{"id":2023,"date":"2022-12-09T11:30:26","date_gmt":"2022-12-09T10:30:26","guid":{"rendered":"https:\/\/mxth.dk\/?p=2023"},"modified":"2026-04-28T22:53:13","modified_gmt":"2026-04-28T20:53:13","slug":"eksponentielle-funktioner","status":"publish","type":"post","link":"https:\/\/mxth.dk\/?p=2023","title":{"rendered":"Eksponentielle funktioner"},"content":{"rendered":"\n<p class=\" eplus-wrapper\">En funktion er en operation der overs\u00e6tter et s\u00e6t af tal (definitionsm\u00e6ngden, x-v\u00e6rdier, den uafh\u00e6ngige variabel) til et andet s\u00e6t af tal (v\u00e6rdim\u00e6ngden, y-v\u00e6rdier, den afh\u00e6ngige variabel). Du kan eventuelt l\u00e6se mere om det <a href=\"https:\/\/mxth.dk\/?p=3826\" data-type=\"post\" data-id=\"3826\">her<\/a>.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Men lad os lige kendkalde hvad vi allerede ved fra line\u00e6re funktioner. En line\u00e6r funktion har formen<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">\\(y=f(x)=a\\cdot x+b\\)<\/p>\n\n\n\n<p class=\" eplus-wrapper\">og kunne for eksempel v\u00e6re<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">\\(f(x)=2x+3\\)<\/p>\n\n\n\n<p class=\" eplus-wrapper\">hvor \\(a\\) er h\u00e6ldningstallet (hvor meget g\u00e5r vi op\/ned, n\u00e5r \\(x\\) \u00f8ges med 1) i vores eksempel er det 2. V\u00e6rdien af funktionen \u00f8ges med 2 hvor gang vi \u00f8ger \\(x\\) med 1. V\u00e6rdien \\(b\\) kan vi huske er sk\u00e6ringen med \\(y\\)-aksen, man kunne ogs\u00e5 kalde den for begyndelsesv\u00e6rdien da det er den \\(y\\)-v\u00e6rdi funktionen har n\u00e5r \\(x\\) er nul.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">For den line\u00e6re funktion er <strong>h\u00e6ldningstallet<\/strong> det man ville kalde en <strong>absolut tilv\u00e6kst<\/strong>, da det er den samme tilv\u00e6st, den samme v\u00e6rdie der l\u00e6gges til, uanset hvor henne p\u00e5 funktionen vi er. Dette er dog ikke altid tilf\u00e6ldet for alle situationer som vi gerne ville beskrive med en funktion. Lad os se p\u00e5 et eksempel.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Vi starter med et eksempel i biologiens verden. Celler kan dele sig s\u00e5ledes at \u00e9n celle bliver til to. <\/p>\n\n\n\n<figure class=\" wp-block-video eplus-wrapper\"><video height=\"1080\" style=\"aspect-ratio: 1920 \/ 1080;\" width=\"1920\" autoplay controls loop muted src=\"https:\/\/mxth.dk\/wp-content\/uploads\/2022\/12\/CellDivision-2.mp4\" playsinline><\/video><\/figure>\n\n\n\n<p class=\" eplus-wrapper\">For hver ny generation bliver antallet fordoblet, s\u00e5ledes at vi til at starte med har 1 celle og herefter f\u00e5r vi 2, s\u00e5 4, herefter 8, 16 32 osv.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">For hver generation ganges der med to. S\u00e5 hvis vi starter med \u00e9n celle s\u00e5 vil der efter \u00e9n generation v\u00e6re \\(1\\cdot 2=2\\) celler. Efter to generationer er der \\(2\\cdot2=4\\) celler, efter tre generationer er der \\(4\\cdot2=8\\) celler og efter fire generationer vil der v\u00e6re \\(8\\cdot2=16\\) celler osv. Da generation to er to gange antallet i generation \u00e9t s\u00e5 kan vi skrive \\(1\\cdot2\\cdot2=4\\). For generation tre ville det blive \\(1\\cdot2\\cdot2\\cdot2=8\\) og for generation fire \\(1\\cdot2\\cdot2\\cdot2\\cdot2=16\\). Vi kan se at vi for antallet af celler i en generation ville kunne skrive det som<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">\\(\\text{antal celler i generation n}=1\\cdot2^n\\)<\/p>\n\n\n\n<p class=\" eplus-wrapper\">da vi ganger med 2 for hver generation og derfor kan skrive det som en potens. <\/p>\n\n\n\n<p class=\" eplus-wrapper\">En funktion hvor den uafh\u00e6ngige variabel st\u00e5 i potens kaldes for en <strong>eksponentiel funktion<\/strong> og kan generelt skrives som<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">\\(f(x)=b\\cdot a^x\\)<\/p>\n\n\n\n<p class=\" eplus-wrapper\">hvor <strong>b<\/strong> kaldes for <strong>begyndelsesv\u00e6rdien<\/strong> og er den y-v\u00e6rdi <em>hvor funktionen sk\u00e6rer y-aksen<\/em>, lige som vi kender det fra den line\u00e6re funktion.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">V\u00e6rdien <strong>a<\/strong> kaldes for <strong>fremskrivningsfaktoren<\/strong> og beskriver hvor meget en funktion vokser med n\u00e5r x \u00f8ges med \u00e9n. <\/p>\n\n\n\n<p class=\" eplus-wrapper\">For vores celledeling fordobles antallet af celler og derfor er fremskrivningsfaktoren 2. Men lad os se p\u00e5 et andet eksempel fra finansverdenen. Hvis vi s\u00e6tter 100 kr. ind p\u00e5 en konto hvor vi f\u00e5r 10 % i rente (ja i matematikken har vi virkelig gode banker). Det vil sige at vi hvert \u00e5r skal ligge 10 % til. Efter \u00e9t \u00e5r har vi<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">\\(100+\\dfrac{100}{100}\\cdot 10=110\\)<\/p>\n\n\n\n<p class=\" eplus-wrapper\">efter to \u00e5r vil vi have<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">\\(110+\\dfrac{110}{100}\\cdot10=121\\)<\/p>\n\n\n\n<p class=\" eplus-wrapper\">og efter tre \u00e5r<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">\\(121+\\dfrac{121}{100}\\cdot 10=133,1\\)<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Hvis vi ser p\u00e5 de tre udregninger kan vi se at<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">\\(\\begin{align*}100+\\dfrac{100}{100}\\cdot10&amp;=100\\cdot(1+\\dfrac{1}{100}\\cdot10)\\\\ &amp;=100\\cdot(1+0,1)\\\\ &amp;=100\\cdot1,1\\\\ &amp;=110\\end{align*}\\)<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">\\(\\begin{align*}110+\\dfrac{110}{100}\\cdot10&amp;=110\\cdot(1+\\dfrac{1}{100}\\cdot10)\\\\ &amp;=110\\cdot(1+0,1)\\\\ &amp;=110\\cdot1,1\\\\ &amp;=121\\end{align*}\\)<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">\\(\\begin{align*}121+\\dfrac{121}{100}\\cdot10&amp;=121\\cdot(1+\\dfrac{1}{100}\\cdot10)\\\\ &amp;=121\\cdot(1+0,1)\\\\ &amp;=121\\cdot1,1\\\\ &amp;=133,1\\end{align*}\\)<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Vi kan her se at for at ligge 10 % til skal vi gange med 1,1 og det skal vi g\u00f8re hvert \u00e5r. Da<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">\\(110=100\\cdot1,1\\)<\/p>\n\n\n\n<p class=\" eplus-wrapper\">kan vi erstatte de 110 i i udregningen for det andet \u00e5r og f\u00e5r<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">\\(100\\cdot1,1\\cdot1,1=100\\cdot1,1^2=121\\)<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Vi kan derfor for det tredje \u00e5r skrive<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">\\(100\\cdot1,1^3=133,1\\)<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Formlen for at udregne hvor mange penge vi har i et vilk\u00e5rligt \u00e5r bliver derfor<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">\\(100\\cdot1,1^x\\)<\/p>\n\n\n\n<p class=\" eplus-wrapper\">hvor \\(x\\) er antallet af \u00e5r der er g\u00e5et, begyndelsesv\u00e6rdien er 100 og fremskrivningsfaktoren er 1,1 som kan beregnes ved hj\u00e6lp af<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">\\(a=1+r\\)<\/p>\n\n\n\n<p class=\" eplus-wrapper\">hvor<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">\\(r=\\dfrac{\\text{renten}}{100}\\)<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Den eksponentielle funktion har derfor forskriften<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">\\(f(x)=b\\cdot a^x\\)<\/p>\n\n\n\n<p class=\" eplus-wrapper\">hvis fremskrivningsfaktoren er kendt eller<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">\\(f(x)=b\\cdot (1+r)^x\\)<\/p>\n\n\n\n<p class=\" eplus-wrapper\">hvis vi kender den procentuelle tilv\u00e6kst. Fremskrivningsfaktoren beskriver alts\u00e5 hvor meget funktionen vokser med hver gang \\(x\\) \u00f8ges med 1 og beskriver derfor den relative tilv\u00e6kst som er beskrevet med \\(r\\). <\/p>\n\n\n\n\n\n\n\n<h4 class=\" wp-block-heading eplus-wrapper\" id=\"opgaver\">Opgaver<\/h4>\n\n\n\n<style>\n  table, th, td {\n    border: 1px solid black;\n    border-collapse: collapse;\n  }\n<\/style>\n<table width=\"100%\" border=\"1\">\n  <thead>\n    <tr>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#3CB371\">Gr\u00f8n<\/th>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#FFA500\">Gul<\/th>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#DC143C\">R\u00f8d<\/th>\n    <\/tr>\n  <\/thead>\n  <tbody>\n    <tr>\n      <td>\n        <p><a href=\"https:\/\/matstxab1opgaver.systime.dk\/?id=171#c1161\" taget=\"_blank\" rel=\"noopener\">matAB1stx 5.01<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/matbhtx.systime.dk\/?id=c13753&#038;L=0\" taget=\"_blank\" rel=\"noopener\">matBhtx 8.51<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/matbhtx.systime.dk\/?id=c13752&#038;L=0\" taget=\"_blank\" rel=\"noopener\">matBhtx 8.50<\/a><\/p>\n      <\/td>\n    <\/tr>\n    <tr>\n      <td>\n        <p><a href=\"https:\/\/matstxab1opgaver.systime.dk\/?id=171#c1165\" taget=\"_blank\" rel=\"noopener\">matAB1stx 5.02<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/matstxab1opgaver.systime.dk\/?id=172#c1255\" taget=\"_blank\" rel=\"noopener\">matAB1stx 5.39<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/matstxab1opgaver.systime.dk\/?id=171#c1180\" taget=\"_blank\" rel=\"noopener\">matAB1stx 5.08<\/a><\/p>\n      <\/td>\n    <\/tr>\n    <tr>\n      <td>\n        <p><a href=\"https:\/\/matstxab1opgaver.systime.dk\/?id=171#c1169\" taget=\"_blank\" rel=\"noopener\">matAB1stx 5.03<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"\" taget=\"_blank\" rel=\"noopener\"><\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"\" taget=\"_blank\" rel=\"noopener\"><\/a><\/p>\n      <\/td>\n    <\/tr>\n    <tr>\n      <td>\n        <p><a href=\"https:\/\/matstxab1opgaver.systime.dk\/?id=171#c1173\" taget=\"_blank\" rel=\"noopener\">matAB1stx 5.04<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"\" taget=\"_blank\" rel=\"noopener\"><\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"\" taget=\"_blank\" rel=\"noopener\"><\/a><\/p>\n      <\/td>\n    <\/tr>\n    <tr>\n      <td>\n        <p><a href=\"https:\/\/matbhtx.systime.dk\/?id=c13734&#038;L=0\" taget=\"_blank\" rel=\"noopener\">matBhtx 8.36<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"\" taget=\"_blank\" rel=\"noopener\"><\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"\" taget=\"_blank\" rel=\"noopener\"><\/a><\/p>\n      <\/td>\n    <\/tr>\n    <tr>\n      <td>\n        <p><a href=\"https:\/\/matbhtx.systime.dk\/?id=c13751&#038;L=0\" taget=\"_blank\" rel=\"noopener\">matBhtx 8.49<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"\" taget=\"_blank\" rel=\"noopener\"><\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"\" taget=\"_blank\" rel=\"noopener\"><\/a><\/p>\n      <\/td>\n    <\/tr>\n    <tr>\n      <td>\n        <p><a href=\"https:\/\/matbhtx.systime.dk\/?id=c13754&#038;L=0\" taget=\"_blank\" rel=\"noopener\">matBhtx 8.52<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"\" taget=\"_blank\" rel=\"noopener\"><\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"\" taget=\"_blank\" rel=\"noopener\"><\/a><\/p>\n      <\/td>\n    <\/tr>\n    <tr>\n      <td>\n        <p><a href=\"https:\/\/matstxab1opgaver.systime.dk\/?id=172#c1237\" taget=\"_blank\" rel=\"noopener\">matAB1stx 5.29<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"\" taget=\"_blank\" rel=\"noopener\"><\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"\" taget=\"_blank\" rel=\"noopener\"><\/a><\/p>\n      <\/td>\n    <\/tr>\n    <tr>\n      <td>\n        <p><a href=\"https:\/\/matstxab1opgaver.systime.dk\/?id=172#c1237\" taget=\"_blank\" rel=\"noopener\">matAB1stx 5.30<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"\" taget=\"_blank\" rel=\"noopener\"><\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"\" taget=\"_blank\" rel=\"noopener\"><\/a><\/p>\n      <\/td>\n    <\/tr>\n  <\/tbody>\n<\/table>\n","protected":false},"excerpt":{"rendered":"<p>En funktion er en operation der overs\u00e6tter et s\u00e6t af tal (definitionsm\u00e6ngden, x-v\u00e6rdier, den uafh\u00e6ngige variabel) til et andet s\u00e6t af tal (v\u00e6rdim\u00e6ngden, y-v\u00e6rdier, den afh\u00e6ngige variabel). Du kan eventuelt l\u00e6se mere om det her. Men lad os lige kendkalde hvad vi allerede ved fra line\u00e6re funktioner. En line\u00e6r funktion har formen \\(y=f(x)=a\\cdot x+b\\) og [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"ub_ctt_via":"","editor_plus_copied_stylings":"{}","_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[35,12,3],"tags":[],"class_list":["post-2023","post","type-post","status-publish","format-standard","hentry","category-eksponentiel-funktion","category-funktioner","category-matematik"],"featured_image_src":null,"author_info":{"display_name":"Henriksen","author_link":"https:\/\/mxth.dk\/?author=1"},"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/posts\/2023","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2023"}],"version-history":[{"count":21,"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/posts\/2023\/revisions"}],"predecessor-version":[{"id":4450,"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/posts\/2023\/revisions\/4450"}],"wp:attachment":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2023"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2023"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2023"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}