{"id":49,"date":"2023-09-17T10:54:42","date_gmt":"2023-09-17T10:54:42","guid":{"rendered":"https:\/\/mathority.org\/it\/rappresentazione-delle-funzioni\/"},"modified":"2023-09-17T10:54:42","modified_gmt":"2023-09-17T10:54:42","slug":"rappresentazione-delle-funzioni","status":"publish","type":"post","link":"https:\/\/mathority.org\/it\/rappresentazione-delle-funzioni\/","title":{"rendered":"Rappresentazione delle funzioni"},"content":{"rendered":"<p class=\"has-text-align-left\">In questo articolo vedremo <strong>come rappresentare qualsiasi tipo di funzione su un grafico.<\/strong> Inoltre, troverai esercizi passo passo risolti sulla rappresentazione delle funzioni su un grafico. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"como-representar-una-funcion-en-una-grafica\"><\/span> Come rappresentare una funzione su un grafico<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Per rappresentare una funzione su un grafico \u00e8 necessario eseguire i seguenti passaggi: <\/p>\n<div style=\"background-color:#FFF3E0; padding-top: 23px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 10px; border-radius:30px;\">\n<ol style=\"color:#64B5F6; font-weight: bold;\">\n<li style=\"margin-bottom:16px\"> <span style=\"color:#000000;font-weight: normal;\">Trova il <strong>dominio<\/strong> della funzione.<\/span><\/li>\n<li style=\"margin-bottom:16px;\"> <span style=\"color:#000000;font-weight: normal;\">Calcolare i <strong>punti limite<\/strong> della funzione con gli assi cartesiani.<\/span><\/li>\n<li style=\"margin-bottom:16px;\"> <span style=\"color:#000000;font-weight: normal;\">Calcolare gli <strong>asintoti<\/strong> della funzione.<\/span><\/li>\n<li style=\"margin-bottom:16px;\"> <span style=\"color:#000000;font-weight: normal;\">Studia la monotonia della funzione e trova i suoi <strong>estremi relativi<\/strong> .<\/span><\/li>\n<li style=\"margin-bottom:16px;\"> <span style=\"color:#000000;font-weight: normal;\">Studia la curvatura della funzione e trova i suoi <strong>punti di flesso<\/strong> .<\/span><\/li>\n<li> <span style=\"color:#000000;font-weight: normal;\"><strong>Traccia<\/strong> i punti di interruzione, gli asintoti, gli estremi relativi e i punti di flesso, quindi traccia la funzione.<\/span> <\/li>\n<\/ol>\n<\/div>\n<h2 class=\"estil_titol_H2 wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-de-la-representacion-de-una-funcion\"><\/span> Esempio di rappresentazione di una funzione<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Affinch\u00e9 tu possa vedere come \u00e8 rappresentata graficamente una funzione, risolveremo passo dopo passo il seguente esercizio:<\/p>\n<ul>\n<li> Traccia su un grafico la seguente funzione razionale:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-eb173dfd702785865be0051c9bcb7738_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\cfrac{x^2}{x-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"101\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> La prima cosa da fare \u00e8 <strong>calcolare il dominio della funzione<\/strong> . Questa \u00e8 una funzione razionale, quindi dobbiamo impostare il denominatore uguale a zero per vedere quali numeri non appartengono al dominio della funzione: <\/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-2a57ca6c48b6f646aeb64eb7f05e4840_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x-1=0\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"73\" 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-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>\n<p class=\"has-text-align-left\"> Quindi, quando x \u00e8 1, il denominatore sar\u00e0 0 e quindi la funzione non esister\u00e0. Il dominio della funzione \u00e8 quindi costituito da tutti i numeri reali tranne x=1.<\/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-66d11e82f81cd2425ea2e6641e374baf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{Dom } f= \\mathbb{R}-\\{1 \\}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"138\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Per trovare il <strong>punto di intersezione con l&#8217;asse X<\/strong> , dobbiamo risolvere l&#8217;equazione<\/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-bbb52c33bfaff434771f0e4ddd4cf677_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)= 0.\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"71\" style=\"vertical-align: -5px;\"><\/p>\n<p> Poich\u00e9 la funzione ha sempre un valore pari a 0 sull&#8217;asse X:<\/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-0bce6c022ed0fc63f4659af75888f96c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"67\" 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-598e2ac6e2410e5d89ee067071c1d280_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x^2}{x-1} = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"73\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Il termine<\/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-d6ca0f3c84745bcbcccc5f4ebf219891_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x -1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"40\" style=\"vertical-align: 0px;\"><\/p>\n<p> Ci\u00f2 comporta la divisione dell&#8217;intero lato sinistro, quindi possiamo moltiplicarlo per l&#8217;intero lato destro:<\/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-f086c381edce77135440070151e8ce65_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2 = 0 \\cdot (x-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"117\" 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-97bb5e9fb1f811f609395daafea9e9c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2 = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"50\" 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-69deb06de751e80bf5f09f379ee2bc53_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>Il punto di intersezione con l&#8217;asse OX \u00e8 quindi:<\/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-2c790019bd70403eba876c59c82c0f9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,0)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> E per trovare il <strong>punto di intersezione con l&#8217;asse Y<\/strong> , calcoliamo<\/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-d449eebd1f011aebdf90931f3a66a3b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(0).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<p> Poich\u00e9 x \u00e8 sempre 0 sull&#8217;asse Y:<\/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-7617c5eab838a7e451fef14b9ccce246_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(0)=\\cfrac{0^2}{0-1} = \\cfrac{0}{-1} = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"179\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Pertanto, il punto di taglio con l&#8217;asse OY \u00e8:<\/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-2c790019bd70403eba876c59c82c0f9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,0)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> In questo caso, quando la funzione passa per l&#8217;origine delle coordinate, il punto di intersezione con l&#8217;asse X coincide con il punto di intersezione con l&#8217;asse Y.<\/p>\n<p> Una volta che conosciamo il dominio e i punti limite, dobbiamo <strong>calcolare gli asintoti della funzione<\/strong> .<\/p>\n<p> Per vedere se la funzione ha asintoti verticali, dobbiamo calcolare il limite della funzione nei punti che non appartengono al dominio (in questo caso x=1). E se il risultato \u00e8 infinito, \u00e8 un asintoto verticale. Ancora:<\/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-d1933bd0c8df1e0e994ff71304ce3627_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to 1} \\ \\cfrac{x^2}{x-1} = \\cfrac{1^2}{1-1} = \\cfrac{1}{0} = \\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"220\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Poich\u00e9 il limite della funzione quando x tende a 1 d\u00e0 infinito, x=1 \u00e8 un asintoto verticale: <\/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\/represente-fonctions-vertical-asymptote.webp\" alt=\"rappresentare funzioni, asintoto verticale\" class=\"wp-image-2626\" width=\"550\" height=\"604\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Una volta calcolato l&#8217;asintoto verticale \u00e8 necessario calcolare i limiti laterali della funzione rispetto ad esso. Poich\u00e9 non sappiamo se la funzione tender\u00e0 a -\u221e o +\u221e quando si avvicina a x=1 da sinistra, e non sappiamo quando si avvicina a x=1 da destra.<\/p>\n<p> Procediamo quindi a calcolare il limite laterale sinistro della funzione in x=1:<\/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-a2a03f93135e37cc8d84b375dfc5b40e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to 1^{-}} \\cfrac{x^2}{x-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"84\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> Per calcolare numericamente il limite laterale in un punto, dobbiamo sostituire nella funzione un numero che \u00e8 molto vicino al punto. In questo caso, vogliamo un numero molto vicino a 1 a sinistra, come 0,9. Sostituiamo quindi il punto 0.9 nella funzione:<\/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-9828df62f758cce355c916237e82b766_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{0,9^2}{0,9-1}=\\cfrac{0,81}{-0,1}=-81\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"176\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> I limiti laterali ad un asintoto possono dare solo +\u221e o -\u221e. E poich\u00e9 sostituendo nella funzione un numero molto vicino a 1 a sinistra abbiamo ottenuto un risultato negativo, il limite a sinistra \u00e8 -\u221e:<\/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-e6a013ed3f883f6822c94fb274360b7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to 1^{-}} \\cfrac{x^2}{x-1} = \\bm{-\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"139\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> Ora eseguiamo la stessa procedura con il confine del lato destro:<\/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-1aef4f24697f29600025e161323e07dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to 1^{+}} \\cfrac{x^2}{x-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"84\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> Sostituiamo un numero molto vicino a 1 a destra nella funzione. Ad esempio il punto 1.1:<\/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-93573af45a33dd1b799499d4568686f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{1,1^2}{1,1-1}=\\cfrac{1,21}{0,1}=+12,1\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"187\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> In questo caso, il risultato del limite laterale \u00e8 un numero positivo. Il limite a destra \u00e8 quindi +\u221e:<\/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-8800ea3138e573b9285eef458f08fa91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to 1^{+}} \\cfrac{x^2}{x-1} = \\bm{+\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"138\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> In conclusione, per x=1 la funzione tende verso meno infinito a sinistra, e pi\u00f9 infinito a destra: <\/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\/fonctions-graphiques-asymptote-verticale.webp\" alt=\"funzioni grafiche, asintoto verticale\" class=\"wp-image-2627\" width=\"550\" height=\"602\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> D&#8217;altra parte, l&#8217;asintoto orizzontale della funzione sar\u00e0 il risultato del limite infinito della funzione. Ancora: <\/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-df700cb3b219bb9b3dff71de849ac381_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\ \\cfrac{x^2}{x-1} = \\cfrac{+\\infty}{+\\infty } =+\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"217\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<div style=\"background-color:#FFFDE7; padding-top: 23px; padding-bottom: 0.5px; padding-right: 40px; padding-left: 30px; border: 2.5px dashed #FFB74D; border-radius:20px;\">\n<p> <strong>Ricorda<\/strong> come calcolare i limiti infiniti delle funzioni razionali:<\/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-2c969e4b99985b44006e57d554ff0247_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to \\pm \\infty}}\\frac{a_nx^r+a_{n-1}x^{r-1}+a_{n-2}x^{r-2}+\\dots}{b_nx^s+b_{n-1}x^{s-1}+b_{n-2}x^{s-2}+\\dots}=\\left\\{ \\begin{array}{lcl} 0 &amp; \\text{si} &amp; r<s \\\\[3ex]=&quot;&quot; \\cfrac{a_n}{b_n}=&quot;&quot; &amp;=&quot;&quot; \\text{si}=&quot;&quot; r=&quot;s&quot; \\\\[5ex]=&quot;&quot; \\pm=&quot;&quot; \\infty=&quot;&quot;>s \\end{array}\\right.&#8221; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;139&#8243; width=&#8221;767&#8243; style=&#8221;vertical-align: 0px;&#8221;><\/p>\n<\/p>\n<\/div>\n<p> Il limite infinito della funzione ci ha dato +\u221e, quindi la funzione non ha asintoto orizzontale.<\/p>\n<p> Calcoliamo ora l&#8217;asintoto obliquo. Gli asintoti obliqui sono della forma<\/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-8e4adcc4368f6296906b6231bf17a6a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=mx+n\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"91\" style=\"vertical-align: -4px;\"><\/p>\n<p> . E<\/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> Si calcola con la seguente formula:<\/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-9dcdd1bef23f97f1397a19964de98fa6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m = \\lim_{x \\to +\\infty} f(x):x\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"147\" style=\"vertical-align: -13px;\"><\/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-2be9bdf7dce3b2e7d79079b5528fe177_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m = \\lim_{x \\to +\\infty} \\cfrac{x^2}{x-1}:x\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"156\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> La x \u00e8 come se avesse 1 come denominatore:<\/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-2f799c07d0e78080d55bc31bc5278446_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m = \\lim_{x \\to +\\infty} \\cfrac{x^2}{x-1}:\\cfrac{x}{1}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"158\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> \u00c8 una divisione di frazioni, quindi le moltiplichiamo trasversalmente:<\/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-66422a93cd9b73474054b6370ecbdc76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m = \\lim_{x \\to +\\infty} \\cfrac{x^2 \\cdot 1 }{(x-1) \\cdot x}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"168\" style=\"vertical-align: -17px;\"><\/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-529870b7f0ef84afc97448cbc7855056_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m = \\lim_{x \\to +\\infty} \\cfrac{x^2 }{x^2-x}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"140\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> E calcoliamo il limite:<\/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-e82cefa1bb58cc251d600848a5a0a57d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m = \\lim_{x \\to +\\infty} \\cfrac{x^2 }{x^2-x} =  \\cfrac{+\\infty}{+\\infty } = \\cfrac{1}{1} = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"271\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> Quindi m=1. Ora calcoliamo<\/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-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"n\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p> con la seguente formula:<\/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-81d8f8b6af95602f96372b8abe4af497_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to +\\infty} \\bigl[f(x)-mx\\bigr]\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"174\" style=\"vertical-align: -13px;\"><\/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-10dfa8fdcfbf0c978e02374654a66b7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to +\\infty} \\left[\\cfrac{x^2}{x-1}-1x\\right] = \\cfrac{+\\infty}{+\\infty} -(+\\infty) = +\\infty - \\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"413\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<p> Ma otteniamo l&#8217;indeterminazione infinito meno infinito, quindi dobbiamo ridurre i termini a un denominatore comune. Per fare ci\u00f2, moltiplichiamo e dividiamo il termine x per il denominatore della frazione:<\/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-70026c2aed1bb58a120f8c18423d9ef5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to +\\infty}\\left[\\cfrac{x^2}{x-1}-x\\right]  = \\lim_{x \\to +\\infty} \\left[\\cfrac{x^2}{x-1}-\\cfrac{x\\cdot (x-1)}{x-1} \\right] = \\lim_{x \\to +\\infty} \\left[\\cfrac{x^2}{x-1}-\\cfrac{x^2-x}{x-1}\\right]\" title=\"Rendered by QuickLaTeX.com\" height=\"109\" width=\"582\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<p> Ora che i due termini hanno lo stesso denominatore possiamo raggrupparli:<\/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-7702287a02af6d8e3dddaa3f0c6eb1b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to +\\infty} \\left[\\cfrac{x^2-(x^2-x)}{x-1}  \\right] =\\lim_{x \\to +\\infty} \\left[\\cfrac{x}{x-1} \\right]\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"340\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<p> E infine risolviamo il limite:<\/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-feb5faa9dc5d3b68d3273ad4d75d2bb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n =\\lim_{x \\to +\\infty} \\left[\\cfrac{x}{x-1} \\right] = \\cfrac{+\\infty}{+\\infty} = \\cfrac{1}{1} = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"278\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<p> Quindi n = 1. L\u2019asintoto obliquo \u00e8 quindi: <\/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-6fbe1cc5f3362ddbd80ed0b29c0bb4ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y = mx+n\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"91\" style=\"vertical-align: -4px;\"><\/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-dd133b92b6c5b350ce4383147df52e3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y = 1x+1\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"82\" style=\"vertical-align: -4px;\"><\/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-dcfc739c1fd18f6fd834ff3e59b009e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y = x+1}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"73\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Una volta calcolato l&#8217;asintoto obliquo, lo rappresentiamo sullo stesso grafico realizzando una tabella di valori:<\/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-73556269fc16e4cae71ddfde0ff51632_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=x+1\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"73\" style=\"vertical-align: -4px;\"><\/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-6e696990e6cea37c0267d01c4553240f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c|c} x &amp; y \\\\ \\hline 0 &amp; 1 \\\\ 1 &amp; 2 \\\\ 2 &amp; 3 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"90\" width=\"52\" style=\"vertical-align: -40px;\"><\/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\/representent-fonctions-oblique-asymptote.webp\" alt=\"rappresentare funzioni, asintoto obliquo\" class=\"wp-image-2632\" width=\"501\" height=\"551\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Ora che conosciamo tutti gli asintoti della funzione, dobbiamo analizzare la <strong>monotonicit\u00e0 della funzione<\/strong> . Dobbiamo cio\u00e8 studiare in quali intervalli la funzione aumenta e in quali intervalli diminuisce. Calcoliamo quindi la derivata prima della funzione:<\/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-370529412c7f94fbe43e8d844520a185_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\cfrac{x^2}{x-1} \\ \\longrightarrow \\ f'(x)= \\cfrac{2x\\cdot (x-1) - x^2 \\cdot 1}{(x-1)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"364\" style=\"vertical-align: -17px;\"><\/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-eafd7df834025eab179670d70e631871_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)= \\cfrac{2x^2-2x - x^2}{(x-1)^2}  = \\cfrac{x^2-2x}{(x-1)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"260\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> E ora impostiamo la derivata uguale a 0 e risolviamo l&#8217;equazione:<\/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-36700780d306ccf4975387990b1949fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" 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-6f17b3ce9d9690b68738698290f1b33f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x^2-2x}{(x-1)^2}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"95\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Il termine<\/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-cc1f4cc53676f0eb98290b3478031fef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(x-1\\right)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"60\" style=\"vertical-align: -5px;\"><\/p>\n<p> Ci\u00f2 comporta la divisione dell&#8217;intero lato sinistro, quindi possiamo moltiplicarlo per l&#8217;intero lato destro:<\/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-3d8bb0359e60db0b26d9bfce1b349e9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2-2x=0\\cdot \\left(x-1\\right)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"165\" 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-62138ee9fb8dc604ee836f1703379032_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2-2x=0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"91\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Estraiamo il fattore comune per risolvere l&#8217;equazione quadratica:<\/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-b243129a0d8853ec8716beb6d2d5c504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x(x-2)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"97\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Perch\u00e9 la moltiplicazione sia uguale a 0, uno dei due elementi della moltiplicazione deve essere zero. Pertanto, impostiamo ciascun fattore uguale a 0 e otteniamo entrambe le soluzioni dell&#8217;equazione:<\/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-55127e675ce8f7742db17d565c2ae507_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle x\\cdot(x-2) =0   \\longrightarrow  \\begin{cases} \\bm{x=0} \\\\[2ex] x-2=0 \\ \\longrightarrow \\ \\bm{x= 2} \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"329\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Rappresentiamo ora sulla retta numerica tutti i punti critici trovati, cio\u00e8 i punti che non appartengono al dominio (x=1) e quelli che annullano la derivata (x=0 e x=2): <\/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\/ligne-numerique-0-1-2.webp\" alt=\"\" class=\"wp-image-2443\" width=\"390\" height=\"77\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> E valutiamo il segno della derivata in ciascun intervallo, per sapere se la funzione aumenta o diminuisce. Prendiamo quindi un punto in ogni intervallo (mai i punti critici) e guardiamo che segno ha la derivata in quel punto: <\/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-8c77f1f797549bb4663fca07fcea2302_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=\\cfrac{x^2-2x}{\\left(x-1\\right)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"128\" style=\"vertical-align: -20px;\"><\/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-171fa182722405650545d6e7fe14d5b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(-1) = \\cfrac{(-1)^2-2(-1)}{\\left((-1)-1\\right)^2} =\\cfrac{+3}{+4} = +0,75 \\  \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"369\" style=\"vertical-align: -20px;\"><\/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-3daba13acad48408dfadae7c683d62d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(0,5) = \\cfrac{0,5^2-2\\cdot 0,5}{\\left(0,5-1\\right)^2} = \\cfrac{-0,75}{+0,25} = -3  \\  \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"363\" style=\"vertical-align: -20px;\"><\/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-bd76e394ebfc759caaedca3a6ff66762_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1,5) = \\cfrac{1,5^2-2\\cdot 1,5}{\\left(1,5-1\\right)^2} = \\cfrac{-0,75}{+0,25} = -3  \\  \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"363\" style=\"vertical-align: -20px;\"><\/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-e8d15f1093f1455f39042681fd9ab133_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(3) = \\cfrac{3^2-2\\cdot 3}{\\left(3-1\\right)^2} =\\cfrac{+3}{+4} = +0,75 \\  \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"313\" style=\"vertical-align: -20px;\"><\/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\/ligne-numerique-0-1-2-positif-negatif-positif.webp\" alt=\"\" class=\"wp-image-2444\" width=\"390\" height=\"136\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Se la derivata \u00e8 positiva significa che la funzione \u00e8 crescente, se la derivata \u00e8 negativa significa che la funzione \u00e8 decrescente. Pertanto gli intervalli di crescita e declino sono:<\/p>\n<p class=\"has-text-align-center\"> <strong>Crescita:<\/strong><\/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-11ebeca24ba262661dd73042a326110c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-\\infty, 0)\\cup (2,+\\infty)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"142\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>Diminuire:<\/strong><\/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-206ab3f38b17a58b25209bf269265919_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,1)\\cup (1,2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"97\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Inoltre, per x=0 la funzione passa da crescente a decrescente, quindi x=0 \u00e8 un massimo relativo della funzione. E in x=2, la funzione passa da decrescente ad crescente, quindi x=2 \u00e8 un minimo relativo della funzione.<\/p>\n<p> Infine, sostituiamo gli estremi trovati nella funzione originale per trovare la coordinata Y dei punti:<\/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-d8bb02550f4c83abce02040f9e9ab495_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(0)=\\cfrac{0^2}{0-1} = \\cfrac{0}{-1} = 0 \\ \\longrightarrow \\ (0,0)\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"268\" 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-74333ede5561c728c68899d68b31ee62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(2)=\\cfrac{2^2}{2-1} = \\cfrac{4}{1} = 4 \\ \\longrightarrow \\ (2,4)\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"254\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Gli estremi relativi della funzione sono quindi:<\/p>\n<p class=\"has-text-align-center\"> <strong>Massimo punto<\/strong><\/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-2c790019bd70403eba876c59c82c0f9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,0)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>Minimo da puntare<\/strong><\/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-a59b564601b4cd9f2bc149baa80c44a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(2,4)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Rappresentiamo il massimo e il minimo sul grafico: <\/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\/representent-les-fonctions-maximum-et-minimum.webp\" alt=\"rappresentano le funzioni massimo e minimo\" class=\"wp-image-2633\" width=\"548\" height=\"604\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Infine \u00e8 sufficiente <strong>studiare la curvatura della funzione<\/strong> , cio\u00e8 studiare gli intervalli di concavit\u00e0 e convessit\u00e0 della funzione. Per fare ci\u00f2 calcoliamo la sua derivata seconda: <\/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-06628d8d896e24d462c90c9d6a47fdfc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=\\cfrac{x^2-2x}{(x-1)^2} \\ \\longrightarrow \\ f''(x)= \\cfrac{(2x-2)\\cdot (x-1)^2- (x^2-2x)\\cdot 2(x-1)\\cdot 1}{\\left(\\left(x-1\\right)^2\\right)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"62\" width=\"575\" style=\"vertical-align: -33px;\"><\/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-9c854f6cd07cc084869bbb880365c1d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= \\cfrac{(2x-2)\\cdot (x-1)^2- (x^2-2x)\\cdot 2(x-1)}{(x-1)^4}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"377\" style=\"vertical-align: -17px;\"><\/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-15214503897d52ddd5c4f2865cc8a3b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= \\cfrac{(2x-2)\\cdot (x-1)^{\\cancel{2}}- (x^2-2x)\\cdot 2\\cancel{(x-1)}}{(x-1)^{\\cancelto{3}{4}}} = \\cfrac{(2x-2)\\cdot (x-1)- (x^2-2x)\\cdot 2}{(x-1)^3}\" title=\"Rendered by QuickLaTeX.com\" height=\"93\" width=\"582\" style=\"vertical-align: -17px;\"><\/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-c72f4714446d0cb718dd19637001f822_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= \\cfrac{2x^2-2x-2x+2- (2x^2-4x)}{(x-1)^3}  =\\cfrac{2x^2-2x-2x+2- 2x^2+4x}{(x-1)^3}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"563\" style=\"vertical-align: -17px;\"><\/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-c3f75e315c71b8f2e66c226c897c6585_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x) =\\cfrac{2}{(x-1)^3}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"131\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> E ora impostiamo la derivata seconda uguale a zero e risolviamo l&#8217;equazione: <\/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-981d85a257dd56afdb3fc7eb53d5eadf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"75\" 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-06d4145cca14c05a89eeea18d0cb9bf0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{2}{(x-1)^3} =0\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"95\" style=\"vertical-align: -17px;\"><\/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-15603629285ab47e50ccf28d5ab28607_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2=0\\cdot \\left(x-1\\right)^3\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"116\" 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-e6f185609f51e8f3ae79e0e459644dc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2=0\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> 2 non sar\u00e0 mai uguale a 0, quindi l&#8217;equazione<\/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-981d85a257dd56afdb3fc7eb53d5eadf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"75\" style=\"vertical-align: -5px;\"><\/p>\n<p> Non c&#8217;\u00e8 soluzione.<\/p>\n<p> Rappresentiamo ora sulla retta numerica tutti i punti critici trovati, cio\u00e8 i punti che non appartengono al dominio (x=1) e quelli che annullano la derivata seconda (in questo caso non ce ne sono has): <\/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\/nombre-ligne-1.webp\" alt=\"\" class=\"wp-image-2564\" width=\"226\" height=\"88\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> E valutiamo il segno della derivata in ogni intervallo, per sapere se la funzione \u00e8 convessa o concava. Prendiamo quindi un punto in ogni intervallo (mai i punti singolari) e guardiamo che segno ha la derivata in questo punto: <\/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-c3f75e315c71b8f2e66c226c897c6585_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x) =\\cfrac{2}{(x-1)^3}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"131\" style=\"vertical-align: -17px;\"><\/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-d986c4a85ebad90dba014c4958a2d6fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(0) =\\cfrac{2}{(0-1)^3} = \\cfrac{2}{-1}=-2 \\  \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"275\" style=\"vertical-align: -17px;\"><\/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-b42e050afcc2170bb221768109f1f839_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(2) =\\cfrac{2}{(2-1)^3} = \\cfrac{2}{1}=2 \\  \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"247\" style=\"vertical-align: -17px;\"><\/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\/droite-numerique-1-concave-convexe.webp\" alt=\"\" class=\"wp-image-2565\" width=\"227\" height=\"151\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> E infine deduciamo gli intervalli di concavit\u00e0 e convessit\u00e0 della funzione. Se la derivata seconda \u00e8 positiva significa che la funzione \u00e8 convessa.<\/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-49efa6d9ab88562f20df743cb7d267f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cup})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> , e se la derivata seconda \u00e8 negativa significa che la funzione \u00e8 concava<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-59e636042d77445b1534260d9d7309a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cap})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> . Gli intervalli di concavit\u00e0 e convessit\u00e0 sono quindi:<\/p>\n<p class=\"has-text-align-center\"> <strong>Convesso<\/strong><\/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-49efa6d9ab88562f20df743cb7d267f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cup})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> <strong>:<\/strong><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-87d6843b66a0ebea6c769017a30a8d75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(1,+\\infty)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"61\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>Concavo<\/strong><\/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-59e636042d77445b1534260d9d7309a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cap})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> <strong>:<\/strong><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-36b85798b30f125fea3702a0671c77ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-\\infty,1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"60\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Tuttavia, anche se c&#8217;\u00e8 un cambiamento nella curvatura in x=1, non \u00e8 un punto di flesso. Perch\u00e9 x=1 non appartiene al dominio della funzione.<\/p>\n<p> Quindi possiamo finire di rappresentare la funzione utilizzando tutto ci\u00f2 che abbiamo calcolato: <\/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\/representation-des-fonctions.webp\" alt=\"rappresentazione delle funzioni\" class=\"wp-image-2634\" width=\"550\" height=\"608\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> La funzione rappresentata nel grafico si presenta quindi cos\u00ec: <\/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\/representation-graphique-de-la-fonction-rationnelle.webp\" alt=\"rappresentazione grafica della funzione razionale\" class=\"wp-image-2635\" width=\"546\" height=\"598\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-representacion-de-funciones\"><\/span> Esercizi risolti per rappresentare le funzioni<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Esercizio 1<\/h3>\n<p> Rappresentare graficamente la seguente funzione polinomiale: <\/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-4fbbd639713355d58e743cb927faeee0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)=x^3-3x^2+4\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"155\" style=\"vertical-align: -5px;\"><\/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>Vedi la soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> La prima cosa da fare \u00e8 calcolare il dominio di definizione della funzione. Questa \u00e8 una funzione polinomiale, quindi il dominio \u00e8 costituito solo da numeri reali:<\/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-4f565027fd5d2a4381e3a23d183c9f76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{Dom } f= \\mathbb{R}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"90\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Per trovare il punto di intersezione con l&#8217;asse X, risolviamo <\/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-bbb52c33bfaff434771f0e4ddd4cf677_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)= 0.\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"71\" 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-0bce6c022ed0fc63f4659af75888f96c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"67\" 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-32372d448fb254237a89bd11fa071711_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^3-3x^2+4=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"129\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Questa \u00e8 un&#8217;equazione di grado maggiore di 2. Pertanto, fattorizziamo l&#8217;equazione:<\/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-f40f79838a9f4eb8ef8092860e41bfe2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x+1)(x^2-4x+4)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"189\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Quindi x=-1 \u00e8 una soluzione. E calcoliamo le altre soluzioni risolvendo l&#8217;equazione quadratica risultante:<\/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-e79a2a2f6650c4095c0dca52188c40c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}x &amp; =\\cfrac{-b \\pm \\sqrt{b^2-4ac}}{2a} =\\cfrac{-(-4) \\pm \\sqrt{(-4)^2-4\\cdot 1 \\cdot 4}}{2\\cdot 1} \\\\[2ex] &amp;=\\cfrac{+4 \\pm \\sqrt{16-16}}{2} =\\cfrac{4 \\pm \\sqrt{0}}{2} = \\cfrac{4 }{2 } = 2\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"105\" width=\"406\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> I punti di intersezione con l\u2019asse X sono quindi:<\/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-d9a1ceadc45948f3a942ccb21109ccf2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-1,0)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"51\" style=\"vertical-align: -5px;\"><\/p>\n<p> E<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9109f132c97f810054198982440ac8c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(2,0)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E per trovare il punto di intersezione con l&#8217;asse Y, calcoliamo<\/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-d449eebd1f011aebdf90931f3a66a3b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(0).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<p> Poich\u00e9 x \u00e8 sempre 0 sull&#8217;asse Y:<\/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-2a6de1788b17877aa807a31eae47e8a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(0)=0^3-3\\cdot0^2+4 = 4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"196\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Il punto di intersezione con l&#8217;asse Y \u00e8 quindi:<\/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-d0aa7bcc7acd9b70190168abbe3d05d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,4)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Per vedere se la funzione ha asintoti verticali, dobbiamo calcolare il limite della funzione nei punti che non appartengono al dominio. In questo caso il dominio comprende tutti i numeri reali. La funzione quindi non ha asintoto verticale.<\/p>\n<p class=\"has-text-align-left\"> D&#8217;altra parte, l&#8217;asintoto orizzontale della funzione sar\u00e0 il risultato del limite infinito della funzione. Ancora:<\/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-ea1063bd9a29fc7ccfd470464efcd868_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\ x^3-3x^2+4 =(+\\infty)^3 = +\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"283\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Il limite infinito della funzione ci ha dato +\u221e, quindi la funzione non ha asintoto orizzontale.<\/p>\n<p class=\"has-text-align-left\"> Calcoliamo ora l&#8217;asintoto obliquo. Gli asintoti obliqui sono della forma<\/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-ad313410fc976bc53709807aa8aed8e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=mx+n.\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"95\" style=\"vertical-align: -4px;\"><\/p>\n<p> E<\/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> Si calcola con la seguente formula: <\/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-df30ba17002e63ee33654a94955bbac9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m = \\lim_{x \\to +\\infty} f(x):x = \\lim_{x \\to +\\infty} \\left( x^3-3x^2+4\\right): x =\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"376\" style=\"vertical-align: -13px;\"><\/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-7b5d944c651bb86db8c00770059ebea7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle = \\lim_{x \\to +\\infty} \\cfrac{x^3-3x^2+4}{x} = \\cfrac{+\\infty}{+\\infty} = +\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"286\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Il limite ci ha dato +\u221e, quindi anche la funzione non ha asintoto obliquo.<\/p>\n<p class=\"has-text-align-left\"> Per studiare la monotonicit\u00e0 della funzione, dobbiamo prima calcolare la sua derivata:<\/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-f9453115e5750d874ed1f96014f8481b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)= x^3-3x^2+4 \\ \\longrightarrow \\ f'(x)= 3x^2-6x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"335\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ora impostiamo la derivata uguale a 0 e risolviamo l&#8217;equazione: <\/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-4890b9dfeb634c4d7a349351be73b5d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" 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-f360aca5f393d3a9c7e882e09f37fa7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3x^2-6x=0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"100\" 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-7c36141e8c1b73c1c73f9fea0dc115cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x(3x-6)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"106\" 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-d23e2b378508baca9f51117fc8767e90_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle x\\cdot(3x-6) =0 \\longrightarrow \\begin{cases} \\bm{x=0} \\\\[2ex] 3x-6=0 \\ \\longrightarrow \\ x= \\cfrac{6}{3} = 2 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"381\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Rappresentiamo ora sulla retta numerica tutti i punti singolari ottenuti, cio\u00e8 i punti che non appartengono al dominio (in questo caso lo appartengono tutti) e quelli che annullano la derivata (x=0 e x=2) : <\/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\/ligne-numerique-0-2.webp\" alt=\"\" class=\"wp-image-2638\" width=\"285\" height=\"75\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> E valutiamo il segno della derivata in ciascun intervallo, per sapere se la funzione aumenta o diminuisce. Prendiamo quindi un punto in ogni intervallo (mai i punti singolari) e guardiamo che segno ha la derivata in questo punto: <\/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-0412923f1191d192d18528b63a0e57ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(-1)=3(-1)^2-6(-1)= 3+6 = 9\\ \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"343\" 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-e29cd090a269505f244837bcbd11be75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1)=3\\cdot 1^2-6\\cdot 1= 3-6 = -3\\ \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"314\" 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-b1e444ac85ebaf441a325b83eb48714d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(3)=3\\cdot 3^2-6\\cdot 3= 27-18 = 9\\ \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"317\" 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\/ligne-numerique-0-2-monotonie.webp\" alt=\"\" class=\"wp-image-2639\" width=\"286\" height=\"135\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Se la derivata \u00e8 positiva significa che la funzione \u00e8 crescente, se la derivata \u00e8 negativa significa che la funzione \u00e8 decrescente. Pertanto gli intervalli di crescita e declino sono:<\/p>\n<p class=\"has-text-align-center\"> <strong>Crescita:<\/strong><\/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-b16ae35401c1d15b6d08334338f92172_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-\\infty,0)\\cup (2,+\\infty)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"142\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>Diminuire:<\/strong><\/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-6928e231184fd28bd944d9531a322d74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> La funzione passa da crescente a decrescente in x=0, quindi x=0 \u00e8 il massimo della funzione. E la funzione passa da decrescente ad crescente in x=2, quindi x=2 \u00e8 il minimo della funzione.<\/p>\n<p class=\"has-text-align-left\"> Infine, sostituiamo gli estremi trovati nella funzione originale per trovare le coordinate Y dei punti: <\/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-27cd3181606a77a6e28a89ac0e82f545_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(0)=0^3-3\\cdot 0^2+4 = 4 \\ \\longrightarrow \\ (0,4)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"285\" 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-a261755ca09d967ba2a2b3cbc84c1d8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(2)=2^3-3\\cdot 2^2+4 = 8-3 \\cdot 4 +4 = 0 \\ \\longrightarrow \\ (2,0)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"400\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Gli estremi relativi della funzione sono quindi:<\/p>\n<p class=\"has-text-align-center\"> <strong>Massimo punto<\/strong><\/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-d0aa7bcc7acd9b70190168abbe3d05d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,4)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>Minimo da puntare<\/strong><\/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-9109f132c97f810054198982440ac8c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(2,0)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Per studiare la curvatura della funzione, calcoliamo la sua derivata seconda:<\/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-aec39b1d4c0168da2b76e56643593a08_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)= 3x^2-6x \\ \\longrightarrow \\ f''(x)= 6x-6\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"296\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ora impostiamo la derivata seconda uguale a 0 e risolviamo l&#8217;equazione: <\/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-f618f4961c18c45be60fc496ad4896e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"75\" 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-905af0a687269e0a0775640143c2ea2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"6x-6=0\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"82\" 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-c9c29e31dbe2952c3f947c7999aaea32_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"6x=6\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"52\" 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-1b7492f8f450a70fe2916618ad1021b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= \\cfrac{6}{6} = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"76\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Rappresentiamo sulla retta tutti i punti singolari trovati, cio\u00e8 i punti che non appartengono al dominio (in questo caso lo appartengono tutti) e quelli che annullano la derivata (x=1): <\/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\/nombre-ligne-1.webp\" alt=\"\" class=\"wp-image-2564\" width=\"201\" height=\"78\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> E ora valutiamo il segno della derivata seconda in ogni intervallo, per sapere se la funzione \u00e8 concava o convessa. Prendiamo quindi un punto in ogni intervallo (mai i punti singolari) e guardiamo che segno ha la derivata seconda in questo punto: <\/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-e82925d4c4cfc1e0d51b97ea177d8508_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(0)=6\\cdot 0-6= -6 \\ \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"225\" 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-d532f146f034f820e7a7a72860a8bc54_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(2)=6\\cdot 2-6= 12-6= 6 \\ \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"283\" 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\/droite-numerique-1-concave-convexe.webp\" alt=\"\" class=\"wp-image-2565\" width=\"206\" height=\"138\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Se la derivata seconda \u00e8 positiva significa che la funzione \u00e8 convessa.<\/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-49efa6d9ab88562f20df743cb7d267f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cup})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> , e se la derivata seconda \u00e8 negativa significa che la funzione \u00e8 concava<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-59e636042d77445b1534260d9d7309a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cap})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> . Gli intervalli di concavit\u00e0 e convessit\u00e0 sono quindi:<\/p>\n<p class=\"has-text-align-center\"> <strong>Convesso<\/strong><\/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-49efa6d9ab88562f20df743cb7d267f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cup})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> <strong>:<\/strong><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-87d6843b66a0ebea6c769017a30a8d75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(1,+\\infty)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"61\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>Concavo<\/strong><\/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-59e636042d77445b1534260d9d7309a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cap})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> <strong>:<\/strong><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-36b85798b30f125fea3702a0671c77ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-\\infty,1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"60\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Inoltre, la funzione cambia da concava a convessa in x=1, quindi x=1 \u00e8 un punto di flesso della funzione.<\/p>\n<p class=\"has-text-align-left\"> Infine, sostituiamo i punti di flesso trovati nella funzione originale per trovare la coordinata Y dei punti:<\/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-ad643a4b4579b1f3a7484efda7d37f51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(1)=1^3-3\\cdot 1^2+ 4= 1 -3 +4 =2 \\ \\longrightarrow \\ (1,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"379\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> I punti di svolta della funzione sono quindi:<\/p>\n<p class=\"has-text-align-center\"> <strong>Punti di svolta:<\/strong><\/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-47f62d161c48f3aa8d8c81141352f01c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(1,2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Infine, sulla base di tutte le informazioni che abbiamo calcolato, rappresentiamo graficamente la funzione: <\/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\/representation-graphique-fonction-polynomiale.webp\" alt=\"rappresentazione grafica della funzione polinomiale\" class=\"wp-image-2640\" width=\"419\" height=\"464\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Esercizio 2<\/h3>\n<p> Rappresentare graficamente la seguente funzione razionale: <\/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-1e778797ead4e88a87997bf75163584e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)=\\frac{x^2+2}{x^2-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"109\" style=\"vertical-align: -12px;\"><\/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>Vedi la soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Per trovare il dominio della funzione, impostiamo il denominatore uguale. porta la frazione a zero e risolvi l&#8217;equazione risultante: <\/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-8818c0eafadfe429ce54f48546e7c06c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2-1= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"81\" 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-959000af33497314f9a59a9bed2a19c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2=1\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"49\" 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-003a71cb0ec797dfcd0cca915b03a795_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{x^2}=\\sqrt{1}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"79\" style=\"vertical-align: -2px;\"><\/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-b2e5d1349000e44cc1988f98254e0389_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\pm 1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" 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-8b555e0164cc0f3755093da248489375_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{Dom } f= \\mathbb{R}-\\{-1, +1 \\}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"182\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> In secondo luogo, determiniamo le soglie della funzione con l&#8217;asse x uguale all&#8217;espressione algebrica della funzione. acciaio: <\/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-0bce6c022ed0fc63f4659af75888f96c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"67\" 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-f4fd6078af8da3dbdaadfef085a088a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x^2+2}{x^2-1}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"81\" 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-396d8c2a41a4ee6344c8a7ac4824d785_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2+2=0\\cdot (x^2-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"155\" 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-5655aa321be25373afe5d61e410eb5cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2+2=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"81\" style=\"vertical-align: -2px;\"><\/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-23644022351ee7016d72eda1f084c02d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2=-2\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"63\" 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-07d269376ad563a1d6aa25499b471829_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\sqrt{-2} \\quad \\color{red}\\bm{\\times}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"127\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Non esiste la radice quadrata di un numero negativo. Pertanto, la funzione non interseca l&#8217;asse X.<\/p>\n<p class=\"has-text-align-left\"> E per trovare il punto di intersezione con l&#8217;asse del computer, valutiamo la funzione in x=0.<\/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-f88af2bf1737fdf34e3aed57447206b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(0)=\\cfrac{0^2+2}{0^2-1}= \\cfrac{2}{-1} = -2\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"200\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Il punto di intersezione con l&#8217;asse Y \u00e8 quindi:<\/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-b1761f5972412ad38102968850bf6220_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,-2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"52\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Per vedere se la funzione ha asintoti verticali, dobbiamo calcolare il limite della funzione nei punti che non appartengono al dominio (in questo caso x=-1 e x=+1). E se il risultato \u00e8 infinito, \u00e8 un asintoto verticale. Ancora:<\/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-9213a1057c2accea4cd10e01b44e0a0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to -1} \\cfrac{x^2+2}{x^2-1} = \\cfrac{(-1)^2+2}{(-1)^2-1} =\\cfrac{1+2}{1-1}= \\cfrac{3}{0} = \\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"333\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Poich\u00e9 il limite della funzione quando x si avvicina a -1 d\u00e0 infinito, x=-1 \u00e8 un asintoto verticale.<\/p>\n<p class=\"has-text-align-left\"> Calcoliamo i limiti laterali dell&#8217;asintoto x=-1 sostituendo nella funzione un numero molto vicino ad esso: <\/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-3734e234658371143046355d62d2b15c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(-1,1)=\\cfrac{(-1,1)^2+2}{(-1,1)^2-1} =+15,29 \\longrightarrow \\lim_{x \\to -1^{-}} \\ \\cfrac{x^2+2}{x^2-1} = +\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"463\" style=\"vertical-align: -17px;\"><\/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-97f3b163c528f25bd56635fd28639ecb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(-0,9)=\\cfrac{(-0,9)^2+2}{(-0,9)^2-1} =-14,79 \\longrightarrow \\lim_{x \\to -1^{+}} \\ \\cfrac{x^2+2}{x^2-1} = -\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"463\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ora vediamo se x=+1 \u00e8 un asintoto verticale:<\/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-19d299b6cd11eea52abec214e2d5dbf5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +1} \\cfrac{x^2+2}{x^2-1} = \\cfrac{1^2+2}{1^2-1} =\\cfrac{1+2}{1-1}= \\cfrac{3}{0} = \\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"305\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Poich\u00e9 il limite della funzione quando x si avvicina a +1 d\u00e0 infinito, x=+1 \u00e8 un asintoto verticale.<\/p>\n<p class=\"has-text-align-left\"> Calcoliamo i limiti laterali dell&#8217;asintoto x=1 sostituendo nella funzione un numero molto vicino ad esso: <\/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-1e721a23e7ead77eb1974f5f4ccb781e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(0,9)=\\cfrac{(0,9)^2+2}{(0,9)^2-1} =-14,79 \\longrightarrow \\lim_{x \\to +1^{-}} \\ \\cfrac{x^2+2}{x^2-1} = -\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"435\" style=\"vertical-align: -17px;\"><\/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-e5caabec3166aad7e5c7d6a8d25b2388_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(1,1)=\\cfrac{(1,1)^2+2}{(1,1)^2-1} =+15,29 \\longrightarrow \\lim_{x \\to +1^{+}} \\ \\cfrac{x^2+2}{x^2-1} = +\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"435\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> D&#8217;altra parte, l&#8217;asintoto orizzontale della funzione sar\u00e0 il risultato del limite infinito della funzione. Ancora:<\/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-7aa229147046645987101dfc3b4cfec1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\ \\cfrac{x^2+2}{x^2-1} = \\cfrac{+\\infty}{+\\infty } =\\cfrac{1}{1} = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"236\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Il limite infinito della funzione ci ha dato 1, quindi la funzione ha un asintoto orizzontale in y=1.<\/p>\n<p class=\"has-text-align-left\"> Poich\u00e9 la funzione ha un asintoto orizzontale, non avr\u00e0 un asintoto obliquo.<\/p>\n<p class=\"has-text-align-left\"> Differenziamo la funzione quindi studiamo gli intervalli di crescita e diminuzione: <\/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-5641380d832933540291b6dbcb8e151a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\cfrac{x^2+2}{x^2-1}  \\ \\longrightarrow \\ f'(x)= \\cfrac{2x \\cdot (x^2-1) -(x^2+2) \\cdot 2x}{\\left(x^2-1 \\right)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"433\" style=\"vertical-align: -20px;\"><\/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-5c2348d02afd4187badfc0b04346dc34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)= \\cfrac{2x^3-2x - (2x^3+4x) }{\\left(x^2-1 \\right)^2} = \\cfrac{-6x}{\\left(x^2-1 \\right)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"330\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ora impostiamo la derivata uguale a 0 e risolviamo l&#8217;equazione: <\/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-4890b9dfeb634c4d7a349351be73b5d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" 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-b60ec42d7180d4f5c76674c6a8915e4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{-6x}{\\left(x^2-1 \\right)^2}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"102\" style=\"vertical-align: -20px;\"><\/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-e8c7d7a359a454599e1662d8f4c64859_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-6x=0\\cdot\\left(x^2-1 \\right)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"149\" style=\"vertical-align: -7px;\"><\/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-430a0f35bef91d5e64798d69fc210c41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-6x= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"64\" 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-fd71758c91c04f304eae7e7fd164eb1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\cfrac{0}{-6} = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"91\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Rappresentiamo sulla retta tutti i punti critici calcolati, ovvero i punti che non appartengono al dominio (x=-1 e x=+1) e quelli che annullano la derivata (x=0): <\/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\/number-line-1-0-1.webp\" alt=\"\" class=\"wp-image-2643\" width=\"397\" height=\"77\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> E valutiamo il segno della derivata in ciascun intervallo, per sapere se la funzione aumenta o diminuisce. Prendiamo quindi un punto in ogni intervallo (mai i punti singolari) e guardiamo che segno ha la derivata in questo punto: <\/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-8110ed8ec9e600c9ec17fa5ffa3c088f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(-2)= \\cfrac{-6(-2)}{\\left((-2)^2-1 \\right)^2} = \\cfrac{12}{9} =1,33 \\ \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"48\" width=\"327\" style=\"vertical-align: -20px;\"><\/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-4b982f336007ee0be2dcee5d52be0105_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(-0,5)= \\cfrac{-6(-0,5)}{\\left((-0,5)^2-1 \\right)^2} = \\cfrac{3}{0,56} =5,33 \\ \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"48\" width=\"377\" style=\"vertical-align: -20px;\"><\/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-82b12d15e99f6b63fe4d10bfb77b32e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(0,5)= \\cfrac{-6\\cdot 0,5}{\\left(0,5^2-1 \\right)^2} = \\cfrac{-3}{0,56} =-5,33 \\ \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"350\" style=\"vertical-align: -20px;\"><\/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-7166a97e07562fc86f89483322af2efd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(2)= \\cfrac{-6\\cdot 2}{\\left(2^2-1 \\right)^2} = \\cfrac{-12}{9} =-1,33 \\ \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"321\" style=\"vertical-align: -20px;\"><\/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\/number-line-1-0-1-croissant-decroissant.webp\" alt=\"\" class=\"wp-image-2645\" width=\"400\" height=\"141\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> La funzione aumenta dove la derivata \u00e8 positiva e diminuisce dove la funzione \u00e8 negativa:<\/p>\n<p class=\"has-text-align-center\"> <strong>Crescita:<\/strong><\/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-14d2fb1f352a42d7f0e8b6c91776fe24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-\\infty,-1)\\cup (-1,0)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"147\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>Diminuire:<\/strong><\/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-9df00656c8f1b8a5b5660489739aecc9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,1)\\cup (1,+\\infty)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"120\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> La funzione passa da crescente a decrescente in x=0, quindi x=0 \u00e8 un massimo locale della funzione.<\/p>\n<p class=\"has-text-align-left\"> Sostituiamo l&#8217;estremo trovato nella funzione originale per trovare la coordinata Y del punto:<\/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-7a0031b1f78a601b773cab39f1f54d4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(0)=\\cfrac{0^2+2}{0^2-1}= \\cfrac{2}{-1} = -2 \\ \\longrightarrow \\ (0,-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"303\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Gli estremi relativi della funzione sono quindi:<\/p>\n<p class=\"has-text-align-center\"> <strong>Massimo punto<\/strong><\/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-b1761f5972412ad38102968850bf6220_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,-2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"52\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Per studiare la curvatura della funzione, calcoliamo la sua derivata seconda:<\/p>\n<p class=\"ql-center-displayed-equation\" style=\"line-height: 280px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-802852beb818dd5a0dce2f30374f3a88_l3.png\" height=\"280\" width=\"687\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"f'(x)=\\cfrac{-6x}{\\left(x^2-1 \\right)^2}  \\ \\longrightarrow <span class=&quot;ql-right-eqno&quot;>   <\/span><span class=&quot;ql-left-eqno&quot;>   <\/span><img src=&quot;https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-273969cf60ee8cf3413ee2f8b1db7688_l3.png&quot; height=&quot;129&quot; width=&quot;476&quot; class=&quot;ql-img-displayed-equation quicklatex-auto-format&quot; alt=&quot;\\[f''(x)= \\cfrac{-6 \\cdot \\left(x^2-1 \\right)^2 - (-6x) \\cdot 2(x^2-1) \\cdot 2x}{ \\left(\\left(x^2-1 \\right)^2\\right)^2}$$ f''(x)= \\cfrac{-6 \\left(x^2-1 \\right)^2 -(-6x)\\cdot 4x(x^2-1)}{\\left(x^2 -1\\right)^4} =\\]&quot; title=&quot;Rendered by QuickLaTeX.com&quot;\/> \\cfrac{-6 \\left(x^2-1 \\right)^2 + 24x^2(x^2-1)}{\\left(x^2 -1\\right)^4}&#8221; title=&#8221;Rendered by QuickLaTeX.com&#8221;><\/p>\n<\/p>\n<p class=\"has-text-align-left\">Tutti i termini hanno<\/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-2050d569319abb9789111cc5f49b21cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x^2-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"60\" style=\"vertical-align: -5px;\"><\/p>\n<p> , possiamo quindi semplificare la frazione: <\/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-675ece9a985c5d3a11397a0fc84d7b5c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= \\cfrac{-6 \\left(x^2-1 \\right)^{\\cancel{2}} + 24x^2\\cancel{(x^2-1)}}{\\left(x^2-1 \\right)^\\cancelto{3}{4}}  =\\cfrac{-6 \\left(x^2-1 \\right) + 24x^2}{\\left(x^2 -1\\right)^3}\" title=\"Rendered by QuickLaTeX.com\" height=\"52\" width=\"475\" style=\"vertical-align: -20px;\"><\/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-3dba9e99bfb4fcfd547c2d2edaafb4b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= \\cfrac{-6x^2+6  + 24x^2}{\\left(x^2 -1\\right)^3} =\\cfrac{18x^2+6}{\\left(x^2 -1\\right)^3}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"300\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ora impostiamo la derivata seconda uguale a 0 e risolviamo l&#8217;equazione: <\/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-f618f4961c18c45be60fc496ad4896e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"75\" 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-f373044cf16a3d9191973dbd6a9a21f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{18x^2+6}{\\left(x^2 -1\\right)^3}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"102\" style=\"vertical-align: -20px;\"><\/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-c431872e1278c9771b294701b56bf632_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"18x^2+6=0\\cdot \\left(x^2 -1\\right)^3\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"182\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\">\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f760897a45aa19dc1c2d5cbd6864de12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"18x^2+6= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"98\" style=\"vertical-align: -2px;\"><\/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-9ffbd0e0d0183993e003026edffe4f0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"18x^2=-6\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"81\" 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-2753f9ca0b7d0a2054856746d7ba3c26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2=\\cfrac{-6}{18}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"74\" 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-b4cd1f1648c3b49fe6be0a39600890b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2=-0,33\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"90\" style=\"vertical-align: -4px;\"><\/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-c3ab76c31e735be0675b2a6c095c213e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\sqrt{-0,33} \\quad \\color{red}\\bm{\\times}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"153\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Non esiste la radice quadrata di un numero negativo. Quindi non ha senso che corrisponda<\/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-981d85a257dd56afdb3fc7eb53d5eadf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"75\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Rappresentiamo ora sulla retta tutti i punti singolari trovati, cio\u00e8 i punti che non appartengono al dominio (x=-1 e x=+1) e quelli che annullano la derivata seconda (in questo caso non ci sono Qualunque): <\/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\/nombre-ligne-1-1.webp\" alt=\"\" class=\"wp-image-2596\" width=\"294\" height=\"75\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> E valutiamo il segno della derivata seconda in ogni intervallo, per sapere se la funzione \u00e8 concava o convessa. Prendiamo quindi un punto in ogni intervallo (mai i punti singolari) e guardiamo che segno ha la derivata seconda in questo punto: <\/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-b84333a96a947c0a9c2ef605e8426f77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(-2)=\\cfrac{18(-2)^2+6}{\\left((-2)^2 -1\\right)^3}= \\cfrac{78}{27}  = 2,89 \\ \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"331\" style=\"vertical-align: -20px;\"><\/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-2ec750b2e7e3b0b47e650695ad1f4259_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(0)=\\cfrac{18\\cdot 0^2+6}{\\left(0^2 -1\\right)^3}= \\cfrac{6}{-1}  = -6 \\ \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"292\" style=\"vertical-align: -20px;\"><\/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-7f079ac6677ac2cfc283b9b4ff817a13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(2)=\\cfrac{18\\cdot 2^2+6}{\\left(2^2 -1\\right)^3}= \\cfrac{78}{27}  = 2,89 \\ \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"299\" style=\"vertical-align: -20px;\"><\/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\/ligne-numerique-1-1-positif-negatif-positif.webp\" alt=\"\" class=\"wp-image-2597\" width=\"297\" height=\"132\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Se la derivata seconda \u00e8 positiva significa che la funzione \u00e8 convessa.<\/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-49efa6d9ab88562f20df743cb7d267f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cup})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> , e se la derivata seconda \u00e8 negativa significa che la funzione \u00e8 concava<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-59e636042d77445b1534260d9d7309a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cap})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> . Gli intervalli di concavit\u00e0 e convessit\u00e0 sono quindi:<\/p>\n<p class=\"has-text-align-center\"> <strong>Convesso<\/strong><\/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-49efa6d9ab88562f20df743cb7d267f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cup})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> <strong>:<\/strong><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dc2d3b8519f698562d39a2807dc7a906_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-\\infty, -1) \\cup (1, +\\infty)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"156\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>Concavo<\/strong><\/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-59e636042d77445b1534260d9d7309a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cap})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> <strong>:<\/strong><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3c8937d362a3ba07fa9068381afe74a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-1,1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"51\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Tuttavia, sebbene vi sia un cambiamento nella curvatura in x=-1 e in x=1, questi non sono punti di flesso. Perch\u00e9 non appartengono al dominio della funzione.<\/p>\n<p class=\"has-text-align-left\"> E infine, rappresentiamo graficamente la funzione utilizzando tutti i calcoli eseguiti: <\/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\/fonctions-graphiques-exercice-resolu.webp\" alt=\"esercizio sulle funzioni grafiche risolto\" class=\"wp-image-2646\" width=\"436\" height=\"439\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Esercizio 3<\/h3>\n<p> Traccia su un grafico la seguente funzione razionale: <\/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-3ac9ccc5e8540cca38f599ed36507792_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)=\\frac{x^3}{x^2-4}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"109\" style=\"vertical-align: -12px;\"><\/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>Vedi la soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Questa \u00e8 una funzione razionale, quindi dobbiamo impostare il denominatore uguale a 0 per vedere quali numeri non appartengono al dominio della funzione: <\/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-a34dfe78d673534873a2013c16e1b353_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2-4= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"81\" 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-c269e23a1070b3e5556abece040af75a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2=4\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"50\" 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-c4c32359b264b28ac80f2606c09d5a2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{x^2}=\\sqrt{4}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"79\" style=\"vertical-align: -2px;\"><\/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-c06a55e3acdd1e283973786926b27716_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\pm 2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" 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-e666f828709575f965b5120fbdda085e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{Dom } f= \\mathbb{R}-\\{-2, +2 \\}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"182\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Per trovare il punto di intersezione con l&#8217;asse X, risolviamo<\/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-bbb52c33bfaff434771f0e4ddd4cf677_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)= 0.\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"71\" style=\"vertical-align: -5px;\"><\/p>\n<p> Poich\u00e9 la funzione ha sempre un valore pari a 0 sull&#8217;asse X: <\/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-0bce6c022ed0fc63f4659af75888f96c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"67\" 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-4fb6c030af3bbc674466d58da3704303_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x^3}{x^2-4}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"81\" 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-d913fc1b7622c590b24cb0bcff8f07a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^3=0\\cdot (x^2-4)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"125\" 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-085b61b70fdf3a335b744ed8bc4f06a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^3=0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"50\" 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-432867a310687d633fea4e3e72197b03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\sqrt[3]{0}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"91\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Il punto di intersezione con l&#8217;asse X \u00e8 quindi:<\/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-2c790019bd70403eba876c59c82c0f9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,0)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E per trovare il punto di intersezione con l&#8217;asse Y, calcoliamo<\/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-d449eebd1f011aebdf90931f3a66a3b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(0).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<p> Poich\u00e9 x \u00e8 sempre 0 sull&#8217;asse Y:<\/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-21000bf141667a94b3deec79162d963f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(0)=\\cfrac{0^3}{0^2-4} = \\cfrac{0}{-4} = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"187\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Il punto di intersezione con l&#8217;asse Y \u00e8 quindi:<\/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-2c790019bd70403eba876c59c82c0f9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,0)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> In questo caso il punto di intersezione con l&#8217;asse X coincide con il punto di intersezione con l&#8217;asse Y, poich\u00e9 la funzione passa per l&#8217;origine delle coordinate.<\/p>\n<p class=\"has-text-align-left\"> Per vedere se la funzione ha asintoti verticali, dobbiamo calcolare il limite della funzione nei punti che non appartengono al dominio (in questo caso x=-2 e x=+2). E se il risultato \u00e8 infinito, \u00e8 un asintoto verticale. Ancora:<\/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-08adb9903ed9589da031a215aca7cf82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to -2} \\cfrac{x^3}{x^2-4} = \\cfrac{(-2)^3}{(-2)^2-4} =\\cfrac{-8}{4-4}= \\cfrac{-8}{0} = \\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"354\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Poich\u00e9 il limite della funzione quando x si avvicina a -2 d\u00e0 infinito, x=-2 \u00e8 un asintoto verticale.<\/p>\n<p class=\"has-text-align-left\"> Calcoliamo i limiti laterali dell&#8217;asintoto x=-2 sostituendo nella funzione un numero molto vicino ad esso: <\/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-91d1a99bc96ac49d3b83d2ba9459558a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(-2,1)=\\cfrac{(-2,1)^3}{(-2,1)^2-4} =-22,59 \\longrightarrow \\lim_{x \\to -2^{-}}  \\cfrac{x^3}{x^2-4} = -\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"457\" style=\"vertical-align: -17px;\"><\/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-a430ef18ddb27cf657aa3e810c0a5e24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(-1,9)=\\cfrac{(-1,9)^3}{(-1,9)^2-4} =+17,59 \\longrightarrow \\lim_{x \\to -2^{+}}  \\cfrac{x^3}{x^2-4} = +\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"457\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ora vediamo se x=+2 \u00e8 un asintoto verticale:<\/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-38e9b49183f8680c3ee84f96454334d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +2} \\cfrac{x^3}{x^2-4} = \\cfrac{(2)^3}{(2)^2-4} =\\cfrac{8}{4-4}= \\cfrac{8}{0} = \\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"319\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Poich\u00e9 il limite della funzione quando x si avvicina a +2 d\u00e0 infinito, x=+2 \u00e8 un asintoto verticale.<\/p>\n<p class=\"has-text-align-left\"> Calcoliamo i limiti laterali dell&#8217;asintoto x=2 sostituendo nella funzione un numero molto vicino ad esso: <\/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-28d87cfb93cd7a3673f853003fcd158d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(1,9)=\\cfrac{1,9^3}{1,9^2-4} =-17,59 \\longrightarrow \\lim_{x \\to 2^{-}}  \\cfrac{x^3}{x^2-4} = -\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"405\" style=\"vertical-align: -16px;\"><\/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-be984dcc8e4149888e44944c44675d87_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(2,1)=\\cfrac{2,1^3}{2,1^2-4} =22,59 \\longrightarrow \\lim_{x \\to 2^{+}}  \\cfrac{x^3}{x^2-4} = +\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"391\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> D&#8217;altra parte, l&#8217;asintoto orizzontale della funzione sar\u00e0 il risultato del limite infinito della funzione. Ancora:<\/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-aa6c8c730e78a8f4ea067adbd5f487be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\ \\cfrac{x^3}{x^2-4} = \\cfrac{+\\infty}{+\\infty } =+\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"225\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Il limite infinito della funzione ci ha dato +\u221e, quindi la funzione non ha asintoto orizzontale.<\/p>\n<p class=\"has-text-align-left\"> Calcoliamo ora l&#8217;asintoto obliquo. Gli asintoti obliqui sono della forma<\/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-ad313410fc976bc53709807aa8aed8e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=mx+n.\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"95\" style=\"vertical-align: -4px;\"><\/p>\n<p> E<\/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> Si calcola con la seguente formula: <\/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-8bac34a30967593c9e1d8d1dc6ff816c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m = \\lim_{x \\to +\\infty} f(x):x = \\lim_{x \\to +\\infty} \\cfrac{x^3}{x^2-4}: x =\\lim_{x \\to +\\infty} \\cfrac{x^3}{x^2-4}: \\frac{x}{1}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"446\" style=\"vertical-align: -13px;\"><\/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-8f5c82ccc7edabf4fd047ea738ac8124_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m =\\lim_{x \\to +\\infty}\\cfrac{x^3\\cdot 1}{(x^2-4)\\cdot x} =\\lim_{x \\to +\\infty}\\cfrac{x^3}{x^3-4x}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"309\" style=\"vertical-align: -17px;\"><\/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-d10814f4c377eb6df5e7ef2eb1ba1e1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m =\\lim_{x \\to +\\infty}\\cfrac{x^3}{x^3-4x} = \\frac{+\\infty}{+\\infty} = \\frac{1}{1} = \\bm{1}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"276\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Una volta conosciuta la pendenza dell&#8217;asintoto obliquo, determiniamo l&#8217;intercetta utilizzando la seguente formula: <\/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-de4326a40acf34b64a28c9da8250bf00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to +\\infty} \\left[f(x)-mx\\right] = \\lim_{x \\to +\\infty} \\left[ \\cfrac{x^3}{x^2-4}-1x\\right]\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"355\" style=\"vertical-align: -23px;\"><\/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-13478ac6f6fac958ec8b2a714c28bc3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to +\\infty} \\left[ \\cfrac{x^3}{x^2-4}-x\\right] = \\cfrac{+\\infty}{+\\infty} - (+\\infty) = \\bm{+\\infty - \\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"412\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ma otteniamo l&#8217;indeterminazione \u221e \u2013 \u221e. \u00c8 quindi necessario ridurre i termini ad un denominatore comune. Per fare ci\u00f2, moltiplichiamo e dividiamo la x per il denominatore della frazione: <\/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-5e939b43a3405ba644d4b60bb4bacadb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to +\\infty} \\left[ \\cfrac{x^3}{x^2-4}-\\cfrac{x \\cdot (x^2-4)}{(x^2-4)}\\right] =\\lim_{x \\to +\\infty} \\left[ \\cfrac{x^3}{x^2-4}-\\cfrac{x^3-4x}{x^2-4}\\right]\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"484\" style=\"vertical-align: -23px;\"><\/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-1ce9e619c697b7b25ec55b52ec531fd1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n =  \\lim_{x \\to +\\infty} \\cfrac{x^3- (x^3-4x)}{x^2-4} = \\lim_{x \\to +\\infty} \\cfrac{4x}{x^2-4}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"320\" style=\"vertical-align: -13px;\"><\/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-6395fc2ac5efce4007680594b4e78fa9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n =\\lim_{x \\to +\\infty} \\cfrac{4x}{x^2-4} = \\cfrac{+\\infty}{\\infty} =\\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"231\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> In breve, l\u2019asintoto obliquo \u00e8: <\/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-6fbe1cc5f3362ddbd80ed0b29c0bb4ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y = mx+n\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"91\" style=\"vertical-align: -4px;\"><\/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-b76adff686f2ca940f3054478fa10fc8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y = 1x + 0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -4px;\"><\/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-fca2e14d8d10e98015169017e681c9e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y = x }\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Per studiare la monotonicit\u00e0 della funzione, dobbiamo prima calcolare la sua derivata: <\/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-1397d0e8e73bd7b1d851411dee28daed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\cfrac{x^3}{x^2-4}  \\ \\longrightarrow \\ f'(x)= \\cfrac{3x^2 \\cdot (x^2-4) - x^3 \\cdot 2x }{\\left(x^2-4\\right)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"396\" style=\"vertical-align: -20px;\"><\/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-495f08a881718b2734ef1db17b5f39ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)= \\cfrac{3x^4-12x^2-2x^4}{\\left(x^2-4\\right)^2} = \\cfrac{x^4-12x^2}{\\left(x^2-4\\right)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"298\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ora impostiamo la derivata uguale a 0 e risolviamo l&#8217;equazione: <\/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-4890b9dfeb634c4d7a349351be73b5d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" 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-1d8ad11a06814972752eadc64b861f62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x^4-12x^2}{\\left(x^2-4\\right)^2}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"108\" style=\"vertical-align: -20px;\"><\/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-bd6cbd9bc26086bfc990fdf2152e75f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^4-12x^2=0\\cdot \\left(x^2-4\\right)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"192\" style=\"vertical-align: -7px;\"><\/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-05799ab35c660f110890406329cc4b25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^4-12x^2=0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"108\" 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-be662a07bcc839ad08a7e30c0538fc1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2(x^2-12)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"121\" 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-dc1f64cdcd293da4fee1ef02fff9a588_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle x^2\\cdot(x^2-12) =0 \\longrightarrow \\begin{cases} x^2 =0 \\ \\longrightarrow \\ \\bm{x=0} \\\\[2ex] x^2-12=0 \\ \\longrightarrow \\ x=\\sqrt{12} \\ \\longrightarrow \\ \\bm{x= \\pm 3,46} \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"527\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Rappresentiamo ora sulla retta tutti i punti singolari trovati, cio\u00e8 i punti che non appartengono al dominio (x=-2 e x=+2) e quelli che annullano la derivata (x=0, x=- 3,46 e x= +3,46): <\/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\/number-line-346-2-0.webp\" alt=\"\" class=\"wp-image-2653\" width=\"532\" height=\"70\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> E valutiamo il segno della derivata in ciascun intervallo, per sapere se la funzione aumenta o diminuisce. Prendiamo quindi un punto in ogni intervallo (mai i punti singolari) e guardiamo che segno ha la derivata in questo punto: <\/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-2dc401187921384b6e083fd0d662e404_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(-4)= \\cfrac{(-4)^4-12(-4)^2}{\\left((-4)^2-4\\right)^2} = \\cfrac{64}{144} = 0,44 \\ \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"367\" style=\"vertical-align: -20px;\"><\/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-c2195913b32457357954d053d8f37b91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(-3)= \\cfrac{(-3)^4-12(-3)^2}{\\left((-3)^2-4\\right)^2} = \\cfrac{-27}{25} = -1,08 \\ \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"394\" style=\"vertical-align: -20px;\"><\/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-840f32597f014205b3223516b1557773_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(-1)= \\cfrac{(-1)^4-12(-1)^2}{\\left((-1)^2-4\\right)^2} = \\cfrac{-11}{9} = -1,22 \\ \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"394\" style=\"vertical-align: -20px;\"><\/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-88eb3af4c1a297fdbdeae6470edf7165_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1)= \\cfrac{1^4-12\\cdot1^2}{\\left(1^2-4\\right)^2} = \\cfrac{-11}{9} = -1,22 \\ \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"338\" style=\"vertical-align: -20px;\"><\/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-2fce37cc9db62e965c31ae2ff6d6ad09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(3)= \\cfrac{3^4-12\\cdot 3^2}{\\left(3^2-4\\right)^2} = \\cfrac{-27}{25} = -1,08 \\ \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"338\" style=\"vertical-align: -20px;\"><\/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-688354135682d1cadec94bbdcac7cc46_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(4)= \\cfrac{4^4-12\\cdot 4^2}{\\left(4^2-4\\right)^2} = \\cfrac{64}{144} = 0,44 \\ \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"311\" style=\"vertical-align: -20px;\"><\/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\/ligne-numerique-346-monotonie.webp\" alt=\"\" class=\"wp-image-2654\" width=\"534\" height=\"127\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Se la derivata \u00e8 positiva significa che la funzione \u00e8 crescente, se la derivata \u00e8 negativa significa che la funzione \u00e8 decrescente. Pertanto gli intervalli di crescita e declino sono:<\/p>\n<p class=\"has-text-align-center\"> <strong>Crescita:<\/strong><\/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-fcc905c2730bc3771063bf7280f05002_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-\\infty,-3,46)\\cup (3,46,+\\infty)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"207\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>Diminuire:<\/strong><\/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-6812d1ef2a6df5de54448b0f42751758_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-3,46,-2)\\cup(-2,0)\\cup (0,2) \\cup (2,3,46)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"308\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> La funzione passa da crescente a decrescente in x=-3,46, quindi x=-3,46 \u00e8 il massimo della funzione. E la funzione passa da decrescente ad crescente in x=3,46, quindi x=3,46 \u00e8 il minimo della funzione.<\/p>\n<p class=\"has-text-align-left\"> Determiniamo le coordinate Y dei relativi estremi: <\/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-673718f25b824c11e7777325974ffeb7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(-3,46)=\\cfrac{(-3,46)^3}{(-3,46)^2-4} = \\cfrac{-41,42}{7,97}=-5,20 \\ \\longrightarrow \\ (-3,46,-5,20)\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"529\" style=\"vertical-align: -17px;\"><\/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-bfebef64900ebf04ef1a0fa2f969e1f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(3,46)=\\cfrac{3,46^3}{3,46^2-4} = \\cfrac{41,42}{7,97}=5,20 \\ \\longrightarrow \\ (3,46,5,20)\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"424\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Gli estremi relativi della funzione sono quindi:<\/p>\n<p class=\"has-text-align-center\"> <strong>Massimo punto<\/strong><\/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-6670c6858270b2890ec3a8b85d68cc23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-3,46,-5,20)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"117\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>Minimo da puntare<\/strong><\/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-58e02eb58912f164efba8d6b648e45bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(3,46,5,20)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"89\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Per studiare la curvatura della funzione, calcoliamo la derivata seconda della funzione: <\/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-50df15bb48cacf8f031b640994661e47_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= \\cfrac{\\left(4x^3-24x\\right)\\cdot \\left(x^2-4\\right)^2 - \\left(x^4-12x^2\\right)\\cdot 2\\left(x^2-4\\right)\\cdot 2x }{ \\left(\\left(x^2-4\\right)^2 \\right)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"483\" style=\"vertical-align: -33px;\"><\/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-3b05b5f09c2adbfead593df2cdf2ad29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= \\cfrac{\\left(4x^3-24x\\right)\\cdot \\left(x^2-4\\right)^2 - \\left(x^4-12x^2\\right)\\cdot 4x\\left(x^2-4\\right) }{\\left(x^2-4\\right)^4 }\" title=\"Rendered by QuickLaTeX.com\" height=\"52\" width=\"461\" style=\"vertical-align: -20px;\"><\/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-92e1aa280d06bf8b58045845d5e21f37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= \\cfrac{\\left(4x^3-24x\\right)\\cdot \\left(x^2-4\\right)^{\\cancel{2}} - \\left(x^4-12x^2\\right)\\cdot 4x\\cancel{\\left(x^2-4\\right)} }{\\left(x^2-4\\right)^{\\cancelto{3}{4}} }\" title=\"Rendered by QuickLaTeX.com\" height=\"52\" width=\"458\" style=\"vertical-align: -20px;\"><\/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-a912eff3359969b6ffbef96a3f16932d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= \\cfrac{\\left(4x^3-24x\\right)\\cdot \\left(x^2-4\\right) - \\left(x^4-12x^2\\right)\\cdot 4x}{\\left(x^2-4\\right)^3 }\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"386\" style=\"vertical-align: -20px;\"><\/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-8bc35bdd2b70bbac52fa0f24bbefa261_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= \\cfrac{4x^5-16x^3-24x^3+96x - \\left(4x^5-48x^3\\right) }{\\left(x^2-4\\right)^3 }\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"381\" style=\"vertical-align: -20px;\"><\/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-045971f71cc11ced77ea0df9f2c514fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= \\cfrac{4x^5-16x^3-24x^3+96x - 4x^5+48x^3 }{\\left(x^2-4\\right)^3 }\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"365\" style=\"vertical-align: -20px;\"><\/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-3144a0aa00ee8ec427752f05f0fac40c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= \\cfrac{8x^3+96x  }{\\left(x^2-4\\right)^3 }\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"145\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ora impostiamo la derivata seconda uguale a 0 e risolviamo l&#8217;equazione: <\/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-f618f4961c18c45be60fc496ad4896e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"75\" 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-e8ed519add27a4d51c75b49179e632ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{8x^3+96x  }{\\left(x^2-4\\right)^3 }=0\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"109\" style=\"vertical-align: -20px;\"><\/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-b5c0d1e44accc3b68a67598f5c4d834c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"8x^3+96x =0\\cdot \\left(x^2-4\\right)^3\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"193\" style=\"vertical-align: -7px;\"><\/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-4c52a7448e4acc67488ef5747cc3bed9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"8x^3+96x =0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"109\" style=\"vertical-align: -2px;\"><\/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-4f2884a1c9b8dabf6ea5323f2ac71b2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x(8x^2+96)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"123\" 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-31adba554b44aa92fd7227506440ccaf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle x\\cdot(8x^2+96) =0 \\longrightarrow \\begin{cases} \\bm{x =0} \\\\[2ex] 8x^2+96=0 \\ \\longrightarrow \\ x^2=\\cfrac{-96}{8}} = -12 \\ \\longrightarrow \\ x= \\sqrt{-12} \\ \\color{red}\\bm{\\times} \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"635\" 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-635e2ffa452a5a66a4bcacb0e111c5ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= \\sqrt{-12}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"81\" style=\"vertical-align: -3px;\"><\/p>\n<p> Non esiste una soluzione poich\u00e9 non esiste una radice negativa di un numero reale.<\/p>\n<p class=\"has-text-align-left\"> Rappresentiamo ora sulla retta tutti i punti singolari trovati, cio\u00e8 i punti che non appartengono al dominio (x=-2 e x=+2) e quelli che annullano la derivata seconda (x=0): <\/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\/droite-numerique-2-0-2.webp\" alt=\"\" class=\"wp-image-2399\" width=\"370\" height=\"72\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> E valutiamo il segno della derivata seconda in ogni intervallo, per sapere se la funzione \u00e8 concava o convessa. Prendiamo quindi un punto in ogni intervallo (mai i punti singolari) e guardiamo che segno ha la derivata seconda in questo punto: <\/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-3b5618d1ab96a078d50507f45155595b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(-3)=\\cfrac{8(-3)^3+96(-3)  }{\\left((-3)^2-4\\right)^3 } = \\cfrac{-504}{125}=-4,03 \\ \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"408\" style=\"vertical-align: -20px;\"><\/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-e2e868f1f815d4155a187c55b004cc13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(-1)=\\cfrac{8(-1)^3+96(-1)  }{\\left((-1)^2-4\\right)^3 } = \\cfrac{-104}{-27}=3,85 \\ \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"394\" style=\"vertical-align: -20px;\"><\/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-acc8341ccd6a1c8a3cd3d6a0ce888dba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(1)=\\cfrac{8\\cdot 1^3+96\\cdot 1  }{\\left(1^2-4\\right)^3 } = \\cfrac{104}{-27}=-3,85 \\ \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"348\" style=\"vertical-align: -20px;\"><\/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-54d98824f72954de12bc065471a610e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(3)=\\cfrac{8\\cdot 3^3+96\\cdot 3  }{\\left(3^2-4\\right)^3 } = \\cfrac{504}{125}=4,03 \\ \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"329\" style=\"vertical-align: -20px;\"><\/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\/ligne-numerique-2-0-2-courbure.webp\" alt=\"\" class=\"wp-image-2568\" width=\"371\" height=\"124\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Se la derivata seconda \u00e8 positiva significa che la funzione \u00e8 convessa.<\/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-49efa6d9ab88562f20df743cb7d267f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cup})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> , e se la derivata seconda \u00e8 negativa significa che la funzione \u00e8 concava<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-59e636042d77445b1534260d9d7309a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cap})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> . Gli intervalli di concavit\u00e0 e convessit\u00e0 sono quindi:<\/p>\n<p class=\"has-text-align-center\"> <strong>Convesso<\/strong><\/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-49efa6d9ab88562f20df743cb7d267f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cup})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> <strong>:<\/strong><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0cd739761b2dc845594c0a0696a240c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-2,0)\\cup (2,+\\infty)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"134\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>Concavo<\/strong><\/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-59e636042d77445b1534260d9d7309a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cap})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> <strong>:<\/strong><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5e741ac026627200772655094f921f26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-\\infty,-2)\\cup (0,2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"133\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Tuttavia, sebbene vi sia un cambiamento nella curvatura in x=-2 ex=+2, questi non sono punti di flesso. Perch\u00e9 x=-2 e x=+2 non appartengono al dominio della funzione. Invece in x=0 si verifica un cambiamento di curvatura (la funzione passa da convessa a concava) e questo appartiene alla funzione, quindi x=0 \u00e8 un punto di flesso.<\/p>\n<p class=\"has-text-align-left\"> Sostituiamo i punti di flesso trovati nella funzione originale per trovare l&#8217;altra coordinata del punto di flesso:<\/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-8a1a8a10485a749c631531b80ce05642_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(0) = \\cfrac{0^3}{0^2-4}  = \\cfrac{0}{-4} =0\\ \\longrightarrow \\ (0,0)\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"276\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> I punti di svolta della funzione sono quindi:<\/p>\n<p class=\"has-text-align-center\"> <strong>Punti di svolta:<\/strong><\/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-2c790019bd70403eba876c59c82c0f9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,0)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Infine, sulla base di tutte le informazioni che abbiamo calcolato, rappresentiamo la funzione: <\/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\/representation-graphique-des-fonctions-exercices-resolus.webp\" alt=\"rappresentazione grafica delle funzioni risolte, esercizi\" class=\"wp-image-2655\" width=\"449\" height=\"518\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Commento: si noti che la funzione attraversa l&#8217;asintoto obliquo nel punto<\/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-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> Gli asintoti obliqui infatti determinano soprattutto il comportamento della funzione quando x tende verso +\u221e e -\u221e, infatti la funzione non attraversa mai l&#8217;asintoto obliquo a destra del grafico (x\u2192+\u221e) e a sinistra di il grafico (x\u2192-\u221e). Tuttavia \u00e8 molto raro che la funzione attraversi al centro l&#8217;asintoto obliquo, \u00e8 un caso molto particolare.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>In questo articolo vedremo come rappresentare qualsiasi tipo di funzione su un grafico. Inoltre, troverai esercizi passo passo risolti sulla rappresentazione delle funzioni su un grafico. Come rappresentare una funzione su un grafico Per rappresentare una funzione su un grafico \u00e8 necessario eseguire i seguenti passaggi: Trova il dominio della funzione. Calcolare i punti limite &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/it\/rappresentazione-delle-funzioni\/\"> <span class=\"screen-reader-text\">Rappresentazione delle funzioni<\/span> Leggi altro &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":[17],"tags":[],"class_list":["post-49","post","type-post","status-publish","format-standard","hentry","category-rappresentazione-delle-funzioni"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Rappresentazione delle funzioni -<\/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\/it\/rappresentazione-delle-funzioni\/\" \/>\n<meta property=\"og:locale\" content=\"it_IT\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Rappresentazione delle funzioni -\" \/>\n<meta property=\"og:description\" content=\"In questo articolo vedremo come rappresentare qualsiasi tipo di funzione su un grafico. 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