>f(x+\u0394x)=a(x+\u0394x)+c<\/span><\/p><p class=\"Normal\"><span class=\"T54\">= ax +&#x200b;&#x200b;&nbsp;<\/span><span class=\"T55\">a<\/span><span>\u0394<\/span><span class=\"T56\">x + c&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"Normal\"><span class=\"T57\">= f(x)+a<\/span><span>\u0394<\/span><span class=\"T58\">x<\/span><\/p><p class=\"Normal\"><span>S\u00e5 n\u00e5r x vokser med <\/span>&#x200b;&#x200b;&nbsp;<span>\u0394x, vokser f(x) med a\u0394x<\/span><\/p><p class=\"Normal\">&nbsp;<\/p><h3 class=\"Overskrift3\"><span id=\"_Toc449013659\"\/><span>Type II<\/span><\/h3><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mfrac><mml:mrow><mml:mi>d<\/mml:mi><mml:mi>y<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>d<\/mml:mi><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfrac><mml:mo>=<\/mml:mo><mml:mi>a<\/mml:mi><mml:mi>*<\/mml:mi><mml:mi>y<\/mml:mi><\/mml:math><\/span><\/span><span class=\"T59\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"Normal\"><span>Vi ganger med dx p\u00e5 begge sider:<\/span><\/p><p class=\"Normal\"><span>dy = a*y*dx<\/span><\/p><p class=\"Normal\"><span>Vi dividerer med y p\u00e5 begge sider:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mfrac><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mi>y<\/mml:mi><\/mml:mrow><\/mml:mfrac><mml:mi>d<\/mml:mi><mml:mi>y<\/mml:mi><mml:mo>=<\/mml:mo><mml:mi>a<\/mml:mi><mml:mi>d<\/mml:mi><mml:mi>x<\/mml:mi><\/mml:math><\/span><\/span>&#x200b;&#x200b;&nbsp;<\/p><p class=\"Normal\"><span>Og ved simpel integration f\u00e5s s\u00e5:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mo>\u222b<\/mml:mo><mml:mfrac><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mi>y<\/mml:mi><\/mml:mrow><\/mml:mfrac><mml:mi>d<\/mml:mi><mml:mi>y<\/mml:mi><mml:mo>=<\/mml:mo><mml:mo>\u222b<\/mml:mo><mml:mi>a<\/mml:mi><mml:mi>d<\/mml:mi><mml:mi>x<\/mml:mi><\/mml:math><\/span><\/span>&#x200b;&#x200b;&nbsp;<\/p><p class=\"Normal\"><span>S\u00e5 tager vi stamfunktionen til ovenst\u00e5ende:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mi>l<\/mml:mi><mml:mi>n<\/mml:mi><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:mi>y<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>=<\/mml:mo><mml:mi>a<\/mml:mi><mml:mi>x<\/mml:mi><mml:mo>+<\/mml:mo><mml:mi>c<\/mml:mi><\/mml:math><\/span><\/span>&#x200b;&#x200b;&nbsp;<\/p><p class=\"Normal\"><span>S\u00e5 benytter vi e<\/span>&#x200b;&#x200b;&nbsp;<span>for at fjerne ln:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:mi>y<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>=<\/mml:mo><mml:msup><mml:mrow><mml:mi>e<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>a<\/mml:mi><mml:mi>x<\/mml:mi><mml:mo>+<\/mml:mo><mml:msub><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:msup><mml:mo>=<\/mml:mo><mml:msup><mml:mrow><mml:mi>e<\/mml:mi><\/mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:msup><mml:msup><mml:mrow><mml:mi>e<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>a<\/mml:mi><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:math><\/span><\/span>&#x200b;&#x200b;&nbsp;<\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span>S\u00e5 benytter vi<\/span>&#x200b;&#x200b;&nbsp;<span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u00b1<\/mml:mo><\/mml:math><\/span><\/span>&#x200b;&#x200b;&nbsp;<span>for at oph\u00e6ve y numerisk:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:msup><mml:mrow><mml:mi>y<\/mml:mi><mml:mo>=<\/mml:mo><mml:mo>\u00b1<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:mi>e<\/mml:mi><\/mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:msup><mml:msup><mml:mrow><mml:mi>e<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>a<\/mml:mi><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo>=<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:mo>\u00b1<\/mml:mo><mml:msub><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:msup><mml:mrow><mml:mi>e<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>a<\/mml:mi><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:math><\/span><\/span>&#x200b;&#x200b;&nbsp;<\/p><p class=\"Normal\"><span>Hvor<\/span>&#x200b;&#x200b;&nbsp;<\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:msub><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>\u2208<\/mml:mo><mml:mi>R<\/mml:mi><mml:mo>\u2216<\/mml:mo><mml:mo>{<\/mml:mo><mml:mn>0<\/mml:mn><mml:mo>}<\/mml:mo><\/mml:math><\/span><\/span>&#x200b;&#x200b;&nbsp;<\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span>Idet<\/span>&#x200b;&#x200b;&nbsp;<span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:msup><mml:mrow><mml:mi>e<\/mml:mi><\/mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:msup><\/mml:math><\/span><\/span>&#x200b;&#x200b;&nbsp;<span>aldrig er 0<\/span><\/p><p class=\"Normal\"><span>Sammen med l\u00f8sningen:<\/span><\/p><p class=\"Normal\"><span>f(x) = 0 = 0e<\/span><span class=\"T60\">ax<\/span><\/p><p class=\"Normal\"><span>ser vi, at samtlige l\u00f8sninger til differentialligningen netop er funktionerne:<\/span><\/p><p class=\"Normal\"><span>y = f(x) = c*e<\/span><span class=\"T61\">ax<\/span><span>, hvor c er en reel konstant<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span class=\"T62\">Hvo<\/span><span class=\"T63\">r mange procent vokser y med, n\u00e5r x vokser med&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u2206<\/mml:mo><mml:mi mathvariant=\"bold-italic\">x<\/mml:mi><mml:mo>?<\/mml:mo><\/mml:math><\/span><\/span><\/p><p class=\"P64\"><span>Vi starter med at s\u00e6tte x+\u0394x p\u00e5 x plads<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mi>f<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><mml:mo>+<\/mml:mo><mml:mo>\u2206<\/mml:mo><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>=<\/mml:mo><mml:mi>c<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:msup><mml:mrow><mml:mi>e<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>a<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><mml:mo>+<\/mml:mo><mml:mo>\u2206<\/mml:mo><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:msup><\/mml:math><\/span><\/span><span class=\"T65\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"P66\"><span>Dern\u00e6st ganger vi a ind i parentesen<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mo>=<\/mml:mo><mml:mi>c<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:msup><mml:mrow><mml:mi>e<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>a<\/mml:mi><mml:mi>x<\/mml:mi><mml:mo>+<\/mml:mo><mml:mi>a<\/mml:mi><mml:mo>\u2206<\/mml:mo><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:math><\/span><\/span><span class=\"T67\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span class=\"T68\">nu bruger vi potensreglen&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:msup><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>x<\/mml:mi><mml:mo>+<\/mml:mo><mml:mi>y<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo>=<\/mml:mo><mml:msup><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo>\u00b7<\/mml:mo><mml:msup><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>y<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:math><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mo>=<\/mml:mo><mml:mi>c<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:msup><mml:mrow><mml:mi>e<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>a<\/mml:mi><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo>\u00b7<\/mml:mo><mml:msup><mml:mrow><mml:mi>e<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>a<\/mml:mi><mml:mo>\u2206<\/mml:mo><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:math><\/span><\/span><span class=\"T69\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span class=\"T70\">nu har vi&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mi>c<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:msup><mml:mrow><mml:mi>e<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>a<\/mml:mi><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:math><\/span><\/span><span class=\"T71\">&#x200b;&#x200b;&nbsp;og da det er lig f(x) erstatter vi&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mi>c<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:msup><mml:mrow><mml:mi>e<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>a<\/mml:mi><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mi>\u00a0<\/mml:mi><mml:mi>m<\/mml:mi><mml:mi>e<\/mml:mi><mml:mi>d<\/mml:mi><mml:mi>\u00a0<\/mml:mi><mml:mi>f<\/mml:mi><mml:mo>(<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:math><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mo>=<\/mml:mo><mml:mi>f<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mi>*<\/mml:mi><mml:msup><mml:mrow><mml:mi>e<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>a<\/mml:mi><mml:mo>\u2206<\/mml:mo><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:math><\/span><\/span><span class=\"T72\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"Normal\"><span class=\"T73\">Det ses at y vokser med (e<\/span><span class=\"T74\">a\u0394x&#x200b;&#x200b;&nbsp;<\/span><span class=\"T75\">- 1)%, n\u00e5r x vokser<\/span><span class=\"T76\">&#x200b;&#x200b;&nbsp;med \u0394x.<\/span><\/p><p class=\"P77\">&nbsp;<\/p><h3 class=\"Overskrift3\"><span id=\"_Toc449013660\"\/><span>Type III<\/span><\/h3><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mfrac><mml:mrow><mml:mi>d<\/mml:mi><mml:mi>y<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>d<\/mml:mi><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfrac><mml:mo>=<\/mml:mo><mml:mi>a<\/mml:mi><mml:mi>*<\/mml:mi><mml:mfrac><mml:mrow><mml:mi>y<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfrac><\/mml:math><\/span><\/span><span class=\"T78\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"P79\"><span>Vi starter med at dividere med y p\u00e5 begge sider:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfrac><mml:mrow><mml:mi>d<\/mml:mi><mml:mi>y<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>y<\/mml:mi><\/mml:mrow><\/mml:mfrac><mml:mo>=<\/mml:mo><mml:mi>a<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:mfrac><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfrac><mml:mo>\u00b7<\/mml:mo><mml:mi>d<\/mml:mi><mml:mi>x<\/mml:mi><\/mml:math><\/span><\/span><span class=\"T80\">&#x200b;&#x200b;&nbsp;<\/span><span class=\"T81\"><br\/><\/span><span class=\"T82\">derefter tager vi integralerne p\u00e5 begge sider:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mo>\u222b<\/mml:mo><mml:mfrac><mml:mrow><mml:mi>d<\/mml:mi><mml:mi>y<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>y<\/mml:mi><\/mml:mrow><\/mml:mfrac><mml:mo>=<\/mml:mo><mml:mo>\u222b<\/mml:mo><mml:mi>a<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:mfrac><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfrac><mml:mo>\u00b7<\/mml:mo><mml:mi>d<\/mml:mi><mml:mi>x<\/mml:mi><\/mml:math><\/span><\/span><span class=\"T83\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"Normal\"><span class=\"T84\">S\u00e5 finder vi stamfunktionerne til begge sider:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P85\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">ln<\/mml:mi><\/mml:mrow><mml:mo>\u2061<\/mml:mo><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:mi>y<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mrow><mml:mo>=<\/mml:mo><mml:mi>a<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">ln<\/mml:mi><\/mml:mrow><mml:mo>\u2061<\/mml:mo><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mrow><mml:mo>+<\/mml:mo><mml:mi>c<\/mml:mi><\/mml:math><\/span><\/span><\/p><p class=\"Normal\"><span class=\"T86\">S\u00e5 tager vi e<\/span><span class=\"T87\">x<\/span><span class=\"T88\">&#x200b;&#x200b;&nbsp;for at fjerne ln:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P89\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:mi>y<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>=<\/mml:mo><mml:msup><mml:mrow><mml:mi>e<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo>\u00b7<\/mml:mo><mml:msup><mml:mrow><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:math><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span class=\"T90\">Nu har kun fundet den numeriskv\u00e6rdi til y, og vi skal<\/span><span class=\"T91\">&#x200b;&#x200b;&nbsp;finde alle v\u00e6rdier, derfor skal vi tage +- p\u00e5 begge sider<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>:<\/mml:mo><\/mml:math><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P92\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mi>y<\/mml:mi><mml:mo>=<\/mml:mo><mml:mo>\u00b1<\/mml:mo><mml:msup><mml:mrow><mml:mi>e<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo>\u00b7<\/mml:mo><mml:msup><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:math><\/span><\/span><\/p><p class=\"P93\"><span>Nu vil vi tjekke om 0 ogs\u00e5 er en l\u00f8sning:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P94\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mn>0<\/mml:mn><mml:mo>=<\/mml:mo><mml:mi>a<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:mfrac><mml:mrow><mml:mn>0<\/mml:mn><\/mml:mrow><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfrac><\/mml:math><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span class=\"T95\">da 0 ogs\u00e5 er en l\u00f8sning kan vi erstatte&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u00b1<\/mml:mo><mml:msup><mml:mrow><mml:mi>e<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:math><\/span><\/span><span class=\"T96\">&#x200b;&#x200b;&nbsp;med en ny konstant 0:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P97\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mi>y<\/mml:mi><mml:mo>=<\/mml:mo><mml:mi>k<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:msup><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:math><\/span><\/span><\/p><p class=\"P98\"><span>Hvor mange procent vokser y med, n\u00e5r x vokser med r procent?<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span class=\"T99\">Vi starter med at s\u00e6tte&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mi>x<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:mfenced separators=\"|\"><mml:mrow><mml:mn>1<\/mml:mn><mml:mo>+<\/mml:mo><mml:mi>r<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T100\">&#x200b;&#x200b;&nbsp;ind<\/span><span class=\"T101\">&#x200b;&#x200b;&nbsp;p\u00e5 x plads:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mi>f<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:mfenced separators=\"|\"><mml:mrow><mml:mn>1<\/mml:mn><mml:mo>+<\/mml:mo><mml:mi>r<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mfenced><mml:mo>=<\/mml:mo><mml:mi>k<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:msup><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:mfenced separators=\"|\"><mml:mrow><mml:mn>1<\/mml:mn><mml:mo>+<\/mml:mo><mml:mi>r<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mfenced><\/mml:mrow><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:math><\/span><\/span><span class=\"T102\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span class=\"T103\">Vi bruger potensreglen&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:msup><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:mi>y<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo>=<\/mml:mo><mml:msup><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo>\u00b7<\/mml:mo><mml:msup><mml:mrow><mml:mi>y<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:math><\/span><\/span><span class=\"T104\">:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mi>f<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:mfenced separators=\"|\"><mml:mrow><mml:mn>1<\/mml:mn><mml:mo>+<\/mml:mo><mml:mi>r<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mfenced><mml:mo>=<\/mml:mo><mml:mi>k<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:msup><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo>\u00b7<\/mml:mo><mml:msup><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mn>1<\/mml:mn><mml:mo>+<\/mml:mo><mml:mi>r<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:math><\/span><\/span><span class=\"T105\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span class=\"T106\">da&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mi>f<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>=<\/mml:mo><mml:mi>k<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:msup><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:math><\/span><\/span><span class=\"T107\">&#x200b;&#x200b;&nbsp;erstatter vi&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mi>k<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:msup><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:math><\/span><\/span><span class=\"T108\">&#x200b;&#x200b;&nbsp;med&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mi>f<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T109\">:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mi>f<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:mfenced separators=\"|\"><mml:mrow><mml:mn>1<\/mml:mn><mml:mo>+<\/mml:mo><mml:mi>r<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mfenced><mml:mo>=<\/mml:mo><mml:mi>f<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>\u00b7<\/mml:mo><mml:msup><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mn>1<\/mml:mn><mml:mo>+<\/mml:mo><mml:mi>r<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:math><\/span><\/span><span class=\"T110\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"Normal\"><span class=\"T111\">Det kan s\u00e5 ses her at n\u00e5r x stiger med r procent, s\u00e5 stiger y med ((1+r)<\/span><span class=\"T112\">a<\/span><span class=\"T113\">-1) %<\/span><\/p><p class=\"Normal\">&nbsp;<\/p><h3 class=\"Overskrift3\"><span id=\"_Toc449013661\"\/><span>Type IV<\/span><\/h3><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mfrac><mml:mrow><mml:mi>d<\/mml:mi><mml:mi>y<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>d<\/mml:mi><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfrac><mml:mo>=<\/mml:mo><mml:mfrac><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfrac><\/mml:math><\/span><\/span><span class=\"T114\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"P115\"><span>Vi starter med at gange med dx p\u00e5 begge sider:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mi>d<\/mml:mi><mml:mi>y<\/mml:mi><mml:mo>=<\/mml:mo><mml:mfrac><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfrac><mml:mo>\u00b7<\/mml:mo><mml:mi>d<\/mml:mi><mml:mi>x<\/mml:mi><\/mml:math><\/span><\/span><span class=\"T116\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"P117\"><span>vi taget integralerne til begge sider:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mo>\u222b<\/mml:mo><mml:mi>d<\/mml:mi><mml:mi>y<\/mml:mi><mml:mo>=<\/mml:mo><mml:mo>\u222b<\/mml:mo><mml:mfrac><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfrac><mml:mo>\u00b7<\/mml:mo><mml:mi>d<\/mml:mi><mml:mi>x<\/mml:mi><\/mml:math><\/span><\/span><span class=\"T118\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span class=\"T119\">Vi tager<\/span><span class=\"T120\">&#x200b;&#x200b;&nbsp;stamfunktionerne til begge sider:<\/span><span class=\"T121\"><br\/><\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mi>y<\/mml:mi><mml:mo>=<\/mml:mo><mml:mi>a<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">ln<\/mml:mi><\/mml:mrow><mml:mo>\u2061<\/mml:mo><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mrow><mml:mo>+<\/mml:mo><mml:mi>c<\/mml:mi><\/mml:math><\/span><\/span><span class=\"T122\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"P123\">&nbsp;<\/p><p class=\"P124\"><span>Hvor meget vokser y med, n\u00e5r x vokser med r procent?<\/span><\/p><p class=\"Normal\"><span>Vi starter med at s\u00e6tte x*(1+r) in p\u00e5 x plads:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mi>f<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:mfenced separators=\"|\"><mml:mrow><mml:mn>1<\/mml:mn><mml:mo>+<\/mml:mo><mml:mi>r<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mfenced><mml:mo>=<\/mml:mo><mml:mi>a<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">ln<\/mml:mi><\/mml:mrow><mml:mo>\u2061<\/mml:mo><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:mfenced separators=\"|\"><mml:mrow><mml:mn>1<\/mml:mn><mml:mo>+<\/mml:mo><mml:mi>r<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mrow><mml:mo>+<\/mml:mo><mml:mi>c<\/mml:mi><\/mml:math><\/span><\/span><span class=\"T125\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span class=\"T126\">Her bruger vi logaritmereglen&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">ln<\/mml:mi><\/mml:mrow><mml:mo>\u2061<\/mml:mo><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:mi>y<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mrow><mml:mo>=<\/mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">ln<\/mml:mi><\/mml:mrow><mml:mo>\u2061<\/mml:mo><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mrow><mml:mo>\u00b7<\/mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">ln<\/mml:mi><\/mml:mrow><mml:mo>\u2061<\/mml:mo><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>y<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mrow><\/mml:math><\/span><\/span><span class=\"T127\">:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mi>f<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:mfenced separators=\"|\"><mml:mrow><mml:mn>1<\/mml:mn><mml:mo>+<\/mml:mo><mml:mi>r<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mfenced><mml:mo>=<\/mml:mo><mml:mi>a<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">ln<\/mml:mi><\/mml:mrow><mml:mo>\u2061<\/mml:mo><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mrow><mml:mo>\u00b7<\/mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">ln<\/mml:mi><\/mml:mrow><mml:mo>\u2061<\/mml:mo><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mn>1<\/mml:mn><mml:mo>+<\/mml:mo><mml:mi>r<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mrow><mml:mo>+<\/mml:mo><mml:mi>c<\/mml:mi><\/mml:math><\/span><\/span><span class=\"T128\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span class=\"T129\">Nu har vi a*ln(x)+c og da det er lig med f(x) inds\u00e6tter vi f(x) i stedet for a*ln(<\/span><span class=\"T130\">x)+c<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>:<\/mml:mo><\/mml:math><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mi>f<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:mfenced separators=\"|\"><mml:mrow><mml:mn>1<\/mml:mn><mml:mo>+<\/mml:mo><mml:mi>r<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mfenced><mml:mo>=<\/mml:mo><mml:mi>f<\/mml:mi><mml:mo>(<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo>)<\/mml:mo><mml:mo>\u00b7<\/mml:mo><mml:mi mathvariant=\"normal\">l<\/mml:mi><mml:mi mathvariant=\"normal\">n<\/mml:mi><mml:mo>\u2061<\/mml:mo><mml:mo>(<\/mml:mo><mml:mn>1<\/mml:mn><mml:mo>+<\/mml:mo><mml:mi>r<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T131\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"Normal\"><span class=\"T132\">Det ses at n\u00e5r x stiger med r procent s\u00e5 stiger y med (ln(1+r)-1) %<\/span><\/p><p class=\"P133\">&nbsp;<\/p><p class=\"P134\"><span>Bestem for a=2 den l\u00f8sning der g\u00e5r gennem punktet (3,15) for hver af de 4 differentialligninger<\/span><\/p><h3 class=\"Overskrift3\"><span id=\"_Toc449013662\"\/><span>Type I<\/span><\/h3><p class=\"Normal\"><span>Ved type I fandt vi ud af, at den fuldst\u00e6ndige l\u00f8sning er y = ax + c. Vi<\/span>&#x200b;&#x200b;&nbsp;<span>s\u00e6tter s\u00e5 2 ind p\u00e5 a\u2019s plads:<\/span><\/p><p class=\"Normal\"><span>y = 2x + c<\/span><\/p><p class=\"Normal\"><span>Herefter kan vi isolere c:<\/span><\/p><p class=\"Normal\"><span>y - 2x = c<\/span><\/p><p class=\"Normal\"><span>S\u00e5 s\u00e6tter vi (3,15) ind:<\/span><\/p><p class=\"Normal\"><span>15 - 2 * 3 = c<\/span><\/p><p class=\"Normal\"><span>Vi udregner:<\/span><\/p><p class=\"Normal\"><span>9 = c<\/span><\/p><p class=\"Normal\"><span>Og nu har vi l\u00f8sning der g\u00e5r gennem punktet (3,15):<\/span><\/p><p class=\"Normal\"><span>f(x) = 2x + 9<\/span><\/p><p class=\"Normal\">&nbsp;<\/p><h3 class=\"Overskrift3\"><span id=\"_Toc449013663\"\/><span>Type II<\/span><\/h3><p class=\"P135\"><span>Ved type to fandt vi f\u00f8lgende fuldst\u00e6ndige<\/span>&#x200b;&#x200b;&nbsp;<span>l\u00f8sning:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mi>y<\/mml:mi><mml:mo>=<\/mml:mo><mml:mi>c<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:msup><mml:mrow><mml:mi>e<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>a<\/mml:mi><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:math><\/span><\/span><span class=\"T136\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span class=\"T137\">Vi inds\u00e6tter s\u00e5 a = 2 og punktet (3,15) i denne:<\/span><span class=\"T138\">&#x200b;&#x200b;&nbsp;<\/span><span class=\"T139\"><br\/><\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mn>15<\/mml:mn><mml:mo>=<\/mml:mo><mml:mi>c<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:msup><mml:mrow><mml:mi>e<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><mml:mo>\u00b7<\/mml:mo><mml:mn>3<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:math><\/span><\/span><span class=\"T140\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"Normal\"><span class=\"T141\">Vi dividerer med e<\/span><span class=\"T142\">2*3<\/span><span class=\"T143\">&#x200b;&#x200b;&nbsp;p\u00e5 hver side af lighedstegnet:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mfrac><mml:mrow><mml:mn>15<\/mml:mn><\/mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>e<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>6<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:mfrac><mml:mo>=<\/mml:mo><mml:mi>c<\/mml:mi><\/mml:math><\/span><\/span><span class=\"T144\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"P145\"><span>Herefter kan vi s\u00e5 beregne c:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mi>c<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>0,03718<\/mml:mn><\/mml:math><\/span><\/span><span class=\"T146\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"P147\"><span>Og nu har vi l\u00f8sningen der g\u00e5r gennem punktet (3,15):<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mi>y<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>0,03718<\/mml:mn><mml:mo>\u00b7<\/mml:mo><mml:msup><mml:mrow><mml:mi>e<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:math><\/span><\/span><span class=\"T148\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"P149\">&nbsp;<\/p><h3 class=\"Overskrift3\"><span id=\"_Toc449013664\"\/><span>Type III<\/span><\/h3><p class=\"Normal\"><span>Her tager vi udgangspunkt i f\u00f8lgende:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mi>y<\/mml:mi><mml:mo>=<\/mml:mo><mml:mi>k<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:msup><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:math><\/span><\/span><span class=\"T150\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"P151\"><span>Hern\u00e6st inds\u00e6tter vi nok engang 2 p\u00e5 a\u2019s plads samt punktet (3,15) p\u00e5 henholdsvis x og y\u2019s plads.<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mn>15<\/mml:mn><mml:mo>=<\/mml:mo><mml:mi>k<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:msup><mml:mrow><mml:mn>3<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:math><\/span><\/span><span class=\"T152\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"P153\"><span>S\u00e5 isolerer vi k:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mfrac><mml:mrow><mml:mn>15<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>9<\/mml:mn><\/mml:mrow><\/mml:mfrac><mml:mo>=<\/mml:mo><mml:mi>k<\/mml:mi><\/mml:math><\/span><\/span><span class=\"T154\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"P155\"><span>Og vi kan nu finde l\u00f8sningen:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span class=\"T156\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mi>y<\/mml:mi><mml:mo>=<\/mml:mo><mml:mfrac><mml:mrow><mml:mn>15<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>9<\/mml:mn><\/mml:mrow><\/mml:mfrac><mml:mo>\u00b7<\/mml:mo><mml:msup><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:math><\/span><\/span><\/p><h3 class=\"Overskrift3\">&nbsp;<\/h3><h3 class=\"Overskrift3\"><span id=\"_Toc449013665\"\/><span>Type IV<\/span><\/h3><p class=\"Normal\"><span>Ved type IV fandt vi f\u00f8lgende:<\/span>&#x200b;&#x200b;&nbsp;<\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mi>y<\/mml:mi><mml:mo>=<\/mml:mo><mml:mi>a<\/mml:mi><mml:mo>\u00b7<\/mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">ln<\/mml:mi><\/mml:mrow><mml:mo>\u2061<\/mml:mo><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mrow><mml:mo>+<\/mml:mo><mml:mi>c<\/mml:mi><\/mml:math><\/span><\/span><span class=\"T157\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"P158\"><span>Vi inds\u00e6tter s\u00e5 2 samt (3,15) heri:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mn>15<\/mml:mn><mml:mo>=<\/mml:mo><mml:mn>2<\/mml:mn><mml:mo>\u00b7<\/mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">ln<\/mml:mi><\/mml:mrow><mml:mo>\u2061<\/mml:mo><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mn>3<\/mml:mn><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mrow><mml:mo>+<\/mml:mo><mml:mi>c<\/mml:mi><\/mml:math><\/span><\/span><span class=\"T159\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"P160\"><span>S\u00e5 isolerer vi c:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mfrac><mml:mrow><mml:mn>15<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><mml:mo>\u00b7<\/mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">ln<\/mml:mi><\/mml:mrow><mml:mo>\u2061<\/mml:mo><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mn>3<\/mml:mn><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mrow><\/mml:mrow><\/mml:mfrac><mml:mo>=<\/mml:mo><mml:mi>c<\/mml:mi><\/mml:math><\/span><\/span><span class=\"T161\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"P162\"><span>Vi<\/span>&#x200b;&#x200b;&nbsp;<span>beregner c:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mi>c<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>6,8269<\/mml:mn><\/mml:math><\/span><\/span><span class=\"T163\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"P164\"><span>Og dermed har vi l\u00f8sningen som bliver:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"left\"><mml:mi>y<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>2<\/mml:mn><mml:mo>\u00b7<\/mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">ln<\/mml:mi><\/mml:mrow><mml:mo>\u2061<\/mml:mo><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mrow><mml:mo>+<\/mml:mo><mml:mn>6,8269<\/mml:mn><\/mml:math><\/span><\/span><span class=\"T165\">&#x200b;&#x200b;&nbsp;<\/span><\/p><\/div><\/section><script>(function( $ ) {\n    \/\/footnotes\n    $('.defaultNote').children(\":first\").css('cursor', 'pointer');\n    $('.defaultNote').children(\":first\").click(function(){\n        var gotoID = '#' + $(this).parent().attr('id');\n        $('html, body').animate(\n            {\n              scrollTop: $('a[href=\"' + gotoID + '\"]').offset().top - 60,\n            },\n            250,\n            'linear'\n          )\n    });\n}(jQuery));\n\n\n<\/script><script>\nvar numFormat = \",.\";\n(function( $ ) {\n    \"use strict\";\n    \/\/do some cleaning on load\n    var sortCells = $('td[data-sorting], th[data-sorting]');\n    sortCells.each(function(){\n        var first = $(this).text().trim().charAt(0);\n        if (first == '@'){\n            var oldHTML = $(this).html();\n            var newHTML = oldHTML.replace('@', '');\n            $(this).html(newHTML);\n        } \n    });\n    function 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