{"id":38,"date":"2023-09-17T10:59:38","date_gmt":"2023-09-17T10:59:38","guid":{"rendered":"https:\/\/mathority.org\/cn\/%e5%87%bd%e6%95%b0%e7%9a%84%e5%8f%af%e5%be%ae%e6%80%a7\/"},"modified":"2023-09-17T10:59:38","modified_gmt":"2023-09-17T10:59:38","slug":"%e5%87%bd%e6%95%b0%e7%9a%84%e5%8f%af%e5%be%ae%e6%80%a7","status":"publish","type":"post","link":"https:\/\/mathority.org\/cn\/%e5%87%bd%e6%95%b0%e7%9a%84%e5%8f%af%e5%be%ae%e6%80%a7\/","title":{"rendered":"\u51fd\u6570\u7684\u53ef\u5fae\u6027"},"content":{"rendered":"<p>\u5728\u672c\u6587\u4e2d\uff0c\u60a8\u5c06\u5b66\u4e60\u5982\u4f55\u7814\u7a76\u51fd\u6570\u7684\u53ef\u5fae\u6027\uff0c\u5373\u51fd\u6570\u662f\u5426\u53ef\u5fae\u3002\u6b64\u5916\uff0c\u6211\u4eec\u8fd8\u5c06\u770b\u5230\u51fd\u6570\u7684\u53ef\u5fae\u6027\u548c\u8fde\u7eed\u6027\u4e4b\u95f4\u7684\u5173\u7cfb\u3002\u6700\u540e\uff0c\u6211\u4eec\u5c06\u7814\u7a76\u5206\u6bb5\u51fd\u6570\u7684\u53ef\u5fae\u6027\u3002 <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"derivabilidad-y-continuidad-de-una-funcion\"><\/span>\u51fd\u6570\u7684\u53ef\u5fae\u6027\u548c\u8fde\u7eed\u6027<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u51fd\u6570\u5728\u4e00\u70b9\u7684<strong>\u8fde\u7eed\u6027\u548c\u53ef\u5bfc\u6027<\/strong>\u5173\u7cfb\u5982\u4e0b\uff1a<\/p>\n<ul>\n<li>\u5982\u679c\u51fd\u6570\u5728\u4e00\u70b9\u53ef\u5fae\uff0c\u5219\u8be5\u51fd\u6570\u5728\u8be5\u70b9\u8fde\u7eed\u3002<\/li>\n<li>\u5982\u679c\u51fd\u6570\u5728\u4e00\u70b9\u4e0d\u8fde\u7eed\uff0c\u90a3\u4e48\u5b83\u5728\u8be5\u70b9\u4e5f\u662f\u4e0d\u53ef\u5fae\u7684\u3002<\/li>\n<\/ul>\n<p>\u7136\u800c\uff0c\u8be5\u5b9a\u7406\u7684\u9006\u547d\u9898\u662f\u9519\u8bef\u7684\uff1a\u4ec5\u4ec5\u56e0\u4e3a\u51fd\u6570\u5728\u4e00\u70b9\u8fde\u7eed\uff0c\u5e76\u4e0d\u610f\u5473\u7740\u5b83\u5728\u8be5\u70b9\u603b\u662f\u53ef\u5fae\u7684\u3002<\/p>\n<p>\u60a8\u8fd8\u53ef\u4ee5\u67e5\u770b\u51fd\u6570\u5728\u67d0\u4e2a\u70b9\u4e0e\u5176\u56fe\u5f62\u8868\u793a\u662f\u5426\u53ef\u5fae\uff1a<\/p>\n<ul>\n<li>\u5982\u679c\u662f<strong>\u5e73\u6ed1\u70b9\uff0c<\/strong>\u5219\u51fd\u6570\u5728\u8be5\u70b9\u53ef\u5fae\u3002<\/li>\n<li>\u5982\u679c\u662f<strong>\u89d2\u70b9\uff0c<\/strong>\u5219\u51fd\u6570\u5728\u8be5\u70b9\u8fde\u7eed\u4f46\u4e0d\u53ef\u5fae\u3002 <\/li>\n<\/ul>\n<div class=\"wp-block-columns is-layout-flex wp-container-15\">\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow\">\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\/exercices-resolus-pour-representer-une-fonction-quadratique-incomplete.webp\" alt=\"\" class=\"wp-image-140\" width=\"250\" height=\"279\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-center\"> x=0 \u5904\u7684<span style=\"color:#1976d2;\"><strong>\u5e73\u6ed1\u70b9<\/strong><\/span>\uff1a<br \/>\u6b64\u9636\u6bb5\u7684\u8fde\u7eed\u4e14\u53ef\u5fae\u7684\u51fd\u6570\u3002 <\/p>\n<\/div>\n<div class=\"wp-block-column is-layout-flow\">\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\/comment-representer-graphiquement-une-fonction-avec-valeur-absolue.webp\" alt=\"\" class=\"wp-image-230\" width=\"286\" height=\"305\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-center\"> x=2 \u5904\u7684<span style=\"color:#1976d2;\"><strong>\u89d2\u70b9<\/strong><\/span>\uff1a<br \/>\u51fd\u6570\u5728\u6b64\u9636\u6bb5\u8fde\u7eed\u4f46\u4e0d\u53ef\u5fae\u3002 <\/p>\n<\/div>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"derivabilidad-de-una-funcion-a-trozos\"><\/span>\u5206\u6bb5\u51fd\u6570\u7684\u53ef\u5fae\u6027<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u4e00\u65e6\u6211\u4eec\u77e5\u9053\u4e86\u51fd\u6570\u7684\u8fde\u7eed\u6027\u548c\u53ef\u5fae\u6027\u4e4b\u95f4\u7684\u5173\u7cfb\uff0c\u6211\u4eec\u5c31\u4f1a\u4e86\u89e3\u5982\u4f55\u7814\u7a76\u5206\u6bb5\u5b9a\u4e49\u51fd\u6570\u7684\u53ef\u5fae\u6027\u3002<\/p>\n<p>\u60a8\u53ef\u4ee5\u901a\u8fc7\u8ba1\u7b97\u8be5\u70b9\u7684<strong>\u6a2a\u5411\u5bfc\u6570<\/strong>\u6765\u5224\u65ad\u5206\u6bb5\u51fd\u6570\u5728\u8be5\u70b9\u662f\u5426\u53ef\u5fae\uff1a <\/p>\n<div style=\"background-color:#FFF3E0; padding-top: 23px; padding-bottom: 0.5px; padding-right: 40px; padding-left: 20px; border-radius:20px;\">\n<ul style=\"color:#64B5F6; font-weight: bold;\">\n<li><span style=\"color:#000000;font-weight: normal;\">\u5982\u679c\u67d0\u70b9\u7684\u4fa7\u5bfc\u6570\u4e0d\u76f8\u7b49\uff0c\u5219\u51fd\u6570\u5728\u8be5\u70b9\u4e0d\u53ef\u5fae\uff1a<\/span><\/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-97c60c64dc01a7e0a9084313d15b0886_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x_o^-) \\neq f'(x_o^+) \\ \\longrightarrow\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"163\" style=\"vertical-align: -5px;\"><\/p>\n<p>\u4e0d\u53ef\u6263\u9664<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-67ab403a6241009b92035b251f86c88e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x_o\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<ul style=\"color:#64B5F6; font-weight: bold;\">\n<li><span style=\"color:#000000;font-weight: normal;\">\u5982\u679c\u67d0\u70b9\u7684\u4fa7\u5bfc\u6570\u91cd\u5408\uff0c\u5219\u51fd\u6570\u5728\u8be5\u70b9\u53ef\u5fae\uff1a<\/span><\/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-3f9477828318b9e6392465762f831642_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x_o^-) = f'(x_o^+) \\ \\longrightarrow\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"163\" style=\"vertical-align: -5px;\"><\/p>\n<p>\u662f\u7684\uff0c\u5b83\u53ef\u5fae\u5206\u4e8e<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-67ab403a6241009b92035b251f86c88e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x_o\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<\/div>\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 style=\"text-align:left\"><strong>\u6ce8\u610f\uff1a<\/strong>\u5bf9\u4e8e\u5728\u4e00\u70b9\u53ef\u5fae\u7684\u51fd\u6570\uff0c\u8be5\u51fd\u6570\u5fc5\u987b\u5728\u8be5\u70b9\u8fde\u7eed\u3002\u56e0\u6b64\uff0c\u5728\u8ba1\u7b97\u4fa7\u5bfc\u6570\u4e4b\u524d\uff0c\u6211\u4eec\u9700\u8981\u4fdd\u8bc1\u51fd\u6570\u5728\u8be5\u70b9\u8fde\u7eed\u3002\u5982\u679c\u60a8\u4e0d\u77e5\u9053\u5982\u4f55\u7814\u7a76\u67d0\u4e00\u70b9\u7684\u8fde\u7eed\u6027\uff0c\u60a8\u53ef\u4ee5\u5728\u4ee5\u4e0b\u94fe\u63a5\u4e2d\u4e86\u89e3\u5b83\u662f\u5982\u4f55\u5b8c\u6210\u7684\uff1a<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span><strong>\u8bf7\u53c2\u89c1\uff1a<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/cn\/\u8fde\u7eed\u51fd\u6570-\u51fd\u6570\u7684\u8fde\u7eed\u6027\/\">\u51fd\u6570\u5728\u4e00\u70b9\u7684\u8fde\u7eed\u6027<\/a><\/span><\/p>\n<\/div>\n<p>\u73b0\u5728\u8ba9\u6211\u4eec\u770b\u4e00\u4e2a\u793a\u4f8b\uff0c\u4e86\u89e3\u5982\u4f55\u8ba1\u7b97\u5728\u67d0\u4e00\u70b9\u5206\u6bb5\u5b9a\u4e49\u7684\u51fd\u6570\u7684\u5bfc\u6570\uff1a<\/p>\n<ul>\n<li>\u7814\u7a76\u4ee5\u4e0b\u5728\u70b9 x=2 \u5904\u5206\u6bb5\u5b9a\u4e49\u7684\u51fd\u6570\u7684\u8fde\u7eed\u6027\u548c\u53ef\u5fae\u6027\uff1a<\/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-a98eee72521c68fd394eb6209a7d0a59_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)=  \\left\\{ \\begin{array}{lcl} 3x^2-6x &amp; \\text{si} &amp;  x<2 \\\\[2ex] 6\\ln (x-1) &amp; \\text{si} &amp; x\\geq 2 \\end{array} \\right.\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"249\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u4e24\u90e8\u5206\u7684\u51fd\u6570\u5728\u5404\u81ea\u7684\u533a\u95f4\u5185\u8fde\u7eed\uff0c\u4f46\u9700\u8981\u770b\u51fd\u6570\u5728\u4e34\u754c\u70b9x=2\u5904\u662f\u5426\u8fde\u7eed\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u6c42\u89e3\u51fd\u6570\u5728\u8be5\u70b9\u7684\u6a2a\u5411\u6781\u9650\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dad5bcb0055431aa87a67068c04d2ce2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{x\\to 2^-} f(x) = \\lim\\limits_{x\\to 2^-} \\bigl(3x^2-6x\\bigr) = 3\\cdot2^2-6\\cdot2=12-12=\\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"449\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-31e1ba2ea2c5fd9fa86e5cefed0e5535_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{x\\to 2^+} f(x) = \\lim\\limits_{x\\to 2^+} 6\\ln (x-1) = 6\\ln (2-1)=6 \\ln 1=6 \\cdot 0= \\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"474\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p>\u4e34\u754c\u70b9\u5904\u7684\u6a2a\u5411\u6781\u9650\u7ed9\u4e86\u6211\u4eec\u76f8\u540c\u7684\u7ed3\u679c\uff0c\u56e0\u6b64<strong>\u51fd\u6570\u5728 x=2 \u70b9\u5904\u8fde\u7eed\u3002<\/strong><\/p>\n<p>\u4e00\u65e6\u6211\u4eec\u77e5\u9053\u51fd\u6570\u5728 x=2 \u5904\u8fde\u7eed\uff0c\u6211\u4eec\u5c06\u7814\u7a76\u51fd\u6570\u5728\u8be5\u70b9\u7684\u53ef\u5fae\u6027\u3002\u4e3a\u6b64\uff0c<strong>\u6211\u4eec\u8ba1\u7b97\u5206\u6bb5\u5b9a\u4e49\u7684\u51fd\u6570\u7684\u4fa7\u5bfc\u6570<\/strong>\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3709995609d0f69f382ff651e397c00a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)= \\left\\{ \\begin{array}{lcl} 6x-6 &amp; \\text{si} &amp;  x<2 \\\\[2ex] \\cfrac{6}{x-1} &amp; \\text{si} &amp; x\\geq 2 \\end{array} \\right.\" title=\"Rendered by QuickLaTeX.com\" height=\"76\" width=\"222\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u6211\u4eec\u73b0\u5728\u8bc4\u4f30\u4e34\u754c\u70b9\u5904\u7684\u6bcf\u4e2a\u4fa7\u5bfc\u6570\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ca27189960575c1151402d040bfa76f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(2^-)=6\\cdot2-6=12-6 = \\bm{6}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"239\" 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-058310a16d7d545ea56e99517845842b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(2^+)=\\cfrac{6}{2-1} = \\cfrac{6}{1} = \\bm{6}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"181\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p>\u4e24\u4e2a\u6a2a\u5411\u5bfc\u6570\u7ed9\u51fa\u4e86\u76f8\u540c\u7684\u7ed3\u679c\uff0c\u56e0\u6b64\u51fd\u6570\u5728 x=2 \u5904\u53ef\u5fae\uff0c\u5bfc\u6570\u503c\u4e3a 6\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-aadf046f27916c46f0a302d6e0c34113_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(2^-) = f'(2^+) = 6 \\ \\longrightarrow \\ \\bm{f'(2) = 6}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"275\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p>\u53e6\u4e00\u65b9\u9762\uff0c\u5982\u679c\u6a2a\u5411\u5bfc\u6570\u7ed9\u4e86\u6211\u4eec\u4e0d\u540c\u7684\u7ed3\u679c\uff0c\u5219\u610f\u5473\u7740\u8be5\u51fd\u6570\u5728 x=2 \u5904\u4e0d\u53ef\u5fae\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u6b64\u65f6\u5bfc\u6570\u5c06\u4e0d\u5b58\u5728\u3002<\/p>\n<p>\u6700\u540e\uff0c\u8bf7\u8bb0\u4f4f\uff0c\u6b64\u8fc7\u7a0b\u5bf9\u4e8e\u7814\u7a76\u7edd\u5bf9\u503c\u51fd\u6570\u7684\u53ef\u5fae\u6027\u4e5f\u6709\u6548\uff0c\u56e0\u4e3a\u7edd\u5bf9\u503c\u51fd\u6570\u4e5f\u53ef\u4ee5\u5206\u6bb5\u5b9a\u4e49\u3002\u60a8\u53ef\u4ee5\u5728\u6b64\u5904\u4e86\u89e3\u5982\u4f55\u5c06\u7edd\u5bf9\u503c\u51fd\u6570\u8f6c\u6362\u4e3a\u5757\uff1a<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span><strong>\u8bf7\u53c2\u9605\uff1a<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/cn\/\u5177\u6709\u7edd\u5bf9\u503c\u7684\u51fd\u6570\/\">\u5982\u4f55\u5206\u6bb5\u5b9a\u4e49\u5177\u6709\u7edd\u5bf9\u503c\u7684\u51fd\u6570<\/a><\/span><\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-la-derivabilidad-de-una-funcion\"><\/span>\u89e3\u51b3\u4e86\u51fd\u6570\u53ef\u5fae\u6027\u7684\u7ec3\u4e60<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e601<\/h3>\n<p>\u7814\u7a76\u4ee5\u4e0b\u5206\u6bb5\u51fd\u6570\u7684\u8fde\u7eed\u6027\u548c\u53ef\u5bfc\u6027\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3656065bb8de98bd07da153f26fd326e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)= \\left\\{ \\begin{array}{lcl} x^3-4x^2 + 5 &amp; \\text{si} &amp;  x<1 \\\\[2ex] -x^2+3x &amp; \\text{si} &amp; x\\geq 1 \\end{array} \\right.\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"265\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u4e24\u90e8\u5206\u7684\u51fd\u6570\u662f\u8fde\u7eed\u7684\uff0c\u4f46\u6211\u4eec\u5fc5\u987b\u770b\u51fd\u6570\u5728\u4e34\u754c\u70b9x=1\u5904\u662f\u5426\u8fde\u7eed\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u6c42\u89e3\u51fd\u6570\u5728\u8be5\u70b9\u7684\u6a2a\u5411\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cf8ad0b4baa312a5ae1bb073e5c8ff8b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{x\\to 1^-} f(x) = \\lim\\limits_{x\\to 1^-} \\bigl(x^3-4x^2 + 5\\bigr)=1^3-4\\cdot 1^2 + 5=2\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"413\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f25bd83439fbb5cbd148b8be88f8770b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{x\\to 1^+} f(x) = \\lim\\limits_{x\\to 1^+} \\bigl( -x^2+3x \\bigr)=-1^2+3\\cdot 1=2\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"364\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u4e34\u754c\u70b9\u5904\u7684\u4e24\u4e2a\u6a2a\u5411\u6781\u9650\u7ed9\u51fa\u76f8\u540c\u7684\u7ed3\u679c\uff0c\u56e0\u6b64\u8be5\u51fd\u6570\u5728 x=1 \u5904\u8fde\u7eed\u3002<\/p>\n<p class=\"has-text-align-left\">\u4e00\u65e6\u6211\u4eec\u77e5\u9053\u51fd\u6570\u5728\u4e34\u754c\u70b9\u8fde\u7eed\uff0c\u6211\u4eec\u5c31\u4f1a\u7814\u7a76\u5b83\u5728\u540c\u4e00\u70b9\u662f\u5426\u53ef\u5fae\u3002\u56e0\u6b64\u6211\u4eec\u8ba1\u7b97\u6a2a\u5411\u5bfc\u6570\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-42451fa799527167fe9a2e2259248870_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)= \\left\\{ \\begin{array}{lcl} 3x^2-8x  &amp; \\text{si} &amp;  x<1 \\\\[2ex] -2x+3 &amp; \\text{si} &amp; x\\geq 1 \\end{array} \\right.\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"240\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6211\u4eec\u8bc4\u4f30 x=1 \u5904\u7684\u4e24\u4e2a\u4fa7\u5bfc\u6570\uff1b <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c9afcac5ff2f762d2b471725d4e755fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1^-)=3\\cdot1^2-8\\cdot 1=3-8=-5\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"272\" 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-dd2d9e21e09ecc434cafbf5fb0b5afaa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1^+)=-2\\cdot 1+3=-2+3 =1\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"256\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u4fa7\u5bfc\u6570\u5728 x=1 \u70b9\u5904\u4e0d\u91cd\u5408\uff0c\u56e0\u6b64\u51fd\u6570\u5728\u8be5\u70b9\u4e0d\u53ef\u5fae\u3002 <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-73b9cb3dada6f8aea03ffc1342d0f22f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1^-) \\neq f'(1^+) \\ \\longrightarrow \\ \\cancel{\\exists} \\ f'(1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"225\" 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\">\u7ec3\u4e602<\/h3>\n<p>\u5206\u6790\u5404\u8282\u4e2d\u5b9a\u4e49\u7684\u4ee5\u4e0b\u51fd\u6570\u7684\u53ef\u5fae\u6027\u548c\u8fde\u7eed\u6027\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0d118e3904c810abd15e427e9c7d0504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)= \\left\\{ \\begin{array}{lcl} \\sqrt{4x} &amp; \\text{si} &amp;  x\\leq 1 \\\\[2ex] 2+\\ln x &amp; \\text{si} &amp; x> 1 \\end{array} \\right.&#8221; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;65&#8243; width=&#8221;226&#8243; style=&#8221;vertical-align: 0px;&#8221;><\/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>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u4e24\u6bb5\u51fd\u6570\u5728\u5176\u533a\u95f4\u4e0a\u662f\u8fde\u7eed\u7684\uff0c\u4f46\u8fd8\u9700\u8981\u77e5\u9053\u51fd\u6570\u5728\u5b9a\u4e49 x=1 \u53d8\u5316\u7684\u4e34\u754c\u70b9\u662f\u5426\u8fde\u7eed\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u6b64\u65f6\u5b9a\u4e49\u51fd\u6570\u7684\u6a2a\u5411\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2c4fb7ef0ee9b3feeb5e15654528fc71_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{x\\to 1^-} f(x) = \\lim\\limits_{x\\to 1^-} \\sqrt{4x} = \\sqrt{4\\cdot 1} = \\sqrt{4}=2\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"323\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e509a70bd0e356f44ccac7ba6f075f8b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{x\\to 1^+} f(x) = \\lim\\limits_{x\\to 1^+} \\bigl( 2+\\ln x \\bigr) = 2 + \\ln (1) = 2+0 =2\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"399\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u4e34\u754c\u70b9\u5904\u7684\u4e24\u4e2a\u6a2a\u5411\u6781\u9650\u7ed9\u51fa\u76f8\u540c\u7684\u7ed3\u679c\uff0c\u56e0\u6b64\u8be5\u51fd\u6570\u5728 x=1 \u5904\u8fde\u7eed\u3002<\/p>\n<p class=\"has-text-align-left\">\u73b0\u5728\u6211\u4eec\u901a\u8fc7\u8ba1\u7b97\u4fa7\u5bfc\u6570\u6765\u7814\u7a76\u51fd\u6570\u6b64\u65f6\u662f\u5426\u53ef\u5fae\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8261f3d268b47d9171710997c8cc70bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)= \\left\\{ \\begin{array}{lcl} \\cfrac{4}{2\\sqrt{4x}}  &amp; \\text{si} &amp;  x<1 \\\\[4ex] \\cfrac{1}{x} &amp; \\text{si} &amp; x\\geq 1 \\end{array} \\right.\" title=\"Rendered by QuickLaTeX.com\" height=\"108\" width=\"217\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6211\u4eec\u8ba1\u7b97 x=1 \u5904\u7684\u4e24\u4e2a\u4fa7\u5bfc\u6570\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-de4ee71a175b34be07061d47470afe0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1^-)=\\cfrac{4}{2\\sqrt{4\\cdot1}}=\\cfrac{4}{2\\sqrt{4}}=\\cfrac{4}{2\\cdot 2}=\\cfrac{4}{4}=1\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"309\" 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-594a8d05da432ab8962ff799d62d25a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1^+)=\\cfrac{1}{1}=1\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"115\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u4fa7\u5bfc\u6570\u76f8\u7b49\uff0c\u56e0\u6b64\u51fd\u6570\u5728 x=1 \u5904\u53ef\u5fae\uff0c\u5bfc\u6570\u503c\u4e3a 1\u3002 <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a58ba22b03193839f5070da4e2c1faf5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1^-) = f'(1^+) = 1 \\ \\longrightarrow \\ \\bm{f'(1) = 1}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"274\" 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\">\u7ec3\u4e603<\/h3>\n<p>\u786e\u5b9a\u4ee5\u4e0b\u5206\u6bb5\u51fd\u6570\u5728\u5176\u6574\u4e2a\u57df\u4e0a\u662f\u5426\u8fde\u7eed\u4e14\u53ef\u5fae\uff1a<\/p>\n<pre class=\"ql-errors\"> *** QuickLaTeX cannot compile formula:\n\\displaystyle f(x)= \\left\\{ \\begin{array}{lcl} x^2+2x+1 &amp; \\text{si} &amp; x\\leq -1 \\\\[2ex] 2x+2 &amp; \\text{ si} &amp; -1&lt;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\"&gt;&lt;div class=\"otfm-sp__title\"&gt; &lt;strong&gt;View solution&lt;\/strong&gt;&lt;\/div&gt;&lt; \/div&gt; The functions of all three parts are continuous, but we still need to check if the function is continuous at critical points. We therefore first check the continuity of the function at the point x=-1 by solving the lateral limits at this point:\n\n*** Error message:\nMissing $ inserted.\nleading text: \\displaystyle\nMissing { inserted.\nleading text: ...=\"wp-block-otfm-box-spoiler-start otfm-sp__\nMissing { inserted.\nleading text: ...ox-spoiler-start otfm-sp__wrapper otfm-sp__\nMissing { inserted.\nleading text: ...m-sp__wrapper otfm-sp__box js-otfm-sp-box__\nMissing { inserted.\nleading text: ...fm-sp__box js-otfm-sp-box__closed otfm-sp__\nYou can't use `macro parameter character #' in math mode.\nleading text: ...=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#\nMissing { inserted.\nleading text: ...e=\"text-align:center\"&gt;&lt;div class=\"otfm-sp__\nPlease use \\mathaccent for accents in math mode.\nleading text: ...g&gt;&lt;\/div&gt;&lt;\/div&gt; The functions of the three parts\nPlease use \\mathaccent for accents in math mode.\nleading text: ...are continuous, but we still need to see\n\n<\/pre>\n<p> \\lim\\limits_{x\\to -1^-} f(x) = \\lim\\limits_{x\\to -1^-} \\bigl(x^2+2x+1\\bigr) = (-1)^ 2+2(-1)+1 =0 \\lim\\limits_{x\\to -1^+} f(x) = \\lim\\limits_{x\\to -1^+} \\bigl(2x+2\\bigr ) = 2(-1)+2=0<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fadd0ce26a497a6a6c73bfaa7ed28f4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\" Les deux limites lat\u00e9rales au point x=-1 donnent le m\u00eame r\u00e9sultat, donc la fonction est continue en x=-1. Nous allons maintenant v\u00e9rifier si la fonction est continue ou non au point x=2 : \" title=\"Rendered by QuickLaTeX.com\" height=\"56\" width=\"765\" style=\"vertical-align: 0px;\"><\/p>\n<p> \\lim\\limits_{x\\to 2^-} f(x) = \\lim\\limits_{x\\to 2^-} \\bigl(2x+2\\bigr) = 2\\cdot 2+2=4+2= 6 \\lim\\limits_{x\\to 2^+} f(x) = \\lim\\limits_{x\\to 2^+} \\bigl( -x^2+8x\\bigr) = -2^2+8\\ cdot 2 = -4+16=12<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c93f9d5e367031de798abaf833523710_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\" En revanche, les limites lat\u00e9rales au point x=2 ne donnent pas le m\u00eame r\u00e9sultat, donc la fonction n'est pas continue en x=2. De plus, comme il n'est pas continu \u00e0 ce stade, il ne sera pas non plus d\u00e9rivable \u00e0 x=2. Une fois que l'on a \u00e9tudi\u00e9 la continuit\u00e9 de la fonction, on passe \u00e0 la diff\u00e9rentiabilit\u00e9. On calcule donc les d\u00e9riv\u00e9es lat\u00e9rales :\" title=\"Rendered by QuickLaTeX.com\" height=\"82\" width=\"908\" style=\"vertical-align: -4px;\"><\/p>\n<p> \\displaystyle f'(x)= \\left\\{ \\begin{array}{lcl} 2x+2 &amp; \\text{si} &amp; x\\leq -1 \\\\[2ex] 2 &amp; \\text{si} &amp; -1<\/p>\n<p class=\"has-text-align-left\">\u6211\u4eec\u5df2\u7ecf\u77e5\u9053\u51fd\u6570\u5728 x=2 \u65f6\u4e0d\u53ef\u5fae\uff0c\u56e0\u6b64\u6211\u4eec\u53ea\u9700\u8981\u7814\u7a76\u51fd\u6570\u5728 x=-1 \u65f6\u662f\u5426\u53ef\u5fae\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u8bc4\u4f30\u8be5\u70b9\u7684\u4e24\u4e2a\u4fa7\u5bfc\u6570\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cafb8bbe438865e050973664b6915fa9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(-1^-)=2(-1)+2 = -2+2=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"272\" 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-5781d213b36777be5b2611a84ca95e96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(-1^+)=2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"95\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u4fa7\u5bfc\u6570\u5728 x=-1 \u70b9\u4e0d\u91cd\u5408\uff0c\u56e0\u6b64\u51fd\u6570\u5728\u8be5\u70b9\u4e0d\u53ef\u5fae\u3002 <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f3674d70c292d967d7074a0b4bee230e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(-1^-) \\neq f'(-1^+) \\ \\longrightarrow \\ \\cancel{\\exists} \\ f'(-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"267\" 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\">\u7ec3\u4e604<\/h3>\n<p>\u8ba1\u7b97\u53c2\u6570 a \u548c b \u7684\u503c\uff0c\u4f7f\u4ee5\u4e0b\u5206\u6bb5\u51fd\u6570\u5728\u5176\u6574\u4e2a\u57df\u5185\u8fde\u7eed\u4e14\u53ef\u5fae\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ce34d5d8a949fb3a0b904e9bf7d32f5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)= \\left\\{ \\begin{array}{lcl} 2e^{x-3} + a &amp; \\text{si} &amp;  x< 3 \\\\[2ex](x-b)^2 &amp; \\text{si} &amp; x\\geq 3 \\end{array} \\right.\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"243\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u65e0\u8bba\u672a\u77e5\u6570\u7684\u503c\u662f\u591a\u5c11\uff0c\u51fd\u6570\u5728\u9664 x=3 \u5904\u4ee5\u5916\u7684\u6240\u6709\u70b9\u4e0a\u90fd\u662f\u8fde\u7eed\u4e14\u53ef\u5fae\u7684\uff0c\u5fc5\u987b\u68c0\u67e5\u5176\u8fde\u7eed\u6027\u548c\u53ef\u5fae\u6027\u3002<\/p>\n<p class=\"has-text-align-left\">\u4e3a\u4e86\u4f7f\u51fd\u6570\u5728\u4e00\u70b9\u5904\u8fde\u7eed\uff0c\u8be5\u70b9\u5904\u7684\u4e24\u4e2a\u6a2a\u5411\u6781\u9650\u5fc5\u987b\u91cd\u5408\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u8bc4\u4f30\u4e34\u754c\u70b9\u7684\u6a2a\u5411\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-21920afd0fe6c6a35983a39034b3f9f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{x\\to 3^-} f(x) = \\lim\\limits_{x\\to 3^-} \\bigl(2e^{x-3}+a\\bigr) = 2e^{3-3}+a = 2 \\cdot e^0+a =2\\cdot 1 +a = 2+a\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"566\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f527d4602a1bdc26d763df1064b956d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{x\\to 3^+} f(x) = \\lim\\limits_{x\\to 3^+} (x-b)^2 = (3-b)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"282\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u56e0\u6b64\uff0c\u4ece\u6a2a\u5411\u6781\u9650\u83b7\u5f97\u7684\u4e24\u4e2a\u503c\u5fc5\u987b\u76f8\u7b49\uff0c\u51fd\u6570\u624d\u80fd\u8fde\u7eed\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3bec8303f39289b7f2cd7e6e439703c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2+a = (3-b)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"123\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u73b0\u5728\u6211\u4eec\u5c06\u5206\u6790\u70b9 x=3 \u5904\u7684\u53ef\u5fae\u6027\u3002\u6211\u4eec\u6c42\u51fa\u6a2a\u5411\u5bfc\u6570\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d542fc9488644f0c144059ae1403d961_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)= \\left\\{ \\begin{array}{lcl} 2e^{x-3}  &amp; \\text{si} &amp;  x< 3 \\\\[2ex]2(x-b) &amp; \\text{si} &amp; x\\geq 3 \\end{array} \\right.\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"236\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6211\u4eec\u8bc4\u4f30\u4e34\u754c\u70b9\u5904\u7684\u4e24\u4e2a\u4fa7\u5bfc\u6570\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0fb3afef2352f0f5f96302b987a5de9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(3^-)= 2e^{3-3} =  2e^0 = 2\\cdot 1 = 2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"250\" 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-360474b477f347569ad1e3b64b63cc79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(3^+)=2(3-b) = 6 - 2b\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"205\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u56e0\u6b64\uff0c\u4e3a\u4e86\u4f7f\u51fd\u6570\u5728 x=3 \u5904\u53ef\u5fae\uff0c\u4ece\u4fa7\u5bfc\u6570\u83b7\u5f97\u7684\u503c\u5fc5\u987b\u76f8\u7b49\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8a98a45230845ad86846cd7db486af0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2=6-2b\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"80\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u901a\u8fc7\u89e3\u8fd9\u4e2a\u65b9\u7a0b\u6211\u4eec\u53ef\u4ee5\u627e\u5230 b \u7684\u503c\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7f53d30cae8270029d25ea28322b6986_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2b=6-2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"79\" 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-c263311ac7433ce2b417bb7ad0ef449b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2b=4\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" 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-7dd2f0fd5e5207a815bf5789ada67541_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b=\\cfrac{4}{2} =\\bm{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"74\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6700\u540e\uff0c\u4e00\u65e6\u6211\u4eec\u77e5\u9053\u4e86\u53c2\u6570b\u7684\u503c\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u901a\u8fc7\u6c42\u89e3\u4e4b\u524d\u5728\u6a2a\u5411\u6781\u9650\u4e2d\u5f97\u5230\u7684\u65b9\u7a0b\u6765\u8ba1\u7b97\u53c2\u6570a\u7684\u503c\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3bec8303f39289b7f2cd7e6e439703c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2+a = (3-b)^2\" 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-6468721c373f078ad3e97a290c2d86f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2+a = (3-2)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"124\" 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-d1cc01602cd205cae7b9c9b8ba391760_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2+a =1\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"72\" 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-ce22cc1bba3ee9612a7f8cb2624d2483_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a =1-2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"72\" 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-be5f9e4b074e8b3ba5d0e96f8ae4e2cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{a =-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"55\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u5728\u672c\u6587\u4e2d\uff0c\u60a8\u5c06\u5b66\u4e60\u5982\u4f55\u7814\u7a76\u51fd\u6570\u7684\u53ef\u5fae\u6027\uff0c\u5373\u51fd\u6570\u662f\u5426\u53ef\u5fae\u3002\u6b64\u5916\uff0c\u6211\u4eec\u8fd8\u5c06\u770b\u5230\u51fd\u6570\u7684\u53ef\u5fae\u6027\u548c\u8fde\u7eed\u6027\u4e4b\u95f4\u7684\u5173\u7cfb\u3002\u6700\u540e &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" 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