{"id":83,"date":"2023-09-17T10:58:59","date_gmt":"2023-09-17T10:58:59","guid":{"rendered":"https:\/\/mathority.org\/nl\/raaklijnvergelijking\/"},"modified":"2023-09-17T10:58:59","modified_gmt":"2023-09-17T10:58:59","slug":"raaklijnvergelijking","status":"publish","type":"post","link":"https:\/\/mathority.org\/nl\/raaklijnvergelijking\/","title":{"rendered":"Raaklijnvergelijking"},"content":{"rendered":"<p>In dit artikel zullen we zien <strong>hoe we de vergelijking van de raaklijn aan een curve kunnen vinden<\/strong> . Daarnaast kun je trainen met opgeloste oefeningen van verschillende moeilijkheidsgraden. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ecuacion-de-la-recta-tangente-a-una-funcion-en-un-punto\"><\/span> Vergelijking van de raaklijn aan een functie in een punt <span class=\"ez-toc-section-end\"><\/span><\/h2>\n<div style=\"padding-top: 23px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px dashed #FF9B28; border-radius:20px;\">\n<p style=\"text-align:left\"> De <strong>vergelijking van de raaklijn<\/strong> aan de functie f(x) in het punt x=x <sub>0<\/sub> is:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-326424811181144df35c0b94ce50c462_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y -y_0= m(x-x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p align=\"LEFT\"> Waarbij het punt P(x <sub>0<\/sub> ,y <sub>0<\/sub> ) het punt is waar de raaklijn en de functie samenvallen. En de helling van de raaklijn, m, is gelijk aan de afgeleide van de curve op het punt x <sub>0<\/sub> , dat wil zeggen m=f'(x <sub>0<\/sub> ). <\/p>\n<\/div>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/equation-de-la-tangente-ligne.webp\" alt=\"raaklijnvergelijking\" class=\"wp-image-2306\" width=\"463\" height=\"461\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> In de afbeelding hierboven zie je een curve<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7ee323bc5a3f73ad5e066b13bed5504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -5px;\"><\/p>\n<p> weergegeven in blauw en een oranje lijn die raakt aan de functie<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7ee323bc5a3f73ad5e066b13bed5504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -5px;\"><\/p>\n<p> Over<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bd5304ac1643ba3660a7efa36ade1983_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=x_0\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"51\" style=\"vertical-align: -3px;\"><\/p>\n<p> , omdat ze alleen dit punt gemeen hebben. Welnu, de vergelijking van deze raaklijn is<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-326424811181144df35c0b94ce50c462_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y -y_0= m(x-x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -5px;\"><\/p>\n<p> , en de helling ervan is<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c606eb4b268b71562672c32a0461053_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"85\" style=\"vertical-align: -5px;\"><\/p>\n<p> . <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"como-hallar-la-ecuacion-de-la-recta-tangente\"><\/span> Hoe de raaklijnvergelijking te vinden<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Om de vergelijking van de raaklijn aan een functie in een punt te vinden, moet je het volgende doen:<\/p>\n<ol style=\"color:#FF8A05; font-weight: bold;\">\n<li style=\"margin-bottom:8px\"> <span style=\"color:#101010;font-weight: normal;\">Vind de helling van de raaklijn door de afgeleide van de functie op het raakpunt te berekenen.<\/span><\/li>\n<li style=\"margin-bottom:8px\"> <span style=\"color:#101010;font-weight: normal;\">Bepaal een punt op de raaklijn.<\/span><\/li>\n<li> <span style=\"color:#101010;font-weight: normal;\"><strong>Zoek de vergelijking van de raaklijn<\/strong> met behulp van de berekende helling en het punt van de raaklijn.<\/span> <\/li>\n<\/ol>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-de-la-ecuacion-de-la-recta-tangente-a-una-curva\"><\/span> Voorbeeld van de vergelijking van de raaklijn aan een curve<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Nadat we de theorie over de raaklijnvergelijking hebben gezien, gaan we kijken hoe we de vergelijking van een raaklijn kunnen berekenen door stap voor stap een voorbeeld op te lossen:<\/p>\n<ul>\n<li> Bereken de vergelijking van de raaklijn aan de curve\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ea8efcaacce7a90d3cc483105986c47c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2+x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<p> Over<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3330a01aa4d7d81947b71297d8623d3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\"><\/p>\n<p> .<\/li>\n<\/ul>\n<p> We weten dat de raaklijnvergelijking altijd de volgende vorm heeft:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-326424811181144df35c0b94ce50c462_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y -y_0= m(x-x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Het eerste dat u moet doen, is de helling van de lijn berekenen. Dus de helling van de raaklijn,<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> , zal de waarde zijn van de afgeleide van de curve op het raakpunt x=1, dwz<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a69005ee8bf2d80d73b989ad0cedccd7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(1).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"81\" style=\"vertical-align: -5px;\"><\/p>\n<p> We differenti\u00ebren daarom de functie en berekenen vervolgens <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-eb68a498d3bd60e51d3dc230691f886c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-92c9a5ee4068789701733f793fbac622_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2+x \\quad \\longrightarrow \\quad f'(x)=2x+1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"293\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d96270e9b7d7c3cae8baea602cea53bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1)= 2\\cdot 1+1=2+1=3\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"218\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2443a93bff265c6b8ba692ef8d14f633_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(1)=3\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Zodra we de waarde ervan kennen<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> , we moeten een punt vinden<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8e531a80c13865d1ad612bd3f634efa2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x_0,y_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"54\" style=\"vertical-align: -5px;\"><\/p>\n<p> van de raaklijn om de vergelijking van de raaklijn te voltooien.<\/p>\n<p> De <strong>vergelijking van de raaklijn en de curve hebben altijd een gemeenschappelijk punt<\/strong> , wat in dit geval zo is<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3330a01aa4d7d81947b71297d8623d3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\"><\/p>\n<p> . Daarom, zoals de curve<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7ee323bc5a3f73ad5e066b13bed5504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -5px;\"><\/p>\n<p> door dit punt gaat, kunnen we de andere component van het punt vinden door te berekenen <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-32045357853caad8774629c95963835d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(1):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ea8efcaacce7a90d3cc483105986c47c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2+x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7b2601f20b100f2635bc0342175b4627_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(1)=1^2+1=2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"136\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Het raakpunt is dus:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d26257abb9047188ab3e3887f447e20a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(1,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"52\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Zowel de curve als de raaklijn gaan door dit punt, dus we kunnen deze ook gebruiken om de vergelijking van de raaklijn te vinden.<\/p>\n<p> Het enige dat overblijft is om de gevonden waarden van de helling en het punt van de raaklijn in de vergelijking te vervangen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0321e19825c08a1f47a00b2cf625088f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{c} y -y_0= m(x-x_0) \\\\[3ex] m=3 \\qquad P(1,2) \\end{array} \\right\\} \\longrightarrow \\ y -2= 3(x-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"344\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Kort gezegd is de raaklijnvergelijking: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-037e5c42a4adef3e5ba970a66b8d3459_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y-2=3(x-1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<hr class=\"wp-block-separator has-text-color has-background is-style-wide\" style=\"background-color:#1976d2;color:#1976d2\">\n<p> Je kunt de vergelijking van de raaklijn ook uitdrukken met de expliciete vergelijking van de lijn: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-59097f2ef899c7c608e2527467021b23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y=3x-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"82\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<hr class=\"wp-block-separator has-text-color has-background is-style-wide\" style=\"background-color:#1976d2;color:#1976d2\">\n<p> Hieronder ziet u de curve weergegeven<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ea8efcaacce7a90d3cc483105986c47c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2+x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<p> en de lijn die raakt aan <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e2883f0b53c531552fde7ff189f83165_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=1,\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -4px;\"><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1020651b62576571e0ac9c0cb65dd287_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-2=3(x-1):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"136\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/equation-de-la-tangente-a-une-courbe-en-un-point.webp\" alt=\"vergelijking van de raaklijn aan een kromme in een punt\" class=\"wp-image-2318\" width=\"445\" height=\"434\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Zoals je kunt zien, de curve<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ea8efcaacce7a90d3cc483105986c47c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2+x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<p> en de raaklijn<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3cb3f64ce416f3a5c6cc80c11cae9afb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-2=3(x-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" style=\"vertical-align: -5px;\"><\/p>\n<p> ze hebben alleen het punt gemeen<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-42a32282c97c3b8d9f90b2f1418844d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(1,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<p> , precies zoals we hadden berekend. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-la-ecuacion-de-la-recta-tangente\"><\/span> Opgeloste oefeningen over de raaklijnvergelijking<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Oefening 1<\/h3>\n<p> Bereken de vergelijking van de raaklijn aan de curve<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-99f623c4fede3b664682c5cbc1aab81d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=2x^2-4x+3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"156\" style=\"vertical-align: -5px;\"><\/p>\n<p> Over <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-71e6033606cd14039ab202fb7a18c50e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=2 .\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Zie de oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> De raaklijnvergelijking heeft altijd de volgende vorm:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3441e1da8c7da5805b1133af77b14f60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-y_0=m(x-x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>Stap 1: Bereken de helling van de raaklijn<\/strong><\/p>\n<p class=\"has-text-align-left\"> De helling, <em>m<\/em> , is de waarde van de afgeleide van de curve op het raakpunt. Daarom in dit geval <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c7192bf7bd4300d7d77fe084134d6849_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m = f'(2):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"86\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ab9e97205d5c9f2fbfcb085cbfdbdd75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=2x^2-4x+3 \\ \\longrightarrow \\ f'(x)= 4x-4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"319\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e3cb482f7e68631c8dcc5705ac1257d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(2)= 4\\cdot 2-4=8-4=4\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"218\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b0e7f5cd5a44a120769c9d3a1eae02c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(2)=4\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>Stap 2: Zoek een punt op de raaklijn<\/strong><\/p>\n<p class=\"has-text-align-left\"> De vergelijking van de raaklijn en de curve hebben altijd een gemeenschappelijk punt, wat in dit geval het geval is<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c657687cbbf5ea9a7545edb42190e592_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\"><\/p>\n<p> . Daarom, zoals de curve<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7ee323bc5a3f73ad5e066b13bed5504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -5px;\"><\/p>\n<p> door dit punt gaat, kunnen we de andere component van het punt vinden door te berekenen <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f026e401162db03299777455b748b308_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(2):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-99f623c4fede3b664682c5cbc1aab81d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=2x^2-4x+3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"156\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-67ff5b86b2842fefbdf2ddc7c2df39f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(2)=2\\cdot 2^2-4\\cdot 2+3 =2 \\cdot 4 -8 +3 = 3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"326\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Het punt waar zowel de curve als de raaklijn doorheen gaan, is dus het punt<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d744fa34f41ed1bbe3fdf2c5ad7f55a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(2,3).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"43\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>Stap 3: Schrijf de raaklijnvergelijking<\/strong><\/p>\n<p class=\"has-text-align-left\"> Het enige dat overblijft is om de gevonden waarden van de helling en het punt van de raaklijn in de vergelijking te vervangen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1622c6ecd4d43bb4fc4901b437464652_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{c} y -y_0= m(x-x_0) \\\\[3ex] m=4 \\qquad P(2,3) \\end{array} \\right\\} \\longrightarrow \\ y -3= 4(x-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"344\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> De raaklijnvergelijking is daarom: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c1233301c390a75095fc24bd8765e081_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y -3= 4(x-2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Oefening 2<\/h3>\n<p> Bereken de vergelijking van de raaklijn aan de curve<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d1309dcf6b647174b562cb71ab600c1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)=-3x^2+2x\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"139\" style=\"vertical-align: -5px;\"><\/p>\n<p> op de oorsprong van de co\u00f6rdinaten. <\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Zie de oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> De oorsprong van de co\u00f6rdinaten verwijst naar het punt<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d5f3123d35179a39bd727675fca259c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(0,0).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"43\" style=\"vertical-align: -5px;\"><\/p>\n<p> We moeten daarom de raaklijn aan de functie op het punt berekenen<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-791f3561f68c75b943d5af446c9f988f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(0,0) .\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"43\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Eerst bepalen we de waarde van de helling van de raaklijn door de afgeleide aan de oorsprong van de co\u00f6rdinaten te berekenen: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e3f5e3bd3a06eb5e2831d90e5fc0f31d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=-3x^2+2x \\ \\longrightarrow \\  f'(x)= -6x+2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"315\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d07228d62a796c695cb75841830d0e17_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(0)= -6\\cdot 0+2=2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"168\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-075eebc763002b84e54211e61242356f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(0)=2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> In dit geval kennen we al een punt waar de raaklijn doorheen gaat. Omdat de verklaring ons vertelt dat de lijn de curve moet raken op de oorsprong van de co\u00f6rdinaten, dat wil zeggen op het punt<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d5f3123d35179a39bd727675fca259c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(0,0).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"43\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dus het punt dat de curve en de raaklijn delen, is het punt<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d5f3123d35179a39bd727675fca259c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(0,0).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"43\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Vervang ten slotte eenvoudigweg de gevonden waarden voor de helling en het raakpunt in uw vergelijking:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-de8e4e9dbb7a5bca1d591612abcf7730_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{c} y -y_0= m(x-x_0) \\\\[3ex] m=2 \\qquad P(0,0) \\end{array} \\right\\} \\longrightarrow \\ y -0= 2(x-0)\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"344\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Concluderend is de raaklijnvergelijking: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-19b9a613ed0c41d6c98ab37c6a0a1331_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y -0= 2(x-0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-329a7fc0d44a0b32cbb521e81ee50db6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y = 2x}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"52\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Oefening 3<\/h3>\n<p> Bereken de raaklijn aan de curve<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-33130a168a4e20b536fb742b8ce2a662_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2-2x-1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"146\" style=\"vertical-align: -5px;\"><\/p>\n<p> die evenwijdig is aan de rechterkant<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3b4e35094bc85458a54e2b47228f9c39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-4x-6=0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"113\" style=\"vertical-align: -4px;\"><\/p>\n<p> . <\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Zie de oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> In dit probleem wordt ons verteld dat de raaklijn evenwijdig moet zijn aan de lijn<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-02c5cab1d3747c5baa1ded66f3055f61_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-4x-6=0 .\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"117\" style=\"vertical-align: -4px;\"><\/p>\n<p> En twee lijnen zijn evenwijdig als ze dezelfde helling hebben. De raaklijn moet dus dezelfde helling hebben als de lijn<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3dee6df2aaaef2e062c41a79057de62e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-4x-6=0.\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"117\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dit betekent dat we de helling van de lijn moeten vinden<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-02c5cab1d3747c5baa1ded66f3055f61_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-4x-6=0 .\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"117\" style=\"vertical-align: -4px;\"><\/p>\n<p> Om dit te doen, wissen we de variabele en:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6390ce01aebdda3a7305c4dd1e55d4aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-4x-6=0 \\ \\longrightarrow \\ y =4x+6\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"246\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dus de helling van de lijn<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-824b16de72dde879834460d93bd88610_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=4x+5\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"82\" style=\"vertical-align: -4px;\"><\/p>\n<p> is 4, omdat de helling van een lijn het getal is dat de x vermenigvuldigt wanneer de y helder is.<\/p>\n<p class=\"has-text-align-left\"> Daarom moet de helling van de raaklijn ook 4 zijn, omdat ze, om evenwijdig te zijn, dezelfde helling moeten hebben.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e5072b7479ca854c5e3cdea8ffff2c0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=4\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"48\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> In dit geval vertellen ze ons niet het raakpunt tussen de curve en de raaklijn. Maar we weten dat de afgeleide van de curve op het raakpunt gelijk is aan de helling van de raaklijn, dwz<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c606eb4b268b71562672c32a0461053_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"85\" style=\"vertical-align: -5px;\"><\/p>\n<p> . Hoe kunnen we de waarde van weten?<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> , kunnen we x <sub>0<\/sub> uit de vergelijking vinden<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f79089d702fc8e5b6b7342c0eb2f0c68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(x_0):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"95\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Om dit te doen, berekenen we eerst de afgeleide van <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-40e51e628d64bea41578e16139b71b6a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"43\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ac4abafb3c879a6fd9c906ff9eea94d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)= x^2-2x-1 \\ \\longrightarrow \\ f'(x)=2x-2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"309\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En nu lossen we het op<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c606eb4b268b71562672c32a0461053_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"85\" style=\"vertical-align: -5px;\"><\/p>\n<p> wetende dat <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-421352ccc778c624805a5e2663bb7077_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m = 4 :\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f99f5b23457229b93eb24c214942f41f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m =f'(x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"85\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6a9fc1e268bfa17b6fe04e5fafbaaedc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4 =2(x_0)-2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"103\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-74d7e6af89f911f5a194fee138e70afa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4+2 =2x_0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"89\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-96c892854007fa06281252d3fcc3ae4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"6 =2x_0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"59\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9602ea885075b53499286a5126ad9724_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{6}{2} =x_0\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"49\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f2fb028e2d1cb1011436226f865d5162_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3=x_0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"50\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En zodra we de x-co\u00f6rdinaat van het punt kennen, kunnen we de andere co\u00f6rdinaat van het punt vinden door te berekenen <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4763abfc9310baf690c4bb81c5d8b743_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(3):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9e36063894754424dc75ff41070c42ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(3)=3^2-2\\cdot 3-1= 9-6-1=2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"281\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Het punt waar zowel de curve als de raaklijn doorheen gaan, is dus het punt<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a3558010337a7cdc27dddb44c10f0df1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(3,2).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"43\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Het enige dat overblijft is om de gevonden waarden van de helling en het punt van de raaklijn in de vergelijking te vervangen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8f1f49e9bef505c5c71cffd15f0d29d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{c} y -y_0= m(x-x_0) \\\\[3ex] m=4 \\qquad P(3,2) \\end{array} \\right\\} \\longrightarrow \\ y -2= 4(x-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"344\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En de vergelijking van de raaklijn is: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-89b6fd61abd22d30db13453334da7135_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y -2=4(x-3)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Oefening 4<\/h3>\n<p> Bereken de raaklijn aan de curve<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ab7fb2898ec2a42b558f032b99518338_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=2x^2+5x+1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"155\" style=\"vertical-align: -5px;\"><\/p>\n<p> die een hoek van 45\u00b0 vormt met de X-as. <\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Zie de oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> De probleemstelling vertelt ons dat de raaklijn een hoek van 45\u00b0 moet vormen met de X-as. In deze gevallen moet de volgende formule worden toegepast om de waarde van de helling te vinden: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1e17ba52bc8d7a78aa6abe918856ba28_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m = \\text{tg}\\left(\\alpha\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"82\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c891eec41f7529fbb36d622027b94d46_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m = \\text{tg}\\left(45^{\\text{o}}\\right) = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> De verklaring specificeert niet het raakpunt tussen de curve en de raaklijn. Maar we weten dat de afgeleide van de curve op het raakpunt gelijk is aan de helling van de raaklijn, dwz<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c606eb4b268b71562672c32a0461053_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"85\" style=\"vertical-align: -5px;\"><\/p>\n<p> . We kunnen daarom x <sub>0<\/sub> berekenen door de vergelijking op te lossen<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f79089d702fc8e5b6b7342c0eb2f0c68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(x_0):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"95\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Om dit te doen, berekenen we eerst de afgeleide van <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-40e51e628d64bea41578e16139b71b6a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"43\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d757979ec817338abf9a0d50e4d8838d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=2x^2+5x+1\\ \\longrightarrow \\ f'(x)=4x+5\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"318\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En nu lossen we het op<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c606eb4b268b71562672c32a0461053_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"85\" style=\"vertical-align: -5px;\"><\/p>\n<p> wetende dat <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-32704d9853b0093395b41eb385ebb4e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m = 1 :\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f99f5b23457229b93eb24c214942f41f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m =f'(x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"85\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-eec283acc7af9f75a48ed262d785d7f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"1 =4(x_0)+5\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"102\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4bcad96ea71d673fba2f814bffaee7c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"1-5 =4x_0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"88\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7638e49e3e5eab4f64f4dc439d458ec5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-4 =4x_0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"71\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8561c7a3def54db216a8f1ebf2588e79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{-4}{4} =x_0\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"71\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-64d0311d2aa3328e9ed1f2073e90e4bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-1=x_0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"62\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En zodra we de x-co\u00f6rdinaat van het punt kennen, kunnen we de andere co\u00f6rdinaat van het punt vinden door te berekenen <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ab353a75f8c672950ea7d8376104722d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(-1):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"56\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-33f491d7f9d0eeeb767d846b5650734f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(-1)=2(-1)^2+5(-1)+1=2\\cdot 1  -5 + 1 = -2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"382\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">Het punt waar zowel de curve als de raaklijn doorheen gaan, is dus het punt<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d2ae436ead5cc58de912263561cfbe63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(-1,-2).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"70\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Het enige dat overblijft is om de gevonden waarden van de helling en het punt van de raaklijn in de vergelijking te vervangen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8ed772b3993de50c4c67631a6fd33040_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{c} y -y_0= m(x-x_0) \\\\[3ex] m=1 \\qquad P(-1,-2) \\end{array} \\right\\} \\longrightarrow \\ y -(-2)= 1(x-(-1))\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"414\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En ten slotte voeren we de bewerkingen uit om de vergelijking van de raaklijn te vinden: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-28409d87564a3385166261f1fe92c01e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y -(-2)=1(x-(-1))\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"182\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9ebfc08d88b0dcf9c22ff7f225afdabf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y +2=1(x+1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b1c1b9eabdc99bc1ff5b5e8cdb5baf94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y + 2=x+1}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"103\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>In dit artikel zullen we zien hoe we de vergelijking van de raaklijn aan een curve kunnen vinden . Daarnaast kun je trainen met opgeloste oefeningen van verschillende moeilijkheidsgraden. Vergelijking van de raaklijn aan een functie in een punt De vergelijking van de raaklijn aan de functie f(x) in het punt x=x 0 is: Waarbij &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/nl\/raaklijnvergelijking\/\"> <span class=\"screen-reader-text\">Raaklijnvergelijking<\/span> Lees meer &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[49],"tags":[],"class_list":["post-83","post","type-post","status-publish","format-standard","hentry","category-functie-representatie"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Raaklijnvergelijking -<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/nl\/raaklijnvergelijking\/\" \/>\n<meta property=\"og:locale\" content=\"nl_NL\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Raaklijnvergelijking -\" \/>\n<meta property=\"og:description\" content=\"In dit artikel zullen we zien hoe we de vergelijking van de raaklijn aan een curve kunnen vinden . Daarnaast kun je trainen met opgeloste oefeningen van verschillende moeilijkheidsgraden. Vergelijking van de raaklijn aan een functie in een punt De vergelijking van de raaklijn aan de functie f(x) in het punt x=x 0 is: Waarbij &hellip; Raaklijnvergelijking Lees meer &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/nl\/raaklijnvergelijking\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-09-17T10:58:59+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-326424811181144df35c0b94ce50c462_l3.png\" \/>\n<meta name=\"author\" content=\"Redactioneel Team\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Geschreven door\" \/>\n\t<meta name=\"twitter:data1\" content=\"Redactioneel Team\" \/>\n\t<meta name=\"twitter:label2\" content=\"Geschatte leestijd\" \/>\n\t<meta name=\"twitter:data2\" content=\"6 minuten\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/nl\/raaklijnvergelijking\/\",\"url\":\"https:\/\/mathority.org\/nl\/raaklijnvergelijking\/\",\"name\":\"Raaklijnvergelijking -\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/nl\/#website\"},\"datePublished\":\"2023-09-17T10:58:59+00:00\",\"dateModified\":\"2023-09-17T10:58:59+00:00\",\"author\":{\"@id\":\"https:\/\/mathority.org\/nl\/#\/schema\/person\/19b550cef1a9fbd238be112b7b7bbf64\"},\"breadcrumb\":{\"@id\":\"https:\/\/mathority.org\/nl\/raaklijnvergelijking\/#breadcrumb\"},\"inLanguage\":\"nl-NL\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathority.org\/nl\/raaklijnvergelijking\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathority.org\/nl\/raaklijnvergelijking\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathority.org\/nl\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Raaklijnvergelijking\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathority.org\/nl\/#website\",\"url\":\"https:\/\/mathority.org\/nl\/\",\"name\":\"\",\"description\":\"Waar nieuwsgierigheid en berekening elkaar ontmoeten!\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/mathority.org\/nl\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"nl-NL\"},{\"@type\":\"Person\",\"@id\":\"https:\/\/mathority.org\/nl\/#\/schema\/person\/19b550cef1a9fbd238be112b7b7bbf64\",\"name\":\"Redactioneel Team\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"nl-NL\",\"@id\":\"https:\/\/mathority.org\/nl\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g\",\"caption\":\"Redactioneel Team\"},\"sameAs\":[\"http:\/\/mathority.org\/nl\"]}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Raaklijnvergelijking -","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mathority.org\/nl\/raaklijnvergelijking\/","og_locale":"nl_NL","og_type":"article","og_title":"Raaklijnvergelijking -","og_description":"In dit artikel zullen we zien hoe we de vergelijking van de raaklijn aan een curve kunnen vinden . Daarnaast kun je trainen met opgeloste oefeningen van verschillende moeilijkheidsgraden. Vergelijking van de raaklijn aan een functie in een punt De vergelijking van de raaklijn aan de functie f(x) in het punt x=x 0 is: Waarbij &hellip; Raaklijnvergelijking Lees meer &raquo;","og_url":"https:\/\/mathority.org\/nl\/raaklijnvergelijking\/","article_published_time":"2023-09-17T10:58:59+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-326424811181144df35c0b94ce50c462_l3.png"}],"author":"Redactioneel Team","twitter_card":"summary_large_image","twitter_misc":{"Geschreven door":"Redactioneel Team","Geschatte leestijd":"6 minuten"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/mathority.org\/nl\/raaklijnvergelijking\/","url":"https:\/\/mathority.org\/nl\/raaklijnvergelijking\/","name":"Raaklijnvergelijking -","isPartOf":{"@id":"https:\/\/mathority.org\/nl\/#website"},"datePublished":"2023-09-17T10:58:59+00:00","dateModified":"2023-09-17T10:58:59+00:00","author":{"@id":"https:\/\/mathority.org\/nl\/#\/schema\/person\/19b550cef1a9fbd238be112b7b7bbf64"},"breadcrumb":{"@id":"https:\/\/mathority.org\/nl\/raaklijnvergelijking\/#breadcrumb"},"inLanguage":"nl-NL","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathority.org\/nl\/raaklijnvergelijking\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathority.org\/nl\/raaklijnvergelijking\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathority.org\/nl\/"},{"@type":"ListItem","position":2,"name":"Raaklijnvergelijking"}]},{"@type":"WebSite","@id":"https:\/\/mathority.org\/nl\/#website","url":"https:\/\/mathority.org\/nl\/","name":"","description":"Waar nieuwsgierigheid en berekening elkaar ontmoeten!","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/mathority.org\/nl\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"nl-NL"},{"@type":"Person","@id":"https:\/\/mathority.org\/nl\/#\/schema\/person\/19b550cef1a9fbd238be112b7b7bbf64","name":"Redactioneel Team","image":{"@type":"ImageObject","inLanguage":"nl-NL","@id":"https:\/\/mathority.org\/nl\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g","caption":"Redactioneel Team"},"sameAs":["http:\/\/mathority.org\/nl"]}]}},"yoast_meta":{"yoast_wpseo_title":"","yoast_wpseo_metadesc":"","yoast_wpseo_canonical":""},"_links":{"self":[{"href":"https:\/\/mathority.org\/nl\/wp-json\/wp\/v2\/posts\/83","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathority.org\/nl\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathority.org\/nl\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathority.org\/nl\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathority.org\/nl\/wp-json\/wp\/v2\/comments?post=83"}],"version-history":[{"count":0,"href":"https:\/\/mathority.org\/nl\/wp-json\/wp\/v2\/posts\/83\/revisions"}],"wp:attachment":[{"href":"https:\/\/mathority.org\/nl\/wp-json\/wp\/v2\/media?parent=83"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathority.org\/nl\/wp-json\/wp\/v2\/categories?post=83"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathority.org\/nl\/wp-json\/wp\/v2\/tags?post=83"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}