B\u00e0i to\u00e1n vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh ti\u1ebfp tuy\u1ebfn c\u1ee7a \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1 l\u00e0 1 trong nh\u1eefng b\u00e0i to\u00e1n quan tr\u1ecdng n\u00f3 th\u01b0\u1eddng xu\u1ea5t hi\u1ec7n trong c\u00e1c \u0111\u1ec1 thi t\u1ed1t nghi\u1ec7p, \u0111\u1ec1 thi \u0111\u1ea1i h\u1ecdc nhi\u1ec1u n\u0103m g\u1ea7n \u0111\u00e2y. C\u00e1c b\u1ea1n c\u1ea7n l\u01b0u \u00fd nhi\u1ec1u \u0111\u1ebfn d\u1ea1ng to\u00e1n n\u00e0y \u0111\u1ec3 c\u00f3 th\u1ec3 gi\u00fap c\u00e1c b\u1ea1n c\u00f3 b\u00e0i th\u00ec t\u1ed1t nh\u1ea5t v\u00e0 c\u00f3 nh\u1eefng \u0111i\u1ec3m s\u1ed1 cao nh\u1ea5t.<\/p>\n

1\/ Ki\u1ebfn th\u1ee9c l\u01b0u \u00fd khi gi\u1ea3i b\u00e0i to\u00e1n vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh ti\u1ebfp tuy\u1ebfn.<\/h2>\n

\u0110\u1ecbnh ngh\u0129a ti\u1ebfp tuy\u1ebfn l\u00e0 g\u00ec?. Theo c\u00e1c gi\u1ea3i th\u00edch \u0111\u01a1n gi\u1ea3n v\u00e0 d\u1ec5 hi\u1ec3u th\u00ec:<\/p>\n

Gi\u1ea3 s\u1eed ta c\u00f3 1 h\u00e0m s\u1ed1 y=f(x) c\u00f3 \u0111\u1ed3 th\u1ecb l\u00e0 1t \u0111\u01b0\u1eddng cong k\u00fd hi\u1ec7u l\u00e0 (C), 1 \u0111\u01b0\u1eddng th\u1eb3ng d ti\u1ebfp x\u00fac v\u1edbi \u0111\u01b0\u1eddng cong (C) t\u1ea1i \u0111i\u1ec3m M(x_{0 <\/sub>; y0<\/sub>) \u0111\u01b0\u1ee3c\u00a0g\u1ecdi l\u00e0 ti\u1ebfp tuy\u1ebfn c\u1ee7a (C) t\u1ea1i \u0111i\u1ec3m M \u0111\u00f3.<\/p>\n}

<\/p>\n

Ta c\u00f3 “d ti\u1ebfp x\u00fac v\u1edbi (C)”, v\u1eady ta c\u1ea7n gi\u1ea3i th\u00edch nh\u01b0 th\u1ebf n\u00e0o l\u00e0 ti\u1ebfp x\u00fac? H\u00ecnh v\u1ebd b\u00ean tr\u00ean c\u00f3 th\u1ec3 gi\u00fap ta ph\u00e2n bi\u1ebft gi\u1eefa ti\u1ebfp x\u00fac v\u00e0 c\u1eaft.<\/p>\n

Ta c\u00f3 \u0111\u01b0\u1eddng th\u1eb3ng d ti\u1ebfp x\u00fac v\u1edbi \u0111\u01b0\u1eddng cong (C) t\u1ea1i \u0111i\u1ec3m M v\u00e0 c\u1eaft \u0111\u01b0\u1eddng cong (C) t\u1ea1i \u0111i\u1ec3m N.<\/p>\n

Khi \u0111\u00f3 \u0111i\u1ec3m\u00a0M(x_{0 <\/sub>; y0<\/sub>) \u00a0g\u1ecdi l\u00e0 ti\u1ebfp \u0111i\u1ec3m c\u1ee7a ti\u1ebfp tuy\u1ebfn d v\u00e0 \u0111\u1ed3 th\u1ecb (c). \u0110i\u1ec3n M thu\u1ed9c \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1 y=f(x) n\u00ean ta c\u00f3\u00a0y0<\/sub>= \u0192( x0<\/sub>).<\/p>\n}

Khi \u0111\u00f3\u00a0ta c\u00f3 h\u1ec7 s\u1ed1 g\u00f3c c\u1ee7a ti\u1ebfp tuy\u1ebfn d t\u1ea1i \u0111i\u1ec3m\u00a0y_{0<\/sub>= \u0192( x0<\/sub>)\u00a0ch\u00ednh l\u00e0 \u0111\u1ea1o h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 y=f(x) t\u1ea1i \u0111i\u1ec3m\u00a0 n\u00e0y. V\u1eady ta c\u00f3 \u0111\u01b0\u1ee3c ph\u01b0\u01a1ng tr\u00ecnh ti\u1ebfp tuy\u1ebfn:<\/p>\n}

y – y_{0\u00a0<\/sub>=\u00a0\u0192'( x0<\/sub>)(x-x0<\/sub>)<\/p>\n}

Trong 1 b\u00e0i to\u00e1n v\u1ec1 vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh ti\u1ebfp tuy\u1ebfn, ch\u00fang ta ch\u1ec9 c\u1ea7n t\u00ecm \u0111\u01b0\u1ee3c t\u1ecda \u0111\u1ed9 c\u1ee7a ti\u1ebfp \u0111i\u1ec3m\u00a0(x_{0 <\/sub>; y0<\/sub>)\u00a0v\u00e0 h\u1ec7 s\u1ed1 g\u00f3c\u00a0\u0192'( x0<\/sub>)\u00a0l\u00e0 ta c\u00f3 th\u1ec3 vi\u1ebft \u0111\u01b0\u1ee3c ph\u01b0\u01a1ng tr\u00ecnh ti\u1ebfp tuy\u1ebfn n\u00e0y.<\/p>\n}

2\/ C\u00e1c d\u1ea1ng b\u00e0i to\u00e1n vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh ti\u1ebfp tuy\u1ebfn c\u01a1 b\u1ea3n<\/strong><\/h2>\n

D\u1ea1ng 1:<\/strong>\u00a0Vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh ti\u1ebfp tuy\u1ebfn c\u1ee7a h\u00e0m s\u1ed1 khi bi\u1ebft t\u1ecda \u0111\u1ed9 ti\u1ebfp \u0111i\u1ec3m.<\/h3>\n

V\u1edbi d\u1ea1ng b\u00e0i to\u00e1n n\u00e0y n\u00e0y ta ch\u1ec9 c\u1ea7n t\u00ednh h\u1ec7 s\u1ed1 g\u00f3c l\u00e0 c\u00f3 th\u1ec3 vi\u1ebft \u0111\u01b0\u1ee3c ph\u01b0\u01a1ng tr\u00ecnh ti\u1ebfp tuy\u1ebfn.<\/p>\n

B\u00e0i t\u1eadp v\u00ed d\u1ee5<\/u>: Vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh ti\u1ebfp tuy\u1ebfn c\u1ee7a \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1 sau y = \u0192(x)= x\u00b3 – 2x + 1 t\u1ea1i \u0111i\u1ec3m M(2 ; 5).<\/p>\n

Gi\u1ea3i<\/u><\/p>\n

Khi \u0111\u00f3 ta c\u00f3:\u0192'(x) = 3x\u00b2 – 2<\/p>\n

H\u1ec7 s\u1ed1 g\u00f3c c\u1ee7a ti\u1ebfp tuy\u1ebfn t\u1ea1i \u0111i\u1ec3m\u00a0M(2 ; 5) :\u00a0\u0192'(2) = 3.2\u00b2 – 2 = 10<\/p>\n

Ph\u01b0\u01a1ng tr\u00ecnh ti\u1ebfp tuy\u1ebfn c\u1ee7a \u0111\u1ed3 th\u1ecb l\u00e0: y – 5 = 10(x – 2)\u00a0<=> y = 10x – 15<\/p>\n

D\u1ea1ng 2:<\/strong>\u00a0Vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh ti\u1ebfp tuy\u1ebfn c\u1ee7a h\u00e0m s\u1ed1 khi bi\u1ebft ho\u00e0nh \u0111\u1ed9 giao \u0111i\u1ec3m.<\/h3>\n

V\u1edbi d\u1ea1ng b\u00e0i to\u00e0n n\u00e0y ta \u0111\u00e3 bi\u1ebft \u0111\u01b0\u1ee3c x_{0<\/sub>\u00a0, c\u1ea7n t\u00ecm th\u00eam y0\u00a0<\/sub>v\u00e0 h\u1ec7 s\u1ed1 g\u00f3c\u00a0=\u00a0\u0192'(x0)<\/sub> c\u1ee7a h\u00e0m s\u1ed1<\/p>\n}

B\u00e0i t\u1ea1p v\u00ed d\u1ee5<\/u>: H\u00e3y vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh ti\u1ebfp tuy\u1ebfn c\u1ee7a \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1 sau: y = \u0192(x)= x\u00b3 – 2x + 1 t\u1ea1i 1 \u0111i\u1ec3m c\u00f3 ho\u00e0nh \u0111\u1ed9 b\u1eb1ng 1.<\/p>\n

Gi\u1ea3i<\/u><\/p>\n

Khi \u0111\u00f3 ta c\u00f3:\u00a0\u0192'(x) = 3x\u00b2 – 2<\/p>\n

G\u1ecdi \u0111i\u1ec3m N(x_{0 <\/sub>; y0<\/sub>) l\u00e0 ti\u1ebfp \u0111i\u1ec3m c\u1ee7a ti\u1ebfp tuy\u1ebfn \u0111\u00f3 v\u00e0 \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1.<\/p>\n}

Khi \u0111\u00f3 ta c\u00f3:\u00a0x_{0\u00a0<\/sub>= 1 \u00a0=> y0\u00a0<\/sub>=\u0192(1) =0<\/p>\n}

H\u1ec7 s\u1ed1 g\u00f3c c\u1ee7a ti\u1ebfp tuy\u1ebfn l\u00e0:\u00a0\u0192'(1) = 3.2\u00b9 – 2 = 1<\/p>\n

V\u1eady ta c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh ti\u1ebfp tuy\u1ebfn:\u00a0y – 0\u00a0= 1(x – 1)\u00a0<=> y = x – 1<\/p>\n

D\u1ea1ng 3:<\/strong>\u00a0Vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh ti\u1ebfp tuy\u1ebfn c\u1ee7a h\u00e0m s\u1ed1 bi\u1ebft tung \u0111\u1ed9 ti\u1ebfp \u0111i\u1ec3m.<\/h3>\n

V\u1edbi d\u1ea1ng b\u00e0i to\u00e1n n\u00e0y ta \u0111\u00e3 bi\u1ebft \u0111\u01b0\u1ee3c y_{0<\/sub>\u00a0. Ta c\u1ea7n \u0111i t\u00ecm\u00a0x0<\/sub>\u00a0v\u00e0 h\u1ec7 s\u1ed1 g\u00f3c c\u1ee7a h\u00e0m s\u1ed1.<\/p>\n}

B\u00e0i t\u1eadp v\u00ed d\u1ee5<\/u>: H\u00e3y vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh ti\u1ebfp tuy\u1ebfn c\u1ee7a \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1\u00a0sau: y = \u0192(x)= x\u00b3 +\u00a02x + 1\u00a0t\u1ea1i 1 \u0111i\u1ec3m c\u00f3 tung \u0111\u1ed9 b\u1eb1ng 1.<\/p>\n

Gi\u1ea3i<\/u><\/p>\n

Ta c\u00f3 : \u0192'(x) = 3x\u00b2 +\u00a02<\/p>\n

G\u1ecdi\u00a0\u0111i\u1ec3m M(x_{0 <\/sub>; y0<\/sub>) l\u00e0 ti\u1ebfp \u0111i\u1ec3m c\u1ee7a ti\u1ebfp tuy\u1ebfn v\u1edbi \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1.<\/p>\n}

Theo \u0111\u1ec1 b\u00e0i:<\/p>\n

H\u1ec7 s\u1ed1 g\u00f3c c\u1ee7a ti\u1ebfp tuy\u1ebfn l\u00e0 :\u00a0\u0192'(0) = 3.(0)\u00b2 + 2 = 2<\/p>\n

V\u1eady ta c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh ti\u1ebfp tuy\u1ebfn: y – 1 = 2(x – 0) => y = 2x + 1<\/p>\n

<\/p>\n

D\u1ea1ng 4:<\/strong>\u00a0Vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh ti\u1ebfp tuy\u1ebfn khi bi\u1ebft h\u1ec7 s\u1ed1 g\u00f3c c\u1ee7a ti\u1ebfp tuy\u1ebfn.<\/h3>\n

V\u1edbi d\u1ea1ng to\u00e1n n\u00e0y ta c\u1ea7n t\u00ecm t\u1ecda \u0111\u1ed9 c\u1ee7a ti\u1ebfp \u0111i\u1ec3m \u0111\u1ec3 c\u00f3 th\u1ec3 vi\u1ebft \u0111\u01b0\u1ee3c ph\u01b0\u01a1ng tr\u00ecnh ti\u1ebfp tuy\u1ebfn.<\/p>\n

B\u00e0i t\u1eadp v\u00ed d\u1ee5:<\/u>\u00a0Vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh ti\u1ebfp tuy\u1ebfn \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1 sau y = \u0192(x)= x\u00b3 +\u00a02x + 1\u00a0cho bi\u1ebft h\u1ec7 s\u1ed1 g\u00f3c c\u1ee7a ti\u1ebfp tuy\u1ebfn b\u1eb1ng 5.<\/p>\n

Gi\u1ea3i<\/u><\/p>\n

Ta c\u00f3:\u00a0\u0192'(x) = 3x\u00b2 +\u00a02<\/p>\n

G\u1ecdi \u0111i\u1ec3m M(x_{0 <\/sub>; y0<\/sub>) l\u00e0 ti\u1ebfp \u0111i\u1ec3m c\u1ee7a ti\u1ebfp tuy\u1ebfn v\u1edbi \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1.<\/p>\n}

Khi \u0111\u00f3 ta c\u00f3 h\u1ec7 s\u1ed1 g\u00f3c c\u1ee7a ti\u1ebfp tuy\u1ebfn l\u00e0:\u00a0\u0192'(x_{0<\/sub>) = 5 <=> 3x0<\/sub>\u00b2 + 2 = 5 <=> x0\u00a0<\/sub>=\u00a0\u00b11<\/p>\n}

V\u1edbi x_{0\u00a0<\/sub>= 1 , y0\u00a0<\/sub>= 4 th\u00ec \u00a0ph\u01b0\u01a1ng tr\u00ecnh ti\u1ebfp tuy\u1ebfn: y – 4 = 5(x – 1) => y = 5x – 1<\/p>\n}

V\u1edbi x_{0\u00a0<\/sub>= -1 , y0\u00a0<\/sub>= – 2 th\u00ec ph\u01b0\u01a1ng tr\u00ecnh ti\u1ebfp tuy\u1ebfn: y + \u00a02 = 5(x + 1) => y = 5x + 3<\/p>\n}

L\u01b0u \u00fd:<\/strong>\u00a0v\u1edbi d\u1ea1ng to\u00e1n n\u00e0y c\u00f3 th\u1ec3 cho \u1edf d\u1ea1ng vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh ti\u1ebfp tuy\u1ebfn c\u1ee7a h\u00e0m s\u1ed1 khi bi\u1ebft ti\u1ebfp tuy\u1ebfn song song ho\u1eb7c vu\u00f4ng g\u00f3c v\u1edbi 1 \u0111\u01b0\u1eddng th\u1eb3ng cho tr\u01b0\u1edbc. Khi \u0111\u00f3 ta c\u1ea7n s\u1eed d\u1ee5ng c\u00e1c nh\u1eadn x\u00e9t sau \u0111\u1ec3 t\u00ecm h\u1ec7 s\u1ed1 g\u00f3c c\u1ee7a ti\u1ebfp tuy\u1ebfn n\u00e0y:<\/p>\n

\n
• N\u1ebfu 2 \u0111\u01b0\u1eddng th\u1eb3ng song song v\u1edbi nhau th\u00ec hai h\u1ec7 s\u1ed1 g\u00f3c b\u1eb1ng nhau.<\/li>\n
• N\u1ebfu 2 \u0111\u01b0\u1eddng th\u1eb3ng vu\u00f4ng g\u00f3c v\u1edbi nhau th\u00ec t\u00edch hai h\u1ec7 s\u1ed1 g\u00f3c b\u1eb1ng -1.<\/li>\n<\/ul>\n

  N\u1ebfu \u0111\u01b0\u1eddng th\u1eb3ng c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 y = ax + b th\u00ec ta c\u00f3 h\u1ec7 s\u1ed1 g\u00f3c l\u00e0 k = a.<\/p>\n

  B\u00e0i t\u1eadp v\u00ed d\u1ee5:<\/u>\u00a0H\u00e3y vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh ti\u1ebfp tuy\u1ebfn c\u1ee7a \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1 c\u00f3 d\u1ea1ng y = \u0192(x)= x\u00b3 +\u00a02x + 1 cho bi\u1ebft ti\u1ebfp tuy\u1ebfn vu\u00f4ng g\u00f3c v\u1edbi \u0111\u01b0\u1eddng th\u1eb3ng d c\u00f3 d\u1ea1ng – 2x +\u00a0y – 1 = 0: \u00a0.<\/p>\n

  Gi\u1ea3i<\/u><\/p>\n

  Ta c\u00f3: \u0192'(x) = 3x\u00b2 +\u00a02<\/p>\n

  T\u1eeb \u0111\u01b0\u1eddng th\u1eb3ng d ta c\u00f3:\u00a0– 2x +\u00a0y – 1 = 0 => y= 2x + 1<\/p>\n

  V\u1eady h\u1ec7 s\u1ed1 g\u00f3c c\u1ee7a d l\u00e0\u00a0. k(d) = 2<\/p>\n

  G\u1ecdi \u0111i\u1ec3m\u00a0M(x_{0 <\/sub>; y0<\/sub>) l\u00e0 ti\u1ebfp \u0111i\u1ec3m c\u1ee7a ti\u1ebfp tuy\u1ebfn v\u00e0 \u0111\u1ed3 th\u1ecb. V\u1eady h\u1ec7 s\u1ed1 g\u00f3c c\u1ee7a ti\u1ebfp tuy\u1ebfn l\u00e0\u00a0\u0192'(x0<\/sub>) .<\/p>\n}

  V\u00ec ti\u1ebfp tuy\u1ebfn vu\u00f4ng g\u00f3c v\u1edbi d ta c\u00f3:<\/p>\n

  \u0192'(x_{0<\/sub>).k(d) = – 1 <=>\u00a0\u0192'(x0<\/sub>).2 = – 1 <=>\u00a0\u0192'(x0<\/sub>) = – 1\/2 <=>\u00a03x0<\/sub>\u00b2 +\u00a02 = -1\/2<\/p>\n}

  (ph\u01b0\u01a1ng tr\u00ecnh n\u00e0y v\u00f4 nghi\u1ec7m)<\/p>\n

  V\u1eady kh\u00f4ng c\u00f3 ti\u1ebfp tuy\u1ebfn n\u00e0o th\u1ecfa y\u00eau c\u1ea7u b\u00e0i to\u00e1n.<\/p>\n

  Tr\u00ean \u0111\u00e2y l\u00e0 m\u1ed9t s\u1ed1 d\u1ea1ng to\u00e1n vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh ti\u1ebfp tuy\u1ebfn c\u01a1 b\u1ea3n c\u00e1c b\u1ea1n c\u1ea7n n\u1eafm \u0111\u01b0\u1ee3c tr\u01b0\u1edbc khi ti\u1ebfp c\u1eadn v\u1edbi c\u00e1c d\u1ea1ng to\u00e1n kh\u00f3 h\u01a1n trong tuy\u1ec3n t\u1eadp c\u00e1c \u0111\u1ec1 thi tuy\u1ec3n sinh \u0111\u1ea1i h\u1ecdc.<\/p>\n","protected":false},"excerpt":{"rendered":"

  B\u00e0i to\u00e1n vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh ti\u1ebfp tuy\u1ebfn c\u1ee7a \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1 l\u00e0 1 trong nh\u1eefng b\u00e0i to\u00e1n quan tr\u1ecdng n\u00f3 th\u01b0\u1eddng xu\u1ea5t hi\u1ec7n trong c\u00e1c \u0111\u1ec1 thi t\u1ed1t nghi\u1ec7p,… <\/p>\n","protected":false},"author":9,"featured_media":8672,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[450,469],"tags":[],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/onthitot.com\/wp-json\/wp\/v2\/posts\/8670"}],"collection":[{"href":"https:\/\/onthitot.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/onthitot.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/onthitot.com\/wp-json\/wp\/v2\/users\/9"}],"replies":[{"embeddable":true,"href":"https:\/\/onthitot.com\/wp-json\/wp\/v2\/comments?post=8670"}],"version-history":[{"count":0,"href":"https:\/\/onthitot.com\/wp-json\/wp\/v2\/posts\/8670\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/onthitot.com\/wp-json\/wp\/v2\/media\/8672"}],"wp:attachment":[{"href":"https:\/\/onthitot.com\/wp-json\/wp\/v2\/media?parent=8670"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/onthitot.com\/wp-json\/wp\/v2\/categories?post=8670"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/onthitot.com\/wp-json\/wp\/v2\/tags?post=8670"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}