{"id":209,"date":"2025-07-19T07:56:29","date_gmt":"2025-07-19T07:56:29","guid":{"rendered":"https:\/\/giapsgk.com\/chuyen-de\/?p=209"},"modified":"2025-07-19T09:01:59","modified_gmt":"2025-07-19T09:01:59","slug":"tom-tat-kien-thuc-va-cong-thuc-toan-12-sach-ket-noi-tri-thuc-hk1-va-hk2","status":"publish","type":"post","link":"https:\/\/giapsgk.com\/chuyen-de\/tom-tat-kien-thuc-va-cong-thuc-toan-12-sach-ket-noi-tri-thuc-hk1-va-hk2\/","title":{"rendered":"T\u00f3m t\u1eaft ki\u1ebfn th\u1ee9c v\u00e0 c\u00f4ng th\u1ee9c To\u00e1n 12 s\u00e1ch K\u1ebft n\u1ed1i tri th\u1ee9c hk1 v\u00e0 hk2"},"content":{"rendered":"<p><strong data-start=\"798\" data-end=\"907\">B\u1ea1n \u0111ang t\u00ecm ki\u1ebfm m\u1ed9t t\u00e0i li\u1ec7u \u00f4n t\u1eadp To\u00e1n 12 \u0111\u1ea7y \u0111\u1ee7, d\u1ec5 hi\u1ec3u v\u00e0 b\u00e1m s\u00e1t s\u00e1ch gi\u00e1o khoa K\u1ebft n\u1ed1i tri th\u1ee9c?<\/strong> B\u00e0i vi\u1ebft n\u00e0y s\u1ebd gi\u00fap b\u1ea1n h\u1ec7 th\u1ed1ng to\u00e0n b\u1ed9 ki\u1ebfn th\u1ee9c tr\u1ecdng t\u00e2m v\u00e0 c\u00f4ng th\u1ee9c c\u1ea7n nh\u1edb c\u1ee7a To\u00e1n l\u1edbp 12 \u2013 t\u1eeb \u0111\u1ea1o h\u00e0m, kh\u1ea3o s\u00e1t h\u00e0m s\u1ed1, t\u00edch ph\u00e2n \u0111\u1ebfn x\u00e1c su\u1ea5t v\u00e0 h\u00ecnh h\u1ecdc kh\u00f4ng gian. V\u1edbi c\u00e1ch tr\u00ecnh b\u00e0y logic, d\u1ec5 theo d\u00f5i, \u0111\u00e2y s\u1ebd l\u00e0 c\u1ea9m nang \u00f4n thi l\u00fd t\u01b0\u1edfng cho h\u1ecdc sinh l\u1edbp 12 chu\u1ea9n b\u1ecb k\u1ef3 thi THPT Qu\u1ed1c gia.<\/p>\n<h2><strong>Ch\u01b0\u01a1ng 1. \u1ee8ng d\u1ee5ng \u0111\u1ea1o h\u00e0m \u0111\u1ec3 kh\u1ea3o s\u00e1t v\u00e0 v\u1ebd \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1<\/strong><\/h2>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-222 alignnone size-medium\" src=\"https:\/\/giapsgk.com\/chuyen-de\/wp-content\/uploads\/giapsgk-tom-tat-kien-thuc-va-cong-thuc-toan-12-sach-ket-noi-tri-thuc-hk1-va-hk2-1.png\" alt=\"H\u00ecnh \u1ea3nh - tom-tat-kien-thuc-va-cong-thuc-toan-12-sach-ket-noi-tri-thuc-hk1-va-hk2\" width=\"545\" height=\"436\" \/><\/p>\n<h3><strong>1. T\u00ednh \u0111\u01a1n \u0111i\u1ec7u v\u00e0 c\u1ef1c tr\u1ecb<\/strong><\/h3>\n<p>D\u1ea5u c\u1ee7a \u0111\u1ea1o h\u00e0m quy\u1ebft \u0111\u1ecbnh t\u00ednh \u0111\u01a1n \u0111i\u1ec7u:<\/p>\n<ul>\n<li>$f'(x) &gt; 0 \\Rightarrow f(x)$ \u0111\u1ed3ng bi\u1ebfn<\/li>\n<li>$f'(x) &lt; 0 \\Rightarrow f(x)$ ngh\u1ecbch bi\u1ebfn<\/li>\n<\/ul>\n<p><strong>C\u1ef1c tr\u1ecb:<\/strong> T\u00ecm nghi\u1ec7m c\u1ee7a $f'(x) = 0$, d\u00f2 b\u1ea3ng bi\u1ebfn thi\u00ean \u0111\u1ec3 k\u1ebft lu\u1eadn c\u1ef1c \u0111\u1ea1i\/c\u1ef1c ti\u1ec3u<\/p>\n<h3><strong>2. GTTD &#8211; GTLN, GTNN<\/strong><\/h3>\n<ul>\n<li>Tr\u00ean \u0111o\u1ea1n $[a, b]$: so s\u00e1nh gi\u00e1 tr\u1ecb c\u1ef1c tr\u1ecb v\u00e0 bi\u00ean<\/li>\n<li>Tr\u00ean $\\mathbb{R}$: ph\u1ee5 thu\u1ed9c b\u00e0i to\u00e1n t\u00ednh ch\u1ea5t<\/li>\n<\/ul>\n<h3><strong>3. Ti\u1ec7m c\u1eadn<\/strong><\/h3>\n<ul>\n<li><strong>\u0110\u1ee9ng:<\/strong> x\u00e1c \u0111\u1ecbnh t\u1eeb m\u1eabu = 0<\/li>\n<li><strong>Ngang:<\/strong> $\\lim_{x \\to \\pm\\infty} f(x) = L$<\/li>\n<li><strong>Xi\u00ean:<\/strong> D\u00f9ng c\u00e1c gi\u1edbi h\u1ea1n \u0111\u1ec3 t\u00ednh h\u1ec7 s\u1ed1 a, b<\/li>\n<\/ul>\n<h3><strong>4. Kh\u1ea3o s\u00e1t \u0111\u1ed3 th\u1ecb<\/strong><\/h3>\n<ul>\n<li><strong>C\u00e1c b\u01b0\u1edbc:<\/strong> TX\u0110, ti\u1ec7m c\u1eadn, \u0111\u1ea1o h\u00e0m, BBT, \u0111\u1ed3 th\u1ecb<\/li>\n<li><strong>G\u1eafn v\u1edbi c\u00e1c h\u00e0m:<\/strong> b\u1eadc 3, ph\u00e2n th\u1ee9c, logarit<\/li>\n<\/ul>\n<h3><strong>5. \u1ee8ng d\u1ee5ng \u0111\u1ea1o h\u00e0m trong th\u1ef1c t\u1ebf<\/strong><\/h3>\n<ul>\n<li>B\u00e0i to\u00e1n t\u1ed1i \u01b0u ho\u00e1: di\u1ec7n t\u00edch l\u1edbn nh\u1ea5t, chi ph\u00ed nh\u1ecf nh\u1ea5t, kho\u1ea3ng c\u00e1ch ng\u1eafn nh\u1ea5t&#8230;<\/li>\n<li>C\u1ea7n bi\u1ec3u di\u1ec5n \u0111\u1ea1i l\u01b0\u1ee3ng c\u1ea7n t\u1ed1i \u01b0u \u2192 \u0111\u01b0a v\u1ec1 b\u00e0i to\u00e1n c\u1ef1c tr\u1ecb<\/li>\n<\/ul>\n<h2><strong>Ch\u01b0\u01a1ng 2. Vect\u01a1 v\u00e0 h\u1ec7 t\u1ecda \u0111\u1ed9 kh\u00f4ng gian<\/strong><\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-223 alignnone size-medium aligncenter\" src=\"https:\/\/giapsgk.com\/chuyen-de\/wp-content\/uploads\/giapsgk-tom-tat-kien-thuc-va-cong-thuc-toan-12-sach-ket-noi-tri-thuc-hk1-va-hk2-2.png\" alt=\"H\u00ecnh \u1ea3nh - tom-tat-kien-thuc-va-cong-thuc-toan-12-sach-ket-noi-tri-thuc-hk1-va-hk2\" width=\"487\" height=\"444\" \/><\/p>\n<p>T\u1ecda \u0111\u1ed9 \u0111i\u1ec3m A(x, y, z), vect\u01a1 \u2192AB = (x$_2$ &#8211; x$_1$, y$_2$ &#8211; y$_1$, z$_2$ &#8211; z$_1$)<\/p>\n<ul>\n<li><strong>\u0110\u1ed9 d\u00e0i AB:<\/strong> $AB = \\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2 &#8211; z_1)^2}$<\/li>\n<li><strong>T\u00edch v\u00f4 h\u01b0\u1edbng:<\/strong> $\\vec{a} \\cdot \\vec{b} = a_1b_1 + a_2b_2 + a_3b_3$<\/li>\n<li><strong>T\u00edch c\u00f3 h\u01b0\u1edbng:<\/strong> \u0111\u1ecbnh ngh\u0129a b\u1eb1ng \u0111\u1ecbnh th\u1ee9c c\u1ea5p 3<\/li>\n<li><strong>G\u00f3c gi\u1eefa 2 vect\u01a1<\/strong>: $\\cos\\theta = \\frac{\\vec{a} \\cdot \\vec{b}}{|\\vec{a}||\\vec{b}|}$<\/li>\n<li><strong>\u1ee8ng d\u1ee5ng:<\/strong> x\u00e1c \u0111\u1ecbnh \u0111\u1ed9 d\u00e0i, g\u00f3c, \u0111i\u1ec1u ki\u1ec7n vu\u00f4ng g\u00f3c\/\u0111\u1ed3ng ph\u1eb3ng<\/li>\n<\/ul>\n<h2><strong>Ch\u01b0\u01a1ng 3. Th\u1ed1ng k\u00ea<\/strong><\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-224 alignnone size-medium aligncenter\" src=\"https:\/\/giapsgk.com\/chuyen-de\/wp-content\/uploads\/giapsgk-tom-tat-kien-thuc-va-cong-thuc-toan-12-sach-ket-noi-tri-thuc-hk1-va-hk2-3.png\" alt=\"H\u00ecnh \u1ea3nh - tom-tat-kien-thuc-va-cong-thuc-toan-12-sach-ket-noi-tri-thuc-hk1-va-hk2\" width=\"521\" height=\"288\" \/><\/p>\n<ul>\n<li><strong>Kho\u1ea3ng bi\u1ebfn thi\u00ean:<\/strong> R = max &#8211; min<\/li>\n<li><strong>Kho\u1ea3ng t\u1ee9 ph\u00e2n v\u1ecb:<\/strong> IQR = Q3 &#8211; Q1<\/li>\n<li><strong>Ph\u01b0\u01a1ng sai:<\/strong> $s^2 = \\frac{1}{n} \\sum (x_i &#8211; \\bar{x})^2$<\/li>\n<li><strong>\u0110\u1ed9 l\u1ec7ch chu\u1ea9n:<\/strong> $s = \\sqrt{s^2}$<\/li>\n<li><strong>D\u1ea1ng b\u00e0i:<\/strong> t\u00ednh s$^2$, s t\u1eeb b\u1ea3ng t\u1ea7n s\u1ed1<\/li>\n<\/ul>\n<hr \/>\n<h2><strong>Ch\u01b0\u01a1ng 4. Nguy\u00ean h\u00e0m v\u00e0 t\u00edch ph\u00e2n<\/strong><\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-225 alignnone size-medium aligncenter\" src=\"https:\/\/giapsgk.com\/chuyen-de\/wp-content\/uploads\/giapsgk-tom-tat-kien-thuc-va-cong-thuc-toan-12-sach-ket-noi-tri-thuc-hk1-va-hk2-4.png\" alt=\"H\u00ecnh \u1ea3nh - tom-tat-kien-thuc-va-cong-thuc-toan-12-sach-ket-noi-tri-thuc-hk1-va-hk2\" width=\"630\" height=\"599\" \/><\/p>\n<p><strong>\u0110\u1ecbnh ngh\u0129a nguy\u00ean h\u00e0m:<\/strong> $\\int f(x)dx = F(x) + C$ n\u1ebfu $F'(x) = f(x)$<\/p>\n<p><strong>M\u1ed9t s\u1ed1 c\u00f4ng th\u1ee9c:<\/strong><\/p>\n<ul>\n<li>$\\int x^n dx = \\frac{x^{n+1}}{n+1} + C$ (n \u2260 -1)<\/li>\n<li>$\\int \\frac{1}{x} dx = \\ln|x| + C$<\/li>\n<li>$\\int e^x dx = e^x + C$<\/li>\n<\/ul>\n<p><strong>T\u00edch ph\u00e2n x\u00e1c \u0111\u1ecbnh:<\/strong> $\\int_a^b f(x)dx = F(b) &#8211; F(a)$<\/p>\n<p><strong>Di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng:<\/strong> $A = \\int_a^b |f(x)| dx$<\/p>\n<p><strong>Th\u1ec3 t\u00edch kh\u1ed1i tr\u00f2n xoay:<\/strong> $V = \\pi \\int_a^b [f(x)]^2 dx$<\/p>\n<h2><strong>Ch\u01b0\u01a1ng 5. Ph\u01b0\u01a1ng ph\u00e1p t\u1ecda \u0111\u1ed9 trong kh\u00f4ng gian<\/strong><\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-226 alignnone size-medium aligncenter\" src=\"https:\/\/giapsgk.com\/chuyen-de\/wp-content\/uploads\/giapsgk-tom-tat-kien-thuc-va-cong-thuc-toan-12-sach-ket-noi-tri-thuc-hk1-va-hk2-5.png\" alt=\"H\u00ecnh \u1ea3nh - tom-tat-kien-thuc-va-cong-thuc-toan-12-sach-ket-noi-tri-thuc-hk1-va-hk2\" width=\"480\" height=\"348\" \/><\/p>\n<ul>\n<li><strong>Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng:<\/strong> $A(x &#8211; x_0) + B(y &#8211; y_0) + C(z &#8211; z_0) = 0$<\/li>\n<li><strong>Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng:<\/strong> $\\begin{cases} x = x_0 + at \\\\ y = y_0 + bt \\\\ z = z_0 + ct \\end{cases}$<\/li>\n<li><strong>M\u1eb7t c\u1ea7u:<\/strong> $(x &#8211; a)^2 + (y &#8211; b)^2 + (z &#8211; c)^2 = R^2$<\/li>\n<li><strong>G\u00f3c gi\u1eefa c\u00e1c \u0111\u1ed1i t\u01b0\u1ee3ng h\u00ecnh h\u1ecdc:<\/strong> s\u1eed d\u1ee5ng t\u00edch v\u00f4 h\u01b0\u1edbng<\/li>\n<\/ul>\n<h2><strong>Ch\u01b0\u01a1ng 6. X\u00e1c su\u1ea5t c\u00f3 \u0111i\u1ec1u ki\u1ec7n<\/strong><\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-227 alignnone size-medium aligncenter\" src=\"https:\/\/giapsgk.com\/chuyen-de\/wp-content\/uploads\/giapsgk-tom-tat-kien-thuc-va-cong-thuc-toan-12-sach-ket-noi-tri-thuc-hk1-va-hk2-6.png\" alt=\"H\u00ecnh \u1ea3nh - tom-tat-kien-thuc-va-cong-thuc-toan-12-sach-ket-noi-tri-thuc-hk1-va-hk2\" width=\"777\" height=\"342\" \/><\/p>\n<p>$P(A|B) = \\frac{P(A \\cap B)}{P(B)}$<\/p>\n<ul>\n<li><strong>X\u00e1c su\u1ea5t to\u00e0n ph\u1ea7n:<\/strong> $P(A) = \\sum P(A|B_i) \\cdot P(B_i)$<\/li>\n<li><strong>C\u00f4ng th\u1ee9c Bayes:<\/strong> $P(B_j|A) = \\frac{P(A|B_j) \\cdot P(B_j)}{\\sum P(A|B_i) \\cdot P(B_i)}$<\/li>\n<\/ul>\n<p style=\"text-align: center;\"><a href=\"https:\/\/giapsgk.com\/chuyen-de\/wp-content\/uploads\/GiapSGK.com-Cong-thuc-toan-sach-ket-noi-tri-thuc.pdf\">File pdf C\u00f4ng th\u1ee9c to\u00e1n s\u00e1ch k\u1ebft n\u1ed1i tri th\u1ee9c<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>B\u1ea1n \u0111ang t\u00ecm ki\u1ebfm m\u1ed9t t\u00e0i li\u1ec7u \u00f4n t\u1eadp To\u00e1n 12 \u0111\u1ea7y \u0111\u1ee7, d\u1ec5 hi\u1ec3u v\u00e0 b\u00e1m s\u00e1t s\u00e1ch gi\u00e1o khoa K\u1ebft n\u1ed1i tri th\u1ee9c? B\u00e0i vi\u1ebft n\u00e0y s\u1ebd gi\u00fap b\u1ea1n h\u1ec7 th\u1ed1ng to\u00e0n b\u1ed9 ki\u1ebfn th\u1ee9c tr\u1ecdng t\u00e2m v\u00e0 c\u00f4ng th\u1ee9c c\u1ea7n nh\u1edb c\u1ee7a To\u00e1n l\u1edbp 12 \u2013 t\u1eeb \u0111\u1ea1o h\u00e0m, kh\u1ea3o s\u00e1t h\u00e0m &#8230; <a title=\"T\u00f3m t\u1eaft ki\u1ebfn th\u1ee9c v\u00e0 c\u00f4ng th\u1ee9c To\u00e1n 12 s\u00e1ch K\u1ebft n\u1ed1i tri th\u1ee9c hk1 v\u00e0 hk2\" class=\"read-more\" href=\"https:\/\/giapsgk.com\/chuyen-de\/tom-tat-kien-thuc-va-cong-thuc-toan-12-sach-ket-noi-tri-thuc-hk1-va-hk2\/\" aria-label=\"\u0110\u1ecdc th\u00eam v\u1ec1 T\u00f3m t\u1eaft ki\u1ebfn th\u1ee9c v\u00e0 c\u00f4ng th\u1ee9c To\u00e1n 12 s\u00e1ch K\u1ebft n\u1ed1i tri th\u1ee9c hk1 v\u00e0 hk2\">\u0110\u1ecdc ti\u1ebfp<\/a><\/p>\n","protected":false},"author":1,"featured_media":229,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[],"class_list":["post-209","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\/209","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=209"}],"version-history":[{"count":10,"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/posts\/209\/revisions"}],"predecessor-version":[{"id":219,"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/posts\/209\/revisions\/219"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/media\/229"}],"wp:attachment":[{"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/media?parent=209"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/categories?post=209"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/tags?post=209"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}