S gio dc v o to kon tum

TRNG THPT DUY TN

(((((

ti

MC LC

A. M U......................................Trang 1I. L do chn ti................................................................................1II. Nhim v, phm vi v i tng nghin cu....................................11. Nhim v nghin cu........................................................................12. Phm vi nghin cu...........................................................................13. i tng nghin cu........................................................................14. Phng php nghin cu.................................................................. 2 B. KT QU NGHIN CU .............................................3I. Thc trng ca vn trc khi nghin cu......................................3II. Gii php khc phc..........................................................................3Dng1Tm GTLN, GTNN ca hm hai bin bng phng php th....4Dng 2. Tm GTLN, GTNN ca biu thc c tnh i xng...............6Dng 3. Tm GTLN, GTNN ca biu thc c tnh ng cp.............10Bi tp tng hp..................................................................................12Bi tp t luyn...................................................................................18III. Phn tch so snh tc dng ca ti............................................19 C. KT LUN.................................................20CHUYN - SNG KIN KINH NGHIMGIP HC SINH TM GI TR LN NHT V GI TR NH NHT BNG PHNG PHP DN BIN

A. M UI. L do chn tiTi chn ti Gip hc sinh tm gi tr ln nht v gi tr nh nht bng phng php dn bin vi nhng l do sau y:

- y l vn kh i vi a s hc sinh.- y l vn thng gp trong cc k thi cao ng, i hc v thi hc sinh gii cc cp.- y l vn hay, hp dn kch thch t duy hc sinh.

- y l chuyn gp phn tch cc cho hc sinh vo vic rn luyn t duy logich, linh hot, sng to. Mt yu cu quan trng trong xu hng i mi gio dc hin nay.- y l chuyn gp phn gio dc hc sinh thy c ci hay ci p quyn r ca ton hc to s hng th say m vi mn ton.II. Nhim v, phm vi v i tng nghin cu

1. Nhim v nghin cu

- Gip hc sinh nm c phng php dn bin tm gi tr ln nht v gi tr nh nht ca hm nhiu bin.- Chuyn ny gp phn tch cc cho hc sinh vo vic rn luyn t duy logich, linh hot, sng to.

- Chuyn ny gp phn gio dc hc sinh thy c ci hay ci p quyn r ca ton hc to s hng th say m vi mn ton.2. Phm vi nghin cu

- Tm GTLN v GTNN ca hm 2 bin, hm 3 bin c iu kin rng buc i vi mt s dng c bn thng gp trong cc k thi i hc, cao ng v thi hc sinh gii.

3. i tng nghin cu

Hc sinh kh, gii trng THPT Duy tn qua cc nm hc 2011-2012; 2012-2013.4. Phng php nghin cu4.1.Quan st, iu tra kho st thc trng.4.2 Tng kt kinh nghim, tham kho ti liu, t liu lin quan chuyn xy dng gii php.4.3 Trin khai chuyn .4.4 Kim tra, phng vn thng k nh gi kt qu.B. KT QU NGHIN CUI.Thc trng ca vn trc khi nghin cu.

Hc sinh ch bit tm GTLN v GTNN ca hm s 1 bin trong cc trng hp c bn.

Hc sinh lng tng, khng bit tm GTLN v GTNN ca hm s 2 bin , 3 bin v c iu kiu kin rng buc, trong cc trng hp c bn thng gp.

Hc sinh khng bit dng phng php dn bin a hm nhiu bin v hm mt bin gii quyt.

II. Gii php khc phc.- Gip hc sinh nm c phng php Dn bin a hm nhiu bin v hm mt bin ri dng o hm kho st.

- H thng tng dng bi tp c bn thng gp.

- La chn cc v d mu, c bn, in hnh i vi mi dng.- Phn tch, nh hng gii cho mi v d. - Trnh by li gii chnh thc cho mi v d.- Rt ra phng php chung cho tng dng.- Ra bi tp cng c, luyn tp cho hc sinh qua tng dng.- Ra cc bi tp mang tnh vn dng tng hp cho hc sinh vn dng. Gii php c c th ha qua cc dng, cc v d v bi tp sau y: Dng 1. Tm GTLN, GTNN ca hm hai bin bng phng php th VD1. Cho x, y l hai s thc dng tha mn . Tm GTNN ca biu thc .Phn tch: - Ta thy l hm theo hai bin x v y. Tuy nhin hai bin ny rng buc vi nhau theo quan h bc nht i vi x, y: , nn ta d dng rt c y theo x (biu din y theo x) v th vo khi a v hm theo mt bin x: .

- T gi thit, ta tm min bin thin ca bin x , gi s .

- Dng o hm kho st hm: , vi .

- T suy ra c kt qu cn tm.Gii

T gi thit, ta rt: , th vo P ta c:

T gi thit ta suy ra c:

Dng o hm, lp bng bin thin ca hm s :trn khong

T bng bin thin suy ra:

VD2. Cho hai s thc x,y tha mn . Tm GTNN v GTLN ca biu thc .Phn tch:

- Ta thy l hm theo hai bin x v y. Tuy nhin t gi thit ta d dng rt c y theo x (biu din y theo x) v th vo khi a v hm theo mt bin x: .

- T gi thit, ta tm min ca bin x , gi s .

- Dng o hm kho st hm: , vi .

- T suy ra c kt qu cn tmGii

T gi thit suy ra:

Do :

T gi thit:

Dng o hm, kho st hm s: trn on . T tm c kt qu:

Dng 2. Tm GTLN, GTNN ca biu thc c tnh i xng. VD 1. Cho x, y l hai s thc khng m tha mn . Tm GTLN, GTNN ca biu thc:

Phn tch:

- Ta thy biu thc l i xng i vi 2 bin x v y v biu thc iu kin rng buc cng i xng i vi 2 bin x v y nn ta lun biu din c n v theo hai biu thc i xng c bn l :.- Mt khc theo gi thit ta c tng: nn th vo biu thc ta a c v hm mt bin: , vi

- Da vo gi thit, ta tm c min ca bin bng cch dng bt ng thc (ch rng du = y xy ra c)- Dng o hm kho st hm trn on - T ta tm c .

Gii: Ta c:

M nn

t ta a v hm theo mt bin l

By gi ta cn tm min bin thin ca nh sau: ta c . Vy:

Dng o hm ta tm c GTLN v GTNN ca hm , trn on Do :

VD2. Cho x, y l hai s thc tha mn . Tm GTLN, GTNN ca biu thc

Gii: t

Khi :

Ta c: hm s ng bin

Suy ra:

VD3: Cho x, y l hai s thc tha mn . Tm GTLN, GTNN ca biu thc

Phn tch:

Ta thy biu thc cn tm GTLN, GTNN v biu thc rng buc cc bin gi thit u c tnh i xng i vi hai bin x v y. Do ta biu din c chng v theo hai biu thc i xng c bn l :.

Chn l mt trong hai bin hoc khi ta a c v hm theo mt bin . (dn bin)

T iu kin rng buc ta tm min bin thin ca bin . Gi s

- Dng o hm kho st hm trn min

Suy ra GTLN, GTNN ca P

Gii:

Bc 1. Chn bin . Khi

Chuyn P sang hm theo bin t:

(Chn t=x+y hoc t=xy. y chn t=x+y thun tin hn)Bc 2. Tm min ca bin t : T rng buc:

S dng bt ng thc:

Ta suy ra: . Gii ra c:

Bc 3. Tm max, min ca hm f(t) trn min

Ta c:

Suy ra hm f(t) ng bin Vy: VD4: Cho x, y l hai s thc khng m tha mn . Tm GTLN, GTNN ca biu thc

Phn tch:

Ta thy biu thc cn tm GTLN, GTNN v biu thc rng buc cc bin gi thit u c tnh i xng i vi hai bin x v y. Do ta biu din c chng v theo hai biu thc i xng c bn l :.

Chn l mt trong hai bin hoc khi ta a c v hm theo mt bin . (dn bin)

T iu kin rng buc ta tm min bin thin ca bin . Gi s

- Dng o hm kho st hm trn min

Suy ra GTLN, GTNN ca P

Gii:

Bc 1. Chn bin . Khi t gi thit suy ra , suy ra:

Chuyn P sang hm theo bin t:

(Chn t=x+y hoc t=xy. y chn t=x+y thun tin hn)Bc 2. Tm min ca bin t : S dng bt ng thc:

( Ch : nh gi Sao cho ? v ?? l hng s hoc biu thc cha bin t nhng ch du ng thc phi xy ra c)

Bc 3. Tm max, min ca hm f(t) trn min

Ta c: hm s ng bin

Suy ra:

VD5 Cho cc s thc khng m tho mn .

Tm gi tr ln nht ca biu thc .

Phn tch: Dng hng ng thc ta chuyn i c biu thc theo hai biu thc v . T t

Th a c P v hm theo mt bin

Gii:

t

EMBED Equation.DSMT4 .

Chuyn P v hm theo mt bin :

Tm mim ca bin : T iu kin rng buc:

S dng bt ng thc: , ta suy ra: (v )

Kho st hm :

Ta c (v )Suy ra ng bin trn . Do

Du ng thc xy ra khi

Kt lun:

Dng 3. Tm GTLN, GTNN ca biu thc c tnh ng cp. VD 1. Cho x, y,z l ba s thc thuc on v . Tm GTNN ca biu thc

(Trch thi i hc khi A nm 2011)

Phn tch: Ta thy P l tng ca ba phn thc m t v mu u l cc a thc dng ng cp bc nht theo cc bin. Do chia c t v mu cho mt bin bc nht ta chuyn P v theo ba bin mi , , : C th l:

Dng bt ng thc: vi a,b>0 v

Ta dn c P v hm theo mt bin nh sau:

T ta gii quyt tip!Gii

Trc ht ta chng minh: (*) , vi a,b>0 v

Tht vy, ta c: : ng vi a,b>0 v

ng thc xy ra khi v ch khi a=b hoc ab=1

p dng (*) vi x v y thuc on v

Ta c: ,

ng thc xy ra khi v ch khi hoc

t Khi

Xt hm

. ng thc xy ra khi v ch khi

. T (1) v (2) suy ra ng thc xy ra khi v ch khi x=4, y=1 v z=2

Vy GTNN ca P bng khi x=4, y=1 v z=2

Bi tp tng hp

1) Cho x, y l hai s thc tha mn . Tm GTLN, GTNN ca biu thc

Gii Ta c:

t:

EMBED Equation.DSMT4 . Khi :

Tm min ca t:

Kho st hm: , vi

Ta c: hm s ng bin

Suy ra:

2) Cho x, y l cc s thc thay i tha mn iu kin . Tm gi tr ln nht v gi tr nh nht ca biu thc:

GiiTa c:

t

Vy

.

EMBED Equation.DSMT4 3) Cho x,y( R v x,y>1. Tm gi tr nh nht ca

GiiBin i:

t: . V

Ta c:

p dng BT: , ta c:

Mt khc:

nn:

Kho st hm s: trn min , ta c kt qu:

, t c khi

4) Cho hai s thc thay i tha mn . Tm GTLN ca biu thc

(Trch thi i hc khi A nm 2006)

Gii.

Ta c:

t . Ta c:

t

Khi :

Tm min ca t:

Suy ra: . Vy:

5) Cho . Chng minh:

(Trch thi i hc khi A nm 2003)Gii

t

Ta c:

t: , suy ra:

Tm min ca : . Vy:

Kho st hm: , vi ta d dng suy ra c:

. Do : (pcm)6) Cho ba s thc a, b, c tha: . Tm GTNN ca

Gi Chn bin:

a v hm mt bin: , vi

7) Cho x,y,z l cc s thc khng m tha x+y+z=1. Tm GTLN ca biu thc : P=xy+yz+zx-2xyz. (Thi th i 2012-2013. Trng THPT Kon Tum)

Gii

V vai tr ca x, y, z trong bi ton bnh ng nn c th gi s

Mt khc x+y+z=1 nn say ra

Ta c: m . Suy ra:

Kho st hm:

Ta tm c , khi .8) Cho x,y,z l cc s thc khng m tha mn x+y+z=1. Tm GTNN ca biu thc

Gii. Vai tr x, y z bnh ng nn ta c th gi s .

T gi thit suy ra: v y+z=1-x. p dng BT

Kho st hm: , vi

Ta tm c: , khi hoc hoc cc hon v ca n.9) Cho x, y l cc s thc khng m thay i v tha mn iu kin . Tm gi tr ln nht ca biu thc

(Trch thi th i hc nm 2012-2013. Trng THPT chuyn Nguyn Tt Thnh, Kon Tum)

Gii.

, vi .

T gi thit, ta bin i :

EMBED Equation.DSMT4 suy ra:

Hay:

T :

Mt khc x, y khng m nn

By gi ta i tm max ca , trn min

Bng cch dng o hm v kho st hm f(t) tra c , khi .10) Cho cc s thc dng . Tm gi tr nh nht ca biu thc:

(Trch thi hc sinh gii Ton 12, bng A, tnh Ngh An, nm 2012-2013 )

Gii. p dng bt ng thc AG-MG, ta c:

.

ng thc xy ra khi v ch khi .

Suy ra:

t: . Khi ta c:

Kho st hm s: , vi

Ta tm c: , khi v ch khi

Vy ta c , ng thc xy ra khi v ch khi .

Vy gi tr nh nht ca P l khi v ch khi .

11) Cho . Tm GTLN ca biu thc

GiiTa c:

Gi s: , suy ra: v

Suy ra:

t vi

, vi

Dng o hm kho st hm : , vi

Ta thy: suy ra khi

12) Cho v . Tm GTLN ca

Gii. Vai tr x, y , z trong bi ton bnh ng nn c th gii s . Khi v

ng thc xy ra khi y=1 hoc z=1Xt hm s : , vi

Ta d dng tm c GTLN ca bng 14 khi x=2. Suy ra , chng hn khi x=2, y=1, z=3

Bi tp t luyn

1) Cho . Tm GTNN ca biu thc (S: , khi )2) Cho . Tm GTLN v GTNN ca biu thc (S:)

3) Cho hai s thc x, y tha mn . Tm max, min ca

4) Cho . Tm GTNN ca biu thc (S:, khi )

5) Cho x, y khng m tha . Tm GTLN ca biu thc (S:, khi )

6) Cho . Tm GTLN v GTNN ca biu thc (S:, )

7) Cho x,y,z l cc s thc khng m tha mn x+y+z=1. Chng minh:

8) Cho v . Tm GTLN ca (S: )9) Cho v . Tm GTLN ca

10) Cho x, y l cc s thc tha mn . Tm gi tr nh nht ca biu thc .III. Phn tch so snh tc dng ca ti

Qua qu trnh trin khai chuyn ti thu c mt s thng tin tch cc t pha hc sinh nh sau:

Hc sinh nm c phng php dn bin, vn dng v gii c mt s dng c bn thng gp. Hc sinh rt phn khi thch th , t tin khng cn tm l e ngi nh trc Hc sinh say m, yu thch hc ton nhiu hn. Kt qu kim tra kho st nm hc 2011-2012

LpS sKt qu ban uKt qu th nghimVt so vi ch tiu

GKTBYKGKTBYK

12A431312252310273024,5%

Kt qu kim tra kho st nm hc 2012-2013

LpS sKt qu ban uKt qu th nghimVt so vi ch tiu

GKTBYKGKTBYK

12A1392413200513192030,1%

C. KT LUNTrn y l mt s kinh nghim v gii php ca bn thn ti rt ra c qua qu trnh dy hc v luyn thi cho hc sinh lp 12 trong cc nm hc qua v vn tm gi tr ln nht v gi tr nh nht. Qua chuyn ny cc em hc sinh nm c phng php dn bin tm gi tr ln nht v gi tr nh nht, cc em t tin hn, say m hng th hn vi ton hc v t bit l gip cc em c thm c hi nng cao kt qu trong cc k thi i hc cao ng.

gii quyt vn ny chc chn cn phng php khc v gii php khc tt hn , rt mong c hc hi qu thy c gio v cc bn ng nghip.Chuyn ny khng th trch khi nhng thiu st. Rt mong s gp xy dng ca qu thy c gio. Kon tum , ngy 20 thng 12 nm 2012

Ngi thc hin

Hunh Vn Minh

NH GI V XP LOI TI

nh giv xp loi ca hi ng khoa hc trng THPT Duy Tn

nh giv xp loi ca Hi ng khoa hc s GD&T Kon Tum Sang kin kinh nghim

GIP HC SINH

TM GI TR LN NHT

V GI TR NH NHT

BNG PHNG PHP DN BIN

GV. Hunh Vn Minh

T : Ton Tin

Nm hc : 2012 - 2013

Kon Tum 10 / 2013

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