{"id":338,"date":"2025-07-21T02:33:04","date_gmt":"2025-07-21T02:33:04","guid":{"rendered":"https:\/\/giapsgk.com\/chuyen-de\/?p=338"},"modified":"2025-07-21T05:12:52","modified_gmt":"2025-07-21T05:12:52","slug":"cac-cong-thuc-toan-12","status":"publish","type":"post","link":"https:\/\/giapsgk.com\/chuyen-de\/cac-cong-thuc-toan-12\/","title":{"rendered":"\u2713 C\u00e1ch h\u1ecdc thu\u1ed9c c\u00e1c c\u00f4ng th\u1ee9c To\u00e1n 12 nhanh (c\u1ea3 n\u0103m, s\u00e1ch m\u1edbi)"},"content":{"rendered":"<p>B\u1ea1n c\u1ea7n <strong>c\u00f4ng th\u1ee9c To\u00e1n 12<\/strong> \u0111\u1ec3 \u00f4n thi THPT nhanh, hi\u1ec7u qu\u1ea3? B\u00e0i vi\u1ebft n\u00e0y t\u1ed5ng h\u1ee3p \u0111\u1ea7y \u0111\u1ee7 t\u1eeb s\u00e1ch to\u00e1n <a href=\"https:\/\/giapsgk.com\/chuyen-de\/230-2\/\">c\u00e1nh di\u1ec1u<\/a>; <a href=\"https:\/\/giapsgk.com\/chuyen-de\/he-thong-kien-thuc-toan-12-chan-troi-sang-tao-hk1-va-hk2\/\">ch\u00e2n tr\u1eddi s\u00e1ng t\u1ea1o<\/a> v\u00e0 <a href=\"https:\/\/giapsgk.com\/chuyen-de\/tom-tat-kien-thuc-va-cong-thuc-toan-12-sach-ket-noi-tri-thuc-hk1-va-hk2\/\">s\u00e1ch K\u1ebft n\u1ed1i tri th\u1ee9c<\/a>, chi ti\u1ebft c\u00e1c c\u00f4ng th\u1ee9c to\u00e1n h\u1ecdc quan tr\u1ecdng l\u1edbp 12, t\u1eeb \u1ee9ng d\u1ee5ng \u0111\u1ea1o h\u00e0m, t\u1edbi t\u00ecm nguy\u00ean h\u00e0m t\u00edch ph\u00e2n hay h\u00ecnh h\u1ecdc kh\u00f4ng gian. V\u1edbi c\u00e1ch tr\u00ecnh b\u00e0y d\u1ec5 hi\u1ec3u, v\u00ed d\u1ee5 minh h\u1ecda r\u00f5 r\u00e0ng, b\u1ea1n s\u1ebd nhanh ch\u00f3ng n\u1eafm v\u1eefng ki\u1ebfn th\u1ee9c v\u00e0 t\u1ef1 tin chinh ph\u1ee5c m\u1ecdi b\u00e0i to\u00e1n kh\u00f3. H\u00e3y c\u00f9ng kh\u00e1m ph\u00e1 ngay!<\/p>\n<h2><strong>CHUY\u00caN \u0110\u1ec0 1: \u1ee8NG D\u1ee4NG \u0110\u1ea0O H\u00c0M \u2013 KH\u1ea2O S\u00c1T H\u00c0M S\u1ed0<\/strong><\/h2>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-339 size-medium\" src=\"https:\/\/giapsgk.com\/chuyen-de\/wp-content\/uploads\/image-5-300x218.png\" alt=\"t\u1ed5ng h\u1ee3p c\u00f4ng th\u1ee9c to\u00e1n l\u1edbp 12 m\u1edbi nh\u1ea5t\" width=\"300\" height=\"218\" srcset=\"https:\/\/giapsgk.com\/chuyen-de\/wp-content\/uploads\/image-5-300x218.png 300w, https:\/\/giapsgk.com\/chuyen-de\/wp-content\/uploads\/image-5.png 346w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<h3><strong>1.1. \u0110\u01a1n \u0111i\u1ec7u c\u1ee7a h\u00e0m s\u1ed1<\/strong><\/h3>\n<p><strong>\u0110\u1ecbnh ngh\u0129a:<\/strong><\/p>\n<ul>\n<li>H\u00e0m s\u1ed1 $y = f(x)$ \u0111\u1ed3ng bi\u1ebfn tr\u00ean kho\u1ea3ng $I$ n\u1ebfu $x_1 &lt; x_2 \\Rightarrow f(x_1) &lt; f(x_2)$<\/li>\n<li>Ngh\u1ecbch bi\u1ebfn n\u1ebfu $x_1 &lt; x_2 \\Rightarrow f(x_1) &gt; f(x_2)$<\/li>\n<\/ul>\n<p><strong>Li\u00ean h\u1ec7 v\u1edbi \u0111\u1ea1o h\u00e0m:<\/strong><\/p>\n<ul>\n<li>N\u1ebfu $f'(x) &gt; 0$ tr\u00ean kho\u1ea3ng $I$ \u2192 h\u00e0m s\u1ed1 \u0111\u1ed3ng bi\u1ebfn<\/li>\n<li>N\u1ebfu $f'(x) &lt; 0$ tr\u00ean kho\u1ea3ng $I$ \u2192 h\u00e0m s\u1ed1 ngh\u1ecbch bi\u1ebfn<\/li>\n<\/ul>\n<h3><strong>1.2. C\u1ef1c tr\u1ecb c\u1ee7a h\u00e0m s\u1ed1<\/strong><\/h3>\n<p><strong>\u0110\u1ecbnh ngh\u0129a:<\/strong><\/p>\n<ul>\n<li>$x = a$ l\u00e0 \u0111i\u1ec3m c\u1ef1c \u0111\u1ea1i n\u1ebfu $f(a) &gt; f(x)$ v\u1edbi m\u1ecdi $x$ g\u1ea7n $a$<\/li>\n<li>$x = a$ l\u00e0 \u0111i\u1ec3m c\u1ef1c ti\u1ec3u n\u1ebfu $f(a) &lt; f(x)$ v\u1edbi m\u1ecdi $x$ g\u1ea7n $a$<\/li>\n<\/ul>\n<p><strong>\u0110i\u1ec1u ki\u1ec7n:<\/strong><\/p>\n<ul>\n<li>$f'(a) = 0$ v\u00e0 \u0111\u1ed5i d\u1ea5u quanh $a$<\/li>\n<li>B\u1ea3ng bi\u1ebfn thi\u00ean \u0111\u1ec3 x\u00e1c \u0111\u1ecbnh lo\u1ea1i c\u1ef1c tr\u1ecb<\/li>\n<\/ul>\n<h3><strong>1.3. Gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t \u2013 nh\u1ecf nh\u1ea5t<\/strong><\/h3>\n<p>B\u01b0\u1edbc gi\u1ea3i:<\/p>\n<p><strong>B\u01b0\u1edbc 1.<\/strong> T\u00ednh \u0111\u1ea1o h\u00e0m $f'(x)$<\/p>\n<p><strong>B\u01b0\u1edbc 2.<\/strong> T\u00ecm nghi\u1ec7m $f'(x) = 0$<\/p>\n<p><strong>B\u01b0\u1edbc\u00a03.<\/strong> L\u1eadp b\u1ea3ng bi\u1ebfn thi\u00ean, x\u00e9t tr\u00ean \u0111o\u1ea1n $[a, b]$<\/p>\n<p><strong>B\u01b0\u1edbc 4.<\/strong> So s\u00e1nh $f(a), f(b), f(x_0)$ (v\u1edbi $x_0$ l\u00e0 c\u1ef1c tr\u1ecb) \u2192 ch\u1ecdn GTLN, GTNN<\/p>\n<h3><strong>1.4. \u0110\u01b0\u1eddng ti\u1ec7m c\u1eadn<\/strong><\/h3>\n<p><strong>Ti\u1ec7m c\u1eadn \u0111\u1ee9ng: <\/strong>N\u1ebfu h\u00e0m s\u1ed1 c\u00f3 d\u1ea1ng ph\u00e2n th\u1ee9c v\u00e0 m\u1eabu $Q(x) = 0$ \u2192 c\u00f3 ti\u1ec7m c\u1eadn \u0111\u1ee9ng t\u1ea1i nghi\u1ec7m \u0111\u00f3<\/p>\n<p>V\u00ed d\u1ee5: $\\lim_{x \\to a^\\pm} f(x) = \\pm\\infty$<\/p>\n<p><strong>Ti\u1ec7m c\u1eadn ngang:<\/strong> $\\lim_{x \\to \\infty} f(x) = L \\Rightarrow y = L$ l\u00e0 TCN<\/p>\n<p>So s\u00e1nh b\u1eadc t\u1eed v\u00e0 b\u1eadc m\u1eabu \u0111\u1ec3 t\u00ecm TCN<\/p>\n<h3><strong>1.5. Kh\u1ea3o s\u00e1t v\u00e0 v\u1ebd \u0111\u1ed3 th\u1ecb<\/strong><\/h3>\n<p><strong>T\u1eadp x\u00e1c \u0111\u1ecbnh<\/strong><\/p>\n<p><strong>S\u1ef1 bi\u1ebfn thi\u00ean:<\/strong> \u0111\u1ea1o h\u00e0m, b\u1ea3ng bi\u1ebfn thi\u00ean<\/p>\n<p><strong>C\u1ef1c tr\u1ecb<\/strong><\/p>\n<ul>\n<li>Ti\u1ec7m c\u1eadn<\/li>\n<li>\u0110\u1ed3 th\u1ecb \u0111\u1ed1i x\u1ee9ng tr\u1ee5c\/t\u00e2m n\u1ebfu c\u00f3<\/li>\n<\/ul>\n<h2><strong>CHUY\u00caN \u0110\u1ec0 2: NGUY\u00caN H\u00c0M \u2013 T\u00cdCH PH\u00c2N \u2013 \u1ee8NG D\u1ee4NG<\/strong><\/h2>\n<h3><strong>2.1. Nguy\u00ean h\u00e0m<\/strong><\/h3>\n<p><strong>\u0110\u1ecbnh ngh\u0129a: <\/strong>$F(x)$ l\u00e0 nguy\u00ean h\u00e0m c\u1ee7a $f(x)$ n\u1ebfu $F'(x) = f(x)$<\/p>\n<p><strong>C\u00f4ng th\u1ee9c c\u01a1 b\u1ea3n:<\/strong><\/p>\n<p>$ \\begin{align} \\int x^n dx &amp;= \\frac{x^{n+1}}{n+1} + C \\quad (n \\ne -1) \\\\ \\int \\frac{1}{x} dx &amp;= \\ln|x| + C \\\\ \\int e^x dx &amp;= e^x + C \\\\ \\int \\cos x dx &amp;= \\sin x + C \\\\ \\int \\sin x dx &amp;= -\\cos x + C \\\\ \\int \\frac{1}{1 + x^2} dx &amp;= \\arctan x + C \\end{align} $<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-342 size-full\" src=\"https:\/\/giapsgk.com\/chuyen-de\/wp-content\/uploads\/giapsgk-cac-cong-thuc-toan-12-1.png\" alt=\"c\u00e1ch h\u1ecdc thu\u1ed9c c\u00f4ng th\u1ee9c to\u00e1n l\u1edbp 12 nhanh\" width=\"905\" height=\"682\" \/><\/p>\n<h3><strong>2.2. T\u00edch ph\u00e2n<\/strong><\/h3>\n<p>T\u00edch ph\u00e2n x\u00e1c \u0111\u1ecbnh: $ \\int_a^b f(x)\\, dx = F(b) &#8211; F(a) $<\/p>\n<p>Di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1eefa \u0111\u1ed3 th\u1ecb v\u00e0 tr\u1ee5c ho\u00e0nh<\/p>\n<p>T\u00ednh ch\u1ea5t c\u1ea7n nh\u1edb: $ \\int_a^b f(x) dx = -\\int_b^a f(x) dx, \\quad \\int_a^a f(x) dx = 0 $<\/p>\n<h3><strong>2.3. \u1ee8ng d\u1ee5ng h\u00ecnh h\u1ecdc<\/strong><\/h3>\n<p>T\u00ednh di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng: $ S = \\int_a^b |f(x) &#8211; g(x)| dx $<\/p>\n<p>Th\u1ec3 t\u00edch kh\u1ed1i tr\u00f2n xoay (quanh Ox): $ V = \\pi \\int_a^b [f(x)]^2 dx $<\/p>\n<h2><strong>CHUY\u00caN \u0110\u1ec0 3: H\u00ccNH H\u1eccC KH\u00d4NG GIAN T\u1eccA \u0110\u1ed8<\/strong><\/h2>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-344 size-full\" src=\"https:\/\/giapsgk.com\/chuyen-de\/wp-content\/uploads\/giapsgk-cac-cong-thuc-toan-12-2.png\" alt=\"t\u00e0i li\u1ec7u c\u00f4ng th\u1ee9c to\u00e1n 12 \u00f4n thi \u0111\u1ea1i h\u1ecdc\" width=\"472\" height=\"426\" \/><\/p>\n<h3><strong>3.1. Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng<\/strong><\/h3>\n<ul>\n<li>D\u1ea1ng t\u1ed5ng qu\u00e1t: $ Ax + By + Cz + D = 0 $<\/li>\n<li>VTPT: $\\vec{n} = (A, B, C)$<\/li>\n<\/ul>\n<h3><strong>3.2. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng<\/strong><\/h3>\n<ul>\n<li><strong>D\u1ea1ng tham s\u1ed1<\/strong>: $ \\left\\{ \\begin{array}{l} x = x_0 + at \\\\ y = y_0 + bt \\\\ z = z_0 + ct \\end{array} \\right. $<\/li>\n<li><strong>D\u1ea1ng ch\u00ednh t\u1eafc<\/strong>: $\\frac{x &#8211; x_0}{a} = \\frac{y &#8211; y_0}{b} = \\frac{z &#8211; z_0}{c}$<\/li>\n<\/ul>\n<h3><strong>3.3. Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t c\u1ea7u<\/strong><\/h3>\n<p>D\u1ea1ng: $(x &#8211; a)^2 + (y &#8211; b)^2 + (z &#8211; c)^2 = R^2$<\/p>\n<h3><strong>3.4. G\u00f3c \u2013 Kho\u1ea3ng c\u00e1ch<\/strong><\/h3>\n<p>G\u00f3c gi\u1eefa hai vect\u01a1: $\\cos\\theta = \\frac{\\vec{u} \\cdot \\vec{v}}{|\\vec{u}||\\vec{v}|}$<\/p>\n<p>Kho\u1ea3ng c\u00e1ch t\u1eeb \u0111i\u1ec3m \u0111\u1ebfn m\u1eb7t ph\u1eb3ng: $d = \\frac{|Ax_0 + By_0 + Cz_0 + D|}{\\sqrt{A^2 + B^2 + C^2}}$<\/p>\n<h2><strong>CHUY\u00caN \u0110\u1ec0 4: X\u00c1C SU\u1ea4T \u2013 TH\u1ed0NG K\u00ca<\/strong><\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-345 size-full\" src=\"https:\/\/giapsgk.com\/chuyen-de\/wp-content\/uploads\/giapsgk-cac-cong-thuc-toan-12-3.png\" alt=\"c\u00f4ng th\u1ee9c c\u1ea7n nh\u1edb to\u00e1n 12 thi THPT qu\u1ed1c gia\" width=\"714\" height=\"399\" \/><\/p>\n<h3><strong>4.1. X\u00e1c su\u1ea5t c\u00f3 \u0111i\u1ec1u ki\u1ec7n<\/strong><\/h3>\n<p>$P(A|B) = \\frac{P(A \\cap B)}{P(B)}$<\/p>\n<h3><strong>4.2. C\u00f4ng th\u1ee9c x\u00e1c su\u1ea5t to\u00e0n ph\u1ea7n<\/strong><\/h3>\n<p>$P(A) = \\sum_{i=1}^{n} P(A|B_i) \\cdot P(B_i)$<\/p>\n<h3><strong>4.3. C\u00f4ng th\u1ee9c Bayes<\/strong><\/h3>\n<p>$P(B_j|A) = \\frac{P(A|B_j) \\cdot P(B_j)}{\\sum_{i=1}^{n} P(A|B_i) \\cdot P(B_i)}$<\/p>\n<h3><strong>4.4. Th\u1ed1ng k\u00ea \u2013 m\u1ee9c \u0111\u1ed9 ph\u00e2n t\u00e1n<\/strong><\/h3>\n<ul>\n<li><strong>Ph\u01b0\u01a1ng sai:<\/strong> $ \\sigma^2 = \\frac{1}{n} \\sum_{i=1}^n (x_i &#8211; \\bar{x})^2 $<\/li>\n<li><strong>\u0110\u1ed9 l\u1ec7ch chu\u1ea9n:<\/strong> $ \\sigma = \\sqrt{\\sigma^2} $<\/li>\n<li><strong>Kho\u1ea3ng bi\u1ebfn thi\u00ean:<\/strong> $ R = x_{\\text{max}} &#8211; x_{\\text{min}} $<\/li>\n<li><strong>Kho\u1ea3ng t\u1ee9 ph\u00e2n v\u1ecb:<\/strong> ${I_{QR}} = {Q_3} &#8211; {Q_1}$<\/li>\n<\/ul>\n<h2><strong>K\u1ebeT LU\u1eacN: T\u00d3M T\u1eaeT C\u00c1C CH\u1ee6 \u0110\u1ec0 C\u1ea6N \u00d4N<\/strong><\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-346 size-full\" src=\"https:\/\/giapsgk.com\/chuyen-de\/wp-content\/uploads\/giapsgk-cac-cong-thuc-toan-12-4.png\" alt=\"T\u1ed5ng h\u1ee3p c\u00f4ng th\u1ee9c To\u00e1n 12 theo chuy\u00ean \u0111\u1ec1 \u2013 D\u1ec5 h\u1ecdc, d\u1ec5 nh\u1edb, s\u00e1t \u0111\u1ec1 thi t\u1ed1t nghi\u1ec7p\" width=\"728\" height=\"278\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>B\u1ea1n c\u1ea7n c\u00f4ng th\u1ee9c To\u00e1n 12 \u0111\u1ec3 \u00f4n thi THPT nhanh, hi\u1ec7u qu\u1ea3? B\u00e0i vi\u1ebft n\u00e0y t\u1ed5ng h\u1ee3p \u0111\u1ea7y \u0111\u1ee7 t\u1eeb s\u00e1ch to\u00e1n c\u00e1nh di\u1ec1u; ch\u00e2n tr\u1eddi s\u00e1ng t\u1ea1o v\u00e0 s\u00e1ch K\u1ebft n\u1ed1i tri th\u1ee9c, chi ti\u1ebft c\u00e1c c\u00f4ng th\u1ee9c to\u00e1n h\u1ecdc quan tr\u1ecdng l\u1edbp 12, t\u1eeb \u1ee9ng d\u1ee5ng \u0111\u1ea1o h\u00e0m, t\u1edbi t\u00ecm nguy\u00ean h\u00e0m &#8230; <a title=\"\u2713 C\u00e1ch h\u1ecdc thu\u1ed9c c\u00e1c c\u00f4ng th\u1ee9c To\u00e1n 12 nhanh (c\u1ea3 n\u0103m, s\u00e1ch m\u1edbi)\" class=\"read-more\" href=\"https:\/\/giapsgk.com\/chuyen-de\/cac-cong-thuc-toan-12\/\" aria-label=\"\u0110\u1ecdc th\u00eam v\u1ec1 \u2713 C\u00e1ch h\u1ecdc thu\u1ed9c c\u00e1c c\u00f4ng th\u1ee9c To\u00e1n 12 nhanh (c\u1ea3 n\u0103m, s\u00e1ch m\u1edbi)\">\u0110\u1ecdc ti\u1ebfp<\/a><\/p>\n","protected":false},"author":1,"featured_media":351,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[],"class_list":["post-338","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-ly-thuyet"],"_links":{"self":[{"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/posts\/338","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/comments?post=338"}],"version-history":[{"count":10,"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/posts\/338\/revisions"}],"predecessor-version":[{"id":361,"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/posts\/338\/revisions\/361"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/media\/351"}],"wp:attachment":[{"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/media?parent=338"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/categories?post=338"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/tags?post=338"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}