{"id":245,"date":"2025-07-19T09:52:37","date_gmt":"2025-07-19T09:52:37","guid":{"rendered":"https:\/\/giapsgk.com\/chuyen-de\/?p=245"},"modified":"2025-07-19T10:09:16","modified_gmt":"2025-07-19T10:09:16","slug":"he-thong-kien-thuc-toan-12-chan-troi-sang-tao-hk1-va-hk2","status":"publish","type":"post","link":"https:\/\/giapsgk.com\/chuyen-de\/he-thong-kien-thuc-toan-12-chan-troi-sang-tao-hk1-va-hk2\/","title":{"rendered":"H\u1ec7 th\u1ed1ng ki\u1ebfn th\u1ee9c To\u00e1n 12 ch\u00e2n tr\u1eddi s\u00e1ng t\u1ea1o HK1 v\u00e0 HK2"},"content":{"rendered":"<p>Ch\u01b0\u01a1ng tr\u00ecnh To\u00e1n l\u1edbp 12 theo b\u1ed9 s\u00e1ch <strong data-start=\"1043\" data-end=\"1065\">Ch\u00e2n tr\u1eddi s\u00e1ng t\u1ea1o<\/strong> c\u00f3 nhi\u1ec1u n\u1ed9i dung quan tr\u1ecdng li\u00ean quan tr\u1ef1c ti\u1ebfp \u0111\u1ebfn k\u1ef3 thi t\u1ed1t nghi\u1ec7p THPT. \u0110\u1ec3 gi\u00fap h\u1ecdc sinh d\u1ec5 d\u00e0ng n\u1eafm b\u1eaft v\u00e0 ghi nh\u1edb, b\u00e0i vi\u1ebft n\u00e0y s\u1ebd h\u1ec7 th\u1ed1ng to\u00e0n b\u1ed9 ki\u1ebfn th\u1ee9c To\u00e1n 12 c\u1ee7a <strong data-start=\"1243\" data-end=\"1267\">h\u1ecdc k\u1ef3 1 v\u00e0 h\u1ecdc k\u1ef3 2<\/strong>, b\u00e1m s\u00e1t n\u1ed9i dung s\u00e1ch gi\u00e1o khoa m\u1edbi. Bao g\u1ed3m ph\u1ea7n <strong data-start=\"1319\" data-end=\"1341\">l\u00fd thuy\u1ebft ng\u1eafn g\u1ecdn<\/strong>, <strong data-start=\"1343\" data-end=\"1364\">c\u00f4ng th\u1ee9c c\u1ea7n nh\u1edb<\/strong>, v\u00e0 <strong data-start=\"1369\" data-end=\"1399\">c\u00e1c d\u1ea1ng b\u00e0i t\u1eadp \u0111i\u1ec3n h\u00ecnh<\/strong> theo t\u1eebng ch\u01b0\u01a1ng. \u0110\u00e2y s\u1ebd l\u00e0 t\u00e0i li\u1ec7u h\u1ed7 tr\u1ee3 \u0111\u1eafc l\u1ef1c cho h\u1ecdc sinh trong qu\u00e1 tr\u00ecnh \u00f4n luy\u1ec7n, t\u1ef1 h\u1ecdc v\u00e0 ki\u1ec3m tra \u0111\u1ecbnh k\u1ef3.<\/p>\n<h2><strong>Ch\u01b0\u01a1ng 1. \u1ee8ng d\u1ee5ng \u0111\u1ea1o h\u00e0m \u0111\u1ec3 kh\u1ea3o s\u00e1t h\u00e0m s\u1ed1<\/strong><\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-254 size-full aligncenter\" src=\"https:\/\/giapsgk.com\/chuyen-de\/wp-content\/uploads\/giapsgk-he-thong-kien-thuc-toan-12-chan-troi-sang-tao-hk1-va-hk2-2.png\" alt=\"H\u1ec7 th\u1ed1ng to\u00e0n b\u1ed9 ki\u1ebfn th\u1ee9c To\u00e1n 12 Ch\u00e2n tr\u1eddi s\u00e1ng t\u1ea1o theo t\u1eebng ch\u01b0\u01a1ng\" width=\"662\" height=\"469\" \/><\/p>\n<h3><strong>B\u00e0i 1. T\u00ednh \u0111\u01a1n \u0111i\u1ec7u v\u00e0 c\u1ef1c tr\u1ecb c\u1ee7a h\u00e0m s\u1ed1<\/strong><\/h3>\n<p>T\u00ednh \u0111\u01a1n \u0111i\u1ec7u:<\/p>\n<ul>\n<li>H\u00e0m s\u1ed1 \u0111\u1ed3ng bi\u1ebfn tr\u00ean kho\u1ea3ng n\u1ebfu $f'(x) &gt; 0$.<\/li>\n<li>H\u00e0m s\u1ed1 ngh\u1ecbch bi\u1ebfn n\u1ebfu $f'(x) &lt; 0$.<\/li>\n<\/ul>\n<p>C\u1ef1c tr\u1ecb: $f'(x_0) = 0$ v\u00e0 \u0111\u1ed5i d\u1ea5u \u21d2 $f(x_0)$ l\u00e0 c\u1ef1c tr\u1ecb.<\/p>\n<h3><strong>B\u00e0i 2. Gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t, nh\u1ecf nh\u1ea5t<\/strong><\/h3>\n<ul>\n<li>T\u00ecm c\u00e1c \u0111i\u1ec3m c\u1ef1c tr\u1ecb v\u00e0 gi\u00e1 tr\u1ecb t\u1ea1i bi\u00ean \u0111\u1ec3 x\u00e9t GTLN &#8211; GTNN tr\u00ean \u0111o\u1ea1n.<\/li>\n<li>So s\u00e1nh c\u00e1c gi\u00e1 tr\u1ecb: $f(a), f(b), f(x_i)$.<\/li>\n<\/ul>\n<h3><strong>B\u00e0i 3. \u0110\u01b0\u1eddng ti\u1ec7m c\u1eadn c\u1ee7a \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1<\/strong><\/h3>\n<ul>\n<li><strong>Ti\u1ec7m c\u1eadn \u0111\u1ee9ng:<\/strong> $\\lim_{x \\to a^\\pm} f(x) = \\pm\\infty$<\/li>\n<li><strong>Ti\u1ec7m c\u1eadn ngang:<\/strong> $\\lim_{x \\to \\pm\\infty} f(x) = L$<\/li>\n<li><strong>Ti\u1ec7m c\u1eadn xi\u00ean (n\u1ebfu c\u00f3):<\/strong> $y = ax + b$, v\u1edbi $a = \\lim_{x \\to \\infty} \\frac{f(x)}{x},\\ b = \\lim_{x \\to \\infty} (f(x) &#8211; ax)$<\/li>\n<\/ul>\n<h3><strong>B\u00e0i 4. Kh\u1ea3o s\u00e1t v\u00e0 v\u1ebd \u0111\u1ed3 th\u1ecb<\/strong><\/h3>\n<ul>\n<li>X\u00e9t t\u1eadp x\u00e1c \u0111\u1ecbnh, \u0111\u1ea1o h\u00e0m, c\u1ef1c tr\u1ecb, ti\u1ec7m c\u1eadn, b\u1ea3ng bi\u1ebfn thi\u00ean.<\/li>\n<li>C\u00f4ng c\u1ee5 v\u1ebd: \u0110\u1ea1o h\u00e0m b\u1eadc nh\u1ea5t, b\u1eadc hai, ti\u1ec7m c\u1eadn, GTLN\/NN.<\/li>\n<\/ul>\n<h2><strong>Ch\u01b0\u01a1ng 2. Vect\u01a1 v\u00e0 h\u1ec7 t\u1ecda \u0111\u1ed9 trong kh\u00f4ng gian<\/strong><\/h2>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-255 size-full\" src=\"https:\/\/giapsgk.com\/chuyen-de\/wp-content\/uploads\/giapsgk-he-thong-kien-thuc-toan-12-chan-troi-sang-tao-hk1-va-hk2-3.png\" alt=\"Ki\u1ebfn th\u1ee9c tr\u1ecdng t\u00e2m To\u00e1n l\u1edbp 12 Ch\u00e2n tr\u1eddi s\u00e1ng t\u1ea1o h\u1ecdc k\u1ef3 1 v\u00e0 h\u1ecdc k\u1ef3 2\" width=\"445\" height=\"296\" \/><\/p>\n<h3><strong>B\u00e0i 1. Vect\u01a1 v\u00e0 c\u00e1c ph\u00e9p to\u00e1n<\/strong><\/h3>\n<ul>\n<li>Ph\u00e9p c\u1ed9ng, tr\u1eeb, nh\u00e2n vect\u01a1 v\u1edbi s\u1ed1 th\u1ef1c.<\/li>\n<li>T\u00ednh ch\u1ea5t giao ho\u00e1n, k\u1ebft h\u1ee3p.<\/li>\n<\/ul>\n<h3><strong>B\u00e0i 2. T\u1ecda \u0111\u1ed9 vect\u01a1<\/strong><\/h3>\n<ul>\n<li>Trong h\u1ec7 tr\u1ee5c $Oxyz$, vect\u01a1 $\\vec{a} = (a_1; a_2; a_3)$<\/li>\n<li>\u0110\u1ed9 d\u00e0i: $|\\vec{a}| = \\sqrt{a_1^2 + a_2^2 + a_3^2}$<\/li>\n<\/ul>\n<h3><strong>B\u00e0i 3. Bi\u1ec3u th\u1ee9c t\u1ecda \u0111\u1ed9 c\u00e1c ph\u00e9p to\u00e1n<\/strong><\/h3>\n<ul>\n<li>$\\vec{a} + \\vec{b} = (a_1 + b_1, a_2 + b_2, a_3 + b_3)$<\/li>\n<li>T\u00edch v\u00f4 h\u01b0\u1edbng: $\\vec{a} \\cdot \\vec{b} = a_1b_1 + a_2b_2 + a_3b_3$<\/li>\n<\/ul>\n<h2><strong>Ch\u01b0\u01a1ng 3. C\u00e1c s\u1ed1 \u0111\u1eb7c tr\u01b0ng \u0111o \u0111\u1ed9 ph\u00e2n t\u00e1n<\/strong><\/h2>\n<h3><strong>B\u00e0i 1. Kho\u1ea3ng bi\u1ebfn thi\u00ean v\u00e0 t\u1ee9 ph\u00e2n v\u1ecb<\/strong><\/h3>\n<ul>\n<li>Kho\u1ea3ng bi\u1ebfn thi\u00ean: $R = x_{\\text{max}} &#8211; x_{\\text{min}}$<\/li>\n<li>T\u1ee9 ph\u00e2n v\u1ecb: Q1, Q2, Q3 (d\u1ef1a v\u00e0o t\u1ea7n s\u1ed1 gh\u00e9p nh\u00f3m)<\/li>\n<\/ul>\n<h3><strong>B\u00e0i 2. Ph\u01b0\u01a1ng sai \u2013 \u0110\u1ed9 l\u1ec7ch chu\u1ea9n<\/strong><\/h3>\n<p>$\\text{Ph\u01b0\u01a1ng sai: } s^2 = \\frac{1}{n} \\sum f_i(x_i &#8211; \\bar{x})^2$<\/p>\n<p>$\\text{\u0110\u1ed9 l\u1ec7ch chu\u1ea9n: } s = \\sqrt{s^2}$<\/p>\n<h2><strong>Ho\u1ea1t \u0111\u1ed9ng th\u1ef1c h\u00e0nh v\u00e0 tr\u1ea3i nghi\u1ec7m<\/strong><\/h2>\n<h3><strong>B\u00e0i 1. V\u1ebd \u0111\u1ed3 th\u1ecb b\u1eb1ng GeoGebra<\/strong><\/h3>\n<p>S\u1eed d\u1ee5ng ph\u1ea7n m\u1ec1m GeoGebra \u0111\u1ec3 tr\u1ef1c quan h\u00f3a \u0111\u1ed3 th\u1ecb.<\/p>\n<h3><strong>B\u00e0i 2. D\u00f9ng m\u00e1y t\u00ednh c\u1ea7m tay<\/strong><\/h3>\n<p>D\u00f9ng CALC, TABLE ho\u1eb7c GRAPH \u0111\u1ec3 t\u00ecm GTLN, GTNN.<\/p>\n<hr \/>\n<h2><strong>Ch\u01b0\u01a1ng 4. Nguy\u00ean h\u00e0m \u2013 T\u00edch ph\u00e2n<\/strong><\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-256 size-full aligncenter\" src=\"https:\/\/giapsgk.com\/chuyen-de\/wp-content\/uploads\/giapsgk-he-thong-kien-thuc-toan-12-chan-troi-sang-tao-hk1-va-hk2-4.png\" alt=\"T\u00f3m t\u1eaft l\u00fd thuy\u1ebft v\u00e0 c\u00f4ng th\u1ee9c To\u00e1n 12 Ch\u00e2n tr\u1eddi s\u00e1ng t\u1ea1o theo ch\u01b0\u01a1ng tr\u00ecnh m\u1edbi\" width=\"563\" height=\"390\" \/><\/p>\n<h3><strong>B\u00e0i 1. Nguy\u00ean h\u00e0m<\/strong><\/h3>\n<p>$\\int f(x)\\,dx = F(x) + C$<\/p>\n<p>B\u1ea3ng nguy\u00ean h\u00e0m c\u01a1 b\u1ea3n:<\/p>\n<ul>\n<li>$\\int x^n\\,dx = \\frac{x^{n+1}}{n+1} + C \\ (n \\ne -1)$<\/li>\n<li>$\\int \\frac{1}{x}\\,dx = \\ln|x| + C$<\/li>\n<\/ul>\n<h3><strong>B\u00e0i 2. T\u00edch ph\u00e2n<\/strong><\/h3>\n<p>$\\int_a^b f(x)\\,dx = F(b) &#8211; F(a)$<\/p>\n<p>T\u00ednh di\u1ec7n t\u00edch v\u00f9ng gi\u1edbi h\u1ea1n b\u1edfi \u0111\u01b0\u1eddng cong.<\/p>\n<h3><strong>B\u00e0i 3. \u1ee8ng d\u1ee5ng h\u00ecnh h\u1ecdc<\/strong><\/h3>\n<p><strong>Di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng:<\/strong> $S = \\int_a^b |f(x) &#8211; g(x)|\\,dx$<\/p>\n<p><strong>Th\u1ec3 t\u00edch khi quay quanh tr\u1ee5c Ox: <\/strong>$V = \\pi \\int_a^b [f(x)]^2 dx$<\/p>\n<h2><strong>Ch\u01b0\u01a1ng 5. M\u1eb7t ph\u1eb3ng, \u0111\u01b0\u1eddng th\u1eb3ng, m\u1eb7t c\u1ea7u<\/strong><\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-257 size-full\" src=\"https:\/\/giapsgk.com\/chuyen-de\/wp-content\/uploads\/giapsgk-he-thong-kien-thuc-toan-12-chan-troi-sang-tao-hk1-va-hk2-5.png\" alt=\"\u00d4n thi THPT m\u00f4n To\u00e1n l\u1edbp 12 theo s\u00e1ch Ch\u00e2n tr\u1eddi s\u00e1ng t\u1ea1o\" width=\"470\" height=\"344\" \/><\/p>\n<h3><strong>B\u00e0i 1. Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng<\/strong><\/h3>\n<ul>\n<li>$ax + by + cz + d = 0$<\/li>\n<li>M\u1eb7t ph\u1eb3ng \u0111i qua \u0111i\u1ec3m $M_0(x_0, y_0, z_0)$ v\u00e0 vu\u00f4ng g\u00f3c $\\vec{n}$<\/li>\n<\/ul>\n<h3><strong>B\u00e0i 2. \u0110\u01b0\u1eddng th\u1eb3ng trong kh\u00f4ng gian<\/strong><\/h3>\n<p><strong>D\u1ea1ng tham s\u1ed1:<\/strong> $ \\begin{cases} x = x_0 + at \\\\ y = y_0 + bt \\\\ z = z_0 + ct \\end{cases} $<\/p>\n<h3><strong>B\u00e0i 3. M\u1eb7t c\u1ea7u<\/strong><\/h3>\n<p>$(x &#8211; a)^2 + (y &#8211; b)^2 + (z &#8211; c)^2 = R^2$<\/p>\n<h2><strong>Ch\u01b0\u01a1ng 6. X\u00e1c su\u1ea5t c\u00f3 \u0111i\u1ec1u ki\u1ec7n<\/strong><\/h2>\n<h3><strong>B\u00e0i 1. X\u00e1c su\u1ea5t c\u00f3 \u0111i\u1ec1u ki\u1ec7n<\/strong><\/h3>\n<p>$P(A|B) = \\frac{P(A \\cap B)}{P(B)}$<\/p>\n<h3><strong>B\u00e0i 2. X\u00e1c su\u1ea5t to\u00e0n ph\u1ea7n \u2013 Bayes<\/strong><\/h3>\n<ul>\n<li>To\u00e0n ph\u1ea7n: $P(B) = \\sum P(B|A_i)P(A_i)$<\/li>\n<li>Bayes: $P(A_k|B) = \\frac{P(B|A_k)P(A_k)}{\\sum P(B|A_i)P(A_i)}$<\/li>\n<\/ul>\n<p style=\"text-align: center;\"><a href=\"https:\/\/giapsgk.com\/chuyen-de\/wp-content\/uploads\/GiapSGK.com-Cong-thuc-toan-CHAN-TROI-SANG-TAO.pdf\">T\u1ea3i file pdf th\u1ed1ng ki\u1ebfn th\u1ee9c To\u00e1n 12 ch\u00e2n tr\u1eddi s\u00e1ng t\u1ea1o<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Ch\u01b0\u01a1ng tr\u00ecnh To\u00e1n l\u1edbp 12 theo b\u1ed9 s\u00e1ch Ch\u00e2n tr\u1eddi s\u00e1ng t\u1ea1o c\u00f3 nhi\u1ec1u n\u1ed9i dung quan tr\u1ecdng li\u00ean quan tr\u1ef1c ti\u1ebfp \u0111\u1ebfn k\u1ef3 thi t\u1ed1t nghi\u1ec7p THPT. \u0110\u1ec3 gi\u00fap h\u1ecdc sinh d\u1ec5 d\u00e0ng n\u1eafm b\u1eaft v\u00e0 ghi nh\u1edb, b\u00e0i vi\u1ebft n\u00e0y s\u1ebd h\u1ec7 th\u1ed1ng to\u00e0n b\u1ed9 ki\u1ebfn th\u1ee9c To\u00e1n 12 c\u1ee7a h\u1ecdc k\u1ef3 1 &#8230; <a title=\"H\u1ec7 th\u1ed1ng ki\u1ebfn th\u1ee9c To\u00e1n 12 ch\u00e2n tr\u1eddi s\u00e1ng t\u1ea1o HK1 v\u00e0 HK2\" class=\"read-more\" href=\"https:\/\/giapsgk.com\/chuyen-de\/he-thong-kien-thuc-toan-12-chan-troi-sang-tao-hk1-va-hk2\/\" aria-label=\"\u0110\u1ecdc th\u00eam v\u1ec1 H\u1ec7 th\u1ed1ng ki\u1ebfn th\u1ee9c To\u00e1n 12 ch\u00e2n tr\u1eddi s\u00e1ng t\u1ea1o HK1 v\u00e0 HK2\">\u0110\u1ecdc ti\u1ebfp<\/a><\/p>\n","protected":false},"author":1,"featured_media":253,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[],"class_list":["post-245","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\/245","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=245"}],"version-history":[{"count":6,"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/posts\/245\/revisions"}],"predecessor-version":[{"id":251,"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/posts\/245\/revisions\/251"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/media\/253"}],"wp:attachment":[{"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/media?parent=245"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/categories?post=245"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/tags?post=245"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}