{"id":2827,"date":"2026-10-18T23:58:13","date_gmt":"2026-10-18T16:58:13","guid":{"rendered":"https:\/\/navi.edu.vn\/blog\/?p=2827"},"modified":"2026-10-19T00:31:51","modified_gmt":"2026-10-18T17:31:51","slug":"de-cuong-on-tap-giua-ki-1-toan-12-2026-2027","status":"publish","type":"post","link":"https:\/\/navi.edu.vn\/blog\/de-cuong-on-tap-giua-ki-1-toan-12-2026-2027\/","title":{"rendered":"\u0110\u1ec1 c\u01b0\u01a1ng \u00f4n t\u1eadp Gi\u1eefa k\u00ec 1 To\u00e1n 12 (2026-2027) & \u0110\u1ec1 m\u1eabu c\u00f3 \u0111\u00e1p \u00e1n"},"content":{"rendered":"

K\u1ef3 thi gi\u1eefa h\u1ecdc k\u00ec 1 To\u00e1n 12 ch\u00ednh l\u00e0 d\u1ea5u m\u1ed1c quan tr\u1ecdng \u0111\u1ea7u ti\u00ean quy\u1ebft \u0111\u1ecbnh \u0111i\u1ec3m s\u1ed1 h\u1ecdc b\u1ea1 l\u1edbp 12 (ph\u1ee5c v\u1ee5 x\u00e9t tuy\u1ec3n s\u1edbm \u0111\u1ea1i h\u1ecdc), \u0111\u1ed3ng th\u1eddi l\u00e0 \u0111\u1ee3t t\u1ed5ng duy\u1ec7t ki\u1ec3m tra m\u1ee9c \u0111\u1ed9 th\u00edch \u1ee9ng c\u1ee7a h\u1ecdc sinh v\u1edbi c\u1ea5u tr\u00fac \u0111\u1ec1 thi T\u1ed1t nghi\u1ec7p THPT. \u0110\u1ec3 gi\u00fap c\u00e1c s\u0129 t\u1eed \u00f4n luy\u1ec7n \u0111\u00fang tr\u1ecdng t\u00e2m v\u00e0 l\u00e0m ch\u1ee7 \u0111i\u1ec3m s\u1ed1 trong ch\u01b0\u01a1ng tr\u00ecnh h\u1ecdc To\u00e1n l\u1edbp 12<\/strong><\/span>, NaviEdu g\u1eedi t\u1edbi b\u1ed9 \u0111\u1ec1 c\u01b0\u01a1ng \u00f4n t\u1eadp gi\u1eefa k\u00ec 1 to\u00e1n 12<\/b> chu\u1ea9n h\u00f3a: t\u00f3m t\u1eaft \u0111\u1ea7y \u0111\u1ee7 c\u00f4ng th\u1ee9c c\u1ed1t l\u00f5i, ph\u00e2n lo\u1ea1i ph\u01b0\u01a1ng ph\u00e1p gi\u1ea3i t\u1eebng d\u1ea1ng b\u00e0i, ma tr\u1eadn \u0111\u1ec1 thi m\u1eabu k\u00e8m link t\u1ea3i PDF c\u00f3 \u0111\u00e1p \u00e1n chi ti\u1ebft!<\/p>\n

<\/p>\n

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C\u1ea5u tr\u00fac ma tr\u1eadn \u0111\u1ec1 thi gi\u1eefa k\u00ec 1 To\u00e1n 12 (\u0110\u1ecbnh d\u1ea1ng \u0111\u1ec1 thi m\u1edbi)<\/h2>\n
\"T\u00ecm
T\u00ecm hi\u1ec3u v\u1ec1 c\u1ea5u tr\u00fac ma tr\u1eadn \u0111\u1ec1 thi gi\u1eefa HK1 To\u00e1n 12 theo \u0111\u1ecbnh d\u1ea1ng \u0111\u1ec1 thi m\u1edbi<\/figcaption><\/figure>\n

B\u01b0\u1edbc sang ch\u01b0\u01a1ng tr\u00ecnh GDPT m\u1edbi, \u0111\u1ec1 ki\u1ec3m tra gi\u1eefa h\u1ecdc k\u00ec 1 m\u00f4n To\u00e1n 12 (th\u1eddi gian l\u00e0m b\u00e0i 90 ph\u00fat) \u0111\u00e3 thay \u0111\u1ed5i to\u00e0n di\u1ec7n ph\u01b0\u01a1ng th\u1ee9c \u0111\u00e1nh gi\u00e1: chuy\u1ec3n t\u1eeb 50 c\u00e2u tr\u1eafc nghi\u1ec7m 4 l\u1ef1a ch\u1ecdn truy\u1ec1n th\u1ed1ng sang c\u1ea5u tr\u00fac 3 ph\u1ea7n t\u00edch h\u1ee3p<\/strong>. \u0110\u1ecbnh d\u1ea1ng \u0111\u1ec1 m\u1edbi \u0111\u00f2i h\u1ecfi h\u1ecdc sinh ph\u1ea3i hi\u1ec3u s\u00e2u b\u1ea3n ch\u1ea5t to\u00e1n h\u1ecdc, \u0111\u1ecdc \u0111\u1ed3 th\u1ecb nh\u1ea1y b\u00e9n v\u00e0 tri\u1ec7t ti\u00eau ho\u00e0n to\u00e0n kh\u1ea3 n\u0103ng “khoanh l\u1ee5i” may r\u1ee7i.<\/p>\n

1. T\u1ef7 l\u1ec7 ph\u00e2n b\u1ed5 m\u1ea1ch ki\u1ebfn th\u1ee9c<\/h3>\n
    \n
  • \u0110\u1ea1i s\u1ed1 & Gi\u1ea3i t\u00edch (~70% t\u1ed5ng \u0111i\u1ec3m – t\u01b0\u01a1ng \u0111\u01b0\u01a1ng 7.0 \u0111i\u1ec3m):<\/strong> Tr\u1ecdng t\u00e2m xoay quanh Ch\u01b0\u01a1ng I – \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 (t\u00ednh \u0111\u01a1n \u0111i\u1ec7u, c\u1ef1c tr\u1ecb, gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t – nh\u1ecf nh\u1ea5t, \u0111\u01b0\u1eddng ti\u1ec7m c\u1eadn, nh\u1eadn di\u1ec7n \u0111\u1ed3 th\u1ecb v\u00e0 c\u00e1c b\u00e0i to\u00e1n th\u1ef1c t\u1ebf t\u1ed1i \u01b0u h\u00f3a).<\/li>\n
  • H\u00ecnh h\u1ecdc kh\u00f4ng gian (~30% t\u1ed5ng \u0111i\u1ec3m – t\u01b0\u01a1ng \u0111\u01b0\u01a1ng 3.0 \u0111i\u1ec3m):<\/strong> T\u1eadp trung v\u00e0o Ch\u01b0\u01a1ng II – Vect\u01a1 v\u00e0 h\u1ec7 tr\u1ee5c t\u1ecda \u0111\u1ed9 trong kh\u00f4ng gian Oxyz (c\u00e1c quy t\u1eafc vect\u01a1 trong kh\u00f4ng gian, t\u1ecda \u0111\u1ed9 \u0111i\u1ec3m, t\u1ecda \u0111\u1ed9 vect\u01a1 v\u00e0 bi\u1ec3u th\u1ee9c t\u1ecda \u0111\u1ed9 c\u1ee7a c\u00e1c ph\u00e9p to\u00e1n).<\/li>\n<\/ul>\n

    2. C\u1ea5u tr\u00fac 3 ph\u1ea7n theo chu\u1ea9n \u0111\u1ecbnh d\u1ea1ng m\u1edbi (34 l\u1ec7nh h\u1ecfi)<\/h3>\n
    \n

    Ph\u1ea7n I: C\u00e2u tr\u1eafc nghi\u1ec7m nhi\u1ec1u ph\u01b0\u01a1ng \u00e1n l\u1ef1a ch\u1ecdn (3.0 \u0111i\u1ec3m)<\/p>\n

      \n
    • Quy m\u00f4:<\/strong> G\u1ed3m 12 c\u00e2u h\u1ecfi (t\u1eeb c\u00e2u 1 \u0111\u1ebfn c\u00e2u 12), m\u1ed7i c\u00e2u c\u00f3 4 ph\u01b0\u01a1ng \u00e1n A, B, C, D v\u00e0 ch\u1ec9 ch\u1ecdn 1 \u0111\u00e1p \u00e1n \u0111\u00fang duy nh\u1ea5t.<\/li>\n
    • C\u00e1ch t\u00ednh \u0111i\u1ec3m:<\/strong> M\u1ed7i c\u00e2u tr\u1ea3 l\u1eddi \u0111\u00fang \u0111\u01b0\u1ee3c 0.25 \u0111i\u1ec3m<\/strong>.<\/li>\n
    • M\u1ee9c \u0111\u1ed9:<\/strong> T\u1eadp trung v\u00e0o c\u1ea5p \u0111\u1ed9 Nh\u1eadn bi\u1ebft v\u00e0 Th\u00f4ng hi\u1ec3u, ki\u1ec3m tra ph\u1ea3n x\u1ea1 \u0111\u1ecdc b\u1ea3ng bi\u1ebfn thi\u00ean, \u0111\u1ed3 th\u1ecb v\u00e0 c\u00f4ng th\u1ee9c t\u1ecda \u0111\u1ed9 c\u01a1 b\u1ea3n.<\/li>\n<\/ul>\n<\/div>\n
      \n

      Ph\u1ea7n II: C\u00e2u tr\u1eafc nghi\u1ec7m \u0110\u00fang\/Sai (4.0 \u0111i\u1ec3m – Ph\u00e2n h\u00f3a s\u00e2u)<\/p>\n

        \n
      • Quy m\u00f4:<\/strong> G\u1ed3m 4 c\u00e2u h\u1ecfi l\u1edbn (3 c\u00e2u Gi\u1ea3i t\u00edch, 1 c\u00e2u H\u00ecnh h\u1ecdc Oxyz). M\u1ed7i c\u00e2u g\u1ed3m 4 m\u1ec7nh \u0111\u1ec1 nh\u1ecf a), b), c), d) n\u00e2ng d\u1ea7n \u0111\u1ed9 kh\u00f3 t\u1eeb Nh\u1eadn bi\u1ebft \u2192 Th\u00f4ng hi\u1ec3u \u2192 V\u1eadn d\u1ee5ng.<\/li>\n
      • Quy t\u1eafc t\u00ednh \u0111i\u1ec3m l\u0169y ti\u1ebfn:<\/strong> Trong m\u1ed7i c\u00e2u h\u1ecfi l\u1edbn:\n
          \n
        • Tr\u1ea3 l\u1eddi \u0111\u00fang 1 \u00fd: ch\u1ec9 \u0111\u01b0\u1ee3c 0.1 \u0111i\u1ec3m<\/strong>.<\/li>\n
        • Tr\u1ea3 l\u1eddi \u0111\u00fang 2 \u00fd: \u0111\u01b0\u1ee3c 0.25 \u0111i\u1ec3m<\/strong>.<\/li>\n
        • Tr\u1ea3 l\u1eddi \u0111\u00fang 3 \u00fd: \u0111\u01b0\u1ee3c 0.5 \u0111i\u1ec3m<\/strong>.<\/li>\n
        • Tr\u1ea3 l\u1eddi \u0111\u00fang tr\u1ecdn v\u1eb9n c\u1ea3 4 \u00fd: \u0111\u1ea1t t\u1ed1i \u0111a 1.0 \u0111i\u1ec3m<\/strong>.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n
          \n

          Ph\u1ea7n III: C\u00e2u tr\u1eafc nghi\u1ec7m tr\u1ea3 l\u1eddi ng\u1eafn (3.0 \u0111i\u1ec3m – Ch\u1ed1ng \u0111o\u00e1n m\u00f2)<\/p>\n

            \n
          • Quy m\u00f4:<\/strong> G\u1ed3m 6 c\u00e2u h\u1ecfi (t\u1eeb c\u00e2u 1 \u0111\u1ebfn c\u00e2u 6). Kh\u00f4ng c\u00f3 ph\u01b0\u01a1ng \u00e1n g\u1ee3i \u00fd s\u1eb5n; h\u1ecdc sinh ph\u1ea3i t\u1ef1 gi\u1ea3i nh\u01b0 t\u1ef1 lu\u1eadn v\u00e0 \u0111i\u1ec1n k\u1ebft qu\u1ea3 s\u1ed1 v\u00e0o \u00f4 tr\u1ed1ng.<\/li>\n
          • C\u00e1ch t\u00ednh \u0111i\u1ec3m:<\/strong> M\u1ed7i c\u00e2u \u0111i\u1ec1n k\u1ebft qu\u1ea3 \u0111\u00fang \u0111\u01b0\u1ee3c 0.5 \u0111i\u1ec3m<\/strong>.<\/li>\n
          • M\u1ee9c \u0111\u1ed9:<\/strong> C\u00e1c b\u00e0i to\u00e1n V\u1eadn d\u1ee5ng v\u00e0 V\u1eadn d\u1ee5ng cao (to\u00e1n th\u1ef1c t\u1ebf t\u1ed1i \u01b0u chi ph\u00ed\/l\u1ee3i nhu\u1eadn\/th\u1ec3 t\u00edch, t\u00ecm tham s\u1ed1 m, b\u00e0i to\u00e1n th\u1ef1c ti\u1ec5n t\u1ecda \u0111\u1ed9 kh\u00f4ng gian).<\/li>\n<\/ul>\n<\/div>\n

            3. B\u1ea3ng ma tr\u1eadn ph\u00e2n b\u1ed5 c\u00e2u h\u1ecfi tham kh\u1ea3o<\/h3>\n
            \n\n\n\n\n\n\n\n\n\n\n\n
            \u0110\u1ecbnh d\u1ea1ng ph\u1ea7n thi<\/th>\nM\u1ea1ch ki\u1ebfn th\u1ee9c<\/th>\nNh\u1eadn bi\u1ebft<\/th>\nTh\u00f4ng hi\u1ec3u<\/th>\nV\u1eadn d\u1ee5ng<\/th>\nV\u1eadn d\u1ee5ng cao<\/th>\n\u0110i\u1ec3m s\u1ed1<\/th>\n<\/tr>\n<\/thead>\n
            Ph\u1ea7n I: Nhi\u1ec1u l\u1ef1a ch\u1ecdn
            \n(12 c\u00e2u – 0.25\u0111\/c\u00e2u)<\/span><\/td>\n            		\u1ee8ng d\u1ee5ng \u0111\u1ea1o h\u00e0m & Kh\u1ea3o s\u00e1t h\u00e0m s\u1ed1<\/td>\n5 c\u00e2u<\/td>\n3 c\u00e2u<\/td>\n–<\/td>\n–<\/td>\n2.0 \u0111i\u1ec3m<\/td>\n<\/tr>\n
            Vect\u01a1 & H\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz<\/td>\n3 c\u00e2u<\/td>\n1 c\u00e2u<\/td>\n–<\/td>\n–<\/td>\n1.0 \u0111i\u1ec3m<\/td>\n<\/tr>\n
            Ph\u1ea7n II: \u0110\u00fang \/ Sai
            \n(4 c\u00e2u \/ 16 \u00fd – 1.0\u0111\/c\u00e2u)<\/span><\/td>\n            		\u1ee8ng d\u1ee5ng \u0111\u1ea1o h\u00e0m & Kh\u1ea3o s\u00e1t h\u00e0m s\u1ed1<\/td>\n3 \u00fd<\/td>\n5 \u00fd<\/td>\n4 \u00fd<\/td>\n–<\/td>\n3.0 \u0111i\u1ec3m<\/td>\n<\/tr>\n
            Vect\u01a1 & H\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz<\/td>\n1 \u00fd<\/td>\n2 \u00fd<\/td>\n1 \u00fd<\/td>\n–<\/td>\n1.0 \u0111i\u1ec3m<\/td>\n<\/tr>\n
            Ph\u1ea7n III: Tr\u1ea3 l\u1eddi ng\u1eafn
            \n(6 c\u00e2u – 0.5\u0111\/c\u00e2u)<\/span><\/td>\n            		To\u00e1n th\u1ef1c t\u1ebf t\u1ed1i \u01b0u & Tham s\u1ed1 (Gi\u1ea3i t\u00edch)<\/td>\n–<\/td>\n–<\/td>\n2 c\u00e2u<\/td>\n2 c\u00e2u<\/td>\n2.0 \u0111i\u1ec3m<\/td>\n<\/tr>\n
            B\u00e0i to\u00e1n th\u1ef1c ti\u1ec5n Vect\u01a1 & Oxyz<\/td>\n–<\/td>\n–<\/td>\n1 c\u00e2u<\/td>\n1 c\u00e2u<\/td>\n1.0 \u0111i\u1ec3m<\/td>\n<\/tr>\n
            T\u1ed4NG T\u1ef6 L\u1ec6 \/ \u0110I\u1ec2M S\u1ed0<\/td>\n~30%<\/td>\n~30%<\/td>\n~30%<\/td>\n~10%<\/td>\n10.0 \u0111i\u1ec3m<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n

            <\/p>\n

            \n
            \n

            Ph\u1ea7n I: \u0110\u1ea0I S\u1ed0 & Gi\u1ea3i T\u00edch 12 – \u1ee8ng D\u1ee5ng \u0110\u1ea1o H\u00e0m \u0110\u1ec3 Kh\u1ea3o S\u00e1t H\u00e0m S\u1ed1<\/h2>\n<\/div>\n

            <\/p>\n

            1. \u1ee8ng d\u1ee5ng \u0111\u1ea1o h\u00e0m<\/h2>\n

            1.1. \u00dd ngh\u0129a h\u00ecnh h\u1ecdc v\u00e0 \u00fd ngh\u0129a v\u1eadt l\u00fd c\u1ee7a \u0111\u1ea1o h\u00e0m<\/h3>\n
              \n
            • \u00dd ngh\u0129a h\u00ecnh h\u1ecdc:<\/strong> \u0110\u1ea1o h\u00e0m f'(x0<\/sub>)<\/code> l\u00e0 h\u1ec7 s\u1ed1 g\u00f3c c\u1ee7a ti\u1ebfp tuy\u1ebfn v\u1edbi \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1 y = f(x)<\/code> t\u1ea1i \u0111i\u1ec3m M(x0<\/sub>; f(x0<\/sub>))<\/code>. Ph\u01b0\u01a1ng tr\u00ecnh ti\u1ebfp tuy\u1ebfn t\u1ea1i M l\u00e0: y = f'(x0<\/sub>)(x - x0<\/sub>) + f(x0<\/sub>)<\/code>.<\/li>\n
            • \u00dd ngh\u0129a v\u1eadt l\u00fd & th\u1ef1c ti\u1ec5n:<\/strong>\n
                \n
              • N\u1ebfu ph\u01b0\u01a1ng tr\u00ecnh chuy\u1ec3n \u0111\u1ed9ng c\u1ee7a ch\u1ea5t \u0111i\u1ec3m l\u00e0 s = s(t)<\/code> th\u00ec v\u1eadn t\u1ed1c t\u1ee9c th\u1eddi t\u1ea1i th\u1eddi \u0111i\u1ec3m t l\u00e0: v(t) = s'(t)<\/strong>.<\/li>\n
              • Gia t\u1ed1c t\u1ee9c th\u1eddi c\u1ee7a chuy\u1ec3n \u0111\u1ed9ng t\u1ea1i th\u1eddi \u0111i\u1ec3m t l\u00e0 \u0111\u1ea1o h\u00e0m c\u1ee7a v\u1eadn t\u1ed1c: a(t) = v'(t) = s”(t)<\/strong>.<\/li>\n
              • T\u1ed5ng qu\u00e1t: \u0110\u1ea1o h\u00e0m f'(x)<\/code> bi\u1ec3u th\u1ecb t\u1ed1c \u0111\u1ed9 thay \u0111\u1ed5i t\u1ee9c th\u1eddi<\/strong> c\u1ee7a \u0111\u1ea1i l\u01b0\u1ee3ng y = f(x)<\/code> theo \u0111\u1ea1i l\u01b0\u1ee3ng x<\/code> (t\u1ed1c \u0111\u1ed9 t\u0103ng tr\u01b0\u1edfng d\u00e2n s\u1ed1, t\u1ed1c \u0111\u1ed9 truy\u1ec1n nhi\u1ec7t, chi ph\u00ed bi\u00ean trong kinh t\u1ebf).<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

                1.2. B\u1ea3ng \u0111\u1ea1o h\u00e0m c\u01a1 b\u1ea3n v\u00e0 h\u00e0m s\u1ed1 h\u1ee3p th\u01b0\u1eddng g\u1eb7p<\/h3>\n
                \n\n\n\n\n\n\n
                \u0110\u1ea1o h\u00e0m h\u00e0m c\u01a1 b\u1ea3n<\/th>\n\u0110\u1ea1o h\u00e0m h\u00e0m s\u1ed1 h\u1ee3p u = u(x)<\/th>\nC\u00f4ng th\u1ee9c t\u00ednh nhanh ph\u00e2n th\u1ee9c<\/th>\n<\/tr>\n<\/thead>\n
                (xn<\/sup>)’ = n \u00b7 xn-1<\/sup>
                \n(\u221ax)’ = 1 \/ (2\u221ax)
                \n(1\/x)’ = -1 \/ x2<\/sup><\/td>\n                		(un<\/sup>)’ = n \u00b7 un-1<\/sup> \u00b7 u’
                \n(\u221au)’ = u’ \/ (2\u221au)
                \n(1\/u)’ = -u’ \/ u2<\/sup><\/td>\n                		1. B\u1eadc 1 \/ B\u1eadc 1:<\/strong>
                \ny = (ax + b)\/(cx + d)<\/code>
                \n\u21d2 y' = (ad - bc) \/ (cx + d)2<\/sup><\/code>2. B\u1eadc 2 \/ B\u1eadc 1:<\/strong>
                \ny = (ax2<\/sup> + bx + c)\/(mx + n)<\/code>
                \n\u21d2 y' = (amx2<\/sup> + 2anx + bn - cm) \/ (mx + n)2<\/sup><\/code><\/td>\n<\/tr>\n
                (ex<\/sup>)’ = ex<\/sup> ; (ax<\/sup>)’ = ax<\/sup>\u00b7ln a
                \n(ln x)’ = 1\/x ; (sin x)’ = cos x
                \n(cos x)’ = -sin x<\/td>\n                		(eu<\/sup>)’ = u’ \u00b7 eu<\/sup>
                \n(ln u)’ = u’ \/ u
                \n(sin u)’ = u’ \u00b7 cos u<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

                <\/p>\n

                2. T\u00ednh \u0111\u01a1n \u0111i\u1ec7u c\u1ee7a h\u00e0m s\u1ed1<\/h2>\n

                2.1. \u0110\u1ecbnh l\u00fd & \u0110i\u1ec1u ki\u1ec7n c\u1ea7n v\u00e0 \u0111\u1ee7<\/h3>\n

                Cho h\u00e0m s\u1ed1 y = f(x)<\/code> c\u00f3 \u0111\u1ea1o h\u00e0m tr\u00ean kho\u1ea3ng K<\/code>:<\/p>\n

                  \n
                • H\u00e0m s\u1ed1 \u0111\u1ed3ng bi\u1ebfn<\/strong> tr\u00ean kho\u1ea3ng K \u21d4 f'(x) \u2265 0, \u2200x \u2208 K<\/strong> (d\u1ea5u “=” ch\u1ec9 x\u1ea3y ra t\u1ea1i h\u1eefu h\u1ea1n \u0111i\u1ec3m tr\u00ean K).<\/li>\n
                • H\u00e0m s\u1ed1 ngh\u1ecbch bi\u1ebfn<\/strong> tr\u00ean kho\u1ea3ng K \u21d4 f'(x) \u2264 0, \u2200x \u2208 K<\/strong> (d\u1ea5u “=” ch\u1ec9 x\u1ea3y ra t\u1ea1i h\u1eefu h\u1ea1n \u0111i\u1ec3m tr\u00ean K).<\/li>\n
                • L\u01b0u \u00fd \u0111\u1eb7c bi\u1ec7t v\u1edbi h\u00e0m b\u1eadc nh\u1ea5t tr\u00ean b\u1eadc nh\u1ea5t<\/em> y = (ax + b)\/(cx + d)<\/code>: Do t\u1eed s\u1ed1 c\u1ee7a y'<\/code> l\u00e0 h\u1eb1ng s\u1ed1 ad - bc<\/code> kh\u00f4ng ch\u1ee9a x, n\u00ean khi t\u00ecm \u0111i\u1ec1u ki\u1ec7n \u0111\u01a1n \u0111i\u1ec7u ch\u1ec9 d\u00f9ng d\u1ea5u ng\u1eb7t (y’ > 0<\/strong> ho\u1eb7c y’ < 0<\/strong>), tuy\u1ec7t \u0111\u1ed1i kh\u00f4ng c\u00f3 d\u1ea5u “=”.<\/li>\n<\/ul>\n

                  2.2. C\u00e1c d\u1ea1ng b\u00e0i tr\u1ecdng t\u00e2m<\/h3>\n
                    \n
                  • D\u1ea1ng 1: \u0110\u1ecdc b\u1ea3ng bi\u1ebfn thi\u00ean (BBT) ho\u1eb7c \u0111\u1ed3 th\u1ecb \u0111\u1ec3 t\u00ecm kho\u1ea3ng \u0111\u01a1n \u0111i\u1ec7u:<\/strong>\n
                      \n
                    • N\u1ebfu \u0111\u1ec1 cho \u0111\u1ed3 th\u1ecb c\u1ee7a y = f(x)<\/strong>: Nh\u00ecn chi\u1ec1u \u0111i l\u00ean (\u0111\u1ed3ng bi\u1ebfn) ho\u1eb7c \u0111i xu\u1ed1ng (ngh\u1ecbch bi\u1ebfn) c\u1ee7a n\u00e9t v\u1ebd t\u1eeb tr\u00e1i sang ph\u1ea3i.<\/li>\n
                    • N\u1ebfu \u0111\u1ec1 cho \u0111\u1ed3 th\u1ecb c\u1ee7a \u0111\u1ea1o h\u00e0m y = f'(x)<\/strong>: Ph\u1ea7n \u0111\u1ed3 th\u1ecb f'(x)<\/code> n\u1eb1m ph\u00eda tr\u00ean tr\u1ee5c ho\u00e0nh Ox<\/strong> t\u01b0\u01a1ng \u1ee9ng v\u1edbi f'(x) > 0<\/code> (h\u00e0m \u0111\u1ed3ng bi\u1ebfn); ph\u1ea7n n\u1eb1m ph\u00eda d\u01b0\u1edbi tr\u1ee5c Ox<\/strong> t\u01b0\u01a1ng \u1ee9ng v\u1edbi f'(x) < 0<\/code> (h\u00e0m ngh\u1ecbch bi\u1ebfn).<\/li>\n<\/ul>\n<\/li>\n
                    • D\u1ea1ng 2: T\u00ecm tham s\u1ed1 m \u0111\u1ec3 h\u00e0m s\u1ed1 \u0111\u01a1n \u0111i\u1ec7u tr\u00ean kho\u1ea3ng cho tr\u01b0\u1edbc:<\/strong>\n
                        \n
                      • H\u00e0m b\u1eadc ba \u0111\u01a1n \u0111i\u1ec7u tr\u00ean \u211d:<\/em> \u0110\u1ed3ng bi\u1ebfn tr\u00ean \u211d \u21d4 a > 0<\/code> v\u00e0 \u0394y'<\/sub> \u2264 0<\/code>; Ngh\u1ecbch bi\u1ebfn tr\u00ean \u211d \u21d4 a < 0<\/code> v\u00e0 \u0394y'<\/sub> \u2264 0<\/code>.<\/li>\n
                      • \u0110\u01a1n \u0111i\u1ec7u tr\u00ean kho\u1ea3ng (a; b) b\u1eb1ng ph\u01b0\u01a1ng ph\u00e1p c\u00f4 l\u1eadp m:<\/em> Bi\u1ebfn \u0111\u1ed5i b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh f'(x) \u2265 0<\/code> v\u1ec1 d\u1ea1ng m \u2265 g(x), \u2200x \u2208 (a; b) \u21d4 m \u2265 max g(x)<\/code> tr\u00ean [a; b]<\/code>; ho\u1eb7c m \u2264 g(x), \u2200x \u2208 (a; b) \u21d4 m \u2264 min g(x)<\/code> tr\u00ean [a; b]<\/code>.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

                        <\/p>\n

                        3. C\u1ef1c tr\u1ecb c\u1ee7a h\u00e0m s\u1ed1<\/h2>\n

                        3.1. Kh\u00e1i ni\u1ec7m & Hai d\u1ea5u hi\u1ec7u nh\u1eadn bi\u1ebft c\u1ef1c tr\u1ecb<\/h3>\n
                          \n
                        • D\u1ea5u hi\u1ec7u 1 (Qua s\u1ef1 \u0111\u1ed5i d\u1ea5u c\u1ee7a \u0111\u1ea1o h\u00e0m c\u1ea5p 1):<\/strong> \u0110i\u1ec3m x0<\/sub><\/code> l\u00e0 \u0111i\u1ec3m c\u1ef1c tr\u1ecb c\u1ee7a h\u00e0m s\u1ed1 khi v\u00e0 ch\u1ec9 khi f'(x)<\/code> \u0111\u1ed5i d\u1ea5u<\/strong> khi x \u0111i qua x0<\/sub><\/code>:\n
                            \n
                          • \u0110\u1ed5i d\u1ea5u t\u1eeb D\u01b0\u01a1ng (+) sang \u00c2m (-)<\/strong> \u21d2 x0<\/sub><\/code> l\u00e0 \u0111i\u1ec3m c\u1ef1c \u0111\u1ea1i (xC\u0110<\/sub>)<\/strong>.<\/li>\n
                          • \u0110\u1ed5i d\u1ea5u t\u1eeb \u00c2m (-) sang D\u01b0\u01a1ng (+)<\/strong> \u21d2 x0<\/sub><\/code> l\u00e0 \u0111i\u1ec3m c\u1ef1c ti\u1ec3u (xCT<\/sub>)<\/strong>.<\/li>\n
                          • (L\u01b0u \u00fd: \u0110i\u1ec3m c\u1ef1c tr\u1ecb l\u00e0 nghi\u1ec7m \u0111\u01a1n ho\u1eb7c nghi\u1ec7m b\u1ed9i l\u1ebb c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh f'(x) = 0; nghi\u1ec7m b\u1ed9i ch\u1eb5n kh\u00f4ng l\u00e0m f'(x) \u0111\u1ed5i d\u1ea5u n\u00ean kh\u00f4ng ph\u1ea3i l\u00e0 c\u1ef1c tr\u1ecb).<\/em><\/li>\n<\/ul>\n<\/li>\n
                          • D\u1ea5u hi\u1ec7u 2 (D\u00f9ng \u0111\u1ea1o h\u00e0m c\u1ea5p 2):<\/strong> N\u1ebfu f'(x0<\/sub>) = 0<\/code> v\u00e0:\n
                              \n
                            • f''(x0<\/sub>) < 0<\/code> \u21d2 H\u00e0m s\u1ed1 \u0111\u1ea1t c\u1ef1c \u0111\u1ea1i<\/strong> t\u1ea1i x0<\/sub><\/code>.<\/li>\n
                            • f''(x0<\/sub>) > 0<\/code> \u21d2 H\u00e0m s\u1ed1 \u0111\u1ea1t c\u1ef1c ti\u1ec3u<\/strong> t\u1ea1i x0<\/sub><\/code>.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n
                              \n

                              \u26a0\ufe0f C\u1ea3nh b\u00e1o b\u1eaby thu\u1eadt ng\u1eef trong c\u00e2u h\u1ecfi \u0110\u00fang\/Sai:<\/p>\n

                                \n
                              • \u0110i\u1ec3m c\u1ef1c tr\u1ecb c\u1ee7a h\u00e0m s\u1ed1:<\/strong> L\u00e0 gi\u00e1 tr\u1ecb ho\u00e0nh \u0111\u1ed9 x = x0<\/sub><\/strong>.<\/li>\n
                              • Gi\u00e1 tr\u1ecb c\u1ef1c tr\u1ecb (C\u1ef1c \u0111\u1ea1i \/ C\u1ef1c ti\u1ec3u c\u1ee7a h\u00e0m s\u1ed1):<\/strong> L\u00e0 gi\u00e1 tr\u1ecb tung \u0111\u1ed9 y = f(x0<\/sub>)<\/strong>.<\/li>\n
                              • \u0110i\u1ec3m c\u1ef1c tr\u1ecb c\u1ee7a \u0110\u1ed2 TH\u1eca h\u00e0m s\u1ed1:<\/strong> L\u00e0 t\u1ecda \u0111\u1ed9 \u0111i\u1ec3m M(x0<\/sub>; f(x0<\/sub>))<\/strong> g\u1ed3m c\u1ea3 ho\u00e0nh \u0111\u1ed9 v\u00e0 tung \u0111\u1ed9.<\/li>\n<\/ul>\n<\/div>\n

                                3.2. D\u1ea1ng b\u00e0i tr\u1ecdng t\u00e2m<\/h3>\n
                                  \n
                                • \u0110\u1ebfm s\u1ed1 \u0111i\u1ec3m c\u1ef1c tr\u1ecb qua \u0111\u1ed3 th\u1ecb f'(x):<\/strong> S\u1ed1 \u0111i\u1ec3m c\u1ef1c tr\u1ecb b\u1eb1ng s\u1ed1 giao \u0111i\u1ec3m m\u00e0 t\u1ea1i \u0111\u00f3 \u0111\u1ed3 th\u1ecb y = f'(x)<\/code> c\u1eaft xuy\u00ean qua<\/strong> tr\u1ee5c ho\u00e0nh Ox (b\u1ecf qua c\u00e1c \u0111i\u1ec3m ti\u1ebfp x\u00fac).<\/li>\n
                                • C\u1ef1c tr\u1ecb h\u00e0m b\u1eadc ba ch\u1ee9a tham s\u1ed1 m:<\/strong> H\u00e0m s\u1ed1 y = ax3<\/sup> + bx2<\/sup> + cx + d<\/code> c\u00f3 2 \u0111i\u1ec3m c\u1ef1c tr\u1ecb \u21d4 ph\u01b0\u01a1ng tr\u00ecnh y' = 0<\/code> c\u00f3 2 nghi\u1ec7m ph\u00e2n bi\u1ec7t \u21d4 b2<\/sup> – 3ac > 0<\/strong>. K\u1ebft h\u1ee3p \u0111\u1ecbnh l\u00fd Vi-\u00e9t (x1<\/sub> + x2<\/sub> = -2b\/3a<\/code>, x1<\/sub>x2<\/sub> = c\/3a<\/code>) \u0111\u1ec3 gi\u1ea3i quy\u1ebft c\u00e1c \u0111i\u1ec1u ki\u1ec7n v\u1ec1 ho\u00e0nh \u0111\u1ed9 c\u1ef1c tr\u1ecb.<\/li>\n<\/ul>\n

                                  <\/p>\n

                                  4. Gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t nh\u1ecf nh\u1ea5t c\u1ee7a h\u00e0m s\u1ed1<\/h2>\n

                                  4.1. Quy t\u1eafc t\u00ecm GTLN (Max) v\u00e0 GTNN (Min)<\/h3>\n
                                    \n
                                  • Quy t\u1eafc t\u00ecm GTLN, GTNN tr\u00ean m\u1ed9t \u0111o\u1ea1n [a; b]:<\/strong>\n
                                      \n
                                    1. T\u00ednh \u0111\u1ea1o h\u00e0m f'(x)<\/code> v\u00e0 t\u00ecm c\u00e1c \u0111i\u1ec3m x1<\/sub>, x2<\/sub>, ... \u2208 (a; b)<\/code> m\u00e0 t\u1ea1i \u0111\u00f3 f'(x) = 0<\/code> ho\u1eb7c kh\u00f4ng x\u00e1c \u0111\u1ecbnh.<\/li>\n
                                    2. T\u00ednh c\u00e1c gi\u00e1 tr\u1ecb t\u1ea1i \u0111i\u1ec3m d\u1eebng v\u00e0 hai \u0111\u1ea7u m\u00fat: f(a), f(x1<\/sub>), f(x2<\/sub>), ..., f(b)<\/code>.<\/li>\n
                                    3. S\u1ed1 l\u1edbn nh\u1ea5t trong c\u00e1c gi\u00e1 tr\u1ecb tr\u00ean l\u00e0 max f(x)<\/code>, s\u1ed1 nh\u1ecf nh\u1ea5t l\u00e0 min f(x)<\/code> tr\u00ean \u0111o\u1ea1n [a; b]<\/code>.<\/li>\n<\/ol>\n<\/li>\n
                                    4. T\u00ecm GTLN, GTNN tr\u00ean kho\u1ea3ng (a; b) ho\u1eb7c n\u1eeda kho\u1ea3ng:<\/strong> B\u1eaft bu\u1ed9c ph\u1ea3i l\u1eadp B\u1ea3ng bi\u1ebfn thi\u00ean<\/strong> tr\u00ean mi\u1ec1n \u0111\u00f3 \u0111\u1ec3 quan s\u00e1t xu h\u01b0\u1edbng gi\u1edbi h\u1ea1n t\u1ea1i hai \u0111\u1ea7u m\u00fat v\u00e0 c\u1ef1c tr\u1ecb tr\u01b0\u1edbc khi k\u1ebft lu\u1eadn.<\/li>\n<\/ul>\n

                                      4.2. D\u1ea1ng b\u00e0i tr\u1ecdng t\u00e2m<\/h3>\n
                                        \n
                                      • T\u00ecm Min\/Max c\u1ee7a h\u00e0m h\u1ee3p b\u1eb1ng ph\u01b0\u01a1ng ph\u00e1p \u0111\u1eb7t \u1ea9n ph\u1ee5:<\/strong> \u0110\u1eb7t t = u(x)<\/code>, kh\u1ea3o s\u00e1t mi\u1ec1n gi\u00e1 tr\u1ecb c\u1ee7a t<\/code> khi x \u2208 [a; b]<\/code> \u0111\u1ec3 t\u00ecm \u0111o\u1ea1n [tmin<\/sub>; tmax<\/sub>]<\/code>, sau \u0111\u00f3 quy v\u1ec1 b\u00e0i to\u00e1n t\u00ecm Min\/Max c\u1ee7a f(t)<\/code> tr\u00ean \u0111o\u1ea1n \u0111\u00f3.<\/li>\n
                                      • B\u00e0i to\u00e1n Min\/Max ch\u1ee9a tr\u1ecb tuy\u1ec7t \u0111\u1ed1i y = |f(x) + m|<\/code> tr\u00ean [a; b]:<\/strong> G\u1ecdi M = max f(x)<\/code> v\u00e0 N = min f(x)<\/code> tr\u00ean [a; b]<\/code>. Khi \u0111\u00f3 gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t c\u1ee7a h\u00e0m tr\u1ecb tuy\u1ec7t \u0111\u1ed1i l\u00e0 max { |M + m|, |N + m| }<\/code>.<\/li>\n<\/ul>\n

                                        <\/p>\n

                                        5. \u0110\u01b0\u1eddng ti\u1ec7m c\u1eadn c\u1ee7a \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1<\/h2>\n

                                        5.1. \u0110\u1ecbnh ngh\u0129a chu\u1ea9n x\u00e1c 3 lo\u1ea1i \u0111\u01b0\u1eddng ti\u1ec7m c\u1eadn<\/h3>\n
                                        \n\n\n\n\n\n\n\n
                                        Lo\u1ea1i ti\u1ec7m c\u1eadn<\/th>\n\u0110\u1ecbnh ngh\u0129a theo gi\u1edbi h\u1ea1n<\/th>\nC\u00e1ch x\u00e1c \u0111\u1ecbnh nhanh<\/th>\n<\/tr>\n<\/thead>\n
                                        1. Ti\u1ec7m c\u1eadn ngang
                                        \ny = y0<\/sub><\/code><\/td>\n                                        		limx\u2192+\u221e<\/sub> f(x) = y0<\/sub>
                                        \nho\u1eb7c limx\u2192-\u221e<\/sub> f(x) = y0<\/sub><\/td>\n                                        		V\u1edbi h\u00e0m ph\u00e2n th\u1ee9c P(x)\/Q(x)<\/code>:
                                        \n\u2022 B\u1eadc T\u1eed < B\u1eadc M\u1eabu \u21d2 TCN: y = 0<\/code>.
                                        \n\u2022 B\u1eadc T\u1eed = B\u1eadc M\u1eabu \u21d2 TCN: y = a\/c<\/code> (t\u1ec9 s\u1ed1 h\u1ec7 s\u1ed1 b\u1eadc cao nh\u1ea5t).<\/td>\n<\/tr>\n
                                        2. Ti\u1ec7m c\u1eadn \u0111\u1ee9ng
                                        \nx = x0<\/sub><\/code><\/td>\n                                        		limx\u2192x0<\/sub>+<\/sup><\/sub> f(x) = \u00b1\u221e
                                        \nho\u1eb7c limx\u2192x0<\/sub>–<\/sup><\/sub> f(x) = \u00b1\u221e<\/td>\n                                        		Gi\u1ea3i nghi\u1ec7m m\u1eabu Q(x) = 0<\/code>. Nghi\u1ec7m x0<\/sub><\/code> kh\u00f4ng b\u1ecb tri\u1ec7t ti\u00eau ho\u00e0n to\u00e0n b\u1edfi nghi\u1ec7m c\u1ee7a t\u1eed P(x)<\/code> th\u00ec x = x0<\/sub><\/code> l\u00e0 TC\u0110.<\/td>\n<\/tr>\n
                                        3. Ti\u1ec7m c\u1eadn xi\u00ean
                                        \ny = ax + b<\/code> (a \u2260 0)<\/td>\n                                        		limx\u2192\u00b1\u221e<\/sub> [f(x) – (ax + b)] = 0<\/td>\nXu\u1ea5t hi\u1ec7n \u1edf h\u00e0m ph\u00e2n th\u1ee9c c\u00f3 B\u1eadc T\u1eed = B\u1eadc M\u1eabu + 1<\/strong>. Th\u1ef1c hi\u1ec7n chia \u0111a th\u1ee9c: y = ax + b + R\/(mx + n)<\/code> \u21d2 TCX l\u00e0 y = ax + b<\/code>.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

                                        5.2. D\u1ea1ng b\u00e0i tr\u1ecdng t\u00e2m<\/h3>\n
                                          \n
                                        • \u0110\u1ebfm s\u1ed1 \u0111\u01b0\u1eddng ti\u1ec7m c\u1eadn t\u1eeb B\u1ea3ng bi\u1ebfn thi\u00ean:<\/strong> Nh\u00ecn t\u1ea1i hai \u0111\u1ea7u -\u221e<\/code> v\u00e0 +\u221e<\/code> c\u1ee7a d\u00f2ng x, n\u1ebfu d\u00f2ng y ra s\u1ed1 h\u1eefu h\u1ea1n th\u00ec \u0111\u00f3 l\u00e0 TCN (t\u1ed1i \u0111a 2 TCN). Nh\u00ecn t\u1ea1i c\u00e1c c\u1ed9t c\u00f3 d\u1ea5u hai g\u1ea1ch s\u1ecdc ||<\/code>, n\u1ebfu \u00edt nh\u1ea5t m\u1ed9t b\u00ean tr\u00e1i ho\u1eb7c ph\u1ea3i c\u1ee7a d\u00f2ng y xu\u1ea5t hi\u1ec7n \u00b1\u221e<\/code> th\u00ec c\u1ed9t \u0111\u00f3 l\u00e0 1 TC\u0110.<\/li>\n
                                        • X\u1eed l\u00fd nghi\u1ec7m tr\u00f9ng gi\u1eefa t\u1eed v\u00e0 m\u1eabu:<\/strong> V\u00ed d\u1ee5 v\u1edbi h\u00e0m y = (x - 1) \/ (x2<\/sup> - 1) = 1 \/ (x + 1)<\/code> (v\u1edbi x \u2260 1), nghi\u1ec7m x = 1<\/code> b\u1ecb r\u00fat g\u1ecdn h\u1ebft n\u00ean \u0111\u1ed3 th\u1ecb ch\u1ec9 c\u00f3 1 TC\u0110 duy nh\u1ea5t l\u00e0 x = -1<\/code>.<\/li>\n<\/ul>\n

                                          <\/p>\n

                                          6. Kh\u1ea3o s\u00e1t v\u00e0 v\u1ebd \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1<\/h2>\n

                                          6.1. S\u01a1 \u0111\u1ed3 5 b\u01b0\u1edbc kh\u1ea3o s\u00e1t chu\u1ea9n<\/h3>\n

                                          1. T\u00ecm t\u1eadp x\u00e1c \u0111\u1ecbnh D \u2192 2. T\u00ednh \u0111\u1ea1o h\u00e0m y’, x\u00e9t d\u1ea5u v\u00e0 t\u00ecm c\u1ef1c tr\u1ecb \u2192 3. T\u00ecm gi\u1edbi h\u1ea1n v\u00e0 c\u00e1c \u0111\u01b0\u1eddng ti\u1ec7m c\u1eadn (n\u1ebfu c\u00f3) \u2192 4. L\u1eadp b\u1ea3ng bi\u1ebfn thi\u00ean \u2192 5. T\u00ecm giao \u0111i\u1ec3m v\u1edbi c\u00e1c tr\u1ee5c t\u1ecda \u0111\u1ed9 v\u00e0 v\u1ebd \u0111\u1ed3 th\u1ecb.<\/p>\n

                                          6.2. M\u1eb9o nh\u1eadn di\u1ec7n d\u1ea5u h\u1ec7 s\u1ed1 c\u1ee7a 3 h\u00e0m s\u1ed1 tr\u1ecdng t\u00e2m<\/h3>\n
                                            \n
                                          • H\u00e0m s\u1ed1 b\u1eadc ba y = ax3<\/sup> + bx2<\/sup> + cx + d (a \u2260 0)<\/code>:<\/strong>\n
                                              \n
                                            • D\u1ea5u c\u1ee7a a:<\/em> Nh\u00ecn nh\u00e1nh ngo\u00e0i c\u00f9ng b\u00ean ph\u1ea3i c\u1ee7a \u0111\u1ed3 th\u1ecb; \u0111i l\u00ean th\u00ec a > 0<\/strong>, \u0111i xu\u1ed1ng th\u00ec a < 0<\/strong>.<\/li>\n
                                            • D\u1ea5u c\u1ee7a d:<\/em> Nh\u00ecn giao \u0111i\u1ec3m c\u1ee7a \u0111\u1ed3 th\u1ecb v\u1edbi tr\u1ee5c tung Oy t\u1ea1i \u0111i\u1ec3m (0; d)<\/code>; n\u1eb1m tr\u00ean g\u1ed1c O th\u00ec d > 0<\/code>, d\u01b0\u1edbi O th\u00ec d < 0<\/code>.<\/li>\n
                                            • D\u1ea5u c\u1ee7a b v\u00e0 c:<\/em> D\u1ef1a v\u00e0o t\u1ed5ng v\u00e0 t\u00edch hai \u0111i\u1ec3m c\u1ef1c tr\u1ecb: x1<\/sub> + x2<\/sub> = -2b\/(3a)<\/code> v\u00e0 x1<\/sub>x2<\/sub> = c\/(3a)<\/code>.<\/li>\n<\/ul>\n<\/li>\n
                                            • H\u00e0m ph\u00e2n th\u1ee9c b\u1eadc nh\u1ea5t tr\u00ean b\u1eadc nh\u1ea5t y = (ax + b)\/(cx + d)<\/code>:<\/strong>\n
                                                \n
                                              • Ti\u1ec7m c\u1eadn \u0111\u1ee9ng: x = -d\/c<\/code> ; Ti\u1ec7m c\u1eadn ngang: y = a\/c<\/code>.<\/li>\n
                                              • Giao v\u1edbi Oy t\u1ea1i (0; b\/d)<\/code> ; Giao v\u1edbi Ox t\u1ea1i (-b\/a; 0)<\/code>.<\/li>\n<\/ul>\n<\/li>\n
                                              • H\u00e0m ph\u00e2n th\u1ee9c b\u1eadc hai tr\u00ean b\u1eadc nh\u1ea5t y = (ax2<\/sup> + bx + c)\/(mx + n)<\/code>:<\/strong>\n
                                                  \n
                                                • Nh\u1eadn giao \u0111i\u1ec3m c\u1ee7a Ti\u1ec7m c\u1eadn \u0111\u1ee9ng x = -n\/m<\/code> v\u00e0 Ti\u1ec7m c\u1eadn xi\u00ean y = px + q<\/code> l\u00e0m t\u00e2m \u0111\u1ed1i x\u1ee9ng.<\/li>\n
                                                • Nh\u00e1nh \u0111\u1ed3 th\u1ecb \u0111i l\u00ean \u1edf ph\u00eda ph\u1ea3i khi h\u1ec7 s\u1ed1 a\/m > 0<\/code> v\u00e0 \u0111i xu\u1ed1ng khi a\/m < 0<\/code>.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

                                                  6.3. Hai ph\u00e9p bi\u1ebfn \u0111\u1ed5i \u0111\u1ed3 th\u1ecb ch\u1ee9a d\u1ea5u gi\u00e1 tr\u1ecb tuy\u1ec7t \u0111\u1ed1i<\/h3>\n
                                                    \n
                                                  • \u0110\u1ed3 th\u1ecb h\u00e0m s\u1ed1 y = |f(x)|<\/code>:<\/strong> Gi\u1eef nguy\u00ean ph\u1ea7n \u0111\u1ed3 th\u1ecb (C)<\/code> n\u1eb1m ph\u00eda tr\u00ean tr\u1ee5c ho\u00e0nh Ox; l\u1ea5y \u0111\u1ed1i x\u1ee9ng ph\u1ea7n \u0111\u1ed3 th\u1ecb n\u1eb1m d\u01b0\u1edbi tr\u1ee5c Ox qua tr\u1ee5c Ox r\u1ed3i x\u00f3a b\u1ecf ph\u1ea7n ph\u00eda d\u01b0\u1edbi c\u0169. (S\u1ed1 \u0111i\u1ec3m c\u1ef1c tr\u1ecb c\u1ee7a h\u00e0m |f(x)|<\/code> b\u1eb1ng t\u1ed5ng s\u1ed1 \u0111i\u1ec3m c\u1ef1c tr\u1ecb c\u1ee7a f(x)<\/code> c\u1ed9ng v\u1edbi s\u1ed1 giao \u0111i\u1ec3m c\u1eaft xuy\u00ean qua tr\u1ee5c Ox).<\/li>\n
                                                  • \u0110\u1ed3 th\u1ecb h\u00e0m s\u1ed1 y = f(|x|)<\/code>:<\/strong> Gi\u1eef nguy\u00ean ph\u1ea7n \u0111\u1ed3 th\u1ecb (C)<\/code> n\u1eb1m b\u00ean ph\u1ea3i tr\u1ee5c tung Oy (\u1ee9ng v\u1edbi x \u2265 0<\/code>); x\u00f3a ph\u1ea7n b\u00ean tr\u00e1i Oy r\u1ed3i l\u1ea5y \u0111\u1ed1i x\u1ee9ng ph\u1ea7n b\u00ean ph\u1ea3i qua tr\u1ee5c tung Oy.<\/li>\n<\/ul>\n

                                                    <\/p>\n

                                                    7. B\u00e0i to\u00e1n th\u1ef1c t\u1ebf \u1ee9ng d\u1ee5ng \u0111\u1ea1o h\u00e0m<\/h2>\n

                                                    Trong \u0111\u1ecbnh d\u1ea1ng \u0111\u1ec1 thi m\u1edbi, b\u00e0i to\u00e1n th\u1ef1c t\u1ebf \u1ee9ng d\u1ee5ng \u0111\u1ea1o h\u00e0m th\u01b0\u1eddng n\u1eb1m \u1edf Ph\u1ea7n II (\u0110\u00fang\/Sai)<\/strong> v\u00e0 Ph\u1ea7n III (Tr\u1ea3 l\u1eddi ng\u1eafn)<\/strong>, chi\u1ebfm t\u1eeb 1.5 \u0111\u1ebfn 2.0 \u0111i\u1ec3m<\/strong> v\u00e0 \u0111\u00f3ng vai tr\u00f2 ph\u00e2n lo\u1ea1i h\u1ecdc sinh \u0111\u1ea1t m\u1ee9c 8.5+.<\/p>\n

                                                    7.1. Hai m\u00f4 h\u00ecnh to\u00e1n th\u1ef1c t\u1ebf xu\u1ea5t hi\u1ec7n nhi\u1ec1u nh\u1ea5t<\/h3>\n
                                                      \n
                                                    • M\u00f4 h\u00ecnh b\u00e0i to\u00e1n Kinh t\u1ebf (T\u1ed1i \u01b0u l\u1ee3i nhu\u1eadn \u2013 Doanh thu \u2013 Chi ph\u00ed):<\/strong>\n
                                                        \n
                                                      • H\u00e0m doanh thu: R(x) = x \u00b7 p(x)<\/code> (trong \u0111\u00f3 x<\/code> l\u00e0 s\u1ea3n l\u01b0\u1ee3ng, p(x)<\/code> l\u00e0 gi\u00e1 b\u00e1n m\u1ed9t \u0111\u01a1n v\u1ecb s\u1ea3n ph\u1ea9m).<\/li>\n
                                                      • H\u00e0m l\u1ee3i nhu\u1eadn: P(x) = R(x) – C(x)<\/strong> (trong \u0111\u00f3 C(x)<\/code> l\u00e0 t\u1ed5ng chi ph\u00ed s\u1ea3n xu\u1ea5t x \u0111\u01a1n v\u1ecb). \u0110\u1ec1 b\u00e0i th\u01b0\u1eddng y\u00eau c\u1ea7u t\u00ecm x ho\u1eb7c m\u1ee9c gi\u00e1 b\u00e1n gi\u1ea3m gi\u00e1 sao cho l\u1ee3i nhu\u1eadn P(x)<\/code> \u0111\u1ea1t gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t.<\/li>\n<\/ul>\n<\/li>\n
                                                      • M\u00f4 h\u00ecnh b\u00e0i to\u00e1n H\u00ecnh h\u1ecdc & V\u1eadt l\u00fd (T\u1ed1i \u01b0u di\u1ec7n t\u00edch, th\u1ec3 t\u00edch, \u0111\u01b0\u1eddng \u0111i):<\/strong>\n
                                                          \n
                                                        • C\u1eaft t\u1ea5m t\u00f4n h\u00ecnh vu\u00f4ng\/ch\u1eef nh\u1eadt \u0111\u1ec3 g\u1eadp th\u00e0nh h\u00ecnh h\u1ed9p kh\u00f4ng n\u1eafp c\u00f3 th\u1ec3 t\u00edch l\u1edbn nh\u1ea5t.<\/li>\n
                                                        • Thi\u1ebft k\u1ebf b\u1ec3 ch\u1ee9a n\u01b0\u1edbc, lon n\u01b0\u1edbc ng\u1ecdt c\u00f3 th\u1ec3 t\u00edch c\u1ed1 \u0111\u1ecbnh nh\u01b0ng t\u1ed1n \u00edt di\u1ec7n t\u00edch v\u1eadt li\u1ec7u (di\u1ec7n t\u00edch to\u00e0n ph\u1ea7n nh\u1ecf nh\u1ea5t).<\/li>\n
                                                        • B\u00e0i to\u00e1n t\u00ecm l\u1ed9 tr\u00ecnh di chuy\u1ec3n ng\u1eafn nh\u1ea5t ho\u1eb7c chi ph\u00ed l\u1eafp \u0111\u1eb7t d\u00e2y c\u00e1p\/\u0111\u01b0\u1eddng \u1ed1ng qua s\u00f4ng v\u00e0 tr\u00ean b\u1edd th\u1ea5p nh\u1ea5t.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

                                                          7.2. Quy tr\u00ecnh 4 b\u01b0\u1edbc gi\u1ea3i b\u00e0i to\u00e1n t\u1ed1i \u01b0u th\u1ef1c t\u1ebf<\/h3>\n
                                                            \n
                                                          1. B\u01b0\u1edbc 1 (Ch\u1ecdn \u1ea9n & \u0111\u1eb7t \u0111i\u1ec1u ki\u1ec7n):<\/strong> G\u1ecdi x<\/code> l\u00e0 \u0111\u1ea1i l\u01b0\u1ee3ng bi\u1ebfn thi\u00ean c\u1ea7n t\u00ecm, x\u00e1c \u0111\u1ecbnh mi\u1ec1n gi\u00e1 tr\u1ecb th\u1ef1c t\u1ebf x \u2208 D<\/code> (d\u1ef1a v\u00e0o k\u00edch th\u01b0\u1edbc h\u00ecnh h\u1ecdc ho\u1eb7c gi\u1edbi h\u1ea1n s\u1ea3n l\u01b0\u1ee3ng).<\/li>\n
                                                          2. B\u01b0\u1edbc 2 (Thi\u1ebft l\u1eadp h\u00e0m m\u1ee5c ti\u00eau 1 bi\u1ebfn):<\/strong> D\u1ef1a v\u00e0o d\u1eef ki\u1ec7n b\u00e0i to\u00e1n, r\u00fat c\u00e1c bi\u1ebfn ph\u1ee5 theo x<\/code> \u0111\u1ec3 bi\u1ec3u di\u1ec5n \u0111\u1ea1i l\u01b0\u1ee3ng c\u1ea7n t\u1ed1i \u01b0u th\u00e0nh m\u1ed9t h\u00e0m s\u1ed1 m\u1ed9t bi\u1ebfn duy nh\u1ea5t y = f(x)<\/code>.<\/li>\n
                                                          3. B\u01b0\u1edbc 3 (Kh\u1ea3o s\u00e1t b\u1eb1ng \u0111\u1ea1o h\u00e0m):<\/strong> T\u00ednh f'(x)<\/code>, gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh f'(x) = 0<\/code> \u0111\u1ec3 t\u00ecm \u0111i\u1ec3m t\u1edbi h\u1ea1n x0<\/sub> \u2208 D<\/code>.<\/li>\n
                                                          4. B\u01b0\u1edbc 4 (K\u1ebft lu\u1eadn & l\u00e0m tr\u00f2n):<\/strong> L\u1eadp b\u1ea3ng bi\u1ebfn thi\u00ean ho\u1eb7c so s\u00e1nh gi\u00e1 tr\u1ecb \u0111\u1ec3 k\u1ebft lu\u1eadn GTLN \/ GTNN, ch\u00fa \u00fd l\u00e0m tr\u00f2n k\u1ebft qu\u1ea3 theo \u0111\u00fang y\u00eau c\u1ea7u c\u1ee7a c\u00e2u tr\u1ea3 l\u1eddi ng\u1eafn (h\u00e0ng \u0111\u01a1n v\u1ecb, h\u00e0ng ph\u1ea7n m\u01b0\u1eddi ho\u1eb7c ph\u1ea7n tr\u0103m).<\/li>\n<\/ol>\n

                                                            <\/p>\n

                                                            \n

                                                            V\u00ed d\u1ee5 m\u1eabu (C\u00e2u tr\u1ea3 l\u1eddi ng\u1eafn): B\u00e0i to\u00e1n t\u1ed1i \u01b0u h\u00ecnh h\u1ecdc<\/p>\n

                                                            \u0110\u1ec1 b\u00e0i:<\/strong> T\u1eeb m\u1ed9t t\u1ea5m b\u00eca h\u00ecnh vu\u00f4ng c\u1ea1nh 12 cm, ng\u01b0\u1eddi ta c\u1eaft b\u1ecf \u1edf b\u1ed1n g\u00f3c b\u1ed1n h\u00ecnh vu\u00f4ng b\u1eb1ng nhau c\u00f3 c\u1ea1nh l\u00e0 x (cm) r\u1ed3i g\u1eadp l\u1ea1i th\u00e0nh m\u1ed9t chi\u1ebfc h\u1ed9p ch\u1eef nh\u1eadt kh\u00f4ng c\u00f3 n\u1eafp. T\u00ecm x \u0111\u1ec3 th\u1ec3 t\u00edch c\u1ee7a chi\u1ebfc h\u1ed9p thu \u0111\u01b0\u1ee3c l\u00e0 l\u1edbn nh\u1ea5t.<\/p>\n

                                                            L\u1eddi gi\u1ea3i:<\/strong><\/p>\n

                                                            \u2022 \u0110i\u1ec1u ki\u1ec7n th\u1ef1c t\u1ebf:<\/strong> Do c\u1eaft \u1edf hai \u0111\u1ea7u c\u1ea1nh 12 cm n\u00ean 2x < 12 \u21d4 0 < x < 6<\/code>.<\/p>\n

                                                            \u2022 Khi g\u1eadp l\u1ea1i, h\u1ed9p c\u00f3 chi\u1ec1u cao l\u00e0 x<\/code>, \u0111\u00e1y l\u00e0 h\u00ecnh vu\u00f4ng c\u00f3 c\u1ea1nh b\u1eb1ng 12 - 2x<\/code>. Th\u1ec3 t\u00edch kh\u1ed1i h\u1ed9p l\u00e0:<\/p>\n

                                                            V(x) = x \u00b7 (12 – 2x)2<\/sup> = 4x3<\/sup> – 48x2<\/sup> + 144x<\/p>\n

                                                            \u2022 T\u00ednh \u0111\u1ea1o h\u00e0m tr\u00ean kho\u1ea3ng (0; 6):<\/p>\n

                                                            V'(x) = 12x2<\/sup> – 96x + 144 = 0
                                                            \n\u21d4 x2<\/sup> – 8x + 12 = 0 \u21d4 x = 2 (nh\u1eadn) ho\u1eb7c x = 6 (lo\u1ea1i)<\/p>\n

                                                            \u2022 L\u1eadp b\u1ea3ng bi\u1ebfn thi\u00ean tr\u00ean (0; 6), ta th\u1ea5y V'(x)<\/code> \u0111\u1ed5i d\u1ea5u t\u1eeb (+) sang (-) khi qua x = 2<\/code>. Khi \u0111\u00f3 max V(x) = V(2) = 2 \u00b7 (12 - 4)2<\/sup> = 128 (cm3<\/sup>)<\/code>.<\/p>\n

                                                            K\u1ebft qu\u1ea3 \u0111i\u1ec1n \u00f4 tr\u1ea3 l\u1eddi ng\u1eafn:<\/strong> 2<\/strong><\/p>\n<\/div>\n<\/div>\n

                                                            <\/p>\n

                                                            \n
                                                            \n

                                                            Ph\u1ea7n II: H\u00ecnh H\u1ecdc Kh\u00f4ng Gian – Vect\u01a1 V\u00e0 H\u1ec7 T\u1ecda \u0110\u1ed9 Oxyz<\/h2>\n<\/div>\n

                                                            <\/p>\n

                                                            1. Vect\u01a1 trong kh\u00f4ng gian<\/h2>\n

                                                            1.1. \u0110\u1ecbnh ngh\u0129a v\u00e0 c\u00e1c quy t\u1eafc h\u00ecnh h\u1ecdc c\u1ed1t l\u00f5i<\/h3>\n

                                                            Vect\u01a1 trong kh\u00f4ng gian l\u00e0 m\u1ed9t \u0111o\u1ea1n th\u1eb3ng c\u00f3 h\u01b0\u1edbng. M\u1ecdi \u0111\u1ecbnh ngh\u0129a v\u1ec1 hai vect\u01a1 c\u00f9ng ph\u01b0\u01a1ng, c\u00f9ng h\u01b0\u1edbng, b\u1eb1ng nhau, \u0111\u1ed1i nhau \u0111\u1ec1u ho\u00e0n to\u00e0n t\u01b0\u01a1ng t\u1ef1 nh\u01b0 trong m\u1eb7t ph\u1eb3ng nh\u01b0ng \u0111\u01b0\u1ee3c m\u1edf r\u1ed9ng ra kh\u00f4ng gian 3 chi\u1ec1u:<\/p>\n