giáo trình một số điểm cần lưu ý khi dạy học đại số và giải tích 11 nâng cao

TRIING II c} s}TUIWffi BS:l; uu#.ilr @ffih@M PHAM oUc uor GV trudng THCS NguyOn Luong Adng, Thonh MiQn, Hoi Duong Trong Sdch gi6o khoa llinh hoc 9 t6p hai trang 105 c6 bli to6n sd 9, n6i

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TRIING II c(} s(}

TUIWffi BS:l; uu#.ilr

@ffih@M

PHAM oUc uor

(GV trudng THCS NguyOn Luong Adng,

Thonh MiQn, Hoi Duong)

Trong Sdch gi6o khoa llinh hoc 9 t6p hai

trang 105 c6 bli to6n sd 9, n6i dung nhu sau

Tr€n canh AC ldy m\t didm M vd vd dtdng

trbn V, dudng kinh MC Dudng thdng BM cdt

V, tqi D Dudng thdng DA cdt W, tqi S Chtng

minh rdng:

a) ABCD b trt gidc ndi tiip.

b) CA ld tia phdn gidc cil.a g6c SCB.

Ldi gitii a) Do tam gi6c ABC vuOng tai A nOn

D nam tr6n dudng trbn 31 nen fii = 90' (2)

Tt (1) vn (2) suy ra tir gi6c ABCD nQi tidp

dudng trbn ?" dudng kinh BC

b) Ta c6 frA = fiA GO" nQi tidp cirng ch6n

+) Trudng hqp D trtng vdi S Khi d6 AD

h tidp tuydn cria dudng trdn (%) Ta c6

frA=ffe,=fus.a

Khi giii c6u b) nhidu ban thudng chi x6t mot

trudng hqp (tiy thu6c vio hinh v6 ra) Honnfia ch6c it ban dat duoc cdc cdu h6i: Vi trf

cfra didm M th€ nio dd didm D nam trOn cung

nh6 ffi ? Didm S nim ren cung nnb frD t

Didm S trDng vdi D?

Ogai 1,rl6n 2 Cho tam gidc ABC vudng tai A

Didm M di dQng tr€n canh AC, vd dudng trdn

ff, drdng kinh MC, BM cdt V, tqi D, dudng

thdng AD cat fi tai S ){dc dinh vi tri cila didm

M tr€n canh AC dd:

a) Didm S ndm ftan cung nhd frD .

b) Didm D ndm tr€n cung nhd ffi

Ldi gitii a) (h 3) S thu6c cung nh6 frD yni

vi chi khi m fr60 .

Do ha.i tam gi6c OMS vd OSD ddu cdn tai O

n6n suy ra OMS > OMD o OMS > AMB (6)

TOff{ FiA(

' cfr*di$ si x61 l7-zoo7l I

DHinh 3

Ndu ggi giao didm cia Vt v6i BC le / thi

frie = 900 din ddn tr1 gi6c ABIM noi tidp duo.c,

nen frc = tfri (ctng bir vot ,qui ) (7)

Lai do ICM =MCS (biLi to6n 1 cau b) vi'MC

id dudng kinh cira dudng ttdn Vr nOn 1 vi S

ddi xrlng nhau qua MC c6 ifra =eES (8)

Tt (7) vn (8) d6n ddn ffi =trEa (e)

<> 90" - frB > 90" - Tnfu o frtw frn.

b) Lap 1u6n tuong tu ta c6 kdt qui D thu6c cung

nh6 ffi khi vd chi khi frilrfri u tt

thu6c canh AC n6n cdn c6 didu kiQn AC > AB' A

TiI bai bdn2 d6n ddn suY nghi:

Khi ndo hai didm S vh D tring nhau? Tt d6 ta

cd bii to6n sau.

Ogai to6n 3 Cho tam gidc ABC vubng tai A,

Didm M thuAc canh AC, vd dtdng trdn V1

dudng kinh MC, BM cdt Vrtai D ){rtc dinh vi

tri cria didm M ffan canh AC dd AD ld tidp

ruyAh cfia dudng trdn ffr

L6p luan tudng tu bdi to6n 2 ta thdy AD la tidp

tuydn cfra dudng trbn V, khi vd chi khi

cdn c6 didu ki6n AC > AB J

Ddn day ta thdy d6ng phii suy-nghr vd"vi tri

ctra didm M Do didlrr M c6 thd thay ddi tron

canh AC nOn nhidu didm kh6c cfing thay ddi

theo C6c ban hdy quan s6t tia AI, n6 thay ddi

n6n giao didm K cira n5 v6i dudng trdn Vt

cfing thay ddi theo DAn ta ddn bbi to6n sau'

T6##t e+G{

2 Sd 361 (7-2007) & cigjige

Ogai !,rldn 4 Cho tam gidc ABC vuing tai A.Didm M thay ddi ffan canh AC Dung dtdngtrdn Vt dtdng kinh CM Goi giao didm cila BC

idi duang trdn Wt ld I, giao didm cfia AI vdi

tich didm D khi M chuydn ddng trdn canh AC ld

cung nh6 tri ciadudng trbn dudng kinh BC'

Tt d6 ta d1r do6n qu! tich didm K khi M

chuydn dQng tr6n canh AC li hinh ddi xung

vdi cung nh6 iD qua canh AC Qu! tfch niy

li m6t cung trbn cira dudng ttdn V2 ddi xrlng

vdi dudng trdn dudng kinh BC qua canh AC'

Tt d6 taxic dinh duoc dudng lrdn W, nhu sau:

-Ldy B' ddi xrlng vdi B qua canh AC;

- DUng dudng trdn W, dudng kinh B'C

Srl dung kdt qui cita ciic bli tdp tren ta d6

dlng'ch'rlng minh duoc B', M, K thing hing

suy ra B' KC = MKC = 90"

V4y qu! tich didm K khi M cbuydn d6ng trOncanhAC 1I cung rrho ID cira dudng trdn dudng

kinhB'C D

Nhu vay ndl quan siit tdt, nghi0n crlu ki thi chi

tt mOt bii tap trong s6ch gi6o khoa, chring ta

c6 thd khai ih6c duoc nhidu kdt qui thri vi'

Chfc c6c ban gat h6i duoc nhi6u kdt qui'

rdu GL\n mrA flFtrfi cmot\ Hoc sNFil G]roflrdp s

g(p.766 @hiJllinh

Cflu 1 U) piAu kiQn x2 - 4 > o,hay x > 2hoic x < -2.

Fiat J* -+ : y A > 0) Phuong trinh cl5 cho

Ding thirc x6y ra khi vi chi khi

CAu 3 g) Ta thdy x, y, z Y,hLc 0 He da

clugc vi6t lai dudi d4ng

11s

xy12 115 yzlS

1 I 13

zx36Giei hQ ndy tim dugc ;r = 4, ! = 6, z : 9.

PT ndy vd d6i chi5u v6'i di0u kiQn tr€n ta

3-5duoc v = a DAn d6n x: +: (thoa m6n

kiQn).

luqn: Phuong trinh tld cho c6 hai nghiQm

55 22CAu 4 Nhqn xdt: MQt s6 chinh phuong 16 khichia cho 8 sE c6 15 au la t

Thflt viy, xdt s5 chinh phuongld m2 (m e Z),

m lil s614, dflt m = 2n * 1 Lric nity m2 : (2n + 1)2

: 4nz * 4n + I = 4n(n+ l) + 1 ohia cho 8 du i.

Trd lai bdi to5n, ta c6 biQt thric A : b2 - 4ac.

Do b 16 n6n theo nhfn xdt tr€n b2 chia cho 8

du 1 D[t b2 =8k + I (/r e Z) L4i vi a,cld

nln ac le, dAt ac = 2l - | (l e Z).

Khi d6 A = 8k+ I - 4(21 - l) = 8(t- /) + 5chia cho 8 du 5 Tir nhfln xdt tr6n ta th6y A

kh6ng ph6i ld s6 chinh phuong

NghiQm PT de cho (ndu c6) ld x : -

:

4d

nghiQm thi c6c nghiQm 6y kh6ng tfr€ ta sO hfru

dleuXet

Mat kh6c do OM li dudng trung binh cira hinh

thang ABDC nOn OMllAC, md AC I CD suY

nira dudng trdn t6m 9.YdY c6c <li6m M, C, D

cirng nirn-tr€n titip tuy6n ctra ntra dulng trdn

vu6ng tai N, c6 ACB= 45o,

(2) Tri (1) vd (2) ta thu dusc(dpcm)

NGUYEN rAx roaN gP Hi chi Minh)suu tAm vd gi6i thiQu

1aN= 4!

JZ

TOAN HQC (Ti€p trang 3o)

T

cHurtll MnlI cffirill

TrOn mOt bing c6 25 b6ng dEn nhu A hinh 1,

m6i dEn c6 miu D6 ho[c mhu Xanh MQt bing

didu khidn c6 cdc c6ng t6c dugc ghi theo toa dO

dEn Khi dn c6ng tic cfia mdt dEn nio ddy thi

ddn d6 vdtdt ch c6c ddn tidp xric v6i dEn d6 ddu

ddi miu Ching han tr6n hinh 1 ndu dn c}rrgtic

D2 thi c6c ddn D2,DL,D3,E2, C2 ddu ddi miu

Dinh cho ban doc

1) Tr0n hinh 1 c6 8 dEn xdp thinh hinh vu6ng

miu D6, c6c ddn cdn lai miu Xanh Hdy dat

mdt lOnh (mQt d6y s6p thrl tu su dn c1ngtic cdc

ddn) dd sau khi thlrc hiQn lenh d6 thi tdt ch cic

dBn ddu c6 miu D6

2) Trdn hinh 2 c6 5 ddn xdp thinh hinh chfr T(vidt t6t cira To6n Tudi tr6) miu D6, c5c ddn cdn

lai mhu Xanh H6y d{t mqt lQnh dd sau

khi thuc hien lonh d5 thi tdt ce cdc ddn ddu c6

CB

A

E

D

CBA

K

uffi m fifij$ u#p t,oj Tr a:Ulfiffi pfl

b) Tim rn d6 phuong trinh (l) vO nghiQm.

CAu 2 a) Gi6i b6t phuong trinh

| (-x + 3)(-r -r)l-21 , - I l< x2 -7 .

b) Giei hC phuong trinh

[yJi* 2*Ji =tyJ41.

CAu 3 a) Cho a, b ld hai sti thr,rc th6a rndn

Thdi gi.an ld,m bd;i: 150 phdt

Cha 4 Cho tam gi6c ABC nhon c6 tryc t6m li

H vit ilC: 60o Ggi M, N, P l6n luqt ld ch6nc5c dudng cao k0 ti A, B, C cuatam gi6c ABC

vi / ld trung cli6m cta BC

a) Chring minh ring tam gi6c 1NP d6u.

b) Gqi E vd K lin luqt ld trung.di6m cria PB

vd NC Cht?ng minh rdng c6c di6m I, M, E, K

cr)ng thuQc mdt dudng trdn

c) GiA sri /l ld ph6n giric ctta friP Hay tinh

s6 do cta g6c BCP

CAu 5 MQt cdng.ti may giao cho td A may

16800 s6n phdm, td B may 16500 sAn phAm vd

bat Oiu thUc hiQn c6ng viQc cirng mQt lirc N5u

sau s6u ngiy, tt5 A dugc h5 trq th6m 10 c6ng

nhdn may thi ho hoin thdnh cdng viQc cing

lirc v6i tO g N6u td A duqc h5 trq th6m 10

c6ng nhdn may ngay tir etAu thi hq s€ hodn

thinh c6ng.viQc sdm hon tis B mOt ngiy.H6.yx5c tlinh s6 c6ng nhAn ban tliu cria m5l tO.

Bi5t ring m5i c6ng nhdn m5i ngiy may dugc

20 s6n phAm.

NGUYEN DUc TAN (Tp Hd chi Minh)

suu tAm vd gi6i thiQu

BINH LUAN (Ti€p trans to)

I

tinh Jrlr;or.

0Bdi luyQn tSp Tinh t6ng

t =|."r, **"7, *I"i, * *-L-c*.

ehudn bi

:1 Ntiu hC PT ba hn x,y, z kh6ng thay ddi khi

noan vi ulng quurh OOi ,Oi x, !, z th\ kh6ng m6t

tinh t6ng quit c6 th6 gin thi6t x = max (x, y, z).

Nghia ld x > !, x) 7 (xem thi du 3)

ViQc su dpng khio s6t bi6n thi6n cria HS dC

eiii hoic bie; ludn mOt s5 hC PT tgo n6n sg

pnong phf vC th6 loai vd phuong ph6p gidito6n, phir hqp v6i c5c ki thi tuy6n sinh vio

Dai hqc Sau ddy ld mQt sO ttri du minh hqa.

(1)

(2)

Giei h0 phucrng trinh

MQt sii luu y chung

1) Phuong trinhflx) : m co nghiQm khi vd chi

d,i , tf,rQ" tflp gi6 tri cria hdm sti y: f(x) vit

s6 nghiQm ctra phuong trinh (PT) li sd giao

diOm cria dO thi hdm sti (HS) y = flx) vdi

Nr5u HS y = JU) don diQu, thi tt (1), suy ra

x = y ftri OO bii to6n dua v6 gi6i holc biQn

lufln PT (2) theo 6n x

Ntiu HS r- = lU) c6 mQt cgc tri tqi t = a thi n6

thay cl6i chi"Au.bi6n thi6n mQt iAn \tri qva a.

fil-(f ) suy ra x = y hoflc x, y nim vd hai phia

cria a (xem thi du 2)'

(3)

x6tHS f (t) = et -t ,c6f '(t): d -l> 0, v/>

0-Do ct6 Hsflr) ddng bii5n khi r > 0.

l fG)= f(v) Tt(3)suyra -"\_/ J _ 1"' "'-.=x=Y.

[x>0,y>0

Thay vdo (2) duqc logr;+ log5 4x3 =10log2 x-l+2(2+3log2 x) =10, haylog2 x =l '

HQ c6 nghiQm duy nhAt @ ; Y) = 12 ;2).

theo mQt trong hai

6 sd 361 (z-2002) _ T@J & quditl6 HA(

Tath6y f'(t)=0e/=0.

HS l(0 d6ng bii5n trong (-1 ; 0) vd nghich

bi€n trong (0 ; +oo).

Ta c6 (3) e /(.r) = f (y) Lric d6 x : y hoic

xy < O (n6u x, y thuQc cing mQt khoing clon

diQu thi x : !, trong trudng hqp ngugc lai thi

tl6i v6i x,y,z ndnc6th6gi6thi0t "r)y,x)2.

N6u x > y thi /(x) > fu) - y > z

cirng nghich biOn) thi li lufln nhu tr€n ta suy

1-Ldi gidi DK -1 < x,y <3 .

Tr* theo v6 cria (1) cho (2) vd chuy6n v6, a

8'(x) =-+ +, g'(.r) = 0 <> x - l.

2'lx +1 2tl3 - x

Ta c6 s(-1)=2, S(l)=2Ji, BQ)=2

Tir d6 2< s(x)<zJ,

Vdy he c6 nghiQm khi 2 < m <2J2 .

hQ phuong trinh sau cd nghiQm duy nhiit

fs*'y-2y2 -m=o

(l)

(2)

Ldi gidi Ni5u y -< 0 thi v6 trSi cria (l) 6m,

PT kh6ng thoA mdn, suy ra y > 0 Tuong tg c6

x>0.

Trir theo v6 cria (1) cho (2), ta dugc

3x2y -3y2x +2x2 -2y2 =0

[/(z) = s(x)

TCffi\8 HOC

" cfuonfe sd 361 l7-2oo7l 7

Do d5 v6im> 0, hQ c6 nghiOm duy nhAtx : y> 0

(bpn doc tu vE d6 thi ho4c iQp bnng bi6n thiOn

cira HS AC Uam tra k6t quA tr6n)

bidn ddi

J;.1*{6-r

IIIII

Xin h6i ldi giii bidn ddi nhu tren dring hay sai?

(N.N., I lAt, THPT Lang Giong, Bdc Giong)

Mong nhAn duoc ! kidn tri ldi cia cdc ban gitip

2 ban N.N

vd,ln - dtatfoq*

1 (3.07) Y t<ign 3 li chua chinh x6c Ta ncn v€

tru6c bang brit chi rdi sau t6 lai bang brit muc

Ndu kh6ng lfc ddu vE bang n6t drlt nhfrng ch6

dudng khudt Lim ddn cdu sau m6i xem x6t vEdudng n6t lidn cho chinh x6c

(NhO Vdn Vlnh, t tAt , \HPT TAn YOn tt, Bdc Giang)

DAT MUA tAP cHiroAN HQc vA rud rnl

oAt u4trt TAt cAc co sd Buu DIF:N

rRorvc cA aa6c

fiinhlufrn ui

dTfu WMm

pi thi TSEH m6n Todn tnAt a ndm nay sdt vdi chaong trinh phd th6ng Ti lQ gifra cdc cdu dd,

trung binh, kh6 t.d hqp li H7c sinh khd, gi6i vdi kidn thil:c chdc chdn mdi cd th€ dqt daqc didm

cao Vi(c giai day di vd chinh xdc cdc cdu 11.2, 1V.2 kh6ng phdi don gidn Sau ddy ld ldi gitii

md sii cau trong de thi vd m\t sij nhqn xil vA cdc ldi giai d6.

NhQn xdt l) Khi dAt di6u kiQn cho r, dE mac

sai lAm h chi th6y t > 0 md kh6ng th6y

0 < r < 1 Tir d6 d6n tdi k5t qun sailitm=1frl

J

2) Khi gap PT dpng auz +buv+cv2 =0, c6ch

giii chung li :

N6u v;r 0, chia ci hai vi5 cria PT cho ,', sau

d6 d6t 7 =! , tudugc mQt PT bdc hai dtii v6i r

v

(trudng hgp v : 0 dugc x6t ri6ng).

3) N6u PT c6 d4ng m : /(t), thi PT ndy c6

nghiQm khi rn thuQc tip gi6 tri cria HSI(0.

Bdi luyQn fip Bien ludn sii nghiQm cita PT

Jr{ + +**,[7-z* * z + (m +lJx -z = O

v' x2 +2(m + 1)x + m2 + 4m

m ld tham si5 Ti* m dii hdm sa (t) c6 cvc dqi

cqrc tidu, d67tg thdi cdc diiim cqc tri cila d6 thi"

cirng vdi g6c tog d0 O tqo thdnh mAt tum gidc

Nhfin x/r N6u str dpng dinh li Pythagore cho

tam gi6c vu6ng OAB th\ lli gi6i sE dii, tinh

toSn phirc t4p hon

Bdi tuyQn tfrp Chto hdm si5

x2 +(2m-3)x+3m+5

vd didm PQ; -l) Tim m ae ltam s6 c6 cqrc

.: -: , -.:

dqt cuc tteu, dong tnot cac dtem c$c tr! cua

dO thi cilng vdi diAm P @o thdnh mQt tam gidc

*Ceu lY.l Tinh di€n ttch cila hinh phdng

gidi hqn boi cdc dadng y=(e+1)x,y=Q+d)x

Tinh dugc Iz =1, suy "2 ra S =: - 1 (dvdt) tr

NhQnxdr Ni5u HS dudi dAu tich ph6n c6 dang

p(x).e* , trong d6 p(x) ln mQt da thf'c, thi ta

giAi bdi to6n b6ng phuo'ng ph6p tich ph6n tirng

thay doi vd thoa mdn xyz = l Tim gid tri nhd

nhat cua bidu th*c

,_ x2(y+z) - y2(z+x) *

1- yly+zzlz t ^ t z!z t,+Zx.Jx

Ldi gi,fii Ap,dpng.BDT Cauchy cho hai s6

duong tr6n m6i tu s6, vd tir xyz : 1, ta dugc

khix:y:z:LA

NhPn xdt C6i kh6 cira bdi toSn o ch6 kh6o sil

dpng BDT Bunyakovski dO chirng minh BDT (4).

Bdi tuyQn tQp Cho a, b, I vd m, n ld cdc si5

thryc duong Chang minh ring abc-3

k=0Luu yi ting tt6u k ln s6 chin thi

10

!c5,*ua*

-0Khi k le thi

1

lc5,*oa* 0

, A TI -t

PHUONG PHAP

Trudc hdt xin nhac lai kh6i ni€m GTLN,

GTNN cria him s6.

Cho him sdfl.r) x6c dinlitren midn D (DcR).

Gii srl M, m ldn lugt li GTLN, GTNN cira

Sau dAy ld m6t sd bdi to6n minh hoa.

frnei to6n 3 Gid sft M ld gid tri lon nhdt cua

lbl sao cho 4bx3 +(a-lU) x 31, vdi moi

gid tri crta x.[-t;\) vd vdi moi sd thuc a'

suy ra ux lal < 1 vay M < l a

Dd cirng cd phuong ph6p, mdi c6c ban glbi cdc

bdi to6n sau:

onai h6n 4' cho hdm sd f (x)=ox2 tbx*c

stl M =max f(x); m=min f(x) Chftng minh

trong d6 a ld tham sd thuc tilY Y.

Goi M =,Ii3,1,1 f (*); r=,S-il,l t(-)

lu>t t

12 sd 361 (7-2002) &

PhdG GrfiAm Gh@mn

tI THUYET

SO NOUYEN TO

6t trong nhfrng ngudi duoc giii thu&ng

Fields 2006, Terence Tao, di ddy viec

nghiOn cts ciLc sd nguyOn td di xa hon, bang

c6ch dua ra mot dinh li mdi

Nhi to6n hoc trd Terence Tao, &

trudng D+i hoc Los Angeles,

ngudi nam ngo6i duoc nhdn Huy

chuong Field vdi c6c kdt qui

nghi0n crlu vd cilc s6 nguyOn td,

dd lim cho li thuydt c6c sd nguy0n

td phdt tridn hon nfia Cing vdi

ban ddng su Tamar Ziegler b

trudng Dai hoc Michigan, Terence

Tao dd chirng minh rang ddy sd

nguy6n td chrla nhfrng

"cdp s6 da thric dhi

tilY 1i" rr).

56' nguydn t6'ld s6'tu

nhi€n l6n hon I chi chia h€i

cho I vd chinh s6' d6 Ydi

dinh nghia cuc ki don giin

d6, ngudi ta dd xAy dung nhfing vdn dd to6n

hoc rdt phrlc tap.

Tir thdi vdn minh cd dai, ngudi ta dd, bi6t c6c

sd nguyOn td tao thlnh m6t d6y v6 han Nhftng

sd nguy6n td ddu ti6n li 2, 3, 5,7, ll, L3, 17,

19,23 Khi c6c sd cing l6n thi vi6c x6c dinh

c6c sd nguyOn t6 cdng kh6 khan hon Dudng

nhu khi ci1c sd cing l6n thi sd nguy0n td clng

hidm Hi0n nay cdc nhi to6n hoc chua thd m6

tA 16 rlng c6ch thri'c md c6c sd nguyOn td duoc

phdn bd trong ddy c6c sd rU nhi0n

Nam 2004, cing vdi Ben Green, 6 trudng Dai

hoc Cambridge, Terence Tao dd chrlng minh

mOt trong nhtng kdt qui dd mang lai cho 6ng

Huy chuong Fields: Tdp hqp cdc sd nguy€n tdchtta nhfrng "cdp s6'cing c6 dO ddi tiy y" ttc

ld sd sd hang l6n tny y Vi du, d6y 5, Il, 17,

23 lh mdt cdp sd c6 4 sd nguy6n td hon k6m

nhau6(5+6= 11, 11+6= 17,17 +6=23).

Day ln m6t cdp sd c6ng c6 cOng sai lh 5.

Phii chang tdn tai nhfrng d6y sd khOng phii lncdp sd c6ng nhung c6 thd m6 ti quy luAt vd

khoing cdcb grtra hai sd nguyOn td li0n tidpthu6c ddy?

Dinh li mdi m6 rOng kdt qui tru6c cho nhfrng

cdp sd "da thfc", trlc ld nhtng cdp sd c6

khoing c6ch gifra ci{c sd hang 1i0n tidp kh6ng

nhdt thidt li mdt hdng s6, nhu trong trudng

hang tidp sau duoc x6c

thrlc cing loai, trong d6

chi dDng ddn hing r vh

c6c liy thira cfia n6 Su

chri'ng minh dinh li d6 dua tr6n cirng nhfrng f

tu&ng dd cho ph6p Ben Green vd Terence TaogiAi quy6t trudng hgp cr{c cAp s6 cdng TAp hqp

nhfrng sd nguy€n td duo c nhAn dinh nhu ld m6t

Qp hcr,p con cira tap ho,p nhfrng sd nguy6n goi lh

"gdn nhu nguy0n td" Tr0n tap hap d6, ngudi ta

xdy dqrng mOr ham dic bi6t cho ph6p lim ndi rdm6t sd tinh chdt d[c bi6t cia cdc sd nguyOn td

Nhu vdy c6c sd nguy6n td dd 16 ra th€m chrit it

nhfrng didu bi mAt cria chring

NGUYEN VAN THIEM

(theo "La Recherche" thdng I-2007)

nAm gilng day thi didm tai gdn 50 trudng C6c kidn thfc v<i td hop vd xdc sudt dang

THPT vi tidp thu c6c f kidn t0m huydt cira rdt

nhidu c6n b6, gi6o vi0n trong cA nu6c

Ir)I-Ii?:hJ mE-&

6ch 916o khoa (SGK) Dai sd vd Gidi ttch

11 ndng cao dugc biOn soan theo chuong

trinh mOn Toiin Trung hgc Phd th6ng ban

hdnh theo Quydt dinh sd L6120061QD-BGDDT

ngiy 05-5-2006 cira 86 tru6ng B0 Gi6o duc vd

Dio tao, tron co s& tdng kdt cdc uu vh nhuoc

didm cira cudn SGK thi didm Dai sd vh Giii

tich 11 ban Khoa hoc Tu nhi6n (bO 1) sau hai

3ffiffiru *{W{

Sd 361 i7-2007i & €fr.{trUe

Trong khu6n khd cira bii b6o ndy, v6i ciicquan didm nOu tr0n, chring t6i trinh bdy nhfrng

,en ad trong ydu nhdt md m6i gi6o vien vh hoc

sinh cdn luu f khi sil dung cudn SGK niy

trong nam hoc sap tdi, d5c bi0t la nhfrng didm

mdi vd kh6c so v6i chuong trinh vd SGK ldp

11 chinh li ho p nhdt nam 2000 (SGK2000)

r vB NOr DLrNG cHUoNG rRiNH

So v6i SGK2000, SGK Dai sd vd Gidi tich 11ndng cao c6 nhfing didm kh6c bi6t sau ddy:

- Y6, luong gidc,hoc sinh d6 lim quen vdi c6c

kidn thrlc m& ddu vd 1uo ng giSc b ldp 10' Do d6

phdn luo.ng giSclop 11 chi cbn vdn dd cdc hinn

id tuong gidc vit phuong trinh luong gidc'

Chuongirinh kh6ng tli cQp cdc vdn dd' vd' bdt

phuong trinh vd h€ phaong trinh lrtong gidc'

iiJl f#i;1:, A @ @ @ N W [[a[[g qAO

ddi sdng, thuc ti6n khoa hoc

Truc quan, tfc ld coi truc quan ld phuong

phrip chir dao trong vi0c tidp cdn c6c kh6i

nigr, to6n hoc ; dAn d6t hoc sinh nhdn thrlc tir

truc quan sinh dOng ddn tu duy trtu tugng

Nhe nhhng, tri'c tri x6c dinh nhtng yOu cdu

vta phii AOi vOi hoc sinh ; trdnh him ldm ; cd

g6n[ trinh biy van dd ng6n gon, sfc tich,

khdng g?ty cdng thing cho ngudi hoc

Odi mOi, trlc li c6ch tan cilch trinh bly, ndng

cao tinh su pham ; g6p phdn ddi m6i phuong

ph6p day hoc vi phuong ph6p dSnh gi6

ngiy chng tri n0n quan trong ddi vdi m6i con

ngu-Oi trorg xd h6i hiOn dai Vi vAY, A

nhidu nudc,

nam 1995)

- Cdc vAn dd va hdm sd mfi vd hiim sd lAgaritduoc chuydn sang l6p 12, dlLnh thdi luong cho

chuong "Dao hdm" (gdm kh6i niOm dao hhm

vd c6c c6ng thfc tinh dao hlm, chi trir c6ng

thrlc dao him cira him s6 mfi vi him sd

l6garit) Chuong trinh quy dinh khdng tlua

vio kh6i nidm dao hdm m6t b€n.

Vi0c dua dao hdm cirng vdi c6c n6i dung

-tdhqp vd xdc sudt vdo l6p 11 lI m6t.d4c didm

nham cung cdp nhfing cOng cu to6n hoc cdn

348) th :

mrlc do rdt

th gdn vdi thuc (ChI bi6n SGK Dgisd vd Giditich l l Ndng coo) day ln ldn

;;e ;dod'nhrm ndng cao rinh khi thi cira sudt dttoc dua vho chuong trinh phd th6ng

"t oo.rg trinh vi SGK ;6i ; tidp ctn thuc ti6n (khong kd ddn chuong trinh phAn ban thi didm

14

thidt chudn bi cho viOc hoc t6t mot sd m6n hoc

kh6c (nhu Vat li, Sinh hoc, H6a hoc, Dia Ii, ),

thd hien tinh lion mdn trong toin bO Chuong

trinh gido duc phd thdng

tt vi PHUoNG PHAP rdp cAN vA MOc

o0 vBu cAu

Chuong | (Hdm sd luong gidc vd phuong trinh

luong gidc) ld phdn ndi tidp cira phdn luong

gi6c l6p 10 Khi khdo sdt su bidn thi€n ctn cdc

hdm sd luong gi6c, SGK Dar sd vd Gidi tich

11 ndng cao srl dung phuong ph6p truc

quan, trlc li khio s6t su chuydn d6ng c[ra c6c

didm dd suy ra su tang-giim cfia c6c him sd

luong gi6c Phuong ph6p tinh tidn dd thi (dA

hoc & l6p 10) cflng duoc nh6c lai vi sir dung

dd vE dd thi ctra c6c hdm sd luo ng gi6c

Kh6i ni€m hdm sd tudn hodn khOng coi li nOi

dung trong ydu Do d6 SGK gi6i thiOu dinh

ngfr,a hdm sd tudn hodn rdi dua ra mOt vii vi

du nhu dd khdi qudt cdc tinh chdt dac trung vd

tinh tuen hoin ctra cdc hdm sd luong gi6c di

hoc tru6c d6.

Ya phuong trinh luong gidc, SGK d6 c6 g6ng

giim nhe c6c y€u cdu vi ki ning nhu: kh6ng

xdt cdc phuong trinh luong gi6c chita tham sd,

loai b6 ciic dang phuong trinh c5 bi6n ddi

hoac phAi x6t didu kiOn phrlc tap, khdng gidi

'r)

thiOu phuong ph6p dat dn phu ' r = tan I\2)

Vi0c dua cr{c ki hi6u arcsin, arccos, arctan v}r

arccot nham girip gi6o viOn vi hoc sinh trinh

bdy ldi grii cdc phuong trinh luo ng gi6c duo c

gon, khOng nham gidi thiOu c6c him sd luong

gi6c nguoc

Muc dich cira chuong Il (Td hop vd )fic sudt)

li dd hoc sinh lim quen v6i nhtng vdn dd don

glin c6 n6i dung td hgp thudng gap trong ddi

sdng vi khoa hoc Do d6 hdu hdt c6c vi du

trong SGK ddu duoc ldy tt thuc td cuQc sdng

Hoc sinh cdn hidu vd ph6n biOt duoc c6c kh6i

ni6m, nh6 vi vdn dung duoc c6c quy tltc, cdc

c6ng thrlc vio nhfrng bhi to6n don giin, khOng

dbi h6i suy luAn qua nhiCu budc trung gian

Dac biOt, cdn g6n v6i nhflng bii to6n thuc td

nhu vdn dd chon c6n bO l6p, vin dd didm s6,

vdn dd an tohn giao thdng, van dd din sd,

NOi dung chuong lll (Ddy sd, cdp sd c1ng vd

cdp sd nhdn) vd chuong IY (Gi6i han) vd, co

bin cfrng gidng nhu tru6c ddy Tuy nhiOn c6

nram gidi han vd cltc dttoc hidu theo nghia

gidi han hoac bang +co, hodc -oo, kh6ng ch0p

nhxn gi6i han bang co chung chung nhu trudcday Ching han, ding thrlc lim(-l)nn = a nay

kh6ng duoc chdp nhAn, mic dil n6 dfng theo

dinh nghia cira SGK2000 Sr,r thay ddi d5 c6

nhidu uu didm vd phi ho p v6i SGK cira nhidunu6c trOn thd gi6i Chac ch6n n6 sE gdy ra m6t

so kh6 khan cho gi6o vi6n b&i n6 dbi h6i gi6o

vi6n phii thay ddi trong mdt sd c6ch trinh bdy

Ngoii nhflng dinh li v6 gi6i han md c6c SGK

trudc diy thudng dd cdp, SGK cbn cung cdp

nhfrng quy tac chir ydu nham gifp hoc sinhbidt xdc dinh xem m6t gidi han vd cuc khi nio

thi bdng +"o, khi nio thi bang -.o .

Do c6 thay ddi nhu thd, gi6o vi6n cdn nghiOn

criu ki SGK dd tr6nh c6c so suat do th6i quen

cfi dd lai

Dao hdm li n6i dung kh6 quen thudc trongchuong trinh to6n phd thOng Didu d6ng luu f

Id su tldi mdi c6ch tidp cAn mdt sd vdn dr)

nhu: vi du m& ddu ddn ddn khdi ni€m dao

hdm, y nghTa hinh hoc cia dao hhm, dao him

cira hdm sd hop, vi phdn Ching han, vd f

nghia hinh hoc ctra dao hhm, SGK c5 trinh bhy

16 cdch x6c dinh ddu li "vi tri gi6i han cia cdtttrydn Mslvl khi M di chuydn doc theo dd thi

ddn ddn Ms" md trong ciic s6ch trudc dd kh6ng

dd cflp

Cdc tdc gi|l6t mong nhAn duoc c6c f kidn xaydung cria c5c thdy gii{o, cO gi6o, cdc nhi khoahoc vd ciacdc em hoc sinh dd SGK ngiy cdng

t6t hon

- GfudiUe s6 361 l7-20o7) 15

(GV THCS Hing Bitng, Hdi Phdng)

tsir- T2l36L (Lfp 7) Gqi I/ ld giao diOm ba

dulng cao ctia mQt tam gi6c nhqn ABC vd M

lir trung di6m canh BC' Dudng thing vg6ng

g6c v6i MH b H cbt hai duhng thdng AB vit

AC tuong ring o P vd Q Chring minh ring

HP = IIQ

NGUYEN MINH HA

(GV DHSP Hd N1i)

Bni T3/361 Chring minh ring n6u a, b, c, d

ld c5c sl5 nguy6n duong ddi m6t khSc nhau

sao cho bi6u thric -+.+*J-* a+b b+c c+d d+ad

li mQt sO nguy6n thi tich abcd litmQt s5 chinh

x+y+z+-+-!-=- 'xyz9 rr-1111728

Y\l Y zxlz LtDANG QUdC CHA}I

(GV THCS Nam Son, Nam Trqc, Nam Dlnh)

Bni T61361 Cho hinh binh hdnh ABCD' Di€'m

M nim ctng m[t Phing vli ABCD sao cho

frD) = fril Chirng minh ring hai tam giSc

MAB vit MCD c6 cirng trYc t6m'

NGUYEN DANG KHOI

(GV THCS Bach LiAu, YAn Thdnh, NghQ An)

Be[ T71361 Gqi AD, BE, CF theo thir t'u ld c6c

duhng ph6n gi6c trong tam gi6c ABC (D e BC'

E e CA, F e AB); O vi R theo thrl tU le tam

vd b6n kinh dudng trdn ngopi ti6p tam gi6c

d6 Gqi Ot, Oz vd O: theo thf tu li tim dudng

trdn ngopi ti6p c5c tam giSc ABD, BCE vit

Bni T91361 Tim tAt ch cdc him s5 I R -+ Rthoa mdn di6u kiQn

lU' - y) + 2yQf2@) + Y') : 11Y + flx))

DUONG CHAU DINH

(GV THPT chuvAn LA Qu! D6n' Qudng Tri)