CACDANGTOANLIENQUANKSHS

8/4/2019 CACDANGTOANLIENQUANKSHS

1/15

Cc dng ton lin quan n Kho st hm s

Trn Duy Thi 1

CC DNG TON LIN QUAN NKHO ST HM S

Dng 1: CC BI TON V TIP XC

Cho hm s xfy , th l (C). C ba loi phng trnh tip tuyn nh sau:

Loi 1: Tip tuyn ca hm s ti im 0 0;M x y C .

Tnh o hm v gi tr 0'f x .

Phng trnh tip tuyn c dng: 0 0 0'y f x x x y .

Ch :Tip tuyn ti im 0 0;M x y C c h s gc 0'k f x .

Loi 2: Bit h s gc ca tip tuyn l k. Giiphng trnh: ' f x k , tm nghim 0 0x y .

Phng trnh tip tuyn dng: 0 0y k x x y .

Ch :Cho ng thng : 0 Ax By C , khi :

Nu // :d d y ax b h s gc k= a.

Nu :d d y ax b h s gc1

ka

.

Loi 3:Tip tuyn ca (C) i qua im ;A AA x y C .

Gi dl ng thng quaA v c h s gc l k, khi : A Ad y k x x y

iu kin tip xc ca d v C l hphng trnh sau phi c nghim:

'A Af x k x x y

f x k

Tng qut:Cho hai ng cong :C y f x v ' :C y g x . iu kin hai ng cong tip xc vi

nhau l h sau c nghim.

' '

f x g x

f x g x

.

1. Cho hm s 22 y x x a. kho st v v th (C) ca hm s.b. Vit phng trnh tip tuyn ca (C):

i. Ti im c honh 2x .ii. Ti im c tung y = 3.iii.Tip tuyn song song vi ng thng: 1 : 24 2009 0d x y .iv.Tip tuyn vung gc vi ng thng: 2 : 24 2009 0d x y .

2. Cho hm s 2 31

x xyx

c th l (C).

a. Kho st v v th (C) ca hm s trn.b. Vit phng trnh tip tuyn ca (C):

i. Ti giao im ca (C) vi trc tung.ii. Ti giao im ca (C) vi trng honh.

iii. Bit tip tuyn i qua imA(1;1).iv. Bit h s gc ca tip tuyn k= 13.

3. Cho hm s 2 11

x xy

x

c th (C).


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Trn Duy Thi 2

a. Kho st v v th (C) ca hm s trn.b. Vit phng trnh tip tuyn ca (C) ti imx = 0.c. Vit phng trnh tip tuyn ca (C) ti im c tung y = 0.d. Tm tt c cc im trn trc tung m t k c hai tip tuyn n (C).

4. Cho hm s 2 3 31

x xy

x

c th (C).

a. Kho st v v th hm s (C).b. Chng minh rng qua imM(3;1) k c hai tip tuyn ti th (C) sao cho hai tip tuyn

vung gc vi nhau.

5. Cho hm s: 21

xy

x

c th (C).

a. Kho st v v th hm s.b. TmM(C) sao cho tip tuyn ca (C) tiMvung gc vi ng thng i quaMv tmi xng ca (C).

6. Cho hm sy =x3 + mx2 + 1 c th (Cm). Tm m (Cm) ct d:y = x + 1 ti ba im phn bitA(0;1),B, Csao cho cc tip tuyn ca (Cm) tiB v Cvung gc vi nhau.Li gii:Phng trnh honh giao im ca dv (Cm) l:x

3+ mx

2+ 1 = x + 1 x(x2 + mx + 1) = 0 (*)

t g(x) =x2 + mx + 1 . dct (Cm) ti ba im phn bit g(x) = 0 c hai nghim phn bit khc 0.

2 4 0 2

20 1 0

g m m

mg

.

VxB ,xC l nghim ca g(x) = 01

B C

B C

S x x m

P x x

.

Tip tuyn ca (Cm) tiB v Cvung gc vi nhau nn ta c: 1C B f x f x

3 2 3 2 1 B C B C x x x m x m 29 6 4 1 B C B C B C x x x x m x x m

21 9 6 4 1m m m 22 10m 5m (nhn so vi iu kin)

7. Cho hm s 2 1xyx

. Tm tp hp cc im trn mt phng ta t c th k n (C) hai tip

tuyn vung gc.

Li gii:

GiM(x0;y0). Phng trnh ng thng dquaMc h s gc kly = k(x x0) +y0.

Phng trnh honh giao im ca (C) v d: 2

0 0

1, 0

xk x x y kx

x

2 0 01 1 0 *k x y kx x

dtip xc vi (C):

2

0 0

1

4 1 0

k

y kx k

2 2 20 0 0 0

0 0

1

2 2 4 0 I

k

x k x y k y

y kx

TMv hai tip tuyn n (C) vung gc vi nhau khi (1) c hai nghim phn bit tha mn: 1 2

1 2

, 11

k kk k

0

2

0

20

2

0 0

0

41

0

x

y

x

y x

0

2 20 0

0 0

0

4

x

x y

y x

.

Vy tp hp cc im tha mn yu cu bi ton l mt ng trn: 2 2 4x y loi b bn giao im cang trn vi hai ng tim cn.


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Trn Duy Thi 3

8. Cho hm s 21

xy

x

. (H KhiD 2007)

a. Kho st s bin thin v v th ca hm s cho.b. Tm ta imM thuc (C), bit tip tuyn ca (C) ti Mct Ox, Oy ti A,B v din tch tam gic

OAB bng1

4

S:1

; 2

2

M

v 1;1M .

9. Cho hm s 2 12

x xy

x

. (H KhiB 2006)

a Kho st s bin thin v v th (C) ca hm s cho.b. Vit phng trnh tip tuyn vi th (C) bit tip tuyn vung gc vi tim cn xin.

S: b. 2 2 5y x .

10. Gi (Cm) l th ca hm s: 3 21 13 2 3

m y x x (*) (m l tham s). (H KhiD 2005)

a. Kho st s bin thin v v th ca hm s (*) khi m=2.b. GiMl im thuc (Cm) c honh bng 1. Tm m tip tuyn ca (Cm) tiMsong song ving thng 5 0x y

S: m=4.

11. Cho hm s 3 23 3 m y x mx x m C . nh m mC tip xc vi trc honh.12. Cho hm s 4 3 21 my x x m x x m C . nh m mC tip xc vi trc honh.13. Cho th hm s 2 4:

1

xC y

x

. Tm tp hp cc im trn trc honh sao cho t k c mt tip

tuyn n (C).

14. Cho th hm s 3 2: 3 4C y x x . Tm tp hp cc im trn trc honh sao cho t c th kc 3 tip tuyn vi (C).

15. Cho th hm s 4 2: 2 1C y x x . Tm cc imMnm trn Oy sao cho tMk c 3 tip tuynn (C).

16. Cho th hm s 3: 3 2C y x x . Tm cc im trn ng thngy = 4 sao cho t c th kc 3 tip tuyn vi (C).17. Cho hm sy = 4x3 6x2 + 1 (1) (H KhiB 2008)

a. Kho st s bin thin v v th ca hm s (1).b. Vit phng trnh tip tuyn ca th hm s (1), bit rng tip tuyn i qua imM(1;9).

Li gii:a.D=R,y = 12x

2 12x;y = 0 x = 0 hayx = 1.

BBT :

b. Tip tuyn quaM(1;9) c dngy = k(x + 1) 9.Phng trnh honh tip im quaMc dng :

4x3

6x2

+ 1 = (12x2

12x)(x + 1) 9.

4x3 6x2 + 10 = (12x2 12x)(x + 1) 2x3 3x2 + 5 = 6(x2 x)(x + 1).x = 1 hay 2x2 5x + 5 = 6x2 6x x = 1 hay 4x2 x 5 = 0.

x = 1 hayx =5

4;y(1) = 24;

5 15'

4 4y

.

Vyphng trnh cc tip tuyn quaMl:y = 24x + 15 hayy =15

4x

21

4 .

x 0 1 +y' + 0 0 +y 1 +

1C CT

-1 1

-2

2

x

y

3

2

4

6

1

y

x

x


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Trn Duy Thi 4

Dng 2: CC BI TON V CC TR

Cho hm s xfy , th l (C). Cc vn v cc tr cn nh:

Nghim ca phng trnh ' 0f x l honh ca im cc tr.

Nu

0

0

' 0

'' 0

f x

f x

th hm s t cc i ti 0x x .

Nu

0

0

' 0

'' 0

f x

f x

th hm s t cc tiu ti 0x x .

Mt s dng bi tp v cc tr thng gp

hm s y f x c 2 cc tr'

0

0y

a

.

hm s y f x c hai cc tr nm v 2 pha i vi trc honh . 0C CT y y .

hm s y f x c hai cc tr nm v 2 pha i vi trc tung . 0C CT x x .

hm s y f x c hai cc tr nm pha trn trc honh 0. 0

C CT

C CT

y yy y

.

hm s y f x c hai cc tr nm pha di trc honh0

. 0

C CT

C CT

y y

y y

.

hm s y f x c cc tr tip xc vi trc honh . 0C CT y y .

Cch vit phng trnh ng thng i qua hai im cc tr.

Dng 1: hm s 3 2 y ax bx cx d Lyy chia choy, c thng l q(x) v d l r(x). Khi y = r(x) l ng thng i qua 2 im cc tr.

Dng 2: Hm s2

ax bx cydx e

ng thng qua hai im cc tr c dng

2 ' 2

'

ax bx c a by x

dx e d d

1. Chng minh rng hm s y = 2 2 41 1 x m m x m

x m

lun c c cc tr vi mi m. Tm m sao cho hai

cc tr nm trn ng thngy=2x.

2. Cho hm s 3 21 2 13

y x mx m x . nh m :

a.Hm s lun c cc tr.b.C cc tr trong khong 0; .c.C hai cc tr trong khong 0; .

3. nh m hm s 3 2 2 23 1 2 4 y x mx m x b ac t cc i tix = 2.4. Cho hm s y =x33x2+3mx+3m+4.

a.Kho st hm s khi m = 0.b.nh m hm s khng c cc tr.c.nh m hm s c cc i v cc tiu.

5. Cho hm s 3 23 9 3 5 y x mx x m . nh m th hm s c cc i cc tiu, vitphng trnhng thng i qua hai im cc try.


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Trn Duy Thi 5

6. Cho hm s 2 1 1 x m x myx m

. Chng minh rng th hm s lun c cc i, cc tiu vi mi

m. Hy nh m hai cc tr nm v hai pha i vi trc honh.

7. Cho hm s 3 21 2 2 2 y x m x m x m . nh m th hm s c hai cc tr ng thihonh ca im cc tiu nh hn 1.

8. Cho hm s 2 22 1 3 x mx myx m

. nh m th hm s c hai cc tr nm v hai pha i vi trc

tung.

9. Cho hm s 3 21 2 1 23

m y x mx m x m C . nh m hm s c hai im cc tr cng dng.

10.Cho hm s 2 22 1 42

x m x m my

x

(1). (H KhiA nm 2007)

a. Kho st s bin thin v v th ca th hm (1) s khi m=1.b. Tm m hm s (1) c cc i v cc tiu, ng thi cc im cc tr ca th cng vi gc ta O to thnh tam gic vung ti O.

S: 4 2 6m .

11.Cho hm s 3 2 2 23 3 1 3 1 y x x m x m (1), m l tham s. (H KhiB nm 2007)a. Kho st s bin thin v v th ca th hm (1) s khi m=1.b. Tm m hm s (1) c cc i, cc tiu v cc im cc tr ca th hm s (1) cch u gc ta

.

S : b1

2m .

12.Cho hm s 4 2 29 10 y mx m x (1) (m l tham s).a. Kho st s bin thin v v th ca th hm s khi m=1.b. Tm m th hm s (1) c ba im cc tr. (H KhiB nm 2002)

a.

-5 5

-5

5

10

x

y

b. S :3

0 3

m

m

13. Gi (Cm) l th ca hm s 2 1 11

x m x my

x

(*) (m l tham s)

a. Kho st s bin thin v v th ca th hm s khi m=1.b. Chng minh rng vi m bt k, th (Cm) lun c hai im cc i, cc tiu v khong cch gia

hai im bng 20 .


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Trn Duy Thi 6

a.

-4 -2 2

-2

2

4

x

y

b. C(2;m3), CT(0;m+1) 20MN

Dng 3: CC BI TON V NG BINNGHCH BIN

Cho hm s xfy c tp xc nh l minD.

f(x) ng bin trnD Dxxf ,0' .f(x) nghch bin trnD Dxxf ,0' .(ch xt trng hpf(x) = 0 ti mt s hu hn im trn minD)

Thng dng cc kin thc v xt du tam thc bc hai: 2f x ax bx c .1. Nu 0 thf(x) lun cng du vi a.

2. Nu 0 thf(x) c nghim2

bx

a vf(x) lun cng du vi a khi

2

bx

a .

3. Nu 0 thf(x) c hai nghim, trong khong 2 nghimf(x) tri du vi a, ngoi khong 2 nghimf(x) cngdu vi a.

So snh nghim ca tam thc vi s 0

* 1 2

0

0 0

0

x x P

S

* 1 2

0

0 0

0

x x P

S

* 1 20 0 x x P

1. Cho hm s 3 23 1 3 1 1 y x m x m x . nh m :a. Hm s lun ng bin trnR.b. Hm s lun ng bin trn khong 2; .

2. Xc nh m hm s3 2

2 13 2

x mxy x .

a. ng bin trnR.b. ng bin trn 1; .

3. Cho hm s 3 23 2 1 12 5 2 y x m x m x .

a. nh m hm s ng bin trn khong 2; .

b. nh m hm s nghch bin trn khong ; 1 .

4. Cho hm s2 6 2

2

mx xy

x

. nh m hm s nghch bin trn ;1 .


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Trn Duy Thi 7

Dng 4: CC BI TON V GIAO IM CA 2 NG CONG

Quan h gia s nghim v s giao im

Cho hai hm sy=f(x) c th (C1) vy=g(x) c th (C2). Kho st s tng giao gia hai th

(C1) v (C2) tng ng vi kho st s nghim ca phng trnh:f(x) = g(x) (1). S giao im ca (C1) v

(C2) ng bng s nghim ca phng trnh honh giao im (1).

(1) v nghim (C1) v (C2) khng c im chung.

(1) cn nghim (C1) v (C2) cn im chung.

(1) cnghim nx1 (C1) v (C2)ct nhau tiN(x1;y1).

(1) cnghim kpx0 (C1)tip xc (C2) tiM(x0;y0).

1. Cho hm s 2

1

1

xy

x

c th l (C).

a.Kho st v v th ca hm s.b.Bin lun theo m s nghim ca phng trnh 2 2 1 0 x m x m .

2. Cho hm s 2 21 1 y x x c th l (C).a. Kho st v v th hm s trn.

b. Dng th (C) bin lun theo m s nghim ca phng trnh 2

21 2 1 0x m .

3. Cho hm s 3 2 4 y x kx .a. Kho st hm s trn khi k= 3.

b. Tm cc gi tr ca k phng trnh 3 2 4 0 x kx c nghim duy nht.

4. Cho hm s 3 3 2 y x x . (H KhiD 2006)a. Kho st s bin thin v v th (C) ca hm s cho.b. Gi dl ng thng i qua imA(3;20) c h s gc m. Tm m ng thng dct th (C) ti

ba im phn bit.

S: b. 15 , 244

m m .

5. Cho hm s

2 3 3

2 1

x xy

x

(1) (H KhiA 2004)

a. Kho st hm s (1).b. Tm m ng thngy=m ct th hm s (1) ti hai imA,B sao choAB=1.

S: b.1 5

2m

.

6. Cho hm s 21

mx x my

x

(*) (m l tham s) (H KhiA 2003)

a. Kho st s bin thin v v th ca th hm s khi m=1.b. Tm m th hm s (1) ct trc honh ti hai im phn bit v hai im c honh dng.

S: b.1

02

m .

7. a. Kho st s bin thin v v th ca hm s 2 2 42

x xy

x

(1). (H KhiD 2003)

b. Tm m ng thng : 2 2md y mx m ct th hm s (1) ti hai im phn bit.S: m>1.


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Trn Duy Thi 8

8. Cho hm sy = x3 + 3mx2 + 3(1 m2)x + m3m2 (1) (m l tham s) (H KhiA 2002)a. Kho st s bin thin v v th ca hm s (1) khi m = 1.b. Tm k phng trnh x3 + 3x2 + k3 3k2 = 0 c 3 nghim phn bit.c. Vit phng trnh ng thng i qua hai im cc tr ca th hm s (1).

S: b.1 3

0 2

k

k k

, c. 22 y x m m .

Dng 5: CC BI TON V KHONG CCH

Cc cng thc v khong cch:

Khong cch gia hai im ( di on thng): 2 2

B A B A AB x x y y .

Khong cch t mt im n mt ng thng: Cho ng thng : 0 Ax By C v im

M(x0;y0) khi 0 0

2 2,.

Ax By C d M

A B

.

1. Cho hm s 3 23 3 3 2 m y x mx x m C . nh m mC c cc i cc tiu ng thi khongcch gia chng l b nht.2. Cho hm s 2 2:

1

xC y

x

. Tm ta cc imMnm trn (C) c tng khong cch n hai tim cn l

nh nht.

3. Cho hm s 2 1:1

x xC y

x

. Tm cc im M thuc (C) c tng khong cch n 2 tim cn l nh

nht.

4. Cho hm s 2 2:1

xC y

x

. Tm hai imM, Nthuc hai nhnh khc nhau ca (C) sao cho on MN

nh nht.

5. Cho hm s 2 1:1

x xC y

x

. Tm hai imM, Nthuc 2 nhnh khc nhau ca (C) sao cho onMN

nh nht.

6. Cho hm s 2 2 1:1

x xC y

x

.

a.Tm cc im thuc th (C) c tng khong cch n hai trc ta l nh nht.b.Tm hai imM,Nthuc hai nhnh khc nhau ca (C) sao cho onMNnh nht.

7. Gi (Cm) l th ca hm s: 1 y mxx

(*) (m l tham s) (H KhiA 2005)

a. Kho st s bin thin v v th ca hm s (*) khi m =1

4.

b. Tm m th hm s (*) c cc tr v khong cch t im cc tiu ca (Cm) n tim cn xin

bng1

2

. S: m=1.

Dng 6:CC IM C NH

Phng php:T hm s ,y f x m ta a v dng , ,F x y mG x y . Khi ta im c nh nu c l

nghim ca h phng trnh

, 0

, 0

F x y

G x y

.


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Trn Duy Thi 9

1. Cho hm s 3 23 1 3 2 m y x m x mx C . Chng minh rng mC lun i qua hai im c nhkhi m thay i.

2. Cho hm s 22 6 4:2

m

x m xC y

mx

. Chng minh rng th mC lun i qua mt im c nh

khi m thay i.

3. Cho hm s 4 2: 1 2 3 1mC y m x mx m . Tm cc im c nh ca h th trn.4. Chng minh rng th ca hm s

3 23 3 3 6 1 1

m

y m x m x m x m C lun i qua ba

im c nh.

Dng 7: TH CHA DU GI TR TUYT I

y =f(x) c th (C) y f x c th (C) y f x c th (C)

0, y f x x D . Do ta phi

gi nguyn phn pha trn trc Ox v lyi xng phn pha di trc Ox ln trn.

y f x c f x f x ,x D nn y l hm s chn do

c th i xng qua trc tungOy.

x

y

(C)

x

y

(C')

x

y

(C'')

Ch :i vi hm hu t

1. Cho hm s 2:2 2

x xC y

x

.

a.Kho st hm s.b.nh kphng trnh sau c bn nghim phn bit. 2

2 2

x xk

x

.

-2 2 4

-2

2

4

6

x

y

2

2 2

x xy

x

-2 2

-2

2

4

x

y

2

2 2

x xy

x


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Trn Duy Thi 10

2. Cho hm s 2 3 3:1

x xC y

x

.

a.Kho st v v th hm s.b.Bin lun theo m s nghim ca phng trnh: 2 3 3

1

x xm

x

.

-4 -2 2

-2

2

4

x

y

23 3

1

x xy

x

-4 -2 2

-2

2

4

x

y

23 3

1

x xy

x

3. Cho hm s 24:1

x xC y

x

.

a.Kho st hm s.b.nh m phng trnh 2 4 0 x m x m c bn nghim phn bit.

-2 2

-2

2

4

x

y

24

1

x xy

x

-2 2

-2

2

4

x

y

24

1

x xy

x

4. Cho hm s 2 1:2

x xC y

x

.

1. Kho st hm s.2. nh m phng trnh sau c hai nghim phn bit: 2 1 2 1 0 x m x m .5. a. Kho st s bin thin v v th hm s 3 22 9 12 4 y x x x .

b. Tm m phng trnh sau c su nghim phn bit:3 2

2 9 12 x x x m . (H Khi A2006)

-2 2 4

2

4

6

x

y

3

2

9

y

x

x

-2 2

2

4

6

x

y

3

2

2

9

y

x

x

a. S: b. 4


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Trn Duy Thi 11

Dng 8: CC CP IM I XNG

im 0 0;I x y l tm i xng ca th :C y f x Tn ti hai imM(x;y) vM(x;y)

thuc (C) tha:

0

0

' 2

' 2

x x x

f x f x y

0

0 0

' 2

2 2

x x x

f x f x x y

Vy 0 0;I x y l tm i xng ca (C) 0 02 2 f x y f x x .

1. Cho hm s 22 2 22 3

x x myx

c th mC .

Tm gi tr ca m mC c hai im phn bit i xng nhau qua gc ta O.

2. Cho hm s 2 2 22:1

m

x m x mC y

x

.

nh m mC c hai im phn bit i xng nhau qua gc ta O.

3. Cho hm s 3 23 1 y x x m (m l tham s).a. Tm m th hm s (1) c hai im phn bit i xng vi nhau qua gc ta .b. Kho st v v th hm s (1) khi m=2. (H Khi B2003)

S: a. 0 0 0, 0 f x f x x m>0.

4. Cho hm s 3 2 1133 3

x y x x c th C . Tm trn (C) hai imM, Ni xng nhau qua trc

tung.

5. Cho hm s 3 2 1 y x ax bx c . Xc nh a, b, c th hm s(1) c tm i xng lI(0;1) v iqua imM(1;1).6. Cho hm sy =x3 3x2 + 4 (1) (H Khi D2008)

a. Kho st s bin thin v v th ca hm s (1).b. Chng minh rng mi ng thng i qua imI(1;2) vi h s gc k(k> 3) u ct th ca

hm s (1) ti ba im phn bitI,A,B ng thiIl trung im ca on thngAB.

Li gii:

a. D =R.y' = 3x2 6x = 3x(x 2),y' = 0 x = 0,x = 2.y" = 6x 6,y" = 0 x = 1.

x 0 1 2 +y' + 0 | 0 +y" 0 + +y 4 +

C 2 CT U 0

2. d:y 2 = k(x 1) y = kxk+ 2.

Phng trnh honh giao im:x3

3x2

+ 4 = kxk+ 2 x3

3x2

kx + k+ 2 = 0. (x 1)(x2 2xk 2) = 0 x = 1 g(x) =x2 2xk 2 = 0.V ' > 0 v g(1) 0 (do k> 3) vx1 +x2 = 2xI nn c pcm!.

-2 2

2

4

x

y

O


12/15


Trn Duy Thi 12

Dng 9: MT S BI TON LIN QUAN N TIM CN

1. nh ngha:(d) l tim cn ca (C)

0lim

CM

MMH

2. Cch xc nh tim cna. Tim cn ng: 0:lim

0

xxdxf

xx

.

b. Tim cn ngang: 00 :lim yydyxfx

.

c. Tim cn xin: TCX c phng trnh:y=x+ trong :

xxfx

xf

xx

lim;lim .

Cc trng hp c bit:

*Hm s bc nht trn bc nht (hm nht bin)

nmx

baxy

+TX:D= R\

m

n

+TC: m

nxdy

m

nx

:lim

+TCN: m

ayd

m

ay

x

:lim

-4 -3 -2 -1 1 2 3 4 5

-2

-1

1

2

3

x

y

m

ay

m

nx

I

* Hm s bc hai trn bc nht (hm hu t)

nmx

Ax

nmx

cbxaxy

2

+TX:D= R\

m

n

+TC: m

nxdy

m

nx

:lim

+TCX: 0lim nmx

A

x TCX:y=x+

-4 -3 -2 -1 1 2 3 4 5

-2

-1

1

2

3

x

y

xy

m

nx

I

1. Cho hm s 2 2

3 2 21

3

mx m xy

x m

, vi m l tham s thc.

a. Kho st s bin thin v v th hm s (1) khi m =1.b. Tm cc gi tr ca m gc gia hai ng tim cn ca th hm s (1) bng 450.

(H Khi A2008)Li gii:

a. Khi m =1:2 2 4

23 3

x xy x

x x

.

TX:D R 3

2

2

6 5

3

x xy

x

. 0y

1 1 1

5 5 9

x y

x y

4

2

y

x

(d)

(C)

H

M


13/15


Trn Duy Thi 13

Tim cn:3

limx

y

tim cn ng:x = 3. 4lim 03x x

tim cn xin:y =x 2.

lim , limx x

y y

,3 3

lim , limx x

y y

.

Bng bin thin th:

x -5 -1y ' 0 0

y -9 CT

C -1

-3

b. 2 23 2 2 6 2

23 3

mx m x m y mx

x m x m

Gi (Cm) l th hm s. (Cm) c tim cn ng 1 : 3 0d x m v tim cn xin 2 :d 2 0mx y

10

3m m

.

Theo gi thuyt ta c: 02

cos 451

m

m

2

2

2 1

m

m

2 1m 1m (nhn).

2. Cho hm s 2 2

1 1mx m x m y f x

x

. Tm m sao cho th ca hm sf(x) c tim cn xin i

qua gc ta .

3. Cho hm s 2 (2 1). 3 1, 02

ax a x a y a a

x

c th (C). Chng minh rng th ca hm s

ny c tim cn xin lun i qua mt im c nh.

4. Cho hm s 22 3 2( )1

x xy f xx

c th (C).

a. Chng minh rng tch khong cch t mt imMbt k trn (C) n hai ng ng tim cn l mts khng i.

b. Tm ta imNthuc (C) sao cho tng khong cch tNn hi tim cn nh nht.

5. Cho hm s 22 2( )1

x mxy f x

x

c th (Cm). Tm m ng tim cn xin ca th hm s to

vi hai trc ta mt tam gic c din tch bng 4.

6. Tm m th hamd s2

1

1

xy

x mx

c hai tim cn ng lx=x1 vx=x2 tha mn

1 2

3 3

1 2

5

35

x x

x x

.

7. Cho hm s 11

xy

x

c th (C).

a. Kho st s bin thin v v th (C) ca hm s.b. Tm nhng imMthuc (C) sao cho tng khong cch t n n hai ng tim cn nh nht.

8. Cho hm s 2 12

xy

x

c th (H).

a. Kho st s bin thin v v th (H) ca hm s.b. Vit phng trnh tip tuyn ca (H) ti giao im vi trc tung.c. Tm nhng imN(xN>1) thuc (H) sao cho khong cch tNn tip tuyn ngn nht.

HD cu b, c.

-10 -8 -6 -4 -2 2

-10

-8

-6

-4

-2

2

x

y


14/15


Trn Duy Thi 14

-2 2 4

-6

-4

-2

2

x

y

N(2;-5)

M

H

* Gi M kl giao im ca (C) vi trc tung 0;1M . Phng trnh tip tuyn l 3 1y x hay

3 1 0x y .

* Ly 0 0 0 00

3; ; 2 , 1

1N x y H N x x

x

. Khi

0

0

33 2 1

1,

10

xx

d N

. t 0 00

33 3

1g x x

x

.

min min

,d N g x .

* Kho st hm 0 00

33 2

1g x x

x

trn khong 0; ,

0 2

0

3' 3

1g x

x

, 00

0

0' 0

2

xg x

x

, (lp bng bin thin

)

* Do 0 1x nn ta ch nhn nghim 0 2x thay voNta c

2; 5N . Vy 2; 5N th min

6 10,

5d N .

Dng 10: DIN TCHTH TCH

ng dng tch phn (Dng ny thng xut hin nhiu trong cc thi tt nghip)

a. Din tchCho hai hm sy=f(x) vy=g(x) c th (C1), (C2). Din tch hnh phng gii hn bi (C1), (C2) v

hai ng thngx=a,x=b c tnh bi cng thc:

b

a

S f x g x dx

Ch :Nu din tch thiu cc ng thngx=a,x=bta phi gii phng trnhf(x)=g(x) tm a, b.b. Th tchTh tch do hnh phng gii hn bi{(C):y=f(x),y=0,x=a,x=b} quay quanh Ox

c tnh bi cng thc: b

a

dxxfV2

Th tch do hnh phng gii hn bi {(C):x=(y),x=0,y=c,y=d} quay quanh Oy c tnh bi cng thc:

d

c

dyyV2

Th tch trn xoay do hnh phng gii hn bi hai ngy=f(x),y=g(x) quay quanh Ox

(f(x)g(x), x[a;b]) c tnh bi cng thc: b

a

dxxgxfV22

.

*

* *

x

y

O

f(x)

g(x)

ba

x

y

O

f(x)

(x)

ba

y

x c

d

O


15/15


1. Cho hm s 22 11

m x my

x

(1) (m l tham s). (H KhiD 2002)

a. Kho st s bin thin v v th (C) ca hm s (1) ng vi m=1.b. Tnh din tch hnh phng gii hm bi ng cong (C) v hai trc ta .c. Tm m th hm s (1) tip xc vi ng thngy=x.

S: b.4

1 4ln3

S , c 1m .

2. Cho hm s2

23

x xyx

.

a. Kho st v v th hm s trn.b. Tnh phn din tch hnh phng c gii hn bi th ca hm s v trc honh.

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