{"id":380,"date":"2025-07-21T09:15:28","date_gmt":"2025-07-21T09:15:28","guid":{"rendered":"https:\/\/giapsgk.com\/chuyen-de\/?p=380"},"modified":"2025-07-21T09:31:33","modified_gmt":"2025-07-21T09:31:33","slug":"tong-hop-cac-cong-thuc-toan-11-canh-dieu-hoc-ki-1-va-hoc-ki-2","status":"publish","type":"post","link":"https:\/\/giapsgk.com\/chuyen-de\/tong-hop-cac-cong-thuc-toan-11-canh-dieu-hoc-ki-1-va-hoc-ki-2\/","title":{"rendered":"T\u1ed5ng h\u1ee3p c\u00e1c c\u00f4ng th\u1ee9c to\u00e1n 11 c\u00e1nh di\u1ec1u h\u1ecdc k\u00ec 1 v\u00e0 h\u1ecdc k\u00ec 2"},"content":{"rendered":"<p>B\u1ea1n c\u1ea7n t\u00e0i li\u1ec7u <strong>c\u00f4ng th\u1ee9c To\u00e1n 11 C\u00e1nh Di\u1ec1u<\/strong> \u0111\u1ec3 h\u1ecdc nhanh \u2013 nh\u1edb l\u00e2u?\u00a0 B\u00e0i vi\u1ebft n\u00e0y t\u1ed5ng h\u1ee3p to\u00e0n b\u1ed9 <strong data-start=\"624\" data-end=\"671\">\u0111\u1ecbnh ngh\u0129a, c\u00f4ng th\u1ee9c tr\u1ecdng t\u00e2m c\u1ee7a To\u00e1n 11<\/strong>, b\u00e1m s\u00e1t s\u00e1ch gi\u00e1o khoa C\u00e1nh Di\u1ec1u \u2013 t\u1eeb <strong data-start=\"711\" data-end=\"786\">h\u00e0m l\u01b0\u1ee3ng gi\u00e1c, c\u1ea5p s\u1ed1 c\u1ed9ng, logarit \u0111\u1ebfn \u0111\u1ea1o h\u00e0m v\u00e0 h\u00ecnh h\u1ecdc kh\u00f4ng gian<\/strong>. M\u1ed7i ph\u1ea7n \u0111\u1ec1u \u0111\u01b0\u1ee3c tr\u00ecnh b\u00e0y r\u00f5 r\u00e0ng, ng\u1eafn g\u1ecdn, d\u1ec5 ghi nh\u1edb, gi\u00fap b\u1ea1n <strong data-start=\"856\" data-end=\"911\">\u00f4n thi gi\u1eefa k\u1ef3, cu\u1ed1i k\u1ef3 v\u00e0 chu\u1ea9n b\u1ecb cho k\u1ef3 thi THPT<\/strong> m\u1ed9t c\u00e1ch hi\u1ec7u qu\u1ea3 nh\u1ea5t. T\u00e0i li\u1ec7u ph\u00f9 h\u1ee3p cho c\u1ea3 h\u1ecdc sinh v\u00e0 gi\u00e1o vi\u00ean c\u1ea7n h\u1ec7 th\u1ed1ng ki\u1ebfn th\u1ee9c nhanh g\u1ecdn.<\/p>\n<h2><strong>Ch\u01b0\u01a1ng 1: H\u00e0m s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c v\u00e0 ph\u01b0\u01a1ng tr\u00ecnh l\u01b0\u1ee3ng gi\u00e1c<\/strong><\/h2>\n<h3><strong>1. G\u00f3c l\u01b0\u1ee3ng gi\u00e1c \u2013 Gi\u00e1 tr\u1ecb l\u01b0\u1ee3ng gi\u00e1c<\/strong><\/h3>\n<p>G\u00f3c l\u01b0\u1ee3ng gi\u00e1c: L\u00e0 m\u1ed9t g\u00f3c x\u00e1c \u0111\u1ecbnh b\u1eb1ng s\u1ef1 quay c\u1ee7a m\u1ed9t tia quanh \u0111i\u1ec3m g\u1ed1c, c\u00f3 th\u1ec3 l\u00e0 s\u1ed1 d\u01b0\u01a1ng ho\u1eb7c \u00e2m.<\/p>\n<p>Gi\u00e1 tr\u1ecb l\u01b0\u1ee3ng gi\u00e1c c\u1ee7a g\u00f3c \u03b1:<\/p>\n<p>$ \\sin \\alpha = \\frac{y}{r}, \\quad \\cos \\alpha = \\frac{x}{r}, \\quad \\tan \\alpha = \\frac{y}{x}, \\quad \\cot \\alpha = \\frac{x}{y} $<\/p>\n<p>V\u1edbi $r = \\sqrt{x^2 + y^2}$<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-385 size-full\" src=\"https:\/\/giapsgk.com\/chuyen-de\/wp-content\/uploads\/giapsgk-tong-hop-cac-cong-thuc-toan-11-canh-dieu-hoc-ki-1-va-hoc-ki-2-2.png\" alt=\"c\u00f4ng th\u1ee9c to\u00e1n 11 c\u00e1nh di\u1ec1u \u0111\u1ea7y \u0111\u1ee7\" width=\"636\" height=\"345\" \/><\/p>\n<h3><strong>2. Ph\u00e9p bi\u1ebfn \u0111\u1ed5i l\u01b0\u1ee3ng gi\u00e1c<\/strong><\/h3>\n<p>Cung li\u00ean k\u1ebft: $ \\sin(\\pi &#8211; x) = \\sin x,\\quad \\cos(\\pi &#8211; x) = -\\cos x $<\/p>\n<p>C\u00f4ng th\u1ee9c c\u1ed9ng: $ \\sin(a \\pm b) = \\sin a \\cos b \\pm \\cos a \\sin b \\\\ \\cos(a \\pm b) = \\cos a \\cos b \\mp \\sin a \\sin b $<\/p>\n<h3><strong>3. H\u00e0m s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c<\/strong><\/h3>\n<p>T\u1eadp x\u00e1c \u0111\u1ecbnh:<\/p>\n<ul>\n<li>$\\sin x, \\cos x$: $\\mathbb{R}$<\/li>\n<li>$\\tan x$: $\\mathbb{R} \\setminus \\left\\{ \\frac{\\pi}{2} + k\\pi \\right\\}$<\/li>\n<\/ul>\n<p>T\u00ednh tu\u1ea7n ho\u00e0n: $\\sin, \\cos$ c\u00f3 chu k\u1ef3 $2\\pi$, $\\tan, \\cot$ c\u00f3 chu k\u1ef3 $\\pi$<\/p>\n<h3><strong>4. Ph\u01b0\u01a1ng tr\u00ecnh l\u01b0\u1ee3ng gi\u00e1c c\u01a1 b\u1ea3n<\/strong><\/h3>\n<ul>\n<li>$\\sin x = a \\Rightarrow x = \\arcsin a + k2\\pi $ ho\u1eb7c $x = \\pi &#8211; \\arcsin a + k2\\pi$<\/li>\n<li>$\\cos x = a \\Rightarrow x = \\pm \\arccos a + k2\\pi $<\/li>\n<li>$\\tan x = a \\Rightarrow x = \\arctan a + k\\pi $<\/li>\n<\/ul>\n<h2><strong>Ch\u01b0\u01a1ng 2: D\u00e3y s\u1ed1 \u2013 C\u1ea5p s\u1ed1 c\u1ed9ng v\u00e0 c\u1ea5p s\u1ed1 nh\u00e2n<\/strong><\/h2>\n<h3><strong>1. D\u00e3y s\u1ed1<\/strong><\/h3>\n<p>\u0110\u1ecbnh ngh\u0129a: D\u00e3y s\u1ed1 l\u00e0 m\u1ed9t h\u00e0m x\u00e1c \u0111\u1ecbnh tr\u00ean t\u1eadp N, vi\u1ebft d\u01b0\u1edbi d\u1ea1ng $(u_n)$<\/p>\n<h3><strong>2. C\u1ea5p s\u1ed1 c\u1ed9ng (CSC)<\/strong><\/h3>\n<p>C\u00f4ng th\u1ee9c t\u1ed5ng qu\u00e1t: $ u_n = u_1 + (n &#8211; 1)d $<\/p>\n<p>T\u1ed5ng n s\u1ed1 h\u1ea1ng \u0111\u1ea7u: $ S_n = \\frac{n}{2}(u_1 + u_n) $<\/p>\n<h3><strong>3. C\u1ea5p s\u1ed1 nh\u00e2n (CSN)<\/strong><\/h3>\n<p>C\u00f4ng th\u1ee9c s\u1ed1 h\u1ea1ng: $ u_n = u_1 \\cdot q^{n-1} $<\/p>\n<p>T\u1ed5ng n s\u1ed1 h\u1ea1ng \u0111\u1ea7u (q \u2260 1): $ S_n = u_1 \\cdot \\frac{q^n &#8211; 1}{q &#8211; 1} $<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-386 size-full\" src=\"https:\/\/giapsgk.com\/chuyen-de\/wp-content\/uploads\/giapsgk-tong-hop-cac-cong-thuc-toan-11-canh-dieu-hoc-ki-1-va-hoc-ki-2-3.png\" alt=\"c\u00e1c c\u00f4ng th\u1ee9c quan tr\u1ecdng to\u00e1n 11 c\u00e1nh di\u1ec1u\" width=\"779\" height=\"418\" \/><\/p>\n<h2><strong>Ch\u01b0\u01a1ng 3: Gi\u1edbi h\u1ea1n \u2013 H\u00e0m s\u1ed1 li\u00ean t\u1ee5c<\/strong><\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-387 size-full\" src=\"https:\/\/giapsgk.com\/chuyen-de\/wp-content\/uploads\/giapsgk-tong-hop-cac-cong-thuc-toan-11-canh-dieu-hoc-ki-1-va-hoc-ki-2-4.png\" alt=\"h\u1ecdc nhanh c\u00f4ng th\u1ee9c to\u00e1n 11 theo s\u00e1ch c\u00e1nh di\u1ec1u\" width=\"441\" height=\"357\" \/><\/p>\n<h3><strong>1. Gi\u1edbi h\u1ea1n d\u00e3y s\u1ed1<\/strong><\/h3>\n<p><strong>\u0110\u1ecbnh ngh\u0129a: <\/strong><\/p>\n<p>D\u00e3y $(u_n)$ c\u00f3 gi\u1edbi h\u1ea1n $L$ khi $n \\to \\infty$ n\u1ebfu: $ \\forall \\epsilon &gt; 0, \\exists N: n &gt; N \\Rightarrow |u_n &#8211; L| &lt; \\epsilon $<\/p>\n<h3><strong>2. Gi\u1edbi h\u1ea1n h\u00e0m s\u1ed1<\/strong><\/h3>\n<p>$ \\lim_{x \\to a} f(x) = L \\Leftrightarrow \\text{f(x) ti\u1ebfn g\u1ea7n L khi } x \\to a $<\/p>\n<h3><strong>3. H\u00e0m s\u1ed1 li\u00ean t\u1ee5c<\/strong><\/h3>\n<p>H\u00e0m $f(x)$ li\u00ean t\u1ee5c t\u1ea1i $x = a$ n\u1ebfu: $ \\lim_{x \\to a} f(x) = f(a) $<\/p>\n<h2><strong>Ch\u01b0\u01a1ng 4: Quan h\u1ec7 song song \u2013 H\u00ecnh h\u1ecdc kh\u00f4ng gian<\/strong><\/h2>\n<h3><strong>1. Ki\u1ebfn th\u1ee9c c\u01a1 b\u1ea3n<\/strong><\/h3>\n<p>\u0110\u01b0\u1eddng th\u1eb3ng \u2013 m\u1eb7t ph\u1eb3ng \u2013 kh\u00f4ng gian<\/p>\n<h3><strong>2. Quan h\u1ec7 song song<\/strong><\/h3>\n<ul>\n<li>Hai \u0111\u01b0\u1eddng th\u1eb3ng song song n\u1ebfu c\u00f9ng n\u1eb1m tr\u00ean m\u1eb7t ph\u1eb3ng v\u00e0 kh\u00f4ng c\u1eaft nhau.<\/li>\n<li>\u0110\u01b0\u1eddng th\u1eb3ng song song v\u1edbi m\u1eb7t ph\u1eb3ng: n\u1ebfu song song v\u1edbi \u0111\u01b0\u1eddng th\u1eb3ng n\u1eb1m trong m\u1eb7t ph\u1eb3ng.<\/li>\n<li>Hai m\u1eb7t ph\u1eb3ng song song: kh\u00f4ng c\u00f3 \u0111i\u1ec3m chung.<\/li>\n<\/ul>\n<h3><strong>3. H\u00ecnh l\u0103ng tr\u1ee5 \u2013 h\u00ecnh h\u1ed9p<\/strong><\/h3>\n<p>Th\u1ec3 t\u00edch l\u0103ng tr\u1ee5: $ V = S_{\\text{\u0111\u00e1y}} \\cdot h $<\/p>\n<h3><strong>4. Ph\u00e9p chi\u1ebfu song song<\/strong><\/h3>\n<p>D\u00f9ng \u0111\u1ec3 bi\u1ec3u di\u1ec5n h\u00ecnh 3D tr\u00ean m\u1eb7t ph\u1eb3ng, b\u1ea3o to\u00e0n t\u00ednh song song.<\/p>\n<h2><strong>Ch\u01b0\u01a1ng V: Th\u1ed1ng k\u00ea \u2013 X\u00e1c su\u1ea5t<\/strong><\/h2>\n<h3><strong>1. S\u1ed1 \u0111\u1eb7c tr\u01b0ng trung t\u00e2m<\/strong><\/h3>\n<ul>\n<li>Trung b\u00ecnh c\u1ed9ng: $ \\overline{x} = \\frac{\\sum f_i x_i}{\\sum f_i} $<\/li>\n<li>Trung v\u1ecb, m\u1ed1t \u2013 x\u00e1c \u0111\u1ecbnh qua b\u1ea3ng t\u1ea7n s\u1ed1.<\/li>\n<\/ul>\n<h3><strong>2. X\u00e1c su\u1ea5t<\/strong><\/h3>\n<p>Bi\u1ebfn c\u1ed1 h\u1ee3p\/giao\/\u0111\u1ed9c l\u1eadp<\/p>\n<ul>\n<li>C\u00f4ng th\u1ee9c c\u1ed9ng: $ P(A \\cup B) = P(A) + P(B) &#8211; P(A \\cap B) $<\/li>\n<li>C\u00f4ng th\u1ee9c nh\u00e2n (\u0111\u1ed9c l\u1eadp): $ P(A \\cap B) = P(A) \\cdot P(B) $<\/li>\n<\/ul>\n<h2><strong>Ch\u01b0\u01a1ng VI: H\u00e0m m\u0169 \u2013 Logarit<\/strong><\/h2>\n<h3><strong>1. L\u0169y th\u1eeba m\u0169 th\u1ef1c<\/strong><\/h3>\n<p>$a^x$ x\u00e1c \u0111\u1ecbnh khi $a &gt; 0$<\/p>\n<p>T\u00ednh ch\u1ea5t: $ a^m \\cdot a^n = a^{m+n},\\quad (a^m)^n = a^{mn} $<\/p>\n<h3><strong>2. Logarit<\/strong><\/h3>\n<p>$ \\log_a b = x \\Leftrightarrow a^x = b $<\/p>\n<p>T\u00ednh ch\u1ea5t: $ \\log_a(xy) = \\log_a x + \\log_a y $<\/p>\n<h3><strong>3. H\u00e0m s\u1ed1 m\u0169 \u2013 logarit<\/strong><\/h3>\n<p>H\u00e0m s\u1ed1 m\u0169: $y = a^x$ (a &gt; 1): \u0111\u1ed3ng bi\u1ebfn<\/p>\n<p>H\u00e0m logarit: $y = \\log_a x$ (a &gt; 1): \u0111\u1ed3ng bi\u1ebfn<\/p>\n<h3><strong>4. Ph\u01b0\u01a1ng tr\u00ecnh \u2013 BPT m\u0169, logarit<\/strong><\/h3>\n<p>Gi\u1ea3i b\u1eb1ng \u0111\u01b0a v\u1ec1 c\u00f9ng c\u01a1 s\u1ed1, ho\u1eb7c logarit h\u00f3a hai v\u1ebf.<\/p>\n<h2><strong>Ch\u01b0\u01a1ng VII: \u0110\u1ea1o h\u00e0m<\/strong><\/h2>\n<h3><strong>1. \u0110\u1ecbnh ngh\u0129a \u0111\u1ea1o h\u00e0m<\/strong><\/h3>\n<p>$ f'(x) = \\lim_{\\Delta x \\to 0} \\frac{f(x + \\Delta x) &#8211; f(x)}{\\Delta x} $<\/p>\n<h3><strong>2. Quy t\u1eafc \u0111\u1ea1o h\u00e0m<\/strong><\/h3>\n<ul>\n<li>$(x^n)&#8217; = nx^{n-1}$<\/li>\n<li>$(\\sin x)&#8217; = \\cos x$, $(\\cos x)&#8217; = -\\sin x$<\/li>\n<\/ul>\n<p>Quy t\u1eafc t\u1ed5ng, t\u00edch, th\u01b0\u01a1ng, h\u1ee3p<\/p>\n<h3><strong>3. \u0110\u1ea1o h\u00e0m c\u1ea5p hai<\/strong><\/h3>\n<p>$f&#8221;(x) = \\left( {f'(x)} \\right)&#8217;$<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-388 size-full\" src=\"https:\/\/giapsgk.com\/chuyen-de\/wp-content\/uploads\/giapsgk-tong-hop-cac-cong-thuc-toan-11-canh-dieu-hoc-ki-1-va-hoc-ki-2-5.png\" alt=\"\u00f4n t\u1eadp c\u00f4ng th\u1ee9c to\u00e1n 11 c\u00e1nh di\u1ec1u thi h\u1ecdc k\u1ef3\" width=\"411\" height=\"296\" \/><\/p>\n<h2><strong>Ch\u01b0\u01a1ng VIII: Quan h\u1ec7 vu\u00f4ng g\u00f3c \u2013 Th\u1ec3 t\u00edch kh\u1ed1i<\/strong><\/h2>\n<h3><strong>1. Quan h\u1ec7 vu\u00f4ng g\u00f3c<\/strong><\/h3>\n<p>Hai \u0111\u01b0\u1eddng th\u1eb3ng, \u0111\u01b0\u1eddng v\u1edbi m\u1eb7t, hai m\u1eb7t ph\u1eb3ng.<\/p>\n<h3><strong>2. G\u00f3c trong kh\u00f4ng gian<\/strong><\/h3>\n<ul>\n<li><strong>G\u00f3c gi\u1eefa \u0111\u01b0\u1eddng v\u00e0 m\u1eb7t:<\/strong> x\u00e1c \u0111\u1ecbnh b\u1eb1ng h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c<\/li>\n<li><strong>G\u00f3c nh\u1ecb di\u1ec7n:<\/strong> g\u00f3c gi\u1eefa hai m\u1eb7t<\/li>\n<\/ul>\n<h3><strong>3. Kho\u1ea3ng c\u00e1ch \u2013 Th\u1ec3 t\u00edch<\/strong><\/h3>\n<p>Kho\u1ea3ng c\u00e1ch t\u1eeb \u0111i\u1ec3m \u0111\u1ebfn m\u1eb7t ph\u1eb3ng, gi\u1eefa hai \u0111\u01b0\u1eddng.<\/p>\n<p>Th\u1ec3 t\u00edch:<\/p>\n<ul>\n<li>L\u0103ng tr\u1ee5 \u0111\u1ee9ng: $V = S_{\\text{\u0111\u00e1y}} \\cdot h$<\/li>\n<li>Ch\u00f3p \u0111\u1ec1u: $V = \\frac{1}{3} S_{\\text{\u0111\u00e1y}} \\cdot h$<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>B\u1ea1n c\u1ea7n t\u00e0i li\u1ec7u c\u00f4ng th\u1ee9c To\u00e1n 11 C\u00e1nh Di\u1ec1u \u0111\u1ec3 h\u1ecdc nhanh \u2013 nh\u1edb l\u00e2u?\u00a0 B\u00e0i vi\u1ebft n\u00e0y t\u1ed5ng h\u1ee3p to\u00e0n b\u1ed9 \u0111\u1ecbnh ngh\u0129a, c\u00f4ng th\u1ee9c tr\u1ecdng t\u00e2m c\u1ee7a To\u00e1n 11, b\u00e1m s\u00e1t s\u00e1ch gi\u00e1o khoa C\u00e1nh Di\u1ec1u \u2013 t\u1eeb h\u00e0m l\u01b0\u1ee3ng gi\u00e1c, c\u1ea5p s\u1ed1 c\u1ed9ng, logarit \u0111\u1ebfn \u0111\u1ea1o h\u00e0m v\u00e0 h\u00ecnh h\u1ecdc kh\u00f4ng &#8230; <a title=\"T\u1ed5ng h\u1ee3p c\u00e1c c\u00f4ng th\u1ee9c to\u00e1n 11 c\u00e1nh di\u1ec1u h\u1ecdc k\u00ec 1 v\u00e0 h\u1ecdc k\u00ec 2\" class=\"read-more\" href=\"https:\/\/giapsgk.com\/chuyen-de\/tong-hop-cac-cong-thuc-toan-11-canh-dieu-hoc-ki-1-va-hoc-ki-2\/\" aria-label=\"\u0110\u1ecdc th\u00eam v\u1ec1 T\u1ed5ng h\u1ee3p c\u00e1c c\u00f4ng th\u1ee9c to\u00e1n 11 c\u00e1nh di\u1ec1u h\u1ecdc k\u00ec 1 v\u00e0 h\u1ecdc k\u00ec 2\">\u0110\u1ecdc ti\u1ebfp<\/a><\/p>\n","protected":false},"author":1,"featured_media":384,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[],"class_list":["post-380","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\/380","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=380"}],"version-history":[{"count":4,"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/posts\/380\/revisions"}],"predecessor-version":[{"id":389,"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/posts\/380\/revisions\/389"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/media\/384"}],"wp:attachment":[{"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/media?parent=380"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/categories?post=380"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/tags?post=380"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}