Cấp Tốc Giải 10 Chuyên Đề 10 Điểm Thi Môn Toán - 2

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Nối tiếp nội dung phần 1 cuốn sách "Cấp tốc giải 10 chuyên đề 10 điểm thi môn Toán", phần 2 cung cấp cho người đọc các kiến thức: Nguyên hàm, tích phân và ứng dụng, hình học không gian, phương pháp tọa độ trong không gian, hình học giải tích tọa độ Oxy,... Mời các bạn cùng tham khảo.

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Nội dung Text: Cấp Tốc Giải 10 Chuyên Đề 10 Điểm Thi Môn Toán - 2

Cty TNHH IVITV DVVH Khang Vi§t ca'p t6'c giai 10 chuyfin dg 10 diem thi m6n Toan - Nguygn Phu Khanh Chuyfin IV Ta CO- f'(t) = 3t^ + l + - ^ > 0 voi V t > 0 , do do f(t) luon dong bien trong ^ ' tln2 NGUYfeN H A M , TiCH PHAN VA UNG DyNG khoang (0;+oo), phirang trinh (3) f(x) = f(2y) tux x =-2y De hoc gioi mon tich phan lap 12, cAc em can: 1 1 1. Thuoc bang dgo hdm ( nguyen hdm la ngwo-c v&i dgo ham ) ; Vai X = 2y, phu-o-ng trinh ( 2 ) tra thanh : y^ = - o y = - thoa y > 0 . 2. Hieu rd vd nam ki cdc dinh ly tich phan; 3. Lam nhieu bai tap vt nam vu-ng cdc dgng todn. Vay, he cho c6 nghiem: (x;y) = (4] 1. Dinh nghia: Cho ham so f xac djnh tren K . Ham so F du-gc goi la nguyen ham b. Dieu kien: xy > 0 . ciia f tren K neu F'(x) = f (x) Vx € K . 2. Cactinhchat: Dat t = log2 (xy) => xy = 2', khi do phu-o-ng trinh ( l ) tro thanh: Djnh li 1. Neu F la mot nguyen ham ciia ham f tren K thi moi nguyen ham cua f tren K deu c6 dang F(x) + C, C e l . Do vay F(x) + C goi la ho nguyen 9' - 3 = 2(2')'°''' o 3^' - 3 = 2.3* « 32* - 2.3* - 3 = 0 hamcuaham f tren K va du-gc ki hieu: f(x)dx = F(x) + C. < » ( 3 * + l ) ( 3 * - 3 ) = 0,suyra 3* = 3 tu-c xy = 2 ( 3 ) . Djnh li 2. Moi ham so lien tuc tren K deu c6 nguyen ham tren K Djnh h' 3. Neu f, g la hai ham lien tuc tren K thi: I. Phu-ong trinh (2) « + y^ + 2x + 2y +1 = 0 + j i f ( x ) ± g ( x ) ] d x = j f ( x ) d x ± jg(x)dx. o ( x + y f + 2 ( x + y)-2xy + l = 0(x + y f + 2(x + y ) - 3 = 0 do (3), + k.f(x)dx = k f(x)dx vaimoisothirc k^tO. phu-o-ng trinh nay tu-o-ngdu-ong (x + y - l ) { x + y + 3) = 0 Djnh h' 4. Neu ff (x)dx = F(x) + C thi x + y = - 3 hoac x + y = l •f{u(x)).u'(x)dx = Jf (u(x)).d(u(x)) = F(u(x))+ C. x + y = -3 |x = l „ Jx=2 THI:
cap tOc giai 10 chuyen flg 10 d i l m thi mfln Toan - Nguygn Phu Khanh ' ' f 3X + b Vay, F(x) = ( 4 x ^ + x + l l ) V 2 x - 4 . i(cx-a)(dx-p) Vi du 2. T i m nguyen ham : X Tach phan thu-c trong tich phan t r a thanh: p +q -dx cx-a li = I2 = fsin3xcos5xd}< (x + l f 3 X 4" b • Lay nghiem cua c x - a thay vao - — - ta diro-c p Lo-i gidi dx - p 1 3X +b 1. h-j- -dx = d(x + l ) • Lay nghiem cua dx - p thay vao ta du-p-c q cx-a I 1 1 Dang 1: T i m nguyen h a m bang phu-o-ng phap phan tich - + C. 3{x+iy 4{x+iy Vi du 1. f cos8x cos2x 2. I2 =Jsin3xcos5xdx = ^ J ( s i n 8 x - s i n 2 x ) d x = -^ + +C 8 2 1. Go! F(x) la nguyen ham cua ham so f ( x ) = sin2x.tanx thoa man F Dang 2: T i m nguyen ham bang phu-cng phap doi bien so Tinh F l4j Neu | f ( x ) d x = F(x) + C t h i j f ( u ( x ) ) . u ' ( x ) d x = F ( u ( x ) ) + C". 2. Xac dinh a, b, c sao cho F(x) = (ax^+bx + c ) V 2 x - 4 la 1 nguyen ham cua Gla su- ta can t i m ho nguyen ham 1 = |f ( x ) d x , trong do ta c6 the phan tich ^ . 20x^-29x + 7 ^ /_ V ham so f ( x ) = . — trong ( 2 ; + 0 0 ) . f ( x ) d x = g ( u ( x ) ) u ' ( x ) d x t h i t a thu-c hien phep doi bien so t - = u ( x ) \/2x-4 Lai giki =^ dt = u ' ( x ) d x . Khi do: I = j g ( t ) d t = G ( t ) + C = G ( U ( X ) ) + C 1. T a c o : F ( x ) = f s i n 2 x . t a n x d x = f 2 s i n x . c o s x . - ^ ^ d x = 2 fsin^xdx Chii y: Sau k h i ta t i m du-ac ho nguyen ham theo t t h i ta phal thay t = u ( x ) . ^ ^ J J cosx •' Vi du 1. T i m nguyen ham: F(x)= |(l-cos2x)dx = x - ^ ^ +C 1. Ii = jxVx + ldx 2. I2 = | x ^ V x 2 + 9 d x y(3 7 I 1 . 2 7 C _ N / 3 ^ n Ma: F Lo-i g i i i •• — o s m — + C = — =>C = l3 4 3 2 3 4 2 3 1. I j = |xVx + l d x . r./ \x yj3 n Vay: F(X) = X + W c h l : D a t t = x + l = > x = t - l va dx = dt 1 2 2 3 Khi do I i = J(t - 1 ) V t d t = j t V t d t - JVtdt 7t 1 . „ V3 n_sl3-l % 4. = sin2 2 3" 2 12 2 2 t_l^ x+1 1 4 2 = t ^ V t — t V t + C = 2tN/t + C = 2(x + l ) V ^ + C. 5 3 [3 3 5ax^+(3b-8a)x + c - 4 b 2. T a c o : F ' ( x ) - Cach2: 1^ = j{x + l)yf^dx- j^^dx = J(x + l ) 7 ^ d { x + l ) - J V ; m d ( x + l ) , j -1. «i 5a = 20 a=4 2(x+ifv;^ 2(x+l)V;m ^ ^ ^ + 1 ^^ T a l u o n c o : F'(x) = f ( x ) , V x > 2 khi v a c h i k h i 3b-8a = -29 b= l = ^ + C = 2(x + l ) V x + l +C c-4b = 7 c=l l 127
cap »'c giJi 10 chuyen dg 10 digm thi man Toan - Nguygn Phii Khanh Cty TNHH MTV DVVH Khang Vi$t (-sinx.dx) tanxdx = sinx d x = - f3x-2 = 2(x + l)^yx + l + C. 15 ) •'cosx cosx Ban doc xem cdch gidi sau c6 dung khong?. I Dat a = cosx=>da = - s i n x d x f(-sinx.dx) fda , ^ , D a t t = V ^ = * d t = - y : L = d x hay dx = z V ^ d t = 2 t d t . ! B = -p^ — = - l n a+C2 = - i n c o s x + C-2 , . : 2VX + 1 J cosx •' a 1 ? Khido I i = | ( t 2 - l ) 2 t d t - 2 j ( t 3 - t ) d t = 2 +C Vay ]i = A - B = - t a n x + ln cosx + C . 4 2 1 c(4cosx + 5)sinx.dx x2-l .Dat t = cosx=>dt = - s l n x d x rx+1 + C = (x + 1) -1 +C= +c. 2J cos X + 3COSX + 2 2 2 4t + 5 .3(t + l ) + ( t + 2)^^ f Xi-rt'|fi:J(x)'i|:'- Khi do J2 = - dt = - t2_9 1^+9 t^+3t + 2 ( t - f l ) ( t + 2) 2. Dat / x ^ + 9 = x - t = > x : dx = dt 2t 2t' ^ 3 -+- dt = -31n|t + 2|-ln|t + l l + C •vt + 2 t + l_ -t^-9 .(t^-8lf ^ ^-i-dt = -31n|cosx + 2|-ln|cosx + l | + C. 2t 4t^ 16 tanx , f sinx t'* 6561 3. J3 = J — 3 - d ' ' = dx 1 / 3 _ 1 6 2 65_61 4 l d t = - i - -—1621nt T- + C COS x COS X 16- 16 4 4t* Dat t - COSX => dt = - s i n x d x => sinxdx = - d t -dt- x-V? +9 1 -+ C- Kr- + CT 6561 3.cos^x --1621n X - N / X 2 + 9 +C 16 Dang 3: T i m nguyen ham bang phu-o-ng phap tirng phan Vi du 2. T i m nguyen ham : 5sinx + 2sin2x tanx Cho hai ham so u va v lien tuc tren [a;b] va c6 dao ham lien tuc tren fa;b •dx 3. h = dt = dx cos^x Can phai lira chpn u va dv ho-p l i sao cho ta de dang tim du-o-c v va tich phan A = j _ l _ t a n x d x = jtdt = + Cy - ^ l a n ^ x + vdu de tinh b a n udv . Ta thu-ang gap cat dang sau 129 1 9R
Ca^p tOc giai 10 ChuySn 66 10 die'm thi mOn Toan - Nguyjn Phu Khanh Cty TNHH MTV DVVH Khang Vif t sinx D a i J ^ i . - I = fP(x) d x , trong do P(x) la da thu-c. |sin2x.e^^dx = i e ^ \ s i n 2 x - - j c o s 2 x . e ^ ' ' d x = i e ^ ' ' . s i n 2 x - - l 2 , cosx sinx 3x dx. e Vo-i dang nay, ta dat u = P(x), dv = =>h= -^e^" cos2x + - e ^ ' ' . s i n 2 x - - I 2 =^ I2 = • ^ ( 3 c o s 2 x + 2sin2x) + C . cosx "13 u = ln x du = 2 l ^ d x 3. Dat X u = P(x) dv = (2x + l ) d x V = X + X Vai dang nay, ta dat , trong do P(x) la da thu-c dv = e'*''^''dx I3 = ( x ^ + x ) l n 2 x - 2 J(x^ + x ) l n x d x Dang 3: 1= f p ( x ) l n ( m x + n)dx dx Ui = l n x du^ = u = ln(mx + n) X Dat: Vo-i dang nay, ta dat dvi = ( x + l ) d x dv = P{x)dx 1 2 Vi = - x +x sinx Dang 4: \ e'dx cosx Khi do J(x + l ) l n x d x = i ( x 2 + 2 x ) l n x - i | ( x + 2)dx = ^(x^ + 2 x ) l n x - x + C' sinx sinx u = cosx d e t i n h fvdu ta dat cosx Vay I 3 = f x 2 + x ) l n 2 x - ( x 2 + 2 x ] l n x + — + 2X + C. ! Vo-i dang nay, ta dat dv^e^dx dv = e''dx Dang 4: T i n h tich phan bang phiro-ng phap phan tich [[ Vi du. T i m nguyen ham: = { ^ dx ; 2.U- fcosZx.e^^dx; 3. I3 = f(2x + l ) l n ^ x d x ^ -"l-cosZx •' V i d u . Tinh tich phan: 1 Lo-i giai dx 1. A- 2. B = J- 5 x - 1 3 dx; 3.C^J^>^^^dx. u =x x''+3x + 2 x^ - 5 x + 6 du = dx 0 0 X -4x + 4 1. Dat dx • 1^ = - i x c o t x + - [cotxdx dv = v = -cotx 2 2-' at' Lo-i giai sin^x dx _|(;^ + 2 ) - ( x + l)^^^_V 1 1 1 1 ;d(smx) 1 dx = —xcotx-- — x c o t x — I n sinx + C . J(x + l ) ( x + 2) J (x + l ) ( x + 2) x+1 x+2 sinx 2 2 0 2 2 x+1 , 2 , 1 D e y : A = fcotxdx= ^"^^dx, =(ln|x + l | - l n | x + 2|) = ln ln--ln- = lnl •' •'sinx x+2 3 2 3 V I ( s i n x ) ' = cosx = > d ( s i n x ) = (sinx)'dx = cosxdx 2. T:ic6- I ^ _a(x-3) +b(x-2) fa=3 du = - 2 s i n 2 x d x •(x-3)(x-2) x-3 x-2 (x-3){x-2) ^[h = 2, - u = cos2x \ 2. Dat 1 3x '2 = - 6^^ " cos2x + - fsin2x.e^''dx. ' 1, dv = e^''dx 3 3J V 3 dx = (31n|x-2| + 21n|x-3|) = - l n l 8 . 3 x-3 x-2 d u j =2cos2x u-j^ = s i n 2 x Dat ^' x ^ + x + 2 5x-2 3x dx = 1 +- dx d v i =e^''dx Vi =-e 0x''-4x +4 V X^ - 4 x + 4
ca'p ttfc giii 10 chuy6n dg 10 difi'm thi mfln Toan - Nguygn Phii Khanh Cty TNHH MTV DVVH Khang Vigt • v T d u 1. Tinh tich phan: ,,,,,,„ j ^ ^ , 5x-2 5x-2 A ^ B _ A ( x - 2 ) + B _ fA = 5 CM:' i ^f. Ta c6: 2 -3 x2-4x + 4 (x-lf 5^-2 ( x - 2 ) ^ (x-2)2 [8 = 8 ' 1. I l = -dx • 2.l2= J 1 dx 1i + 7 x ^ xVT^ 5 8 1 C= dx = 51n x - 2 = 51n- + 4 :• . t dx x-2 -+- 0 x-20 2 x^dx (x-2)^ 3. I3 = J 4. L = 2x + l + V4x + l ox + Vx^ + 1 Dang 5: T i n h tich phan bang phirang phap doi bien so 1. Dat t = V x - 1 •t^x = t^ + l o d x = 2tdt 1. Phiro-ng phap doi bien so loai 1 Doi bien: x = l = > t = 0, x = 2 = > t = l . 1 • ( Gia su- can tinh I = f (x)dx , ta thirc hien cac birac sau: Vay, I i = ^ — - : - 2 t d t = 2 ^ -dt = 2 | t ^ - t + 2 - — dt 1 +t 0 t +1 0i[ t+1 Biro-c 1: Dat x = u ( t ) {v&i u ( t ) la ham c6 dgo ham lien tuc tren [a;P], f ( u ( t ) ) ^3 j2 = 2 _ - _ +2t-2ln|t + l =2 ;fdc trer? [a;p] vd u ( a ) = a, u(p) = b ) va xac djnh a, p. = il_4l„2 3 Biro-cZrThayvaotichphanbandautaco: ^.^ 2. Dat t = N/TOC r : > - 2 t d t - dx 1= J f ( u ( t ) ) . u ' ( t ) d t = Jg{t)dt = G(t) P = G ( P ) - G ( a ) . Doi can: x = - 8 = i . t = 3 , x = - 3 = > t = 2 a a : dx i tdt d^_^f(t+l)-(t-l) Vay, I2 = — 7 = d x = -2 7 , Mot SO dang thirang diing phirang phap doi bien so loai 1 dt i i - t ^ t i t ^ - i (t-l)(t+l) * Ham so dirai dau tich phan chu-a Va^-b^x^ ta thu-ang dat x ^sin t b t-1 = ( l n | t - l | - l n | t + l|)^ = In = lni-lni =ln- t+1 3 2 3 * Ham so dirai dau tich phan chu-a „Vb^x^ - a ^ ta thu-ang dat x = bsint f 1 3. Dat t = V4x + l = > t ^ = 4 x + l ^ d x = i t d t * Ham so du-ai dau tich phan chu-a a^ +b^x^ ta thu-ang dat x = ^ t a n t 2 Doi can: x = 2 = > t = 3 , x = 6 = > t = 5 * Ham so du-ai dau tich phan chu-a ^ x ( a - b x ) ta thu-6-ng dat x = -^sin^ t 6 s Do do: I3 - - J ^ f. tdt 1 dt :UAi 2x + 1 + V4X + 1 J|(t + ir t +1 2. Phu-o-ng phap doi bien so loai 2 ( t + 1)^ Tu-ang tu- nhu- nguyen ham, ta c6 the tinh tich phan bang phirang phap doi bien so (ta goi la loai 2) nhu- sau: In t + 1 =i n l - l b 2 12 De tinh tich phan 1= j f ( x ) d x , neu f ( x ) = g [ u ( x ) ] . u ' ( x ) , ta c6 the thu-c hiei x-Vx^ + 1 phep doi bien nhu-sau rdt = u'(x)dx. D 6 i c a n x = a=>t = u(a), x = b=>t = u(b) u(b) = J x 3 N ^ ^ d x - V d x = 1-^ = , _ i , vai J = jx^VTTTdx 0 n 5 ^ 5 J Bu-6'c2:Thayvaotac6 1= fg(t)dt = G{t) b a•
Cap toe (jicii 10 clmyen 6i 10 di6'm thi mfln Toan - Nguyln Phu Khanh Cty TNHH MTV DVVH Khang Vi^t Dat t = Vx^ + 1 => = + 1 => 2tdt = 2xdx => tdt = xdx L a i giSi Doi can: x = 0 = > t = l , x = l = > t = 72. u = Inx dx du = X V2 1, Dat dx • dv = - -1 Khido •(t2-l]t.tdt= f f t ' ^ - t2 ) d t = t t (x + 1)^ V = > V / J \ / V / x+1 1 1 e e - 1 -.Inx , ' dx 4^2 2^21 2V2 2 Khido A = = -l+I X + 1 1 ^/x(x + l ) .5 3 15 15 dx (x + 1 ) - ) ^f3sinx-2cosx 1= -dx = dx = (ln X - I n x + 1) 1 Vi d u 2. Tinh ti'ch phan: I = dx ^ ( x + l) J x(x + l ) X x+1 0 (sinx + cosx) 1 Lo-i giSi e e Dat x = - - t =>dx = - d t , x = 0==>t = - , x = - = > t = 0. 1 In 2 2 2 x+1 ^\n——ln-e_ = lne = l . 1 e+1 1 ^ , e e ^r3sinx-2cosx , ^ f 3 c o s t - 2 s i n t ,. 3cosx-2sinx Vay, 1 = 0 . Suy ra: I = K1X = ! = .2tdt = 2 dt = ^^2 + . ^ dt Dang 6: Ti'nh tich phan bang phu-ang phap d d i tirng phan t^-l t^-l t-1 t+lj t-1 Cho hai ham so u va v lien tuc tren [a;b] va c6 dao ham lien tuc tren [a;b 2t + ln = 2 + ln3-ln2 t +1 Khi do : j u d v = uv Vay, B = 2 0 1 n 2 - 6 1 n 3 - 4 vdu 3. C = J — 1 dx = J — ^ - A i x + j x V 5 ^ . d x = K + H V i du 1. Tinh tich phan: e u = ln(5-x) dx du = - „ 1.1n(5-x) 5-x K= J — L — ^ x . D a t • ^ dx dv = — v--i x X 135
Cty TNHH MTV DWH Khang Vi$t Ca'p tO'c giai 10 chuy6n dg 10 di6'm thi mfln Toan - Nguygn Phu Khanh ln(5-x) lf(x)|dx = •f(x)dx cong thu-c nay chi diing k h i f(x) khong doi dau tren K= - - ^ ^ = ln4--f-ln(5-x) + lnx^ =-ln4 a 4- khoang ( a ; b ) . H = jxyjs^.dx.Dat t = y j s ^ =>2tdt = - d x T;^ b b , Neu: f (x) > 0 , Vx 6 [ a ; b] t h i | f (x)|dx = j f {x)dx Doican: x = l = > t = 2, x = 4 ^ t = l ,..D a a b b 1 164 Neu f ( x ) < 0 , V x e [ a ; b] t h i |f(x)|dx = - J f ( x ) d x H= j(5-t2)t(-2t)dt =2 3 5 15 Chiiy: Neu phu-o-ng trinh f ( x ) = 0 c6 k nghiem phan biet Xi,X2,...,X|^ tren (a; b] Vay, C = K + H = ^ l n 4 + ^ 5 15 thi tren moi khoang (a;xi),(xi;x2)...(xi^;b) bieu thu-c f ( x ) khongdoiddu. °fXln{x + 2)^ Vi du 2. Tinh tich phan: I = / ' dx. Khi do tich phan S = J f { x ) dx du-o-c tinh nhu- sau; 7,i| -1 V4-x2 Liri gi&i b X dx S = | f ( x ) | d x = j f ( x ) d x + Jf(x3dx + ...+ | f ( x ) d x Dat u = ln(x + 2 ) , d v = - p = = d x . K h i d 6 du = — - ,v = -V4-V a V4-x^ ^'^^ Cong thu-c tinh dien tich hinh phang gi6i han bo-i cac du-6-ng: Theo cong thu-c tich phan tirng phan, ta c6 b I = - V 4 - x ^ l n ( x + 2) 0 °-V4^ dx = -21n2 + dx. y = f ( x ) va y = g(x) va hai du-6-ng thang x=:a,x = b(a < b ) : S = J f ( x ) - g ( x ) d x . + x+2 x+2 a -1 -1 -1 Vi du 1, Tinh dien tich S ciia hinh phang H gio-i han bo-i: Dat x = 2sint. Khi do dx = 2costdt. Doi can: x = - l = > t = - - , x = 0=:i>t = 0. 6 1. Do thi ham so: v , true hoanh va du-o-ng thang y = 2 - x . r i l ' 0 2. Do thj ham so: y = (e + l ) x va y = (e" + l ) x . 1= dx = ' Acosh dt = 2 ( l - s i n t ) d t = 2(t + cost) = 2+ ^-7^. X+ 2 2sint + 2 3 Lo-i gi^i 1. Hoanh do giao diem cua do t h i ham so: y = -\fx va du-6-ng t h i n g y = 2 - x la Suyra I = - 2 i n 2 + 2 - 7 3 + j . nghiem cua phu-o-ng t r i n h : - V x = 2 - x\/x = x - 2 x>2 x>2 x = 4 Dang 7: IJng dung tich phan x = x^-4x + 4 [x^-5x + 4 = 0 -yirn ik'i {,,,,3; 2 4 Bki toan 1: Dien tich hinh phang gio-i han Dien tich h i n h phang H : S = Vxdx + J^2 - x + Vx jdx Cho ham so y = f (x) lien tuc tren [a;b]. Khi do dien tich S ciia hinh phang (D) 0 2 gio-i han bai: Do thj ham so y = f ( x ) ; true Ox : ( y = 0 ) va hai du-crng thang 2 ^7x3 + n ^ 2/3 ^ 4V2 (16 2 4V2^ 10 .3 0 2x 2 + 3-Vx^ = +[ 3 3 ," 3 x = a;x = b la: S= | f ( x ) d x . 2 ' 2. Phu-o-ng t r i n h hoanh do giao diem: 137
Cty TNHH MTV DVVH Khang Vigt cap t6"c giai 10 chuySn dg 10 die'm thi mOn Todn - NguySn Phu Khanh x=0 x=0 A = (m + l f - 4 m > 0 (e + l ) x = ( l + e'')x x(e''-e) = 0 e''=e x=l ' in + 1 > 0 0 < m A 1 1 m>o • •••'^ : . , Dien tich hinh phang H : S = J ( e + l ) x - ( l + e ' ' ) x d x = J x ( e - e ' ' ) d x Vo-i 0 < m : ? i l t h i phu-cng t r i n h (2) c6 2 nghiem la t = l , t = m , v i m > l nen Vai V x € [ 0 ; l ] , t a l u 6 n c 6 : x(e-e'')>0 4 nghiem phan biet cua ( l ) theo thir tir tang la: - \ / m " , - 1 , 1 , VnT Theo bai toan, ta c6: Vay, S = dx 1 Vm 0 ^Hl ^ ^ " 2 j ^ ^ - ( m + l ) x ^ + m dx = x'* - ( m + l ) x ^ +m|dx 0 U = X du = dx Dat / \. dv = ( e - e ' ' jdx V = ex - e o x ' * - ( m + l ) x ^ + m dx = - x'^ - ( m + l ) x ^ + m dx 1 1 1 2 S= e ^ j d x ^ - ex e -(-1) ^^-13. 2 — e 2 ^ - ( m + 1) x + m dx = 0 ^ - ( m + l ) — + mx 0 = 0 ln8 0 0 •(! ' + ldx. m m+l ^ „ ^ In3 - + l = 0m = 5 2t 5 3 Dat t = V ? T l < = > t ^ = 6 " + l = > e ' ' = t ^ - l = > e ' ' d x = 2tdt hay dx = dt Vay, m = 5 thoa bai toan. t^-l 2. Do thj ham so cat Ox tai 4 diem phan biet x''^ - ^m^ + 2Jx^ + m^ + 1 = 0 (*) ^ 2t2 t-1 Khido: S= [-F-dt- f 2 + - ^ dt = 2t + ln = 2 + ln hay ^x^ - 1 j^x^ - m ^ - 1 j = 0 c6 4 nghiem phan biet, tire m^O . .^t^-l A t ^ - l j t+1 .2, Vi du 2. Vai m^O t h i phu-ang t r i n h (*) c6 4 nghiem phan biet ± 1 ; ± Vm^ + 1 1. Cho ham so y = x ' ^ - ( m + l ) x 2 + m c6 do t h i ( C ^ ) . X a c d i n h m > l de do thi Dien tieh phan hinh phang giai han bcci (C^) v a i true hoanh phan phia tren (C^) cat true Ox tai 4 diem phan biet sao cho hinh phang giai han bai true hoanh la: (C^) va true Ox c6 dien tich phan phia tren true Ox bang dien tieh phan 1 96 20m^ + 16 96 S = 2 j x ' * - ( m ^ + 2 ) x ^ + m^ + l dx = phia du'o-i true Ox . 15 2. T i m cac gia t r i tham so m e R sao cho: y = x"*^ - |m^ + 2jx^ + m^ + 1 , c6 do B^i to^n 2: The tich vat the tron xoay thj (C^) cat true hoanh tai 4 diem phan biet sao cho h i n h phang giai han Tinh the tich vat the t r o n xoay khi quay mien D du-ac giai han b a i cac du-ang 96 bcci (C^) v a i true hoanh phan phia tren Ox c6 dien tich bang y = f ( x ) ; y = 0;x = a;x = b quanh true Ox . 15 Lcri giSi Thiet dien eiia khoi t r o n xoay cat bcci mat phang vuong goc v a i Ox tai diem c6 b X 1. Do t h i ham so cat Ox tai 4 diem phan biet o x ' ^ - ( m + l ) x ^ + m = 0 ( l ) c6 4 hoanh do bang x la mot hinh t r o n c6 ban nghiem phan biet t ^ - ( m + l ) t + m = 0 (2) c6 2 nghiem du-ang phan biet ki'nh R=f(x) nen dien tich thiet dien bang 139
cap tOc ylal 10 chuyen d6 10 diem thi mOn Toan - NguySn Phu Khinh Cty TNHH MTV DVVH Khang Vift S(x) = TTR^ = Tif^ ( x ) . Vay the ti'ch khoi t r o n xoay du-gc t i n h theo cong thu-c: Dat t = t a n x = > d t = dx b b cos^x V = Js(x)dx = 7 t | f ^ ( x ) d x . p 1 Khi do l2 = J(l + t ^ j d t = t + - t ^ + C = tanx + i t a n ^ x + C. Vi du. Cho hinh phang H gioi han boi cac diro-ng : y = x l n x , y = 0, x = e. Tinh 7t _ X ' -2d the tich cua khoi t r o n xoay tao thanh khi quay hinh H quanh true Ox . 4 2) I3 = J — l - ^ X = j - Lo-i giki • " l + smx •' 1^71 X^ J 2 2cos' 2 cos Phu-ong t r i n h hoanh do giao diem: x i n x = 0 = > x = l . v4 2, U 2) - U x) x^ (n x' Vay, V o , = 7 t ] ( x l n x f d x = T t ] x 2 l n 2 x d x = 7cIi * ^ ^'"'"^ ' 4 tan ^4 2) d = -tan 14 2, 2; 2, Tinh nguyen ham 2inx_, 1 *• du = dx . at: I Dat: |'u = ln2 0, •l = j ^ d- d xx I, =I- cosx dx , I3 = (•V21nx J +3. dx cosxsm x sin^x-5sinx + 6 dv = x^dx V = X dx = — Hiro-ng dan giki cosx . I1 = -dx 1 AH i. 1^ X 1 2 'cosxsin'x - ^ l - s i n ^ x j ssin^x i — In X — fx^lnxdx^ I n v a i 1,= fx^lnxdx lU>' 1 1 1 Dat t = sinx => dt = cosxdx dx du = u = lnx h = dt '0^ Dat: (i-t^jt^ ^(i-t'y Kt' t'-i) dv = x dx V = - f1 1 1, dt- dt = (lnt-l-lnt+l )+C ,t-l t+1 t 2^ ^ -Inx "x3" (e' l] 2e^+l sinx-1 1 1 t-1 3 9 3 9 9 In +c. V 1 V y t 2 t+1 sinx 2 sinx + 1 2 2e^+l (5e^-2) cosx -dx I Vay, Vox = ^ (dvtt). sin^x-5sinx + 6 3 3 9 "^7 Dat t = s i n x = > d t = cosx.dx. t...«!. B^i tap t y luyen f i r 1 _ f( (t t- -22) -) (- t( -t 3- 3 )^^ ' 1. Tinh nguyen ham ^ J ^ ^ i ^ J(t-2)(t-3)'^*=J{t-2)(t-3) = tan^xdx dx I3 = f — 1 — d x .1 -v, "•'• • " l + sinx -dt- —-^dt = ln|t-3|-ln|t-2| + C t-3 Hxr&ng dan giai V21nx + 3 t-3 sinx-3 dx = l n + C = in + C. = Jtan^xdx = J — ^ — 1 d x = — ^ d x - J d x = t a n x - x + C. >3 = J t-2 sinx-2 s 'it't ^^^^ ^ cos X" 2 1 1 f 1 f/ 9 \ Dat t = 21nx + 3 = > d t = - d x = > - d t = - d x . '2=1—T~^^= 1 + tan^x . dx x 2 x cos X ^ ' ' cos^x '"'^ el r 12 r ^ tv/t ^ {21nx + 3)x/21nx + 3 Khi do 13= f i V t d t = - . - t V t + C = ^ + C = -i L +c 140 ^ J2 23 3 3 141
Cty TNHH MTV DVVH Khang Vi§t cap ta'c gi^ii 10 chuyfin dg 10 die'm thi man Join - NguySn Phu Khanh 3. Tinh tich phan: Doi bien: x = 2 = > t = 3 , x = 2^5 => t = 5 1 4x-2 ^-x^dx ^. ^. rlt- 1^ t-2 1, 15 a. 1= dx b. J = Khi do: I = = —In — J(x2 + l)(x + 2) x^-9 t2_4)t 3Jt2-4 4JU-2 t+2/ 4 t+2 4 7 0 Hiro-ng dSn gi^i c, Dat t = l + x =>dt = 2xdx. 4x-2 _ A ^ Bx + C ,.2/ x^(A + B) + x(2B + C) + 2C + A Doi can: x = l = > t = 2, x = N/^=>t = 4. a. Ta c6: (x + 2)(x2 + l ) x+2 x^ + l (x + 2)(x^ + l ) t' , dt l1.t-(t-l), iV 1 1^ Khidol3= p ^= -^dt=- dt A + B= 0 A = -2 2Jt(t-l) 2J t ( t - l ) 2JU-I t j Dong nhat thirc 2 ve, ta diro-c: 2B + C = 4 B=2 t-1 2C + A = 0 C=0 = -(ln|t-l|-ln|t|) = - l n ln--ln- 2 2 4 2 2 2 4x-2 J V 2 2x I :'(;iI) Si Vay, 1 = -dx= +- dx o ( x 2 + l ) ( x + 2) oV ^ + ^ ^^^ + 1 272 3/ 3 2^/2 3 - T - l 2^ r V x - x + 2011X 2011 d. U = dx= I —dx+ j dx = M + N 1 4 21n x + 2 + l n x ^ + 1 - - 2 1 n 3 + In2 + I n 2 - l n l = l n - 0 9 b. Dat t = x^ = > d t = 3 x ^ d x M= j 3 dx.Dat t = 3|-l--l = : i > t ^ = - l - i ^ 3 t 2 d t : = - A d x i , 1 J . ^ 1 -1 f _ 1f _ J l_ \%{(tt + 33))--{ (tt- -33) ^ ^ ^ 1 V 1 dt 1 1 dt ^ ^ 3 J t 2 - 9 = 3 J ( t - 3 ) ( t - f 3 ) = 1 8 J ( t - 3 ) ( t + 3) 18 t-3 t+3j Doi can: x = l o t = 0, x = 2V2 => t = - t-3 ^ 1, 1 = — ( l n t - 3 l - l n t + 3)^ - l l n ^ ^n--lnl —In- 18^ ' 0 18 t+3 18 2 18 2 2V2 3 | - 2 - l 3 2 2 1 ^ S. Tinh tich phan: Khi do M = 3 o 2 J 128 3 „ J3 1 ^ ^ 0 x-3 xdx dx t^ = x^ + 5 =>xdx = tdt 2t^ j tj 2
cap tOc giai 10 chuyen ci6 10 diem thi mOn Toin - NguySn Phil Khanh Cty TNHH MTV DVVH Khang Vi§t 6. Tinh ti'ch phan: l t 4 _ ^ 2 ^ t 4 _ ^ 2 31n2 273 Dat t = N/x + l , k h i d 6 I = - 2 f — dt + 2 f - - -dt = A + B dx dx a. I i = J b.l2= j '3=1 dx e+3 t2+3 0 Js xVx^ +4 -1 x+4 ^t^-t^ Xet: A = 2 dt = 2 -4t + 24 dt V l + 31nxlnx t2+3 t2+3 d. L dx e. lr = dx 0 Vx^ - x + 1 1 V-t^ Hu-d-ng dan giSi •% -ft B = -2 dt = - 2 t^-4+ dt = - 2 --4t -24 -dt t2+3 t2+3 X X 0 0 t'+3 a. Dat u = e3 =:>3du = e 3 d x Dat t = v ' 3 t a n v = > d t = 7 3 ( l + t a n ^ v j d v D 6 i c a n : x = 0 = > u = l , x = 31n2=>u = 2 ,S 2/ r 1 >/3(l + tan^v) 1 3du Khi do: I = du Khi ay, f , - d t = f ,^ , dv = -p-v + C l \ ( u + 2)' i'l4u 4(u + 2) 2(u + 2) 't^+S 3 ( l + tan2v) V3 2 1. 44-18^y3-2^/37t i l n u -—In u + 2 Vay, 1 = 4 4 2(u-f2)J 4 .2. 8 C ^i f ••'VJ!. d. Dat t = V l + 31nx = ^ t 2 = l + 31nx=>2tdt = — < » — = — b. C^ch 1: Dat t = Vx^+4 =^ x^ = - 4,xdx = t d t •!'«'^- 1' x x 3 D6ican:x = 7 5 = > t = 3 , x = 2 v ^ = > t - 4 . Doi can: x = 0 = > t = l , x = l = > t = 0. Khi do: xdx tdt dt Khi do: I = 1 , 0... ."O'^^'+f^i. V . ^5 1_1^_2_ i(t'-4y J(t + 2 ) ( t - 2 ) I - jx^ Vl - x^xdx = j ( l - 1 ^ ) t ( - t d t ) = j(t2 - 1 ' ^ )dt = 3 5 " 15 (t + 2 ) - ( t - 2 ) ^ ^ ^ l Y 1 1 '4 nt dt 2(x2-x)(2x-l) r r — ^:i7 \ (t + 2 ) ( t - 2 ) 4 I. I5 = J^ dx.Dat t = Vx2-x + l = ^ l 5 = 2 | ( t 2 - l ) d t = - U-2 t + 2j 4 0 7x^-x + l t-2 = - ( l n t - 2 - I n t + 2) In - I n — I n - = —In- 7. Tinh tich phan: •t + 2 4l 3 5 4 3 1 1 sin2x , _ ^fsin2x + sinx , Cach 2. Dat t - => dt = — r - d x a. rdx b. I 2 = . =-dX 0 (2 + sinx) Q V l + 3cosx . i . . dt dx 1 = -.Dat u = ln 2t + V 4 t ^ + l dx 4t^ + l h = l- J 3 d. I4 = 3cosx-4sinx + 5 T 4tVsin x.cos X CdchS.Dat x = 2tanv 7t ^fXSinx + ( x + l ) c o s x , (x + cosx)dx f. J = 5^ dx 'r - x V x + 1 J ^fxVx+T e. 1 = x s i n x + cosx c. De thay: dx + -dx x + 4 x + 4 4cos^x + 3sin^x -1
cap tO'c gicii 10 chuygn ai 10 digm thi m6n Toan - NguySn Phu Kh^nh Cty TNHH MTV DWH Khang Vi§t x/3 x , dx 1 l + f^ sinx dx £)at t = t a n - = > d t = = - l + tan^^ dx: dx g- K = - 2 2dt 2t 1-t Hu'O'ng dcin giki • dx = ,sinx = ^ , cosx = 1 + t^ 2• 1 + t^ 1 +t , sinZx , „^ sinxcosx Doi can: x = 0 = ^ t = 0, x = - = > t = l i. Ii = J ^x =2 dx. o(2 + s i n x ) o(2 + s i n x ) 2dt Dat t = 2 + s i n x = > d t =cosxdx l + t2 dt dt 1 1= 2^ t-2 = -e. 1-t 4_2^^5 o 3 - 3 f ^ - 8 t + 5 + 5t'^ o(t-2) Doi bien: x = 0 = > t = 2, x = — =>t = 3 I 2 2 1 + t^ 1 + t^ 2^ o. 3 2 Khido; 1 = 2 ^ - 2 d t = 2 ' ^ l 2 dt = 2 lnt + - = 21n t 2 3 U x + cosx)dx 2 jjjj^ 2 cojxdx e. 1 = + = A+B J 2tdt 4-sin^x •;4-sin^x •;4-sin^x b. Dat t = N/1 + 3COSX => 2tdt = - 3 s i n x d x =>sinxdx = - "2 "2 Doi can: x = 0 = > t = 2, x = - = > t = l xdx xdx 2. xdx 2 + Ti'nh A= -+ ;„2, ''^4-sin^x ^^4-sin'^x o 4 - s i. :n„' 2^,x ^(•2sinxcosx + sinx^^_^l-(2cosx + l ) s i n x d x Vay,l2- \ / l + 3cosx V l + 3cosx xdx Trong Datx =-t.Dgthay j '^'^^ f 2tdt^ ;4-sin^x' 4-sin^x n4-sin^x 2 +1 3 I 3 J 2 'it' tl §• "2 2 I 3 , ~3 9 3] p u y ra A = 0 16 2 f2 1 34 cos xdx dt 2+t — + - - + - ^'Tinh B= f - ^ ' ^ ^ ' i ^ . D a t t = s i n x = . B = 19 3, 27 iln = iln3 9 3 .4 - i i n X 4-t^ 4 2-t 2 -1 -1 c. l3 = dx= I , ——dx Vay: l . i l n 3 7r^/tan^x cos x 7t 71 ^ V cos-^ X r x s i n x + (x + 1) cosx J 4 4 r, r xcosx . Dat t = t a n x = > d t = dx dx= dx+ dx = J i + J 2 x s i n x + cosx cos^x 0 0 •' •'xsinx + cosx Doi can: x = —=>t = l , x = —=>t = A/3 4 4 3 T r o n g d o : Ji = j d x = x | Q 4 : = ^ ; xcosx J2 = -dx J3 3 0 0 xsinx + cosx K h i d o 1= J t 4dt = 4t4 =4' ^ - l ) = 4(^-l) at t = x s i n x + c o s x = > d t = xcosxdx. 146
cty TNHH MTV DVVH Khang Vi§t cap te'c giai 10 chuyen dg 10 die'm thi m6n Toan - NguySn Phu Kh^nh Hu-ang dan giai r2dx' De t h a y , A = e''"dx= [xe''dx + = I + ln = ln2 + I dt , •1 = ln Khi do I2 = —=ln t 14 ; u=x du = dx Vo-i I = xe^dx . D a t dt = -3x^dx 2 Doican: X = - ! = : > t = l , x = 0 = > t = 0 r t , iV 2 1 ^ 1 Dat t = tanx, K = —5 dt = - --^-^--7 dt = - l n 3 — l n 2 rO 1 1 1 1 j2t2+5t +2 3^^1 + 2 2t + l B=f(-,).e' = — [ t . e ' d t = — U v a i 1= [ t e M t *: . 3 3-« 3 •« s u =t • du = dt . 1 -i . 1 1 dx Dat ^ l = e'.t - f e M t = e-e' =l.Vay, B= - - h. L = dv = eMt v = et 0 Jo 0 3 e ee e c. C= Jx^{lnjj + x ) d x = [ x ^ l n x d x + [ x ^ d x 1 ^_J_dt = - ^ i = . d x dt=^^.dx= / ..dx t ^ - l xTl-x^ du = —dx u = lnx x Ii = x Inxdx.Dat 1 dv = x^dx 2 1 ^ v =- L= -dt = 1 dt=.i f^ t+lj lt-1 x^lnx ^fx^ , x^lnx X 5eSl - —dx = 36 36 1 /6 6 • L = - ( l n | t - l | - l n | t + l|) I - 1 ' In t-1 2 _1 , 1 In—In , 7- e 7 e^-1 , „ ^ 5e^ + l e^-1 t + li ^ " 2 I , = fx^dx = — 2 3 2+ ^ .Vay, C = + 7 36 7 1. =-ln 2 , + S^ , = ^-1l n, - 7 + 4^/3 .3x 0 x , 0 -f ^/T^e3x ,2 3(2-73) 2 -
Ca'p tO'c giai 10 chuy6n 6i 10 djgm thi mOn Toan - NguySn Phu KhAnh Cty TNHH MTV D W H Khang Vi§t 2x 1-e -dx = -1 h = -ln3 ,3x dx = -ln3 ,8x -ln3 ,2x ,2x HINH HQC KH6NG GIAN Datt = - L - l ^ d t = -^dx De hoc tot mon toan hinh, cac e m can: 1. Ve hinh chinh xdc vd can than; 80 2. Nam vitng cdc dinh ly can bin hinh hoc; U= f - i . ^ d t - 6 . Vay, D = •91n3 3. Sdng tgo, khong suy nghitheo mdy mdc, loi mon; 2x + l . 4. Lam nhieu bai tap detich lay kinh nghiem. du = -^ dx u = ln(x^ + x + l ) _ x +x+1 Dang 1: The tich khoi da dien e. Dat 2 dv = xdx V =• The ti'ch cua khoi chop c6 dien tich day B va chieu cao h la: V = ^ B h . _ i f i ^ d x = iln3-lj The tich ciia khoi lang t r u c6 dien tich day B va chieu cao h la: V = Bh .E = — l n ( x ^ + x + l ) 0 2Jx^.x + l 2 2 The tich cua khoi hop c6 dien tich day B va chieu cao h la: V = Bh . V The tich khoi hop chir n h a t : V - abc. ^ 2x2+x2 1 2x + l Ta CO => J = dx = 2x-l—. dx 2 x^ + x + 1 2 1 The tich khoi lap phu'ang: V = a^ . ,5 X^ + X + 1 +X+ TI so the tich: Neu A',B',C' thuoc cac canh SA,SB,SC cua hinh chop SBC t h i : x^+x + l ] -dx = — l n 2 - Vs.A'Br _ SA' SB- SC- V / 0 2i^ l Y 3 6^3 •
cap te'c giai 10 chuySn dg 10 digm thi man Toan - Nguy§n Phu Khanh Cty TNHH MTV DVVH Khang Vi0 3a a^y5 Taco: ^ =>BC ± (SAB) Tu'I' ( 1 ) va ( 2 ) , s u y ra: — ^ = — p = = < = > V x +a = [ B C I S A ^ ' — => x = 3 2^fa- + x'- 2 2 ..i.i,t => B C l A M => A M 1 (SBC) => A M 1 S C V a y t h e t i c h k h o i c h o p S.ABCD l a : V = - S H . A B . A D = - . — . a . x = ^ ^ ^ ^ T u - a n g tu-: A N 1 ( S C D ) => A N 1 S C , t i r d o s u y r a : S C 1 ( A M N ) 3 3_J 36 N e n A P l a d i r c r n g cao cua h i n h c h o p S.AMPN " v i d u 2 . Cho h i n h chop S.ABCD c6 day ABCD la h i n h t h o i canh a , SA = SB = SC ^ T i n h SD t h e o a de k h o i c h o p S.ABCD c6 t h e t i c h I a n n h a t . S u y r a : Vs.AMPN = ^ A P . S A M P N Lo-i g i a i >m.': J A p d u n g h e t h i r c lu'o-ng t r o n g t a r n giac v u o n g SAC, t a c 6 : Ggi H la h i n h c h i e u ciia S l e n mat d a y , t a s u y r a H la t a m SP SP.SC SA^ _3^gp_3g^_3aVl4 „J du-ang t r o n n g o a i t i e p t a m giac SC~ SC^ " S A ^ + A C ^ 7 7 1 4 A B C nen H thuoc B D . SA.AB _ aVl5 T r o n g t a r n giac v u o n g S A B , t a c6: A M = BDIAC SB ~ 5 Mat khac •ACl(SBD) SH 1 AC ^ ^ MP SP SP.BC 3a735 DoASBC ^ ASPM: . MP = =:>0 = B D n A C la h i n h chieu BC SB SB 35 cua A l e n m a t p h a n g ( S B D ) , ^ aVn S u y r a : S^MPN = 2 S M M P = A M . M P = m a AS = A B = A D = a => 0 l a t a m du-ang t r o n n g o a i t i e p t a m giac SBD 35 ASBD v u o n g t a i S . D a t SD x 1 3aVl4 a^^/2T_3a^^/6 Vay V< CD e n 1 j'j ! S.AMPN 3" 1 4 • 35 ~ 70 Ta c6: S H . B D = SB.SD => SH = va SAPrn = - A C . B D 2. Ggi H l a t r u n g d i e m cua A B => S H 1 A B N§n Vs ABCD = i . ^ ^ . ^ A C B D = iAB.SD.OA -' ' Ma ( S A B ) 1 ( A B C D ) = > S H 1 ( A B C D ) VJ.ABCD = - S H . S A B C D b.ABCu 3 B D 2 3 Ve H K I A C ^ AC1(SHK)=>S1CD1(SHE)=>SEH la goc X J u^.n'^ ' 2 ^ x ^ + 3 a ' ^ - x ' ^ 3a'' A p d u n g b d t C o - s i , t a c o : xV3a - X < = giQa h a i m a t p h a n g ( S C D ) v a m a t d a y n e n S'EH = 30^ 1 3a2 a3 41 Dat A B = X , t r o n g t a m giac S H E , t a c6: Suy r a : V j ABCD - ^ - ^ - " ^ ^ ~^' ^ ^ " ^ t h i r c x a y r a x'^ = 3a^ - x^ x = ^ SH = H E . t a n 3 0 ° = ^ (1) aV6 Vay VsABCD lo-n nhat < = > S D = ax Ta CO A A K H ^ A A B C ^ — = — = > K H = BC A C Dang 2 : T h e tich khoi lang t r u T r o n g t a m giac SHK t a c o : S H - H K t a n 6 0 ° = — ^ ^ = = = (2) Sir d u n g c o n g thu-c t h e t i c h •V i;! A • T h e t i c h k h o i l a n g t r u : V = B.h T h e t i c h k h o i h o p c h i r n h a t c6 cac c a n h a , b , c : V = abc 1 o
Cty TNHH MTV DVVH Khang Vigt cap ta'c g i i i 10 chuy6n 66 10 digm thi mOn Toan - Nguyin Phu Khanh Vi du 2. - The tich khoi lap phu-ong canh a : V = a^ 1 . Cho lang t r u dirng A B C D . A ' B ' C ' D ' c6 day A B C D la hinh thoi canh 2 a . Mat De ti'nh the ti'ch ciia khoi lang t r u AiA2...An.AiA2...An ta can di ti'nh chieu cao phang ( B ' A C ) tao v a i day mot goc 3 0 ° , khoang each t i r B den mat phang ciia lang t r u va dien tich day. Cac tinh chat ciia lang t r u : a. Hinh lang tru MN.DB' = 0 bang - . Tinh the tich khoi tii-dien A C B ' D ' . ' ' 2 !.. Cac canh ben ciia hinh lang t r u song song va bang nhau; t, i • 2. Cho khoi lang t r u ABC.A'B'C c6 do dai canh ben bang 2a, day ABC la tam L Cac mat ben cua hinh lang t r u la cac hinh binh hanh; giac vuong tai A , AB = a,AC = aVs va hinh chieu vuong goc ciia dinh A' tren . Hai day cua hinh lang t r u la hai da giac bang nhau va nam t r o n g hai mat phang mat phang (ABC) la trung diem ciia canh BC. Tinh theo a the tich khoi song song v o l nhau. chop A'.ABC va tinh cosin ciia goc giu-a hai du-ong t h i n g A A ' va B ' C . - Lang t r u c6 cac canh ben vuong goc hai day dugc gpi la lang t r u dii-ng. * Cac canh ben ciia lang t r u d u n g chinh la d u a n g cao cua no. Lo-i giki * Cac mat ben ciia lang t r u dii-ng la cac hinh chu' nhat. 1 . Goi 0 la giao cua hai du-ong cheo AC va B D , ta c6 AC 1 ( B ' O B ) = > B ' O B = 30° Lang t r u dirng c6 day la da giac deu du'gc goi la lang t r u deu. Goi H la hinh chieu cua B len B'O, suy ra Cac mat ben cua lang t r u deu la cac hinh chir nhat bang nhau. B H = d(B,(B'AC)) = d(B,(D'AC)) = | ^ . - - ' ^ 1 . ' b. Hinh Hop : La hinh lang t r u c6 day la hinh binh hanh 1 C'..,-.--^ - Hinh hop dirng c6 cac canh ben vuong goc v 6 i day. • / BH • D o d o : B0 = - // - Hinh hop d u n g c6 day la hinh chQ- nhat du'oc goi la hinh hop chir nhat. -=a •• / // • sin30" - Hinh hop chCr nhat c6 ba kich thu'oc bang nhau du-o-c goi la hinh lap phu-o-ng. = > O C = V B C ^ - B O ^ =aV3, - Du-ang cheo ciia hinh hop chir nhat CO ba kich thu'O'c a,b,c la d = Va + b + c - Du-ong cheo ciia hinh lap phu-ong canh a la d = a>/5. BB'=:BOtan30° = — 3 Vi d u 1 . Cho lang t r u deu ABC.A'B'C. Biet mat phang (A'BC) tao voi mat 1 2 phang ( A ' B ' C ) mot goc 60° va khoang each tir A den mat phang (A'BC) = > SABCD = - A C . B D = 2B0.C0 = 2aW3 i 3a bang — . Tinh the tich ciia khoi lang t r u da cho. Nen VABCD.A'BX'D' = BB'.SABCD -^.2a^S=2a^ Lo-igiai ^. 1 . Goi M l a t r u n g d i e m B C s u y r a B C l ( A ' M A ) Mat khac VB^ABC = ^ D ' A C D = ^CE'C'D' = ^ A A ' B ' C ^ - ^ ^ A B C D . A ' B ' C D ' nen A m = (/VBC^ABC) = ( A ' ^ C ) , ( A ' ? C ' ) - 6 0 ° 1 2a^ Nen suy ra V A C B ' D ' = 3 VABCD.A'B'C'D' = — • Goi H la hinh chieu cua A len A ' M 2. • Tinh the tich khoi chop A'.ABC. r:>AH = d ( A , ( A ' B C ) ) = y Goi H la trung diem BC AH r T h e o b a i ra ta c6 A ' H l ( A B C ) Trong tarn giac vuong A HM ta c6: AM = ?r = ^^-^ sin60° va AH = i B C = i V A C ^ + AB2 =\4^-v?,^ =a. Trong tam giac vuong A A ' M ta c6: AA' = AMtan60° =3a 2 2 2 ,„ ... AB^y3 2AM „ - 2 Do do A'H^ = A'A^ - AH^ = 33^ ^ A'H = aVs. Vi AM = => AB = — y = - = 2a = > S^ABC = 3 Vay the tich khoi lang tru la: V = AA'.S^BC = 3a^ . 1 Vay the tich cua khoi chop A'.ABC la VA-.ABC = r A'H.S^ABC = ^ •
p tflc giai 10 chuySn d6 10 die'm thi men Toan - Nguyin Phu Khanh CtyTNHH MTV DVVH Khang Vi^t Ti'nh goc giu-a hai dirang t h i n g AA' va B ' C . Do AA' // BB', B'C II BC nen goc . r ; . D a n g 3 : T i so t h e t i c h giu'a hai du-o-ng t h i n g AA' va B'C cung la goc giu'a hai 1. T i so ve d i e n t i c h : Cho tam giac ABC ^ d u a n g t h i n g BB' va BC. • Lay cac diem M, N Ian Ap dung Pitago cho tarn giac lu-ot tren cac du-ang M, A ' B ' H , t a c6: t h i n g AB.AC t h i H B ' = : V A ' B ' ^ + A ' H 2 =2a. ^ SAMN^AM AN Suyra tarn giac B ' B H can tai B'. SABC AB'AC' Do do (p = B ' B H la goc giu-a hai du'ong t h i n g AA' va B ' C • Neu diem M nam t r o n g tam giac ABC, AM cat BC tai A' t h i : BH a 1 ,1 " i Obi.;? t i O-SOJ', Vay cos(p = BB' 2.2a 4 SABC MA M\u 3. Cho hinh hop chu' nhat A B C D . A ' B ' C ' D ' , goc giij-a du'ong cheo A C va I Neu diem AD nam tren canh mat day ( A B C D ) b i n g 30° va A C = a, A C B = cp . Tinh the tich khoi hop chir BC ciia tam giac ABC, BM S, CMt h i ^BAM CAM nhat A B C D . A ' B ' C D ' theo a va 9 . Gia su- a khong doi, t i m 9 de the tich ^BAC ^CAB khoi hop 16-n nhat. Neu G la t r p n g tam tam Lo-i giSi giac ABC, t h i ^ 0 a C C 1 (ABCD) =^ CAC = (AC',(ABCD)) = 30°; AC = ACcos30° = SGBC - S G C A - SCAB - ^^ABC De thay tam giac ABC v u o n g t a i B, s u y r a AB = A C s i n 9 = a s i n 9 Nen BC = \ / A C 2 - A B 2 ¥ Neu M nam tren du-ong t r u n g binh u n g v a i canh B C t h i : ^ =i . ^ABC 2 ^aV3^' 2 . 2 a-J3-4sin^9 -a'^sm'^9=—^ V 2 , Neu M nam t r e n du-ang t h i n g di qua A va song song vo-i BC t h i : '^^^ = 1 SABC 2 2 T i so ve k h o S n g each The tich khoi hop chu- nhat Du-6-ng t h i n g A B cat mat p h i n g s-V., ABCD.A'B'C'D' l a : d(A,(P)) AM (P) a d i ^ m M t h i : \ ll=~. B ^ ' d(B,(p)) BM V . CC.CB.AB = i W 3 - 4 s i n ^ 9 ^ ^ Du-o-ng t h i n g A song song vo-i 9 mot mat p h I n g (P) t h i k h o i n g \ 4sin 9 + 3 - 4 s i n 9 K H T a c o : 2sin9^/3-4sin^9 < = —. Nen suy ra V < each tu- moi diem thuoc du-6-ng 2 4 ^ 32 t h i n g A den mat p h I n g (P) bang nhau. Vay maxV = — k h i 2sin9 = V3-4sin^9 1""^ 9 = acsin 1 - 32 • 1 . •,, ' V0 T i so ve thd' t i c h : Cho khoi chop tam giac S.ABC. Tren ba du-cmg t h i n g SA,SB,SC Ian lu-o-t lay ba diem A',B',C b a t k y . 56 157
Cty TNHH MTV DWH Khang Vijt cap tOc giii 10 chuyen 6& 10 die'm thi m6n Toan - Nguy§n Phii Khanh Vs.AHM _ SH SM _ 9 _ 9 3a^ ^S.AHM ~ 32 ^•'^^C Vs.ABC SB'SC 32 32 Vs.AKM _ S K SM _ 9 _9 _3a^V _ 333a^V3 3a^V3 ; Vay "S.AHM ''S.AKM ~ ' V- Chii y Ta c6 the tinh inh the tich khoi chop S.AHMK theo each sau VJAHMK jS'^AHMK- 3 Vi du 2. 1. Cho hinh chop S.ABCD c6 day la hinh vuong canh a, ASC = 90°, SAlap v a i day goc a (0° < a < 9 0 ° ) va mat phang (SAC) vuong goc v 6 i mat phang (ABC). Tinh khoang each t i r mp(SCD) den (SBC). 2. Cho hinh hop du-ng ABCD.A'B'C'D' c6 day la hinh thoi canh a, tam giac ABD la tam giac deu. Goi M , N Ian lu-gt la trung diem ciia cac canh BC.C'D'. Tinh khoang each t i r D den mat phang ( A M N ) biet rang M N 1 B'D. Vi du 1. Cho hinh chop S.ABCD c6 day ABCD la hinh chu- nhat AB = a , AD = 2a. Lo-i giSi Canh ben SA vuong goc v o i day va SA = a^^. Gpi H,K Ian l u o t la hinh chieu cua A len SB,SD; M la giao diem ciia SC v a i ( A H K ) . Chu-ng m i n h rang T a c o VA.SBC = | d ( A , ( S B C ) ) 5 B c s " e n d ( A , ( S B C ) ) = ^ ^ . ^ ^BCS S C 1 AM va tinh the tich khoi chop S.AHMK. Vi ( S A C ) 1 ( A B C ) n e n g p i H la hinh chieu ciia S tren canh A C t h i S H l ( A B C ) , L a i giii hinh chieu ciia S A tren mat phang ( A B C D ) la A H nen ( S A , ( A B c 3 ) ) = S'AH = a. Ta CO A S C = 90° nen S A = AC.cosa = %^.a.cosa, s DO do SH = S A . s i n a = \/2.a.cosasina Nen Vs.ABc = - S H . S A B C = — - a ^ c o s a s i n a . 6 6 Gpi K la t r u n g diem ciia SC t h i O K la du-o-ng t r u n g binh ciia tam giac S A C Mat khac AH 1 SB nen suy ra AH 1 (SBC) ^ AH 1 SC nen 0 K | | S A = ^ 0 K 1 S C . B D 1 ( S A C ) = > B D 1 SC nen B K 1 SC. Hoan toan tu-cng t u ta chu-ng minh du-o-c A K 1 SC. Ta CO SC = A C . s i n a = \/^.a.sina Tir do, suy ra SC 1 ( A H K ) nen S C I AM . Ap dung he thu-c lu-ang t r o n g tam giac vuong ta c6: nen B K = V B C ^ - C K ^ = a ^ ^ - ^ ^ => S^cs = ^a^sina.V2 - s i n ^ a . SH SA^ ^ SA^ = ^ ^ • SK SA^ _ 3, SM ^ SA^ _ 3 o V2 3 S B ~ S B 2 ~ S A 2 + A B 2 " 4 a 2 ~ 4 ' SD SA^+AD^ 7 ' SC SA^ + AC^ 8 3.—.a .cosasina. n: Vay, d [ A , ( S B C ) Su- dung cong thu-c t i so the tich ta c6 du-gc: 6 v2.a.cosa -a^sina.V2-sin^a V2-sin^a 2
Cty TNHH MTV DVVH Khang Vigt cap tec giai 10 chuvSn a& 10 dle'm thi m6n Toan - NguySn Phu Khanh D a n g 4: Mat c a u 2. D a t AA'' = x , A B = y , A D = z. I. Khai niem mat cau. y,, , •/ • q^n I'z^vivi \:xy\.. T a CO t a m giac A B D l a t a m giac d e u n e n : x.y = x.z = 0 | . z P | | | | < : O S 6 0 ° - y . M a t c a u t a m 0 b a n k i n h R ( t a k i h i e u S ( 0 , R ) ) l a t a p h g p cac d i e m M t r o n g D' C khonggianthoa man S(0,R) = {M|OM = R}. _ T a c o D B ' = b D ' + b C + DA = x + y - z . Neu A B la du-ang k i n h m a t cau S ( 0 , R ) t h i v a i m p i d i e m M t h u o c m a t cau ( V i M , N l a t r u n g d i e m c u a BC,C'D' £ — 1 1 • trir A va B ) t h i A M B = 9 0 ° . . . n e n M'N = M C + C C ' + C ' N h a y 1 / ' • \ t / 1 1 ^ 1 / Ngu-ac l a i v a i m o i d i e m M n a m t r o n g khong gian thoa m a n A M B = 90° t h i M N = i AD + C C V i c ' D ' = i z ^ ' / I 1 1 \ d i e m M t h u o c m a t cau du-ang k i n h A B . /I D H\ \ C 1^ I I . V j t r i tu-o-ng d o i c u a m o t d i e m v a i m a t c a u . Vi M N I B ' D n e n MN.bB' = 0 \ C h o m a t c a u S ( 0 , R ) va m o t d i e m A bat k i t r o n g k h o n g gian. ---- M Do d o ( x + y - z ) = 6 A - N e u OA > R t h i A 6- n g o a i m a t cau - N e u OA = R t h i A a t r e n m a t cau f j .D ; t d ^2 -a. - N e u O A < R t h i A a t r o n g m a t cau III. V i t r i tu-o-ng d o i c u a m o t h i n h p h a n g vo-i m a t c a u . 'm 3V,' D . A M N Taco: d(D,(AMN))- Cho m a t cau S ( 0 , R ) v a m o t m a t p h a n g ( P ) b a t k i t r o n g k h o n g g i a n . SAMN Gpi H l a h i n h c h i e u c u a 0 l e n ( P ) . ,,:a. _,„.....„....,.:.„,...„„: 1 c 2 T a t h a y S ^ M D = 2 ^'^•^^D = ^ A B D - - N e u OH > R t h i ( P ) k h o n g cat m a t cau , N e u O H = R t h i ( P ) v a (S) c6 m o t d i e m c h i i n g d u y n h a t la H . Goi H l a t r u n g d i e m cua C D t h i N H l ( A B C D ) , N H = K h i d o t a n o i : ( P ) t i e p x u c v a i m a t cau v a ( P ) g o i la m a t p h a n g t i e p d i e n , H g o i la t i e p d i e m . nen CO V ^ A M N = ^ N . A M D ^ ^ N H ^ A M D = ^ a l N e u OH < R t h i ( P ) cat m a t cau t h e o m o t du-cmg t r o n ( C ) c6 t a m H b a n k i n h Ke H K I A M ta c6 N K I A M . T h e o d j n h l i h a m so c o s i n r = 7 R ^O H ' 7 J7 A M ^ = B A ^ + BC^ -2BA.BC.cosl20° = - a ^ A M= — a . | N e u 0 n a m t r e n ( P ) t h i ( C ) g o i la du-ang t r o n lo-n v a c6 b a n k i n h R . 4 2 IV. V i t r i t u - c n g d o i c u a m o t d u - a n g t h a n g vo-i m a t c a u 3 T a CO S A H M = ^ A B C D " ( ^ A D H + ^ C H M + ^ A B M ) = g ^ A B C D Cho m a t c a u S ( 0 , R ) v a m o t du-ang d b a t k i t r o n g k h o n g g i a n . G o i H la h i n h chieu cua 0 l e n d . •t ' *• • " N e u OH > R t h i d v a m a t cau k h o n g c6 d i e m c h u n g . - N e u OH = R t h i d va m a t cau (S) c6 m o t d i e m c h u n g d u y n h a t la H . K h i d o ta n o i Suy r a S A M N ^ ^ ^ - A M = do do d ( D , ( A M N ) ) = — a . d t i e p xuc vo-i m a t cau v a d g p i la t i e p t u y e n cau m a t cau, H g p i la t i e p d i e m . • N e u OH < R t h i d v a m a t cau CO d u n g h a i d i e m c h u n g . •J22. V a y k h o a n g each t i r d i e m D d e n m a t p h a n g ( A M N ) l a — a . K h i d o t a n o i d c a t m a t cau t a i h a i d i e m p h a n b i e t . ^- M a t c a u n g o a i t i e p v a h i n h c a u n o i t i e p h i n h d a d i e n . A M a t cau ngoai t i e p h i n h da d i e n la m a t cau d i qua t a t ca cac d i n h cua h i n h da d i e n • I W a t cau n o i tiep h i n h da d i e n la m a t cau tiep xuc v o i t a t ca cac m a t cua h i n h da dien.
cap t6'c giai 10 chuygn de 10 die'm thi m6n Toan - NguyJn Phu Khanh Cty TNHH MTV DWH Khang Vi§t Nhan xet. each 1. - Mot da dien c6 mat cau ngoai tiep t h i tat ca cac mat cua da dien deu c6 Ta CO SAH = (SA,(ABC)) = 6 0 ° , du-ang t r o n ngoai tiep. - Neu tam mat cau ngoai tiep cua da dien thuoc mot mat ciia da dien t h i du-o-ng AH = ^ A M = ^ . ^ = ^ t r o n ngoai tiep ciia da dien do la du'ang t r o n Ian. 3 3 2 3 - Khoang each tCr tam mat cau noi tiep cua da dien den cac mat da dien bang ( M la t r u n g diem canh BC ). nhau va bang ban k i n h mat cau noi tiep da dien do. 73a >SH = A H t a n 6 0 ° = — . V 3 = a , VI. Dien tich mat cau va the tich khoi cau. 3 t. Dien tich hinh cau ban kinh R : S = 47iR^. AH 2V3a SA = - 4 •? cos 60,0 The tich khoi cau ban kinh R : V = -nR^. SI SO Do ASIO ASHA Vi du. SH SA 1. Cho tam giac ABC vuong tai B, D A l ( A B C ) , AB = 3a, BC = 4a, AD = 5a. SA.SI _ SA^ _ 2a =>S0 = Chu-ng m i n h rang 4 diem A, B, C, D cCing nam tren 1 mat cau. Xac dinh SH ~ 2 S H ~ T " tam va ban kinh mat cau do. Vay ban k i n h hinh cau ngoai tiep hinh chop la R = — . 2. Cho hinh chop deu S.ABC c6 canh day AB = a , canh ben ho-p vo-i mat day 3 • mot goc bang 6 0 ° . Xac djnh tam va ban kinh hinh cau ngoai tiep h i n h chop Cach 2. Goi D la giao diem ciia AH v a i mat cau, tam 0 thuoc mp (SAD) nen S.ABC. •f du-o-ng t r o n ngoai tiep ASAD la du-o-ng t r o n Iffn. L a i gi^i V3SA ^/3 2yl3a 2a De thay ASAD la tam giac deu nen ban kinh R = 1. * Taco: DA 1 (ABC) DA 1BC va ABIBC suyra BCIBD Vi du 2. * Khi do DAC-DBC^go". Goi 0 la trung 1. Cho hinh chop S.ABCD c6 day ABCD la hinh thoi canh a, BAD = 60° va cac diem CD t h i : OA = OB = OC = OD canh ben SA = SB = SC. Xac dinh tam va tinh ban kinh mat cau ngoai tiep • Vay, A, B, C, D nam tren mat cau tam 0 hinh tii-dien SBCD biet BSD = 90°. CD 2. Cho hinh chop S.ABCD c6 day la hinh thang vuong tai A,D, AB = AD = a,CD = 2a. Canh ben S D l ( A B C D ) va SD = a. Goi E la t r u n g diem la t r u n g diem CD, ban kinh bang — . _ cua DC. Xac djnh tam va tinh ban kinh mat cau ngoai tiep hinh chop S.BCE. Khi do: R = OA = OB = OC = OD OA = icD = i V A D ^ T A C ^ L a i giM hay OA = V25a2 + 1 6 a 2 + 9 a 2 = ^ . Vay, R = ^ 1- Gpi 0 la giao diem hai du-ang cheo ciia hinh thoi ABCD. 2. Gpi H la hinh chieu ciia S len mat phang (ABC), ta c6 H la tam cua AABC. Theo bai ra ta c6, BD = a . Nen SH la true du-ang t r o n ngoai tiep AABC. Trong ASAH d y n g du-ceng trung Ma tam giac ASBD vuong tai S nen tru-c Ix cua canh SA. Goi 0 = Ix n SH. SB = SD = ^ a , S O = - . OeSH OA=OB=OC > 0 la tam mat cau ngoai tiep hinh chop S.ABC . 2 2 OGIX OS = OA Goi H la hinh chieu ciia S tren mat Ban k i n h R = SO. De tinh ban kinh R ta c6 the thirc hien theo hai each sau. phang day t h i H la tam du-ang t r o n ' ngoai tiep tam giac ABD 162 163