{"id":401,"date":"2025-07-21T10:44:58","date_gmt":"2025-07-21T10:44:58","guid":{"rendered":"https:\/\/giapsgk.com\/chuyen-de\/?p=401"},"modified":"2025-07-21T16:09:30","modified_gmt":"2025-07-21T16:09:30","slug":"cac-cong-thuc-toan-11","status":"publish","type":"post","link":"https:\/\/giapsgk.com\/chuyen-de\/cac-cong-thuc-toan-11\/","title":{"rendered":"T\u1ed5ng h\u1ee3p c\u00e1c c\u00f4ng th\u1ee9c To\u00e1n 11 (c\u1ea3 n\u0103m &#8211; s\u00e1ch m\u1edbi)"},"content":{"rendered":"<p dir=\"ltr\" data-pm-slice=\"1 1 []\">B\u1ea1n \u0111ang t\u00ecm ki\u1ebfm <strong>c\u00e1c c\u00f4ng th\u1ee9c To\u00e1n 11<\/strong> \u0111\u1ec3 \u00f4n thi h\u1ecdc k\u1ef3 hi\u1ec7u qu\u1ea3? B\u00e0i vi\u1ebft n\u00e0y t\u1ed5ng h\u1ee3p \u0111\u1ea7y \u0111\u1ee7 v\u00e0 chi ti\u1ebft c\u00e1c c\u00f4ng th\u1ee9c quan tr\u1ecdng, kh\u00e1i ni\u1ec7m v\u00e0 \u0111\u1ecbnh ngh\u0129a t\u1eeb ba b\u1ed9 s\u00e1ch gi\u00e1o khoa ph\u1ed5 bi\u1ebfn: <a href=\"https:\/\/giapsgk.com\/chuyen-de\/tong-hop-cong-thuc-toan-11-ket-noi-tri-thuc-hoc-ki-1-va-hoc-ki-2\/\"><strong>K\u1ebft n\u1ed1i tri th\u1ee9c v\u1edbi cu\u1ed9c s\u1ed1ng<\/strong><\/a>, <a href=\"https:\/\/giapsgk.com\/chuyen-de\/tong-hop-cac-cong-thuc-toan-11-canh-dieu-hoc-ki-1-va-hoc-ki-2\/\"><strong>C\u00e1nh Di\u1ec1u<\/strong><\/a>, v\u00e0 <a href=\"https:\/\/giapsgk.com\/chuyen-de\/he-thong-cac-cong-thuc-toan-11-chan-troi-sang-tao-hk1-va-hk2\/\"><strong>Ch\u00e2n tr\u1eddi s\u00e1ng t\u1ea1o<\/strong><\/a>. T\u1eeb h\u00e0m s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c, d\u00e3y s\u1ed1, c\u1ea5p s\u1ed1 c\u1ed9ng, c\u1ea5p s\u1ed1 nh\u00e2n, \u0111\u1ebfn \u0111\u1ea1o h\u00e0m, x\u00e1c su\u1ea5t v\u00e0 h\u00ecnh h\u1ecdc kh\u00f4ng gian, m\u1ecdi ki\u1ebfn th\u1ee9c \u0111\u1ec1u \u0111\u01b0\u1ee3c tr\u00ecnh b\u00e0y ng\u1eafn g\u1ecdn, d\u1ec5 hi\u1ec3u, k\u00e8m m\u1eb9o \u00f4n t\u1eadp. H\u00e3y kh\u00e1m ph\u00e1 ngay \u0111\u1ec3 n\u1eafm v\u1eefng ki\u1ebfn th\u1ee9c v\u00e0 t\u1ef1 tin chinh ph\u1ee5c k\u1ef3 thi To\u00e1n 11!<\/p>\n<h2><strong> 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> Kh\u00e1i ni\u1ec7m v\u00e0 \u0111\u1ecbnh ngh\u0129a quan tr\u1ecdng<\/strong><\/h3>\n<p><strong> G\u00f3c l\u01b0\u1ee3ng gi\u00e1c:<\/strong> \u0110o b\u1eb1ng \u0111\u1ed9 ho\u1eb7c radian. M\u1ed9t v\u00f2ng tr\u00f2n l\u00e0 \\(360^\\circ\\) ho\u1eb7c \\(2\\pi\\) radian.<\/p>\n<p><strong> Gi\u00e1 tr\u1ecb l\u01b0\u1ee3ng gi\u00e1c<\/strong> (\\(\\sin, \\cos, \\tan, \\cot\\)):<\/p>\n<ul>\n<li>\\(\\sin \\theta = \\frac{\\text{\u0111\u1ed1i}}{\\text{huy\u1ec1n}}\\), \\(\\cos \\theta = \\frac{\\text{k\u1ec1}}{\\text{huy\u1ec1n}}\\)<\/li>\n<li>\\(\\tan \\theta = \\frac{\\sin \\theta}{\\cos \\theta}\\), \\(\\cot \\theta = \\frac{\\cos \\theta}{\\sin \\theta}\\)<\/li>\n<li>C\u1ea7n nh\u1edb gi\u00e1 tr\u1ecb t\u1ea1i c\u00e1c g\u00f3c \u0111\u1eb7c bi\u1ec7t (\\(0^\\circ, 30^\\circ, 45^\\circ, 60^\\circ, 90^\\circ\\)).<\/li>\n<\/ul>\n<p><strong> H\u00e0m s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c:<\/strong><\/p>\n<ul>\n<li>\\(\\sin x\\): Chu k\u1ef3 \\(2\\pi\\), t\u1eadp gi\u00e1 tr\u1ecb \\([-1, 1]\\).<\/li>\n<li>\\(\\cos x\\): Chu k\u1ef3 \\(2\\pi\\), t\u1eadp gi\u00e1 tr\u1ecb \\([-1, 1]\\).<\/li>\n<li>\\(\\tan x\\): Chu k\u1ef3 \\(\\pi\\), kh\u00f4ng x\u00e1c \u0111\u1ecbnh t\u1ea1i \\(x = \\frac{\\pi}{2} + k\\pi\\).<\/li>\n<li>\\(\\cot x\\): Chu k\u1ef3 \\(\\pi\\), kh\u00f4ng x\u00e1c \u0111\u1ecbnh t\u1ea1i \\(x = k\\pi\\).<\/li>\n<\/ul>\n<p><strong> Ph\u01b0\u01a1ng tr\u00ecnh l\u01b0\u1ee3ng gi\u00e1c c\u01a1 b\u1ea3n:<\/strong><\/p>\n<ul>\n<li>\\(\\sin x = a\\): C\u00f3 nghi\u1ec7m khi \\(|a| \\leq 1\\).<\/li>\n<li>\\(\\cos x = a\\): C\u00f3 nghi\u1ec7m khi \\(|a| \\leq 1\\).<\/li>\n<li>\\(\\tan x = a\\), \\(\\cot x = a\\): Lu\u00f4n c\u00f3 nghi\u1ec7m.<\/li>\n<\/ul>\n<h3><strong> C\u00f4ng th\u1ee9c quan tr\u1ecdng<\/strong><\/h3>\n<p><strong> C\u00f4ng th\u1ee9c l\u01b0\u1ee3ng gi\u00e1c:<\/strong><\/p>\n<p><strong> C\u1ed9ng v\u00e0 tr\u1eeb:<\/strong><\/p>\n<ul>\n<li>\\(\\sin(a \\pm b) = \\sin a \\cos b \\pm \\cos a \\sin b\\)<\/li>\n<li>\\(\\cos(a \\pm b) = \\cos a \\cos b \\mp \\sin a \\sin b\\)<\/li>\n<li>\\(\\tan(a \\pm b) = \\frac{\\tan a \\pm \\tan b}{1 \\mp \\tan a \\tan b}\\)<\/li>\n<\/ul>\n<p><strong> Nh\u00e2n \u0111\u00f4i:<\/strong><\/p>\n<ul>\n<li>\\(\\sin 2a = 2 \\sin a \\cos a\\)<\/li>\n<li>\\(\\cos 2a = \\cos^2 a &#8211; \\sin^2 a = 2\\cos^2 a &#8211; 1 = 1 &#8211; 2\\sin^2 a\\)<\/li>\n<li>\\(\\tan 2a = \\frac{2 \\tan a}{1 &#8211; \\tan^2 a}\\)<\/li>\n<\/ul>\n<p><strong> Bi\u1ebfn \u0111\u1ed5i t\u1ed5ng th\u00e0nh t\u00edch:<\/strong><\/p>\n<ul>\n<li>\\(\\sin a + \\sin b = 2 \\sin \\left(\\frac{a+b}{2}\\right) \\cos \\left(\\frac{a-b}{2}\\right)\\)<\/li>\n<li>\\(\\cos a + \\cos b = 2 \\cos \\left(\\frac{a+b}{2}\\right) \\cos \\left(\\frac{a-b}{2}\\right)\\)<\/li>\n<\/ul>\n<p><strong> H\u1eb1ng \u0111\u1eb3ng th\u1ee9c c\u01a1 b\u1ea3n:<\/strong><\/p>\n<ul>\n<li>\\(\\sin^2 x + \\cos^2 x = 1\\)<\/li>\n<li>\\(1 + \\tan^2 x = \\sec^2 x\\)<\/li>\n<li>\\(1 + \\cot^2 x = \\csc^2 x\\)<\/li>\n<\/ul>\n<p><strong> Ph\u01b0\u01a1ng tr\u00ecnh l\u01b0\u1ee3ng gi\u00e1c c\u01a1 b\u1ea3n:<\/strong><\/p>\n<ul>\n<li>\\(\\sin x = a\\): \\(x = \\arcsin a + k2\\pi\\) ho\u1eb7c \\(x = \\pi &#8211; \\arcsin a + k2\\pi\\)<\/li>\n<li>\\(\\cos x = a\\): \\(x = \\pm \\arccos a + k2\\pi\\)<\/li>\n<li>\\(\\tan x = a\\): \\(x = \\arctan a + k\\pi\\)<\/li>\n<li>\\(\\cot x = a\\): $x = {\\mathop{\\rm arccot}\\nolimits} a + k\\pi $<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-408 size-full\" src=\"https:\/\/giapsgk.com\/chuyen-de\/wp-content\/uploads\/giapsgk-cac-cong-thuc-toan-11-1.png\" alt=\"T\u1ed5ng h\u1ee3p c\u00f4ng th\u1ee9c To\u00e1n 11 c\u1ea7n nh\u1edb theo ch\u01b0\u01a1ng tr\u00ecnh m\u1edbi\" width=\"606\" height=\"410\" \/><\/p>\n<h2><strong> 2. D\u00e3y s\u1ed1, C\u1ea5p s\u1ed1 c\u1ed9ng, C\u1ea5p s\u1ed1 nh\u00e2n<\/strong><\/h2>\n<h3><strong> Kh\u00e1i ni\u1ec7m v\u00e0 \u0111\u1ecbnh ngh\u0129a quan tr\u1ecdng<\/strong><\/h3>\n<p><strong> D\u00e3y s\u1ed1:<\/strong> L\u00e0 t\u1eadp h\u1ee3p c\u00e1c s\u1ed1 \u0111\u01b0\u1ee3c s\u1eafp x\u1ebfp theo quy lu\u1eadt.<\/p>\n<ul>\n<li>H\u1eefu h\u1ea1n (s\u1ed1 l\u01b0\u1ee3ng h\u1eefu h\u1ea1n) ho\u1eb7c v\u00f4 h\u1ea1n.<\/li>\n<li>\u0110\u01a1n \u0111i\u1ec7u (t\u0103ng\/gi\u1ea3m) ho\u1eb7c b\u1ecb ch\u1eb7n.<\/li>\n<\/ul>\n<p><strong> C\u1ea5p s\u1ed1 c\u1ed9ng:<\/strong> D\u00e3y s\u1ed1 m\u00e0 m\u1ed7i s\u1ed1 h\u1ea1ng h\u01a1n k\u00e9m s\u1ed1 h\u1ea1ng tr\u01b0\u1edbc m\u1ed9t h\u1eb1ng s\u1ed1 \\(d\\) (c\u00f4ng sai).<\/p>\n<ul>\n<li>C\u00f4ng th\u1ee9c s\u1ed1 h\u1ea1ng t\u1ed5ng qu\u00e1t: \\(u_n = u_1 + (n-1)d\\)<\/li>\n<li>T\u1ed5ng \\(n\\) s\u1ed1 h\u1ea1ng: \\(S_n = \\frac{n}{2}(u_1 + u_n) = \\frac{n}{2}[2u_1 + (n-1)d]\\)<\/li>\n<\/ul>\n<p><strong> C\u1ea5p s\u1ed1 nh\u00e2n:<\/strong> D\u00e3y s\u1ed1 m\u00e0 m\u1ed7i s\u1ed1 h\u1ea1ng \u0111\u01b0\u1ee3c nh\u00e2n v\u1edbi m\u1ed9t h\u1eb1ng s\u1ed1 \\(q\\) (c\u00f4ng b\u1ed9i).<\/p>\n<ul>\n<li>C\u00f4ng th\u1ee9c s\u1ed1 h\u1ea1ng t\u1ed5ng qu\u00e1t: \\(u_n = u_1 \\cdot q^{n-1}\\)<\/li>\n<li>T\u1ed5ng \\(n\\) s\u1ed1 h\u1ea1ng: \\(S_n = u_1 \\frac{1 &#8211; q^n}{1 &#8211; q}\\) (n\u1ebfu \\(q \\neq 1\\)).<\/li>\n<\/ul>\n<h3><strong> C\u00f4ng th\u1ee9c quan tr\u1ecdng<\/strong><\/h3>\n<p><strong> C\u1ea5p s\u1ed1 c\u1ed9ng:<\/strong><\/p>\n<ul>\n<li>\\(u_n = u_1 + (n-1)d\\)<\/li>\n<li>\\(S_n = \\frac{n}{2}(u_1 + u_n)\\)<\/li>\n<\/ul>\n<p><strong> C\u1ea5p s\u1ed1 nh\u00e2n:<\/strong><\/p>\n<ul>\n<li>\\(u_n = u_1 \\cdot q^{n-1}\\)<\/li>\n<li>\\(S_n = u_1 \\frac{1 &#8211; q^n}{1 &#8211; q}\\)<\/li>\n<li>T\u1ed5ng v\u00f4 h\u1ea1n (\\(|q| &lt; 1\\)): \\(S_\\infty = \\frac{u_1}{1 &#8211; q}\\)<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-409 size-full\" src=\"https:\/\/giapsgk.com\/chuyen-de\/wp-content\/uploads\/giapsgk-cac-cong-thuc-toan-11-2.png\" alt=\"C\u00e1c c\u00f4ng th\u1ee9c To\u00e1n l\u1edbp 11 \u0110\u1ea1i s\u1ed1 v\u00e0 H\u00ecnh h\u1ecdc\" width=\"1153\" height=\"483\" \/><\/p>\n<h2><strong> 3. C\u00e1c s\u1ed1 \u0111\u1eb7c tr\u01b0ng \u0111o xu th\u1ebf trung t\u00e2m c\u1ee7a m\u1eabu s\u1ed1 li\u1ec7u gh\u00e9p nh\u00f3m<\/strong><\/h2>\n<h3><strong> Kh\u00e1i ni\u1ec7m v\u00e0 \u0111\u1ecbnh ngh\u0129a quan tr\u1ecdng<\/strong><\/h3>\n<ul>\n<li><strong>M\u1eabu s\u1ed1 li\u1ec7u gh\u00e9p nh\u00f3m:<\/strong> D\u1eef li\u1ec7u \u0111\u01b0\u1ee3c chia th\u00e0nh c\u00e1c kho\u1ea3ng v\u1edbi t\u1ea7n s\u1ed1 t\u01b0\u01a1ng \u1ee9ng.<\/li>\n<li><strong> S\u1ed1 trung b\u00ecnh (mean):<\/strong> Trung b\u00ecnh c\u1ed9ng c\u00f3 tr\u1ecdng s\u1ed1 c\u1ee7a c\u00e1c \u0111i\u1ec3m gi\u1eefa kho\u1ea3ng.<\/li>\n<li><strong> M\u1ed1t (mode):<\/strong> Gi\u00e1 tr\u1ecb ho\u1eb7c kho\u1ea3ng c\u00f3 t\u1ea7n s\u1ed1 cao nh\u1ea5t.<\/li>\n<li><strong> Trung v\u1ecb (median):<\/strong> Gi\u00e1 tr\u1ecb \u1edf gi\u1eefa khi d\u1eef li\u1ec7u \u0111\u01b0\u1ee3c s\u1eafp x\u1ebfp.<\/li>\n<li><strong> T\u1ee9 ph\u00e2n v\u1ecb:<\/strong> Chia d\u1eef li\u1ec7u th\u00e0nh b\u1ed1n ph\u1ea7n b\u1eb1ng nhau (\\(Q_1, Q_2, Q_3\\)).<\/li>\n<\/ul>\n<h3><strong> C\u00f4ng th\u1ee9c quan tr\u1ecdng<\/strong><\/h3>\n<ul>\n<li><strong> S\u1ed1 trung b\u00ecnh:<\/strong> \\(\\bar{x} = \\frac{\\sum f_i x_i}{\\sum f_i}\\), v\u1edbi \\(x_i\\) l\u00e0 \u0111i\u1ec3m gi\u1eefa c\u1ee7a kho\u1ea3ng th\u1ee9 \\(i\\), \\(f_i\\) l\u00e0 t\u1ea7n s\u1ed1.<\/li>\n<li><strong> Trung v\u1ecb:<\/strong> N\u1eb1m trong kho\u1ea3ng m\u00e0 t\u1ea7n s\u1ed1 t\u00edch l\u0169y \u0111\u1ea1t ho\u1eb7c v\u01b0\u1ee3t \\(\\frac{n}{2}\\).<\/li>\n<li><strong> M\u1ed1t:<\/strong> Kho\u1ea3ng c\u00f3 t\u1ea7n s\u1ed1 cao nh\u1ea5t.<\/li>\n<\/ul>\n<h2><strong> 4. Quan h\u1ec7 song song trong kh\u00f4ng gian<\/strong><\/h2>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-410 size-full\" src=\"https:\/\/giapsgk.com\/chuyen-de\/wp-content\/uploads\/giapsgk-cac-cong-thuc-toan-11-3.png\" alt=\"C\u00f4ng th\u1ee9c gi\u1ea3i nhanh To\u00e1n 11 cho \u00f4n thi h\u1ecdc k\u1ef3\" width=\"553\" height=\"379\" \/><\/p>\n<h3><strong> Kh\u00e1i ni\u1ec7m v\u00e0 \u0111\u1ecbnh ngh\u0129a quan tr\u1ecdng<\/strong><\/h3>\n<ul>\n<li><strong> \u0110\u01b0\u1eddng th\u1eb3ng v\u00e0 m\u1eb7t ph\u1eb3ng:<\/strong> \u0110\u01b0\u1ee3c x\u00e1c \u0111\u1ecbnh b\u1edfi c\u00e1c \u0111i\u1ec3m v\u00e0 vector trong kh\u00f4ng gian.<\/li>\n<li><strong> Hai \u0111\u01b0\u1eddng th\u1eb3ng song song:<\/strong> C\u00f3 vector ch\u1ec9 ph\u01b0\u01a1ng song song.<\/li>\n<li><strong> \u0110\u01b0\u1eddng th\u1eb3ng v\u00e0 m\u1eb7t ph\u1eb3ng song song:<\/strong> Vector ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng vu\u00f4ng g\u00f3c v\u1edbi vector ph\u00e1p tuy\u1ebfn c\u1ee7a m\u1eb7t ph\u1eb3ng.<\/li>\n<li><strong> Hai m\u1eb7t ph\u1eb3ng song song:<\/strong> C\u00f3 vector ph\u00e1p tuy\u1ebfn song song.<\/li>\n<li><strong> Ph\u00e9p chi\u1ebfu song song:<\/strong> Chi\u1ebfu h\u00ecnh kh\u00f4ng gian l\u00ean m\u1eb7t ph\u1eb3ng theo m\u1ed9t h\u01b0\u1edbng c\u1ed1 \u0111\u1ecbnh.<\/li>\n<\/ul>\n<h3><strong> C\u00f4ng th\u1ee9c quan tr\u1ecdng<\/strong><\/h3>\n<p><strong> Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng:<\/strong> \\(\\vec{r} = \\vec{a} + t\\vec{b}\\), v\u1edbi \\(\\vec{b}\\) l\u00e0 vector ch\u1ec9 ph\u01b0\u01a1ng.<\/p>\n<p><strong> Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng:<\/strong> \\(ax + by + cz + d = 0\\).<\/p>\n<p><strong> \u0110i\u1ec1u ki\u1ec7n song song:<\/strong><\/p>\n<ul>\n<li>Hai \u0111\u01b0\u1eddng th\u1eb3ng: Vector ch\u1ec9 ph\u01b0\u01a1ng \\(\\vec{b_1}\\), \\(\\vec{b_2}\\) t\u1ec9 l\u1ec7.<\/li>\n<li>\u0110\u01b0\u1eddng th\u1eb3ng v\u00e0 m\u1eb7t ph\u1eb3ng: \\(\\vec{b} \\cdot \\vec{n} = 0\\), v\u1edbi \\(\\vec{n}\\) l\u00e0 vector ph\u00e1p tuy\u1ebfn.<\/li>\n<li>Hai m\u1eb7t ph\u1eb3ng: Vector ph\u00e1p tuy\u1ebfn \\(\\vec{n_1}\\), \\(\\vec{n_2}\\) t\u1ec9 l\u1ec7.<\/li>\n<\/ul>\n<h2><strong> 5. Gi\u1edbi h\u1ea1n. H\u00e0m s\u1ed1 li\u00ean t\u1ee5c<\/strong><\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-411 size-full\" src=\"https:\/\/giapsgk.com\/chuyen-de\/wp-content\/uploads\/giapsgk-cac-cong-thuc-toan-11-4.png\" alt=\"T\u1ea5t c\u1ea3 c\u00f4ng th\u1ee9c To\u00e1n l\u1edbp 11 theo t\u1eebng ch\u01b0\u01a1ng\" width=\"584\" height=\"432\" \/><\/p>\n<h3><strong> Kh\u00e1i ni\u1ec7m v\u00e0 \u0111\u1ecbnh ngh\u0129a quan tr\u1ecdng<\/strong><\/h3>\n<p><strong> Gi\u1edbi h\u1ea1n c\u1ee7a d\u00e3y s\u1ed1:<\/strong> \\(\\lim_{n \\to \\infty} u_n = l\\) n\u1ebfu d\u00e3y ti\u1ebfn \u0111\u1ebfn \\(l\\).<\/p>\n<p><strong> Gi\u1edbi h\u1ea1n c\u1ee7a h\u00e0m s\u1ed1:<\/strong> \\(\\lim_{x \\to a} f(x) = l\\) n\u1ebfu \\(f(x)\\) ti\u1ebfn \u0111\u1ebfn \\(l\\) khi \\(x \\to a\\).<\/p>\n<p><strong> H\u00e0m s\u1ed1 li\u00ean t\u1ee5c:<\/strong> H\u00e0m \\(f(x)\\) li\u00ean t\u1ee5c t\u1ea1i \\(x = a\\) n\u1ebfu:<\/p>\n<ul>\n<li>\\(f(a)\\) x\u00e1c \u0111\u1ecbnh.<\/li>\n<li>\\(\\lim_{x \\to a} f(x)\\) t\u1ed3n t\u1ea1i.<\/li>\n<li>\\(\\lim_{x \\to a} f(x) = f(a)\\).<\/li>\n<\/ul>\n<h3><strong> C\u00f4ng th\u1ee9c quan tr\u1ecdng<\/strong><\/h3>\n<p><strong> Gi\u1edbi h\u1ea1n th\u00f4ng d\u1ee5ng:<\/strong><\/p>\n<ul>\n<li>\\(\\lim_{x \\to 0} \\frac{\\sin x}{x} = 1\\)<\/li>\n<li>\\(\\lim_{x \\to 0} \\frac{1 &#8211; \\cos x}{x} = 0\\)<\/li>\n<li>\\(\\lim_{x \\to \\infty} \\left(1 + \\frac{1}{x}\\right)^x = e\\)<\/li>\n<\/ul>\n<p><strong> Quy t\u1eafc:<\/strong><\/p>\n<ul>\n<li>T\u1ed5ng: \\(\\lim (f(x) + g(x)) = \\lim f(x) + \\lim g(x)\\)<\/li>\n<li>T\u00edch: \\(\\lim (f(x) \\cdot g(x)) = \\lim f(x) \\cdot \\lim g(x)\\)<\/li>\n<li>Th\u01b0\u01a1ng: \\(\\lim \\frac{f(x)}{g(x)} = \\frac{\\lim f(x)}{\\lim g(x)}\\) (n\u1ebfu \\(\\lim g(x) \\neq 0\\)).<\/li>\n<\/ul>\n<h2><strong> 6. H\u00e0m s\u1ed1 m\u0169 v\u00e0 h\u00e0m s\u1ed1 l\u00f4garit<\/strong><\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-412 size-full\" src=\"https:\/\/giapsgk.com\/chuyen-de\/wp-content\/uploads\/giapsgk-cac-cong-thuc-toan-11-5.png\" alt=\"B\u1ea3ng c\u00f4ng th\u1ee9c To\u00e1n l\u1edbp 11 \u0111\u1ea7y \u0111\u1ee7 v\u00e0 d\u1ec5 nh\u1edb\" width=\"564\" height=\"308\" \/><\/p>\n<h3><strong> Kh\u00e1i ni\u1ec7m v\u00e0 \u0111\u1ecbnh ngh\u0129a quan tr\u1ecdng<\/strong><\/h3>\n<ul>\n<li>L\u0169y th\u1eeba: \\(a^x\\), v\u1edbi \\(a &gt; 0\\), \\(a \\neq 1\\), \\(x\\) l\u00e0 s\u1ed1 th\u1ef1c.<\/li>\n<li>L\u00f4garit: \\(\\log_a x\\), x\u00e1c \u0111\u1ecbnh khi \\(x &gt; 0\\), \\(a &gt; 0\\), \\(a \\neq 1\\).<\/li>\n<li>H\u00e0m s\u1ed1 m\u0169: \\(f(x) = a^x\\), t\u0103ng n\u1ebfu \\(a &gt; 1\\), gi\u1ea3m n\u1ebfu \\(0 &lt; a &lt; 1\\).<\/li>\n<li>H\u00e0m s\u1ed1 l\u00f4garit: \\(f(x) = \\log_a x\\), t\u0103ng n\u1ebfu \\(a &gt; 1\\), gi\u1ea3m n\u1ebfu \\(0 &lt; a &lt; 1\\).<\/li>\n<li>Ph\u01b0\u01a1ng tr\u00ecnh m\u0169 v\u00e0 l\u00f4garit: Gi\u1ea3i b\u1eb1ng c\u00e1ch \u0111\u01b0a v\u1ec1 c\u00f9ng c\u01a1 s\u1ed1 ho\u1eb7c d\u00f9ng l\u00f4garit.<\/li>\n<\/ul>\n<h3><strong> C\u00f4ng th\u1ee9c quan tr\u1ecdng<\/strong><\/h3>\n<p><strong> L\u0169y th\u1eeba:<\/strong><\/p>\n<ul>\n<li>\\(a^m \\cdot a^n = a^{m+n}\\)<\/li>\n<li>\\(\\frac{a^m}{a^n} = a^{m-n}\\)<\/li>\n<li>\\((a^m)^n = a^{mn}\\)<\/li>\n<\/ul>\n<p><strong> L\u00f4garit:<\/strong><\/p>\n<ul>\n<li>\\(\\log_a (xy) = \\log_a x + \\log_a y\\)<\/li>\n<li>\\(\\log_a \\left(\\frac{x}{y}\\right) = \\log_a x &#8211; \\log_a y\\)<\/li>\n<li>\\(\\log_a (x^n) = n \\log_a x\\)<\/li>\n<li>\u0110\u1ed5i c\u01a1 s\u1ed1: \\(\\log_a x = \\frac{\\log_b x}{\\log_b a}\\)<\/li>\n<\/ul>\n<p><strong> Ph\u01b0\u01a1ng tr\u00ecnh m\u0169:<\/strong> \\(a^{f(x)} = a^{g(x)} \\implies f(x) = g(x)\\).<\/p>\n<p><strong> B\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh m\u0169\/l\u00f4garit:<\/strong> D\u1ef1a v\u00e0o t\u00ednh \u0111\u01a1n \u0111i\u1ec7u c\u1ee7a h\u00e0m \u0111\u1ec3 so s\u00e1nh.<\/p>\n<h2><strong> 7. Quan h\u1ec7 vu\u00f4ng g\u00f3c trong kh\u00f4ng gian<\/strong><\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-413 size-full\" src=\"https:\/\/giapsgk.com\/chuyen-de\/wp-content\/uploads\/giapsgk-cac-cong-thuc-toan-11-6.png\" alt=\"C\u00e1ch ghi nh\u1edb c\u00f4ng th\u1ee9c To\u00e1n l\u1edbp 11 hi\u1ec7u qu\u1ea3 nh\u1ea5t\" width=\"348\" height=\"277\" \/><\/p>\n<h3><strong> Kh\u00e1i ni\u1ec7m v\u00e0 \u0111\u1ecbnh ngh\u0129a quan tr\u1ecdng<\/strong><\/h3>\n<ul>\n<li><strong> Hai \u0111\u01b0\u1eddng th\u1eb3ng vu\u00f4ng g\u00f3c:<\/strong> Vector ch\u1ec9 ph\u01b0\u01a1ng th\u1ecfa m\u00e3n \\(\\vec{a} \\cdot \\vec{b} = 0\\).<\/li>\n<li><strong> \u0110\u01b0\u1eddng th\u1eb3ng vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng: V<\/strong>ector ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng song song v\u1edbi vector ph\u00e1p tuy\u1ebfn c\u1ee7a m\u1eb7t ph\u1eb3ng.<\/li>\n<li><strong> Hai m\u1eb7t ph\u1eb3ng vu\u00f4ng g\u00f3c:<\/strong> Vector ph\u00e1p tuy\u1ebfn th\u1ecfa m\u00e3n \\(\\vec{n_1} \\cdot \\vec{n_2} = 0\\).<\/li>\n<li><strong>G\u00f3c nh\u1ecb di\u1ec7n:<\/strong> G\u00f3c gi\u1eefa hai m\u1eb7t ph\u1eb3ng.<\/li>\n<li><strong> Kho\u1ea3ng c\u00e1ch:<\/strong> Kho\u1ea3ng c\u00e1ch gi\u1eefa c\u00e1c \u0111i\u1ec3m, \u0111\u01b0\u1eddng th\u1eb3ng, ho\u1eb7c m\u1eb7t ph\u1eb3ng.<\/li>\n<li><strong> Th\u1ec3 t\u00edch:<\/strong> Th\u1ec3 t\u00edch c\u1ee7a l\u0103ng tr\u1ee5, ch\u00f3p, v\u00e0 c\u00e1c kh\u1ed1i kh\u00e1c.<\/li>\n<\/ul>\n<h3><strong> C\u00f4ng th\u1ee9c quan tr\u1ecdng<\/strong><\/h3>\n<ul>\n<li><strong> T\u00edch v\u00f4 h\u01b0\u1edbng:<\/strong> \\(\\vec{a} \\cdot \\vec{b} = |\\vec{a}| |\\vec{b}| \\cos \\theta\\).<\/li>\n<li><strong> Kho\u1ea3ng c\u00e1ch t\u1eeb \u0111i\u1ec3m \u0111\u1ebfn m\u1eb7t ph\u1eb3ng:<\/strong> \\(d = \\frac{|ax_0 + by_0 + cz_0 + d|}{\\sqrt{a^2 + b^2 + c^2}}\\).<\/li>\n<li><strong> G\u00f3c gi\u1eefa \u0111\u01b0\u1eddng th\u1eb3ng v\u00e0 m\u1eb7t ph\u1eb3ng:<\/strong> \\(\\sin \\theta = \\frac{|\\vec{b} \\cdot \\vec{n}|}{|\\vec{b}| |\\vec{n}|}\\).<\/li>\n<li><strong> Th\u1ec3 t\u00edch l\u0103ng tr\u1ee5:<\/strong> \\(V = S \\cdot h\\), v\u1edbi \\(S\\) l\u00e0 di\u1ec7n t\u00edch \u0111\u00e1y, \\(h\\) l\u00e0 chi\u1ec1u cao.<\/li>\n<li><strong> Th\u1ec3 t\u00edch ch\u00f3p:<\/strong> \\(V = \\frac{1}{3} S \\cdot h\\).<\/li>\n<\/ul>\n<h2><strong> 8. C\u00e1c quy t\u1eafc t\u00ednh x\u00e1c su\u1ea5t<\/strong><\/h2>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-414 size-full\" src=\"https:\/\/giapsgk.com\/chuyen-de\/wp-content\/uploads\/giapsgk-cac-cong-thuc-toan-11-7.png\" alt=\"C\u00f4ng th\u1ee9c To\u00e1n 11 tr\u1ecdng t\u00e2m cho thi gi\u1eefa k\u1ef3\" width=\"538\" height=\"371\" \/><\/p>\n<h3><strong> Kh\u00e1i ni\u1ec7m v\u00e0 \u0111\u1ecbnh ngh\u0129a quan tr\u1ecdng<\/strong><\/h3>\n<ul>\n<li><strong> Bi\u1ebfn c\u1ed1:<\/strong> T\u1eadp con c\u1ee7a kh\u00f4ng gian m\u1eabu.<\/li>\n<li><strong> H\u1ee3p:<\/strong> \\(A \\cup B\\) (\u00edt nh\u1ea5t m\u1ed9t trong \\(A\\) ho\u1eb7c \\(B\\) x\u1ea3y ra).<\/li>\n<li><strong> Giao: \\<\/strong>(A \\cap B\\) (c\u1ea3 \\(A\\) v\u00e0 \\(B\\) x\u1ea3y ra).<\/li>\n<li><strong> \u0110\u1ed9c l\u1eadp:<\/strong> \\(A\\) v\u00e0 \\(B\\) \u0111\u1ed9c l\u1eadp n\u1ebfu \\(P(A \\cap B) = P(A) \\cdot P(B)\\).<\/li>\n<li><strong> X\u00e1c su\u1ea5t:<\/strong> \\(P(A) = \\frac{\\text{S\u1ed1 k\u1ebft qu\u1ea3 thu\u1eadn l\u1ee3i}}{\\text{T\u1ed5ng s\u1ed1 k\u1ebft qu\u1ea3}}\\).<\/li>\n<\/ul>\n<h3><strong> C\u00f4ng th\u1ee9c quan tr\u1ecdng<\/strong><\/h3>\n<ul>\n<li><strong> Quy t\u1eafc c\u1ed9ng:<\/strong> \\(P(A \\cup B) = P(A) + P(B) &#8211; P(A \\cap B)\\).<\/li>\n<li><strong> Quy t\u1eafc nh\u00e2n (\u0111\u1ed9c l\u1eadp): \\<\/strong>(P(A \\cap ects the curve at a point.<\/li>\n<li><strong> \u0110\u1ea1o h\u00e0m c\u1ea5p hai:<\/strong> \u0110\u1ea1o h\u00e0m c\u1ee7a \u0111\u1ea1o h\u00e0m, bi\u1ec3u th\u1ecb \u0111\u1ed9 cong.<\/li>\n<\/ul>\n<h2><strong>9. \u0110\u1ea1o h\u00e0m<\/strong><\/h2>\n<h3><strong> C\u00f4ng th\u1ee9c quan tr\u1ecdng<\/strong><\/h3>\n<p><strong> \u0110\u1ea1o h\u00e0m c\u01a1 b\u1ea3n:<\/strong><\/p>\n<ul>\n<li>\\((x^n)&#8217; = n x^{n-1}\\)<\/li>\n<li>\\((\\sin x)&#8217; = \\cos x\\)<\/li>\n<li>\\((\\cos x)&#8217; =-\\sin x\\)<\/li>\n<li>\\((\\tan x)&#8217; = \\sec^2 x\\)<\/li>\n<li>\\((e^x)&#8217; = e^x\\)<\/li>\n<li>\\((\\ln x)&#8217; = \\frac{1}{x}\\)<\/li>\n<li>\\((a^x)&#8217; = a^x \\ln a\\)<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-415 size-full\" src=\"https:\/\/giapsgk.com\/chuyen-de\/wp-content\/uploads\/giapsgk-cac-cong-thuc-toan-11-8.png\" alt=\"C\u00f4ng th\u1ee9c To\u00e1n 11 \u00f4n thi cu\u1ed1i k\u1ef3 1 v\u00e0 h\u1ecdc k\u1ef3 2\" width=\"657\" height=\"821\" \/><\/p>\n<p><strong> Quy t\u1eafc:<\/strong><\/p>\n<ul>\n<li>T\u1ed5ng: \\((f + g)&#8217; = f&#8217; + g&#8217;\\)<\/li>\n<li>T\u00edch: \\((fg)&#8217; = f&#8217;g + fg&#8217;\\)<\/li>\n<li>Th\u01b0\u01a1ng: \\(\\left(\\frac{f}{g}\\right)&#8217; = \\frac{f&#8217;g &#8211; fg&#8217;}{g^2}\\)<\/li>\n<li>Chu\u1ed7i: \\((f(g(x)))&#8217; = f'(g(x)) \\cdot g'(x)\\)<\/li>\n<\/ul>\n<p><strong> \u0110\u1ea1o h\u00e0m c\u1ea5p hai:<\/strong> \\(f&#8221;(x) = \\frac{d}{dx} (f'(x))\\).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>B\u1ea1n \u0111ang t\u00ecm ki\u1ebfm c\u00e1c c\u00f4ng th\u1ee9c To\u00e1n 11 \u0111\u1ec3 \u00f4n thi h\u1ecdc k\u1ef3 hi\u1ec7u qu\u1ea3? B\u00e0i vi\u1ebft n\u00e0y t\u1ed5ng h\u1ee3p \u0111\u1ea7y \u0111\u1ee7 v\u00e0 chi ti\u1ebft c\u00e1c c\u00f4ng th\u1ee9c quan tr\u1ecdng, kh\u00e1i ni\u1ec7m v\u00e0 \u0111\u1ecbnh ngh\u0129a t\u1eeb ba b\u1ed9 s\u00e1ch gi\u00e1o khoa ph\u1ed5 bi\u1ebfn: K\u1ebft n\u1ed1i tri th\u1ee9c v\u1edbi cu\u1ed9c s\u1ed1ng, C\u00e1nh Di\u1ec1u, v\u00e0 Ch\u00e2n &#8230; <a title=\"T\u1ed5ng h\u1ee3p c\u00e1c c\u00f4ng th\u1ee9c To\u00e1n 11 (c\u1ea3 n\u0103m &#8211; s\u00e1ch m\u1edbi)\" class=\"read-more\" href=\"https:\/\/giapsgk.com\/chuyen-de\/cac-cong-thuc-toan-11\/\" aria-label=\"\u0110\u1ecdc th\u00eam v\u1ec1 T\u1ed5ng h\u1ee3p c\u00e1c c\u00f4ng th\u1ee9c To\u00e1n 11 (c\u1ea3 n\u0103m &#8211; s\u00e1ch m\u1edbi)\">\u0110\u1ecdc ti\u1ebfp<\/a><\/p>\n","protected":false},"author":1,"featured_media":417,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[],"class_list":["post-401","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\/401","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=401"}],"version-history":[{"count":8,"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/posts\/401\/revisions"}],"predecessor-version":[{"id":418,"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/posts\/401\/revisions\/418"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/media\/417"}],"wp:attachment":[{"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/media?parent=401"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/categories?post=401"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/giapsgk.com\/chuyen-de\/wp-json\/wp\/v2\/tags?post=401"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}