Sổ tay giải toán 12 - Nguyễn Đức Thắng

Thy Nguyn c Thng 0969119789 thangnd286@gmail.com Trng PTLC Vinschool S TAY GII TON 12 Thy Nguyn c Thng 0969119789 thangnd286@gmail.com Trng PTLC Vinschool MC LC CH TRANG A KHO ST HM S B LU THA - M - LễGARIT 18 C NGUYấN HM TCH PHN V NG DNG 25 D S PHC 42 E NểN TR-CU 47 F PHNG PHP TO TRONG KHễNG GIAN OXYZ 54 G KHI A DIN 64 H GểC V KHONG CCH 67 I B SUNG MT S KIN THC 77 Trung tõm luyn thi cht lng cao Thnh t Tõy M, Nam T Liờm, H Ni Page Thy Nguyn c Thng 0969119789 thangnd286@gmail.com Trng PTLC Vinschool A KHO ST HM S Tớnh n iu 1.1 Lớ thuyt a) nh ngha: Cho K l mt khong, on hoc na khong Gi s f(x) l mt hm s xỏc nh trờn K - Hm s f(x) gi l ng bin trờn K nu " x1 , x2 ẻ K : x1 < x2 ị f ( x1 ) < f ( x2 ) - Hm s f(x) gi l nghch bin trờn K nu " x1 , x2 ẻ K : x1 < x2 ị f ( x1 ) > f ( x2 ) b iu kin cn Gi s f cú o hm trờn khong K - Hm s f(x) khụng i trờn K "x ẻ K : f '( x ) = - Nu f ng bin trờn khong K thỡ f '( x ) 0, "x ẻ K - Nu f nghch bin trờn khong K thỡ f '( x ) Ê 0, "x ẻ K c iu kin Gi s f cú o hm trờn khong K - Nu f (x) 0, "x ẻ I (fÂ(x) = ti mt s hu hn im) thỡ f ng bin trờn K - Nu f (x) Ê 0, "x ẻ I (fÂ(x) = ti mt s hu hn im) thỡ f nghch bin trờn K - Nu fÂ(x) = 0, "x ẻ I thỡ f khụng i trờn K Mt s khỏc a) nh lớ v du ca tam thc bc hai: g(x ) = ax + bx + c (a 0) + Nu D < thỡ g( x ) luụn cựng du vi a ổ b b ), g ỗ - ữ = 2a ố 2a ứ + Nu D > thỡ g( x ) cú hai nghim x1 , x2 v khong hai nghim thỡ g( x ) khỏc du + Nu D = thỡ g( x ) luụn cựng du vi a (tr x = - vi a, ngoi khong hai nghim thỡ g( x ) cựng du vi a ỡa > ỡa < +) y ' Ê 0, "x ẻ R Chỳ ý: - Nu y ' = ax + bx + c (a 0) thỡ: +) y ' 0, "x ẻ R D Ê ợ ợD Ê - Nu D = hay g( x ) = a ( x - a ) thỡ g(x) khụng i du qua a , du ca g(x) ph thuc du ca a - Nu D > thỡ g(x) i du qua x1 , x2 ( i t+ sang sang +, hoc i t - sang + sang -) b) So sỏnh cỏc nghim x1 , x2 ca tam thc bc hai g( x ) = ax + bx + c vi s 0: ỡD ù +) x1 Ê x2 < P > ùợ S < ỡD ù +) < x1 Ê x2 P > ùợ S > +) x1 < < x2 P < c) Hm s bc hai: y = ax + bx + c (a 0) a>0 th hm s l mt parabol cú nh a0 cha K hm s y = f ( x) nghch bin trờn K no ú thỡ tn ti khong f(x) iu kin hm s f ( x ) = ax + bx + cx + d ng bin trờn R l ; nghch bin trờn ợD Ê ỡa < R l ợD Ê ã Hm s f ( x ) = ax + bx + cx + d ng bin ( nghch bin) trờn K thỡ khong m f '( x ) ( f '( x ) ) ca hm s phi cha K b) Hm s phõn thc dng f ( x ) = ax + b (c 0, ad - bc 0) cx + d Thy Nguyn c Thng ( ad - bc < 0) ã ã 0969119789 thangnd286@gmail.com Trng PTLC Vinschool iu kin hm s ng bin (nghch bin) trờn trờn (a ; +Ơ ) l ỡad - bc > ù d ùa Ê c ợ ( ad - bc < ) ỡad - bc > ù d ùa c ợ ( ad - bc < ) iu kin hm s ng bin (nghch bin) trờn trờn ( -Ơ;a ) l +) i vi hm hp y = f (g( x)) , ú hm u = g( x ) xỏc nh v cú o hm trờn ( a; b ) , ly giỏ tr trờn khong ( c; d ) ; hm y = f (u) xỏc nh ( c; d ) v cú o hm trờn ( c; d ) , ly giỏ tr trờn R ã ùỡ g '( x ) > " x ẻ ( a; b ) ùỡ g '( x ) < " x ẻ ( a; b ) Nu hoc thỡ hm s y = f (g( x)) ng bin ùợ f '(u) > "u ẻ ( c; d ) ùợ f '(u) < "u ẻ ( c; d ) trờn ( a; b ) ã ỡù g '( x ) < " x ẻ ( a; b ) ỡù g '( x ) < " x ẻ ( a; b ) Nu hoc thỡ hm s y = f (g( x)) nghch bin ùợ f '(u) > "u ẻ ( c; d ) ùợ f '(u) > "u ẻ ( c; d ) trờn ( a; b ) CC TR CA HM S 2.1 Lớ thuyt a) nh ngha: Gi s hm s f ( x) xỏc nh trờn D, x0 ẻ D - im x0 gi l im cc tiu ca hm s f(x) nu tn ti s thc dng h cho ( x0 - h; x0 + h ) cha D v f (x) > f ( xo ), x ẻ ( x0 - h; x0 + h ) \ { x0 } Khi ú: + Giỏ tr f ( x0 ) gi l giỏ tr cc tiu ca hm s + im ( x0 ; f ( x0 )) gi l im cc tiu ca th hm s y=f(x) + Hm s t cc tiu ti im x0 - im x0 gi l im cc i ca hm s f(x) nu tn ti s thc dng h cho ( x0 - h; x0 + h ) cha D v f ( x ) < f ( xo ), x ẻ ( x0 - h; x0 + h ) \ { x0 } Khi ú: Giỏ tr f ( x0 ) gi l giỏ tr cc i ca hm s im ( x0 ; f ( x0 )) gi l im cc i ca th hm s y=f(x) + Giỏ tr f ( x0 ) gi l giỏ tr cc i ca hm s + im ( x0 ; f ( x0 ) ) gi l im cc i ca th hm s y=f(x) + Hm s t cc i ti im x0 Chỳ ý: Cc i, cc tiu gi chung l cc tr b) nh lớ: Trung tõm luyn thi cht lng cao Thnh t Tõy M, Nam T Liờm, H Ni Page Thy Nguyn c Thng 0969119789 thangnd286@gmail.com Trng PTLC Vinschool iu kin cn: Nu hm s f(x) t cc tr ti im x0 thỡ hoc khụng tn ti f '(x ) hoc f '( x0 ) = iu kin 1: Gi s tn ti ( a; b ) è D ch x0 , hm s y=f(x) liờn tc trờn (a,b) v cú o hm trờn mi khong ( a; x0 ) , ( x0 ; b ) ã ùỡ f '( x ) < "x ẻ ( a; x0 ) Nu thỡ x0 l mt im cc tiu ca hm s f(x) ùợ f '( x ) > "x ẻ ( x ; b ) ã ùỡ f '( x ) > "x ẻ ( a; x0 ) Nu thỡ x0 l mt im cc i ca hm s f(x) ùợ f '( x ) < "x ẻ ( x ; b ) iu kin 2: Gi s tn ti ( a; b ) è D ch x0 , hm s y=f(x) liờn tc trờn (a,b) v cú o hm cp trờn (a;b) v cú o hm cp hai ti x0 Khi ú: ã ỡ f '( x0 ) = Nu thỡ x0 l mt im cc tiu ca hm s f(x) ợ f ''( x0 ) > ỡ f '( x0 ) = Nu thỡ x0 l mt im cc i ca hm s f(x) ợ f ''( x0 ) < 2.2 Mt s khỏc ã a) Hm s a thc bc ba f ( x ) = ax + bx + cx + d (a 0) : ã ỡ ỡa ùa = ù ù Hm s t cc i ti x0 khi: D f '(x) > hoc ớb < ù c ù ợ f ''( x0 ) < = x0 ùợ 2b ã ỡ ỡa ùa = ù ù Hm s t cc tiu ti x0 khi: D f '(x) > hoc ớb > ù c ù f ''( x ) > 0 ợ = x0 ùợ 2b ã ỡa ỡa = Hm s khụng cú cc tr hoc D Ê ợb = ợ f '(x) ã ã ỡa Hm s cú cc i, cc tiu ợ D f '(x) > Phng trỡnh ng thng i qua hai im cc tr ca th hm s y = ax + bx + cx + d ( a ) Vi iu kin b2 - 3ac > , thc hin phộp chia y cho y ta c y = y(x).g(x) + Ax + B Khi ú, ng thng i qua hai im cc tr l y = Ax + B b) Hm s a thc trựng phng: f ( x ) = ax + bx + c (a 0) TH1: a = *) Nu b > Hm s ch cú cc tiu *) Nu b < Hm s ch cú cc i *) Nu b = Hm s khụng cú cc tr Trung tõm luyn thi cht lng cao Thnh t Tõy M, Nam T Liờm, H Ni Page Thy Nguyn c Thng 0969119789 thangnd286@gmail.com ( TH2: a Khi ú: y ' = 4ax + 2bx = x 2ax + b ) Trng PTLC Vinschool *) Nu a.b0: Hm s cú cc tiu, cc i a0: Hm s cú cc tiu a ù u(x) ùv(x) =ớ Li gii y = v( x) ù u(x) v ( x ) < ù v(x) ợ Suy ( M ) = ( C3 ) ẩ ( C4 ) vi ( C3 ) l phn ca th (C) cú honh tha iu kin v ( x ) > v ( C4 ) l phn i xng qua trc honh ca phn th (C) cú honh tha v ( x ) < Dng T th (C) ca hm s y = u( x) v( x) u(x) , suy cỏch v th (N) ca hm s y = ỡu( x) u( x) ù u(x) ùv( x) v( x) Li gii y = =ớ u( x) v( x) ù u( x) 0) n n x n-1 ( u ) = 2u u ( u ) = n u1  ( ) Hm s lng giỏc / cos x ( cot x ) = - 12 = - + cot x sin x (x)/= x -1 ( (u > 0) ' n n -1 u ' (u > 0) / ( sin x ) = cos x / ( cos x ) = - sin x / ( tanx ) = 12 = + tan2 x / Hm ly tha n / ( sin u ) = cos u.u/ / ( cos u ) = - sin u.u/ / ( tan u ) = 12 u/ ) cos u ( cot u ) = - 12 u/ sin u (u)/= u -1u/ / Trung tõm luyn thi cht lng cao Thnh t Tõy M, Nam T Liờm, H Ni Page 81 Thy Nguyn c Thng 0969119789 thangnd286@gmail.com Trng PTLC Vinschool x x u u Hm s m (e ) = e ( e ) = u e (ax) = axlna ( au) = u au.lna Hm logarớt u' (lnx ) = (x>0) ( lnu) = (u>0) x u u' (ln /x/ ) = (x0) ( ln /u/ ) = (u0) x u u' (x>0, 00, 0