{"id":26,"date":"2023-09-17T11:04:43","date_gmt":"2023-09-17T11:04:43","guid":{"rendered":"https:\/\/mathority.org\/pt\/taxa-media-e-instantanea-de-mudanca\/"},"modified":"2023-09-17T11:04:43","modified_gmt":"2023-09-17T11:04:43","slug":"taxa-media-e-instantanea-de-mudanca","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/taxa-media-e-instantanea-de-mudanca\/","title":{"rendered":"Taxa m\u00e9dia e instant\u00e2nea de mudan\u00e7a"},"content":{"rendered":"<p>Aqui explicamos o que s\u00e3o taxa de mudan\u00e7a, taxa m\u00e9dia de mudan\u00e7a e taxa de mudan\u00e7a instant\u00e2nea. Voc\u00ea poder\u00e1 ver v\u00e1rios exemplos de como calcular a taxa de varia\u00e7\u00e3o e, al\u00e9m disso, poder\u00e1 praticar com exerc\u00edcios passo a passo resolvidos sobre a taxa de varia\u00e7\u00e3o. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-la-tasa-de-variacion\"><\/span> Qual \u00e9 a taxa de mudan\u00e7a?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>Em matem\u00e1tica, a taxa de varia\u00e7\u00e3o (TV) de uma fun\u00e7\u00e3o \u00e9 a diferen\u00e7a nos valores de uma fun\u00e7\u00e3o em dois pontos diferentes.<\/strong> Portanto, para calcular a taxa de varia\u00e7\u00e3o entre dois pontos, os valores da fun\u00e7\u00e3o nesses dois pontos devem ser subtra\u00eddos.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c33fec724dd0a5bf45bfec5a911f5bcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{TV}[a,b]=f(b)-f(a)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"171\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Por exemplo, se duas imagens de uma fun\u00e7\u00e3o s\u00e3o f(2)=1 e f(5)=7, sua taxa de varia\u00e7\u00e3o \u00e9:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8696c8ddc5db28bea4a9b14bc6dc0b9a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{TV}[2,5]=f(5)-f(2)=7-1=6\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"269\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Acabamos de ver o significado matem\u00e1tico de taxa de varia\u00e7\u00e3o, mas em economia o conceito de taxa de varia\u00e7\u00e3o significa o seguinte:<\/p>\n<p> Em economia, a taxa de varia\u00e7\u00e3o entre dois valores \u00e9 a diferen\u00e7a entre eles expressa em percentagem, ou seja, a taxa de varia\u00e7\u00e3o de uma vari\u00e1vel entre diferentes per\u00edodos \u00e9 a sua varia\u00e7\u00e3o relativa. Portanto, para calcular a taxa de varia\u00e7\u00e3o, subtraem-se os valores dos dois per\u00edodos distintos e o resultado obtido \u00e9 dividido pelo valor do per\u00edodo inicial.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3a6392a6712643e5eeddbf7226523e42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{TV}[t,t+n]=\\cfrac{Y_{t+n}-Y_t}{Y_t}\\cdot 100\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"227\" style=\"vertical-align: -15px;\"><\/p>\n<\/p>\n<p> Por exemplo, se o valor de determinadas a\u00e7\u00f5es aumentou de 35\u20ac para 50\u20ac num m\u00eas, a sua taxa de varia\u00e7\u00e3o ser\u00e1:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-17a18b6b77c1e1b21cad5d1ffab56aca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{TV}[t,t+1]=\\cfrac{50-35}{35}\\cdot 100=42,86 \\ \\%\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"296\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Considerando os dois significados poss\u00edveis de taxa de varia\u00e7\u00e3o, neste artigo nos concentraremos na compreens\u00e3o da defini\u00e7\u00e3o matem\u00e1tica de taxa de varia\u00e7\u00e3o. Dois tipos de taxa de varia\u00e7\u00e3o podem ser distinguidos: a taxa de varia\u00e7\u00e3o m\u00e9dia e a taxa de varia\u00e7\u00e3o instant\u00e2nea. Abaixo voc\u00ea tem a explica\u00e7\u00e3o de cada tipo.<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"tasa-de-variacion-media\"><\/span> Taxa m\u00e9dia de mudan\u00e7a<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>A taxa m\u00e9dia de varia\u00e7\u00e3o (TVM) de uma fun\u00e7\u00e3o em um intervalo \u00e9 o n\u00famero de unidades que a fun\u00e7\u00e3o aumenta (ou diminui) para cada unidade que sua vari\u00e1vel independente aumenta.<\/strong> Portanto, a taxa m\u00e9dia de varia\u00e7\u00e3o de uma fun\u00e7\u00e3o \u00e9 calculada dividindo o crescimento da fun\u00e7\u00e3o em um intervalo pela amplitude desse mesmo intervalo.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a08373207642a78dec20dd28a9dafc4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{TVM}[a,b]=\\cfrac{f(b)-f(a)}{b-a}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"191\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Para que voc\u00ea possa ver como \u00e9 calculada a taxa m\u00e9dia de varia\u00e7\u00e3o, resolvemos um exemplo passo a passo abaixo. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-del-calculo-de-la-tasa-de-variacion-media-de-una-funcion\"><\/span> Exemplo de c\u00e1lculo da taxa m\u00e9dia de varia\u00e7\u00e3o de uma fun\u00e7\u00e3o<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<ul>\n<li> Calcule a taxa m\u00e9dia de varia\u00e7\u00e3o no intervalo [2.5] da seguinte fun\u00e7\u00e3o:<\/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-e583988ce73cdf54a12306a64c97cc42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x) = x^2-1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"106\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Primeiro, calculamos o valor da fun\u00e7\u00e3o em x=2 e x=5:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-57a8756268ece6ee3543392da3aa5ee3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(5)= 5^2-1=25-1=24\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"217\" 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-7163ad8fbbfd700e474e6d40164c8cb9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(2)=2^2-1=4-1=3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"200\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> E ent\u00e3o calculamos a taxa m\u00e9dia de varia\u00e7\u00e3o da fun\u00e7\u00e3o no intervalo simplesmente aplicando a f\u00f3rmula:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-aca6688a94500231cb464ce5ca91685c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{TVM} [a,b] = \\cfrac{f(b)-f(a)}{b-a}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"191\" 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-dba3fad93a4b7e04a5d6f1517eb6dc1a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{TVM} [2,5]=\\cfrac{f(5)-f(2)}{5-2} = \\cfrac{24 - 3}{5-2} = \\cfrac{21}{3} = \\bm{7}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"341\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Como o resultado de TVM[2,5] \u00e9 positivo, isso significa que a fun\u00e7\u00e3o cresce no intervalo [2,5]. Por outro lado, se o resultado tivesse sido negativo, isso significaria que a fun\u00e7\u00e3o diminui neste intervalo. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"interpretacion-geometrica-de-la-tasa-de-variacion-media\"><\/span> Interpreta\u00e7\u00e3o geom\u00e9trica da taxa m\u00e9dia de mudan\u00e7a<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Geometricamente, a taxa m\u00e9dia de varia\u00e7\u00e3o de uma fun\u00e7\u00e3o em um intervalo representa a inclina\u00e7\u00e3o da reta que une os pontos extremos do intervalo. <\/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\/taux-de-variation-moyen.webp\" alt=\"taxa m\u00e9dia de mudan\u00e7a\" class=\"wp-image-1678\" width=\"337\" height=\"378\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"tasa-de-variacion-instantanea\"><\/span>Taxa instant\u00e2nea de mudan\u00e7a<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>A taxa de varia\u00e7\u00e3o instant\u00e2nea (TVI) de uma fun\u00e7\u00e3o em um ponto \u00e9 o limite infinitesimal do aumento relativo da fun\u00e7\u00e3o ao longo de um intervalo.<\/strong> Portanto, a taxa instant\u00e2nea de mudan\u00e7a \u00e9 calculada resolvendo o limite do quociente de <em>f(a+h)-f(a)<\/em> <em>quando<\/em> <em>h<\/em> se aproxima de zero.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-605387fef74bd8fe7f7e8a3e75686a7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\text{TVI}(a)=\\lim_{h \\to 0}\\frac{f(a+h)-f(a)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"235\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> O valor da taxa de varia\u00e7\u00e3o instant\u00e2nea pode ser positivo, negativo ou zero, e significa que a fun\u00e7\u00e3o naquele ponto est\u00e1 aumentando, diminuindo ou permanecendo a mesma, respectivamente, naquele ponto. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-del-calculo-de-la-tasa-de-variacion-instantanea-de-una-funcion\"><\/span> Exemplo de c\u00e1lculo da taxa instant\u00e2nea de varia\u00e7\u00e3o de uma fun\u00e7\u00e3o<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<ul>\n<li> Calcule a taxa instant\u00e2nea de varia\u00e7\u00e3o no ponto x=2 da seguinte fun\u00e7\u00e3o:<\/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-a0f6973d1b2b1b370cb20764d5a954a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x) = x^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"75\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Para calcular a taxa de varia\u00e7\u00e3o instant\u00e2nea, precisamos aplicar a f\u00f3rmula:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ef687e3c0bab81c2a8eaa578fcc41b9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{TVI} (a) = \\lim\\limits_{h \\to 0} \\cfrac{f(a+h)-f(a)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"235\" 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-fd6499ec6074ceea794309155c7d788d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{TVI} (2) = \\lim\\limits_{h \\to 0} \\cfrac{f(2+h)-f(2)}{h} =  \\lim\\limits_{h \\to 0} \\cfrac{(2+h)^2-2^2}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"390\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> Resolvemos a identidade not\u00e1vel:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-21ea1067eb0c47009dd7f9c214a16e4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{h \\to 0}  \\cfrac{2^2+h^2+2\\cdot 2 \\cdot h -2^2}{h} = \\lim\\limits_{h \\to 0}  \\cfrac{4+h^2+4h -4}{h} = \\lim\\limits_{h \\to 0}  \\cfrac{h^2+4h}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"496\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> <span style=\"color:#FF9B28;\">\u27a4<\/span> Se voc\u00ea n\u00e3o se lembra mais das <u style=\"text-decoration-color:#FF9B28;\">f\u00f3rmulas para identidades not\u00e1veis<\/u> , encontrar\u00e1 todas as f\u00f3rmulas em nosso site especializado em polin\u00f4mios: <u style=\"text-decoration-color:#FF9B28;\">www.polinomios.org<\/u><\/p>\n<p> Agora vamos tentar resolver o 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-b380a5546aa95d74c4da1a81e0e4665b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{h \\to 0}  \\cfrac{h^2+4h}{h} = \\cfrac{0^2+4\\cdot 0}{0} =\\cfrac{0}{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"221\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> Mas encontramos indetermina\u00e7\u00e3o zero entre zero, portanto:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ae32dac146777b5395b505ebf658af5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{h \\to 0}  \\cfrac{h^2+4h}{h}= \\lim\\limits_{h \\to 0} \\cfrac{\\cancel{h}(h+4)}{\\cancel{h}} = \\lim\\limits_{h \\to 0} (h+4)\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"320\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Veja:<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/zero-entre-zero-0-0-indeterminacao\/\">como resolver um limite com indetermina\u00e7\u00e3o de zero entre zero<\/a><\/span><\/p>\n<p> E finalmente resolvemos o 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-3b124f5a6537497e232294814f1d49b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{h \\to 0} (h+4) = 0 +4 = \\bm{4}\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"179\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> Ainda:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9504f8df3dcdd1c6954399b5a6aa0ef4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\mathbf{TVI} \\bm{(2) = 4}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"93\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Como o resultado de TVI(2) \u00e9 positivo, isso significa que a fun\u00e7\u00e3o aumenta em x=2. Por outro lado, se o resultado tivesse sido negativo, isso significaria que a fun\u00e7\u00e3o est\u00e1 diminuindo nesta fase. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"interpretacion-geometrica-de-la-tasa-de-variacion-instantanea\"><\/span> Interpreta\u00e7\u00e3o geom\u00e9trica da taxa instant\u00e2nea de mudan\u00e7a<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Geometricamente, a taxa instant\u00e2nea de varia\u00e7\u00e3o de uma fun\u00e7\u00e3o em um ponto representa a inclina\u00e7\u00e3o da reta tangente \u00e0 fun\u00e7\u00e3o nesse mesmo ponto. <\/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\/taux-de-changement-instantane.webp\" alt=\"taxa instant\u00e2nea de mudan\u00e7a\" class=\"wp-image-1682\" width=\"336\" height=\"355\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Se voc\u00ea olhar de perto, o significado de taxa de varia\u00e7\u00e3o instant\u00e2nea \u00e9 equivalente ao <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/derivados\/\">conceito de derivada de uma fun\u00e7\u00e3o<\/a><\/span> . Assim, a taxa de varia\u00e7\u00e3o instant\u00e2nea tamb\u00e9m \u00e9 usada para calcular o valor da derivada de uma fun\u00e7\u00e3o em um ponto. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-la-tasa-de-variacion\"><\/span> Exerc\u00edcios resolvidos sobre taxa de varia\u00e7\u00e3o<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Calcule o valor da taxa de varia\u00e7\u00e3o da seguinte fun\u00e7\u00e3o no intervalo [1,3]. <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-52242e9edb560029e6861be262dc3418_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2-5\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"106\" 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>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Primeiro, determinamos o valor da fun\u00e7\u00e3o nas extremidades do intervalo: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a68e7a97d5a6f45ee1d01481fe687da1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(1)=1^2-5=1-5=-4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"214\" 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-a0d6343387ae4a98c94387008c73c791_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(3)=3^2-5=9-5=4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"200\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E agora aplicamos a f\u00f3rmula da taxa de varia\u00e7\u00e3o: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c33fec724dd0a5bf45bfec5a911f5bcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{TV}[a,b]=f(b)-f(a)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"171\" 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-6ab143c57ef36bb306946cbe760653ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{TV}[1,3]=f(3)-f(1)=4-(-4)=4+4=\\bm{8}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"360\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 2<\/h3>\n<p> Calcule a taxa m\u00e9dia de varia\u00e7\u00e3o (TVM) da seguinte fun\u00e7\u00e3o no intervalo [1,4]: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b43717721aa2afd3fbc6d587213bbdef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=2x+1\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"107\" 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>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Primeiro calculamos as imagens da fun\u00e7\u00e3o em x=1 ex=4. <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0877ab6ef765a3944e138081718775cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(4)=2\\cdot4+1=9\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"151\" 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-e2e61642cf8070ce3ae09b2eb3335c4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(1)=2\\cdot 1+1=3\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"151\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E aplicamos a f\u00f3rmula para a taxa m\u00e9dia de varia\u00e7\u00e3o: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-aca6688a94500231cb464ce5ca91685c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{TVM} [a,b] = \\cfrac{f(b)-f(a)}{b-a}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"191\" 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-55ff356ed3f6444ca49c49e39dd46072_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{TVM} [1,4] = \\cfrac{f(4)-f(1)}{4-1} = \\cfrac{9-3}{4-1}=\\cfrac{6}{3} = \\bm{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"323\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 3<\/h3>\n<p> Encontre a taxa m\u00e9dia de varia\u00e7\u00e3o da seguinte fun\u00e7\u00e3o no intervalo [-1,3]: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a04612aafb771e8771b2810a18e18475_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=(x+1)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"120\" 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>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Para determinar a taxa m\u00e9dia de mudan\u00e7a, primeiro precisamos calcular f(-1) e f(3): <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3da577fe4ac481741007375057aa116e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(3)=(3+1)^2=(4)^2=16\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"213\" 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-8b7269aaa1c01fb9a5e218a0a627483b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(-1)=((-1)+1)^2=(0)^2=0\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"246\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora usamos a f\u00f3rmula para a taxa m\u00e9dia de varia\u00e7\u00e3o: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-aca6688a94500231cb464ce5ca91685c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{TVM} [a,b] = \\cfrac{f(b)-f(a)}{b-a}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"191\" 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-080aef07d76e5d8823128118052fb808_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{TVM} [-1,3] = \\cfrac{f(3)-f(-1)}{3-(-1)} = \\cfrac{16-0}{3+1}=\\cfrac{16}{4} = \\bm{4}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"369\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 4<\/h3>\n<p> Calcule a taxa m\u00e9dia de varia\u00e7\u00e3o no intervalo [2,4] da fun\u00e7\u00e3o mostrada no gr\u00e1fico a seguir: <\/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-d-une-fonction-quadratique-ou-parabole.webp\" alt=\"\" class=\"wp-image-137\" width=\"290\" height=\"328\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\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>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Aplicamos a f\u00f3rmula para a taxa m\u00e9dia de varia\u00e7\u00e3o: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-aca6688a94500231cb464ce5ca91685c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{TVM} [a,b] = \\cfrac{f(b)-f(a)}{b-a}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"191\" 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-436fb10e74beb8c2ede9c9ac586d96c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{TVM} [2,4]=\\cfrac{f(4)-f(2)}{4-2}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"192\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Como vemos na f\u00f3rmula, precisamos encontrar o valor de f(4) e f(2). E isso pode ser feito facilmente observando a representa\u00e7\u00e3o gr\u00e1fica da fun\u00e7\u00e3o: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fad0fc656e95d7bc9517693b55a15540_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(4)=5\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"65\" 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-9e7d13ebd46205da0aa8ff41e5133ba3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(2)=1\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"65\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E agora que conhecemos os valores da fun\u00e7\u00e3o, substitu\u00edmos-os na f\u00f3rmula: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9a84929c02a38e91b072239affa6d9f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{TVM}[2,4]=\\cfrac{f(4)-f(2)}{4-2}=\\cfrac{5-1}{4-2}=\\cfrac{4}{2}=\\bm{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"323\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 5<\/h3>\n<p> Calcule a taxa instant\u00e2nea de varia\u00e7\u00e3o da seguinte fun\u00e7\u00e3o no ponto x=2: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9de7457ac9ec3e1599c6f986c5ba57ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=3x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"77\" 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>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Para determinar a taxa instant\u00e2nea de varia\u00e7\u00e3o da fun\u00e7\u00e3o no ponto x=2 aplicamos sua f\u00f3rmula correspondente: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a66c321e96cb1e871b9c7debfee1f4bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{TVI}(a)=\\lim\\limits_{h \\to 0} \\cfrac{f(a+h)-f(a)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"235\" 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-73ee822823f921f75014cb9b50e47f51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\text{TVI}(2)=\\lim\\limits_{h \\to 0} \\cfrac{f(2+h)-f(2)}{h}=\\\\[4ex]=\\lim\\limits_{h \\to 0} \\cfrac{3(2+h)-3\\cdot 2}{h} =\\\\[4ex]=\\lim\\limits_{h \\to 0} \\cfrac{6+3h-6}{h}= \\lim\\limits_{h \\to 0} \\cfrac{3h}{h} =\\\\[4ex]=\\lim\\limits_{h \\to 0} \\cfrac{3\\cancel{h}}{\\cancel{h}}=\\lim\\limits_{h \\to 0} 3 = \\bm{3}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"239\" width=\"250\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 6<\/h3>\n<p> Determine a taxa de varia\u00e7\u00e3o instant\u00e2nea (TVI) da seguinte fun\u00e7\u00e3o no ponto 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-a64e895ef4284c9ac73729d092f47767_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2+1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"106\" 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>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Aplicamos a f\u00f3rmula para a taxa de varia\u00e7\u00e3o instant\u00e2nea: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ef687e3c0bab81c2a8eaa578fcc41b9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{TVI} (a) = \\lim\\limits_{h \\to 0} \\cfrac{f(a+h)-f(a)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"235\" 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-9aae03d6907f804a3015daa4ecca1e9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{TVI} (1) = \\lim\\limits_{h \\to 0} \\cfrac{f(1+h)-f(1)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"233\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ent\u00e3o, calculamos<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-59ab22db5134b4e8bc6bbed7ab09bd5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(1+h)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"64\" 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-32045357853caad8774629c95963835d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(1):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bcec545e7bbd1d55ba069ac58c7862ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(1+h) = (1+h)^2+1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"181\" 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-c5f418d9c95675368ebe7525ad0354ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(1)=1^2+1=1+1=2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"199\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E substitu\u00edmos os valores encontrados no 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-89bedae84e1ece0933ddf522e449e005_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{TVI} (1) = \\lim\\limits_{h \\to 0} \\cfrac{f(1+h)-f(1)}{h}= \\lim\\limits_{h \\to 0} \\cfrac{(1+h)^2+1-2}{h} =\\lim\\limits_{h \\to 0} \\cfrac{(1+h)^2-1}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"563\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Resolvemos o produto not\u00e1vel:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ef36f072cd7bee2d30936dc5a18bdf7a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{h \\to 0} \\cfrac{1^2+h^2+2\\cdot 1 \\cdot h-1}{h}=\\lim\\limits_{h \\to 0} \\cfrac{1+h^2+2h-1}{h}=\\lim\\limits_{h \\to 0} \\cfrac{h^2+2h}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"488\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora vamos tentar resolver o 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-eada4220b55e8f3f27f6e565244fa430_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{h \\to 0} \\cfrac{h^2+2h}{h}=\\cfrac{0^2+2\\cdot 0}{0} = \\cfrac{0}{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"221\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Mas encontramos a forma indeterminada zero dividido por zero, ent\u00e3o fatoramos o polin\u00f4mio do numerador da fra\u00e7\u00e3o e simplificamos:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-89248fe68bc38749f3ad45e65d902840_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{h \\to 0} \\cfrac{h^2+2h}{h}=\\lim\\limits_{h \\to 0} \\cfrac{h(h+2)}{h} = \\lim\\limits_{h \\to 0} \\cfrac{\\cancel{h}(h+2)}{\\cancel{h}}= \\lim\\limits_{h \\to 0}(h+2)\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"442\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <span style=\"color:#FF9B28;\">\u27a4<\/span> Se voc\u00ea n\u00e3o sabe <u style=\"text-decoration-color:#FF9B28;\">como resolver a indetermina\u00e7\u00e3o de zero entre zero<\/u> , pode ver a explica\u00e7\u00e3o completa no link acima sobre <u style=\"text-decoration-color:#FF9B28;\">como resolver um limite com indetermina\u00e7\u00e3o de zero entre zero.<\/u><\/p>\n<p class=\"has-text-align-left\"> E finalmente, resolvemos o 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-5467af069ca6192116dee3e32de90ea7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{h \\to 0}(h+2) = 0+2 =\\bm{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"178\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Em resumo, a taxa instant\u00e2nea de varia\u00e7\u00e3o da fun\u00e7\u00e3o no ponto x=1 \u00e9 igual a 2. <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fbcab0fecbbbe2fbefd80027e8748bda_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\mathbf{TVI} \\bm{(1) = 2}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"92\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 7<\/h3>\n<p> Encontre a taxa de varia\u00e7\u00e3o instant\u00e2nea da seguinte fun\u00e7\u00e3o no ponto x=2: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1b3e71aa0e6f65532f0324d35e827180_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=4x^2-x+3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"147\" 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>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Primeiro usamos a f\u00f3rmula da taxa de varia\u00e7\u00e3o instant\u00e2nea: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ef687e3c0bab81c2a8eaa578fcc41b9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{TVI} (a) = \\lim\\limits_{h \\to 0} \\cfrac{f(a+h)-f(a)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"235\" 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-a6a7c9ceda50eea0255d09898ff47cbb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{TVI} (2) = \\lim\\limits_{h \\to 0} \\cfrac{f(2+h)-f(2)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"233\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> N\u00f3s calculamos<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bd34e7bfca2688fc7047d0cbe546068d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(2+h)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"64\" 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-f026e401162db03299777455b748b308_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(2):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-79281afbca1e53d72c77a132af8593a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(2+h) = 4(2+h)^2-(2+h)+3=4(2+h)^2-h+1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"423\" 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-9260dd17b5ceca44c7ff2d860f0354c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(2) =4\\cdot 2^2-2+3=4\\cdot 4-2+3 =17\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"313\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E substitu\u00edmos os valores encontrados no 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-6ad59f5f751af139656a471bf2a41801_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{TVI} (2) = \\lim\\limits_{h \\to 0} \\cfrac{f(2+h)-f(2)}{h}=\\\\[4ex]=\\lim\\limits_{h \\to 0} \\cfrac{4(2+h)^2-h+1-17}{h}=\\\\[4ex]= \\lim\\limits_{h \\to 0} \\cfrac{4(2+h)^2-h-16}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"190\" width=\"286\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Calculamos a igualdade not\u00e1vel:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e358be77d34e9ec7c27433743001162c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{h \\to 0} \\cfrac{4(2^2+h^2+2\\cdot 2 \\cdot h)-h-16}{h}=\\lim\\limits_{h \\to 0} \\cfrac{4(4+h^2+4h)-h-16}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"500\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Operamos no numerador:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e7c2b63592b070fa6046376f878d45bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{h \\to 0} \\cfrac{16+4h^2+16h-h-16}{h}=\\lim\\limits_{h \\to 0} \\cfrac{4h^2+15h}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"355\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora vamos tentar resolver o 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-b9dda4d5e5c5eab8c018f1b46a3da45e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{h \\to 0} \\cfrac{4h^2+15h}{h}=\\cfrac{4\\cdot0^2+15\\cdot 0}{0} = \\cfrac{0}{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"269\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Mas obtemos a indetermina\u00e7\u00e3o zero dividida por zero, ent\u00e3o fatoramos os polin\u00f4mios e simplificamos:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e0d1213a45968f3137f0c39272cd70dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{h \\to 0} \\cfrac{4h^2+15h}{h}=\\lim\\limits_{h \\to 0} \\cfrac{h(4h+15)}{h}=\\lim\\limits_{h \\to 0} \\cfrac{\\cancel{h}(4h+15)}{\\cancel{h}}= \\lim\\limits_{h \\to 0}(4h+15)\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"513\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E finalmente, resolvemos o 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-5eb4579d04c151866059162480d4c816_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{h \\to 0}(4h+15)=4\\cdot 0+15 =\\bm{15}\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"235\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ainda: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9181cbd259448ba47e80eb081625ee2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\mathbf{TVI} \\bm{(2) = 15}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"101\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Aqui explicamos o que s\u00e3o taxa de mudan\u00e7a, taxa m\u00e9dia de mudan\u00e7a e taxa de mudan\u00e7a instant\u00e2nea. Voc\u00ea poder\u00e1 ver v\u00e1rios exemplos de como calcular a taxa de varia\u00e7\u00e3o e, al\u00e9m disso, poder\u00e1 praticar com exerc\u00edcios passo a passo resolvidos sobre a taxa de varia\u00e7\u00e3o. Qual \u00e9 a taxa de mudan\u00e7a? Em matem\u00e1tica, a taxa &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/taxa-media-e-instantanea-de-mudanca\/\"> <span class=\"screen-reader-text\">Taxa m\u00e9dia e instant\u00e2nea de mudan\u00e7a<\/span> Leia mais &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":[11],"tags":[],"class_list":["post-26","post","type-post","status-publish","format-standard","hentry","category-derivados"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u25b7 Taxa de varia\u00e7\u00e3o m\u00e9dia e instant\u00e2nea (exerc\u00edcios resolvidos)<\/title>\n<meta name=\"description\" content=\"Explicamos o que \u00e9 taxa de varia\u00e7\u00e3o (m\u00e9dia e instant\u00e2nea) e como ela \u00e9 calculada. \u2705Com exerc\u00edcios resolvidos sobre a taxa de varia\u00e7\u00e3o. \u2705\" \/>\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\/pt\/taxa-media-e-instantanea-de-mudanca\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u25b7 Taxa de varia\u00e7\u00e3o m\u00e9dia e instant\u00e2nea (exerc\u00edcios resolvidos)\" \/>\n<meta property=\"og:description\" content=\"Explicamos o que \u00e9 taxa de varia\u00e7\u00e3o (m\u00e9dia e instant\u00e2nea) e como ela \u00e9 calculada. \u2705Com exerc\u00edcios resolvidos sobre a taxa de varia\u00e7\u00e3o. \u2705\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/pt\/taxa-media-e-instantanea-de-mudanca\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-09-17T11:04:43+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c33fec724dd0a5bf45bfec5a911f5bcb_l3.png\" \/>\n<meta name=\"author\" content=\"Equipe Mathoridade\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"Equipe Mathoridade\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. tempo de leitura\" \/>\n\t<meta name=\"twitter:data2\" content=\"7 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/pt\/taxa-media-e-instantanea-de-mudanca\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/pt\/taxa-media-e-instantanea-de-mudanca\/\"},\"author\":{\"name\":\"Equipe Mathoridade\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00\"},\"headline\":\"Taxa m\u00e9dia e instant\u00e2nea de mudan\u00e7a\",\"datePublished\":\"2023-09-17T11:04:43+00:00\",\"dateModified\":\"2023-09-17T11:04:43+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/pt\/taxa-media-e-instantanea-de-mudanca\/\"},\"wordCount\":1341,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathority.org\/pt\/#organization\"},\"articleSection\":[\"Derivados\"],\"inLanguage\":\"pt-BR\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathority.org\/pt\/taxa-media-e-instantanea-de-mudanca\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/pt\/taxa-media-e-instantanea-de-mudanca\/\",\"url\":\"https:\/\/mathority.org\/pt\/taxa-media-e-instantanea-de-mudanca\/\",\"name\":\"\u25b7 Taxa de varia\u00e7\u00e3o m\u00e9dia e instant\u00e2nea (exerc\u00edcios resolvidos)\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/pt\/#website\"},\"datePublished\":\"2023-09-17T11:04:43+00:00\",\"dateModified\":\"2023-09-17T11:04:43+00:00\",\"description\":\"Explicamos o que \u00e9 taxa de varia\u00e7\u00e3o (m\u00e9dia e instant\u00e2nea) e como ela \u00e9 calculada. \u2705Com exerc\u00edcios resolvidos sobre a taxa de varia\u00e7\u00e3o. \u2705\",\"breadcrumb\":{\"@id\":\"https:\/\/mathority.org\/pt\/taxa-media-e-instantanea-de-mudanca\/#breadcrumb\"},\"inLanguage\":\"pt-BR\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathority.org\/pt\/taxa-media-e-instantanea-de-mudanca\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathority.org\/pt\/taxa-media-e-instantanea-de-mudanca\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathority.org\/pt\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Taxa m\u00e9dia e instant\u00e2nea de mudan\u00e7a\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathority.org\/pt\/#website\",\"url\":\"https:\/\/mathority.org\/pt\/\",\"name\":\"Mathority\",\"description\":\"Onde a curiosidade encontra o c\u00e1lculo!\",\"publisher\":{\"@id\":\"https:\/\/mathority.org\/pt\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/mathority.org\/pt\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"pt-BR\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/mathority.org\/pt\/#organization\",\"name\":\"Mathority\",\"url\":\"https:\/\/mathority.org\/pt\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"pt-BR\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png\",\"contentUrl\":\"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png\",\"width\":703,\"height\":151,\"caption\":\"Mathority\"},\"image\":{\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/\"}},{\"@type\":\"Person\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00\",\"name\":\"Equipe Mathoridade\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"pt-BR\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g\",\"caption\":\"Equipe Mathoridade\"},\"sameAs\":[\"http:\/\/mathority.org\/pt\"]}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"\u25b7 Taxa de varia\u00e7\u00e3o m\u00e9dia e instant\u00e2nea (exerc\u00edcios resolvidos)","description":"Explicamos o que \u00e9 taxa de varia\u00e7\u00e3o (m\u00e9dia e instant\u00e2nea) e como ela \u00e9 calculada. \u2705Com exerc\u00edcios resolvidos sobre a taxa de varia\u00e7\u00e3o. \u2705","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mathority.org\/pt\/taxa-media-e-instantanea-de-mudanca\/","og_locale":"pt_BR","og_type":"article","og_title":"\u25b7 Taxa de varia\u00e7\u00e3o m\u00e9dia e instant\u00e2nea (exerc\u00edcios resolvidos)","og_description":"Explicamos o que \u00e9 taxa de varia\u00e7\u00e3o (m\u00e9dia e instant\u00e2nea) e como ela \u00e9 calculada. \u2705Com exerc\u00edcios resolvidos sobre a taxa de varia\u00e7\u00e3o. \u2705","og_url":"https:\/\/mathority.org\/pt\/taxa-media-e-instantanea-de-mudanca\/","article_published_time":"2023-09-17T11:04:43+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c33fec724dd0a5bf45bfec5a911f5bcb_l3.png"}],"author":"Equipe Mathoridade","twitter_card":"summary_large_image","twitter_misc":{"Escrito por":"Equipe Mathoridade","Est. tempo de leitura":"7 minutos"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathority.org\/pt\/taxa-media-e-instantanea-de-mudanca\/#article","isPartOf":{"@id":"https:\/\/mathority.org\/pt\/taxa-media-e-instantanea-de-mudanca\/"},"author":{"name":"Equipe Mathoridade","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00"},"headline":"Taxa m\u00e9dia e instant\u00e2nea de mudan\u00e7a","datePublished":"2023-09-17T11:04:43+00:00","dateModified":"2023-09-17T11:04:43+00:00","mainEntityOfPage":{"@id":"https:\/\/mathority.org\/pt\/taxa-media-e-instantanea-de-mudanca\/"},"wordCount":1341,"commentCount":0,"publisher":{"@id":"https:\/\/mathority.org\/pt\/#organization"},"articleSection":["Derivados"],"inLanguage":"pt-BR","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathority.org\/pt\/taxa-media-e-instantanea-de-mudanca\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathority.org\/pt\/taxa-media-e-instantanea-de-mudanca\/","url":"https:\/\/mathority.org\/pt\/taxa-media-e-instantanea-de-mudanca\/","name":"\u25b7 Taxa de varia\u00e7\u00e3o m\u00e9dia e instant\u00e2nea (exerc\u00edcios resolvidos)","isPartOf":{"@id":"https:\/\/mathority.org\/pt\/#website"},"datePublished":"2023-09-17T11:04:43+00:00","dateModified":"2023-09-17T11:04:43+00:00","description":"Explicamos o que \u00e9 taxa de varia\u00e7\u00e3o (m\u00e9dia e instant\u00e2nea) e como ela \u00e9 calculada. \u2705Com exerc\u00edcios resolvidos sobre a taxa de varia\u00e7\u00e3o. \u2705","breadcrumb":{"@id":"https:\/\/mathority.org\/pt\/taxa-media-e-instantanea-de-mudanca\/#breadcrumb"},"inLanguage":"pt-BR","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathority.org\/pt\/taxa-media-e-instantanea-de-mudanca\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathority.org\/pt\/taxa-media-e-instantanea-de-mudanca\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathority.org\/pt\/"},{"@type":"ListItem","position":2,"name":"Taxa m\u00e9dia e instant\u00e2nea de mudan\u00e7a"}]},{"@type":"WebSite","@id":"https:\/\/mathority.org\/pt\/#website","url":"https:\/\/mathority.org\/pt\/","name":"Mathority","description":"Onde a curiosidade encontra o c\u00e1lculo!","publisher":{"@id":"https:\/\/mathority.org\/pt\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/mathority.org\/pt\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"pt-BR"},{"@type":"Organization","@id":"https:\/\/mathority.org\/pt\/#organization","name":"Mathority","url":"https:\/\/mathority.org\/pt\/","logo":{"@type":"ImageObject","inLanguage":"pt-BR","@id":"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/","url":"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png","contentUrl":"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png","width":703,"height":151,"caption":"Mathority"},"image":{"@id":"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/"}},{"@type":"Person","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00","name":"Equipe Mathoridade","image":{"@type":"ImageObject","inLanguage":"pt-BR","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g","caption":"Equipe Mathoridade"},"sameAs":["http:\/\/mathority.org\/pt"]}]}},"yoast_meta":{"yoast_wpseo_title":"","yoast_wpseo_metadesc":"","yoast_wpseo_canonical":""},"_links":{"self":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/posts\/26","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/comments?post=26"}],"version-history":[{"count":0,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/posts\/26\/revisions"}],"wp:attachment":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/media?parent=26"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/categories?post=26"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/tags?post=26"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}