Tài liệu Thiết kế bài giảng đại số nâng cao lớp 10 - tập 2

TRAN VINH rturso TAP HAI 10 // -2 -1 0 NHA XUAT BAN HA NOI TRAN VINH THIET KE BAI GIANG j^" [)AI SO NANG CAO - TAP HAI NHA XUAT BAN HA NOI CbirdNq IV BAT DANG THlTC VA BAT PHl/OlNC TRINH PHAN1 I. NOI DUNG Ndi dung chinh cua chuong IV : Bat phuong trinh : Bat ding thiic ; Dai cuong ve bat phuong trinh ; Bat phuong trinh va he bat phuong trinh mdt an; Dau cua nhi thiic bae nhat ; Bat phuong trinh va he bat phuong trinh bae nha't hai dn; Dau cua tam thiic bae hai; Bat phirong trinh bae hai; Mdt sd phuong trinh va bat phuong trinh quy ve bae hai. B a t d a n g thaiti On tap ve khai niem bat dang thiic; Bat dang thuc ve trung binh cdng va trung binh nhan (bat ding thiic Cd - si); Bat dang thiic chiia gia tri tuyet ddi. Ndi dung chinh cua phan nay la xay dung quan he thii tu tren tap hgp sd thuc. Sach trinh bay mdt each het siic co ban, he thdng ve bat ding thiic. Trong phan nay ciing dua ra cac tinh chat, quy tdc ve bat ding thiic, tir dd giiip hgc sinh chiing minh dugc cac bat ding thuc don gian. Dac biet trong phan nay cd dua vao hai bat ding thiic : Bat ding thiic v6 trung binh cdng va trung binh nhan; bat ding thiic cd gia tri tuyet ddi B a t p]»<«*iig t r i n l i V6 mat hinh thiic, bat phuong trinh dugc trinh bay co ban nhu' phuong trinh, nhung chi khac d day da xay dung quan he thii tu tren tap hgp sd thuc. Ndi dung co ban la trinh bay ve bat phuong trinh mdt an ma d do cung dua ra cac quy tic co ban de tim nghiem ciia chiing. Dac biet la ndi dung ve kinh dien la dau cua nhi thiic bae nhat va dau tam thuc bae hai, ap dung chung trong viec 3 giai bat phuong trinh bae nhat va bat phuong trinh bae hai. Trong chuong nay cung dua them khai niem ve bat phuong trinh bae hai hai in, d dd cd dang dap ciia bai loan kinh te. bai toan tdi uu. II. MUC TIEU 1. Kien thufc Nim dugc loan bd kien thiic co ban trong chuong da neu tren, cu the : Hieu khai niem bat ding thiic; chiing minh bat ding thirc. Hieu y nghia cac quy tic va tinh chat cua bat ding thiic va van dung dugc trong viec chirng minh bat ding thirc. Nim dugc mdt each chic chin ve khai niem bat phuong trinh, tap nghiem va quan he tuong duong ciia cac bat phuong trinh. Nim dugc cac kien thuc co ban nha't vd dau cua nhi thirc bae nhat va tam thiic bae hai. van dung linh hoat trong viec giai bat phuong trinh Nim dugc cac khai niem Ve bat phuong trinh bae nha't hai in. tap nghiem ciia chiing va van dung trong giai bai toan kinh te. 2. KT nang Chiing minh thanh thao cac bat ding thirc, van dung linh hoat cac quy tic: chuyen ve. nhan hoac chia hai \6 cua mdt bat ding thuc vdi mdt sd. Giai mdt each thanh thao cac bat phuong trinh bae nhat xa bat phuong trinh bae hai. •DBiet xac dinh tap nghiem cua bat phuong trinh bae nhat hai in, tir do manh nha giai cac bai toan kinh te rat don gian. 3. Thai do •DTir giac, tich cue. ddc lap va chu ddng phat hien cQng nhu linh hdi kien thu'c trong qua trinh hoat ddng. Cin than chinh xac trong lap luan va tinh toan. PHAN 2 CAC B A I SOAKT §1. Bat dang" thdc va cluitng m i n h b a t dang^ thu'c (tiet 1, 2, 3) I. MUC TIEU 1. Kien thiic HS nim dugc : Cac khai niem ve bat ding thirc. Cac tinh chat ciia bat ding thiic. Cac bat ding thiic co ban va cac tinh chat cua nd. He thdng dugc cac bat ding thuc, tir dd hinh thanh cac phuong phap chiing minh cac bat ding thuc. Van' dung cac bat ding thiic Cd-si, bat ding thiic cd chua dau gia tri tuyet ddi de giai cac bai tap cd lien quan. Biet tim gia tri Idn nhat va nhd nhat ciia mdt ham sd, mot bieu thiic dua vao bat ding thiic. 2. KT nang HS phai chiing minh dugc cac bat ding thiic don gian. Van dung thanh thao cac tinh chat cua bat ding thirc de bien ddi. tir dd giai dugc cac bai toan \'6 chiing minh bat ding thiic, tim gia tri Idn nhat, nhd nhat cua ham sd, ciia mdt bieu thiic. 3. Thai do Tu giac, tich cue trong hgc tap. Biet phan biet rd cac khai niem co ban, cac tinh chat va van dung trong tirng trudng hgfp cu the. Tu duy cac van di cua toan hgc mdt each logic va he thdng, budc diu CO tu duy cue tri trong qua trinh sang tao. II. CHUAN BI CUA GV VA HS 1. Chuan bi cua GV D6 dat cau hdi cho HS, trong qua trinh thao tac day hgc GV cd the chuin bi mdt sd kien thiic ma HS da hgc d ldp 8 v6 bat ding thiic Chuin bi phan mau \h. mdt sd cdng cu khac. 2. Chuan bi ciia HS Can dn lai mot sd kien thiic da hgc d ldp dudi. III. PHAN PHOI THCH LUONG Bai nay chia lam 3 tiet : Tiet 1 : Tit ddu den het phdn 2. Tiet 2 : Tit phdn 3. Tiet edn Iqi: chda bdi tap. IV TIEN TRINH DAY HOC A. OAT VAN OE Cau hdi 1 So sanh cac sd sau: a) 2007^°°^ va 2006^°°^ b) 5 + V3 va (1 + V3)2 Cau hoi 2 Nhirng ket luan sau day, ket luan nao dung? (a) x~ + X + 1 > 0 vdi mgi x G M. - (b) Vi 3 > 2 nen 3a > 2a Va ^ 0. ( c ) x - - l > 0 Vx G R. (d) 3 > 2 va a > b nen 3a > 2b. GV : Chi co (a) Id diing. B. BAI Mdi HOAT DONG 1 1. On Tap va bo sung tinh chat ciia bat ding thijfc • GV neu van de: HI. Hay neu khai niem ve bat ding thiic. _ H2. The nao la chiing minh bat ding thuc? H3. a +h < -2 cd phai la bat dang thiic hay khdng? neu phai thi day la bat dang thiic diing hay sai? • GV neu dinh nghia: Cdc menh de dang a < b hoac "a > b" "a > b" "a < b" dugc ggi la nhiing bdt ddng thdc. Hoat ddng cua GV Cau hoi 1 Dien vao chd trd'ng: a < b c ^ a - b < .... a-b<0<:^a...b. Cau hoi 2 dien vao chd trd'ng a - b - 1 < 0 <:^ a < ... Hoat dong ciia HS Ggi y tra 161 cau hoi 1 aa-b<0 a-b<0<^a a < b + 1. Tinh chat ciia bat ding thiic GV: Cho HS dgc vd xem xet trang 104 SGK. Sau do chia HS thanh 4 nhom, mdi nhom thao luan vd cii dai dien len bdng thuc hien thao tdc dien vao chS trdng. Nhom 1. Dien ddu > hoac < vao cho tiring. Tinh chdt Dieu kien Ten goi Noi dung a a + c ... b + d (3 > 0, c > 0 aac ... bd Cdng hai ve cua bat ding thuc vdi mot sd Nhan hai ve ciia bat ding thire vdi mdt sd Cdng hai bat ding thuc cung chi6u Nhan hai bat ding thiic cung chieu a > b c^ }fa..Mb Nhom 2. Dien ddu > hoac < vao cho trd'ng. Tinh chdt Ten goi Dieu kien n nguyen duong a>0 Noi dung a >b>0=> a"...b" a < b <=> yla...^b a < b c^ yfa...yfb Nang hai ve ciia bat ding thiic len mot luy thira Khai can hai ve cua mdt bat ding thiic Nhom 3. Dien => hoac c^ vao chS trdng. Tinh chdt Dieu kien Ten goi Noi dung a be a0,c>0 Nhan hai ve ciia bat ding thiic vdi mdt sd + c hoac o vao cho trd'ng. Tinh chdt Dieu kien n nguyen duong a>Q> Ten gpi Noi dung a>h>0 ...a" > b" a < b ... yfa < \/b a b va b > c ^> a > c; a > b C:> a + c > b + c Neu c > 0 thi a > b <:^ a c > be ; Neu c < 0 thi a > b <» ac < be. + Cdc he qua: a > b va c > d = > a + c > b + d ; a + c>b <:^a>b-c; a > b > 0 va c > d > 0 ^ a > b > 0 va a>b>0=5>^/a>^y^; a.c> b.d ; UGN* ^ a">b"; a>h => yfa > ifb • GV neu vi du 1 trong SGK va cho HS lam, ggi y bing cac cau hdi sau: HI. Binh phuong hai ve cua mdt bat ding thirc, ta dugc bat ding thiic ciing chi6u Diing hay sai? H2. Neu sai can bd sung dieu kien gi de dugc khing dinh diing? H3. Gia sit v2 + V3 > 3. hay binh phuong hai ve va so sanh. • GV neu quy udc trong SGK. • Neu vi du 2 trong SGK va cho HS lam, ggi y bing cac cau hdi sau: HI. Hay chuyen ve va dua bat ding thiic ve dang : f(x) > 0 H2. Hay chiing minh f(x) > 0 la bat ding thirc diing. • Neu vi du 3 trong SGK va cho HS lam, ggi y bing cac cau hdi sau: HI. Hay neu mdi quan he giira cac canh trong tam giac. 2 2 2 H2. Giai thich vi sao a > a - (b - c) H3. Giai thich vi sao (a - b + c)(a + b - c) = a^ -(b-c)^ H4. Hay lam tuong tu va chiing minh bat ding thiic da cho. 2. Bat dang thurc ve gia tri tuyet ddi. • GV neu cac bat ding thiic trong SGK -\a\ - a < X < a vdi a > 0. \x\>a <^ x <-a hoac .v > a vaia >0. • GV neu bat ding thirc quan trgng sau: \a\ - \b\ <\a + b\< \a\ + \b\ (vdi mgi a, b e R). GV cho HS chiing minh bat dang thiic tren (ke ca |H1| bing HD sau) 10 Hoat dong cua H S Hoat dong cua G V Cau hoi 1 Hay chiing minh bat dang thiic : \a + b\< \a\ + \b\ bang each binh phirang hai ve. Ggi y tra 161 cau hoi 1 \a + b\ < \a\ + \b\ c^ (a + b)^ < a^ + 2\ab\ + b^ <=> a^ + 2ab + b^ < a^ + 2\ab\ + b^ o Cau hoi 2 Hay chimg minh bat dang thiic : \a\ - \b\ <\a + b\ b i n g each binh phirang hai ve. ab < \ab\ Ggi y tra 161 cau hoi 2 Ifir + ^l > \a\ - \b\ <:^ (a + b)^> a^-2\ab\ o + b^ a^ + 2ab + b^ > a^ - 2\ab\ + h^ Ci> ab > - \ab\ Cau hoi 3 Hay chiing minh bat dang thiic : lai - \b\ <\a + b\ b i n g each sir dung cau hdi 1 Ggi y tra 161 cau hdi 3 Ta cd |a| = |a + Z) - Z?| < |a + Z?| + \-b\ = \a + b\ + \b\ Chuyen ve ta dugc DPCM. HOAT DONG 2 2. Bat d^ng thiifc trung binh cong va trung binh nhan a) Do! vdi hai so khong am • GV neu dinh li. Vdi mgi a > 0, b > 0 ta cd a+b i—r > yjab 2 Ding thirc xay ra khi va chi khi a = b. 11 Sau dd GV neu cac cau hdi HI. Hay phat bieu dinh li bing Idi. De chung minh dinh li, GV neu cac cau hdi sau: H2. Dien cac dau >. < >. < vao chd trdng sau: ^ ' ^ = - - ( a + b- 2 ^ ) 2 2 = - - ( ^ - V^)2...0, 2 H3. Hay ket luan va chi ra trudng hgp dau bing xay ra. H4. Van dung dinh li hay chii'ng minh I tan x + cot x 1> 2 Thuc hien H2 GV cho HS doc va hieu noi dung H2 GV treo hinh 4.1. Sau do thuc hien theo cac thao tac sau: GS-^ thao tdc trong 3 phiit. Hoat dong cua GV Cau hoi 1 Hay xac dinh gdc ACB Cau hdi 2 Hay tinh OD. Cau hdi 3 Hay tinh CH. Cau hdi 4 Hay so sanh OD va CH, tCr dd riit ra bat dang thiic. Hoat dong cua HS Ggi y tra 161 cau hdi 1 ACB =90° Ggi y tra 161 cau hdi 2 0D=^+^ 2 Ggi y tra 161 cau hoi 3 CH = ^ Ggi y tra 161 cau hdi 4 OD>CH tir do t a c d ''^^>V^ 2 • GV neu vi du 4 va hudng din HS lam theo cac cau hdi sau: HI. Hay phan tich \'e trai thanh tong ciia nhCing sd cd dang nghich dao ciia nhau. H2. Hay ap dung bat ding thuc Cd-si. 12 • GV phat bieu he qua bing Idi: HE QUA NcAt hai sd duang thay doi nhung co tong khong ddi thi tich ciia chiing lini nhdt khi hai sd do bdng nhan. Nen hai sd duang thay doi nhung cd ticli khong ddi thi tong ciia chiing nho nhdt khi hai so do bang nhau. Sau dd hudng dan HS chiing minh theo cac cau hdi sau day: S rHl. Chiing minh — > Jxy c 2 H2. xy Idn nhat khi nao? H3. Chung minh x + y > 2 V ^ H4. x+ y nhd nhat khi nao? H5. Ap dung he qua hay tim gia tri nhd nhat cua bi6u thiic ciia bieu thirc : • GV neu iing dung : Trong tdt cd cdc hinh chif nhdt cd cimg chii vi, hinh viiong cd dien tich lan nhdt. Trong td'i cd cdc hinh chit nhdt cd ciing dien tich, hinh viidng cd chu vi nho nhdt. • GV neu vi du 5 va hudng din HS thuc hien theo cac cau hdi sau: HI. Chiing minh ring f(x) > 2V3 H2. Dau bing xay ra khi nao? b) Ddi vdi ba sd khdng am • GV neu dinh li 13 a+b+c ^Jabc <» a = b- c (vdi mgi a, b, c > 0). Sau dd GV dua ra cac cau hdi sau: HI. Hay phat bieu dinh li bing Idi. H2. Neu bd di dieu kien ba sd khdng am thi dinh li edn diing hay khdng? hay neu mdt vi du. • GV neu vi du 6. Sau dd hudng din HS chiing minh bing cac cau hdi sau day: HI. Hay ap dung dinh li ve bat ding thiic Cd-si cho ba sd a, b, c. . 1 1 1 H2. Hay ap dung dinh li v6 bat dang thiic Cd-si cho ba sd a b H3. Hay chiing minh bat ding thiic tren. H4. Dau bing xay ra khi nao? Thuc hien H2 Hoat dong cua GV Hoat dong cua HS Cau hoi 1 Phat bieu ke't qua tirang tir he qua tren cho trudng hgp ba sd duang. Cau hoi 2 Ne'u bd di dieu kien ba so duang thi ke't qua con dung hay khdng? Ggi y tra 161 cau hoi 1 Neu ba sd duong thay ddi nhung co tong khdng ddi thi tich cua chiing Idn nhat khi ba sd do bing nhau. Neu ba sd duong thay ddi nhung cd tich khdng ddi thi tong ciia chiing nhd nhat khi ba sd dd bing nhau. Ggi y tra 161 cau hdi 2 Khdng. 14 TOM TAT BAI H O C 1. Cac bat dang thirc cd dang a < b, a > b, a < b, a > b. 2. Cac tinh chat ciia bat ding thiic. Tinh chdt Dieu kien Noi dung a 0 a < b <^ aa < ba a<0 a ba aa a >0, c> 0 n nguyen duong fl>0 + c ac < bd a \fa 0. xay ra khi vd chi khi a = b. Cac he qua: He qua 1 Ne'u X, y cimg duang vd cd tong khdng ddi thi tich xy Idn nhdt khi vd chi khi x = y. 15 He qua 2 Neu X, V Cling duang va co tich khong doi thi long x + y nho nhdt khi vd chi khi x = v. Bat ding thiic chira dau gia tri tuyet ddi. Noi dung Dieu kien \.\-\ > 0. Ixl > X. |x| > -X x| < a <:> -a < X < a a>0 |x| > a <=> X < - a hoac x > a \a\ -\b\ <\a + b\ < \a\ + \b\ M O T SO C A U H O I T R A C N G H I E M I. Hay di6n cac dau > hoac < vao chd trdng sau day : (a) a" + b^ 2ab; (b) b + c" (c) a + C 2ac; (d) a + b + c 2 2 be; 2 ab + be + ca Tra Idi. Cau (a) (b) (c) (d) Di6n > > > > 2. Trong cac khang dinh sau, hay chgn khang dinh diing vdi mgi x. (a) x^ > X (b) X' = X (c) 2x- > -X (d) 2x- > X- Trd Idi. (d). 3. Hay chgn khing dinh diing trong cac khing dinh sau : (a) X + Ixl > 0 (c)-2x+ x <0 16 (b) X - Ixl > 0 (d)x + 2 X < 0 . Tra lai. (a). 4. Hay dien cac dau (>, <, =) vao cac chd trdng thich hgp sau (a) V2 V3; (c) 72 + Vs 73 + V? ; (b) Vs VV (d) 7I0 722 Tra lai. Cau (a) (b) (c) (d) Di6n < < < < 5. Hay di6n diing - sai vao cac cau sau : (a) 4 > 2 ^ 4a > 2a ; D (b) 4 > 2 => 4a < 2a ; D (c) 4 > 2 ^ 4a > 2a Va > 0; D (d) 4 > 2 ^ 4a < 2a Va < 0. D Tra lai. Cau (a) (b) (c) (d) Di6n S S D D 6. Hay dien diing - sai vao cac cau sau day : (a) 2006 > 2005, a > b => 2006a > 2005b D (b) 2006 > 2005, a < b ^ 2006a < 2005b D (c) 2006 > 2005, a > b => 2006 + a > 2005 + b D (d) 2006 > 2005, a > b =^ ^ 2006 2005 > --— a b ^ U Tra Idi. 2.TKBGOAIS610(NC) 17 Cau (a) (b) (c) (d) Dien D D D S 7. Hay dien dung - sai vao cac cau sau day : (a) 5 > b ^ D 75 > 7 b (b)5>b>0^75>7b D (c) a > b ^ D 7 a > \/^ D (d) a > b ^ a" > b" n e Z Tra Idi. Cau (a) (b) (c) (d) Dien S D D S 8. Cho a, b la 2 sd cung dau. Hay chgn khang dinh diing trong cac khang dinh sau: (a) ^ . ^ ^ 2 ; b a (c) a b —-I- — b b a a b (d) —+ — <0. b a >2 a Tra Idi. (c). 9. Cho a > b > 0. Hay chgn ket qua diing trong cac ke't qua sau : a a+1 (a) - < b b+1 a a+ l (c - > b b+1 Tra Idi. (c). (b) - < b a (d) ca ba cau dtu sai. 10. Hay chgn ket qua diing trong cac ket qua sau (a) a -H 1 > a -K a Va e :i: 18 (b) a"^ + 1 < a'' -h a Va G J<;