NGUYEN O]NH TRi (Chu bien)LE TRQNG VINH - DUONG THljy Vy

(DUNG CHO SINH VII:N cAc TRl10NG CAO DANG)

NHA XUAT SAN GlAo Dl)C

NGUY~N DiNH TRi (Chu bien)

LE TRQNG VINH - DUONG THUY VY

Giao trlnh

TOAN HOC CAO CAPT~P 1

(Stich dung cho cae truong Cao ddng)

NHA XUAT BAN mAo DlJC

U:II NO\ DAU

Sinh vien mot VaG nam hoc tlur nhat eric tnrcng dai hoc, cao dang thtrcng

gap kho khan do phtrong phap day. phtrong phap hoc abac hoc nay co nhieudieu khac biet so voi abac Trung hoc. Toan hoc cao cap lai la mot men hockh6 voi thai luong 100 ella nam thu nhat eric mrong dai hoc, cao dang ki

thuat, nham ren luyen nr duy khoa hoc, cung cap cong Cl,I loan hoc d~ sinh

vien hoc cac moo khoa hoc ki thuat khac va xay dung tiem hrc de riep tuc nrhoc sau nay.

B giao trinh "Toan hoc cao cap" nay dtrcc bien soan can ce VaG chtrcngtrlnh khung da ducc ban hanh. va thuc te giang day ella h cao dang ella mot

s6 tnrong dai hoc ki thuat va can Clc VaG chuang trtnh men Toan hien nay

ella cdc tnrcng Trung hoc Ph6 thong, nbam gulp eho sinh vien h cao dAng

h9C tot men h9C nay.

Do yeu cau dao tao hien nay ella h cao dang, mot s6 phan cua roan hoc cao

cap nlur cau true dai s6, dang roan phtrrmg, tfch phan phu thuoc tham 56, tfch

phan ba lap, tfch phan mat, cbuci Fourier, ... khong duoc dua vao giao trinh

nay. Nhitng khai niem toan hoc co ban, nhitng plurong phap co ban, nhfrng

ker qua co ban cua eric chuang deu duoc trinh bay day duo Mot s6 dinh lf

khong duoc chung minh. nhung )' nghla cua nhtmg dinh li quan trong duoc

giai thfch r6 rang, nhieu vf du minh hoa dtroc dua fa. Nhieu rrng dung cua Ifthuyet vao tfnh gan dung dtroc trlnh bay aday. Rieng vet nhfrng kien thirc vegiiii tich rna sinh vien dtrcc hoc aTrung hoc Ph6 thong, giao trlnh nay chinhac lai mot each h thong cac diem chfnh va trinh bay cac kien thee nang

cao. Phan cau hoi on t~p acuci rnoi chuang nham giup sinh vien hoc t~p vat\! kidm tra ket qua hqc t~p eua mlnh. UUTl nhii'ng bili t~p de fa acu6i mOichuang se giup ngubi hqc hieu sau s~c han cac khlij ni~m toan hqc, ren

luyn ki nang tinh toan va kha nang v~ dl;mg cac khai ni~m fly. cac bai ~p

d6 se dugc gi3.i trong b bai t~p kern theo b giao trlnh nay.

B giao trlnh nay dugc viet thanh 2 t~p va la cong trlnh t~p th~ eua ba nhagiao: Nguyen Dlnh Trf (chu bien), U Trqng Vinh va Duang Thuy Vy. Gng

Le Trqng Vinh viet cac chuang I, II, IV, V. Gog Duang Thuy Vy viet cac

chuang III, VI, VIII, IX. Gng Nguyen Dinh Tri viet cae chuang VII, X, XI.

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Khi xay dung de ctrong cho b giao trlnh nay cling nhtr khi bien soan giaotrtnh, chung toi dfi rham khao kinh nghiem cue nhieu nha giao dfi giang daynhieu narn mon Toan hoc cao cap cho he cao dang cac tnrong dai hoc ki

thuat. Chung rei xin chan thanh cam 0'11 cac ban dong nghiep da doc ban

rhao va cho nhieu ykien quy bau.B giao rrtnh nay duoc viet Ian dau, chac khong tranh het duoc nhfrng khiernkhuyet. Cluing toi chan thanh cam em moi y kien dong gop cua ban doc. Thugop y xin giti ve C6ng ti CO phdn Sach Dai hoc - Day nghe, 25 Han Thuyen,HaNi

Cac tric gia

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MI,IC LI,IC

Uri nol dau 3

Chuang I. Tap hop va dnh xa. S6 thuc va so phuc 7

1. Nhac Iai ve menh de roan hoc va kf hieu logic 72.~~ 8

13. Anh xa 12

4. so thtrc 16IS. S6 pinrc 18

Quh6i~~ 32

~~ 33Dap s6 36

Chuang II. Ham s6 mQt bien s6. GiOi han va lien tuc.

Dau ham va vi phan 39

I. Die khai nem co ban ve ham 56 mot bien 56 39

2. Pharr Ioai ham 56 44

3. Gioi han cua day 56 50

4. Gioi han ella ham 56 59

5. VO cung be va vo cling Ian 64

6. Ham s6 lien tuc 69

17. D~o ham 77

8~~~ ~

Cau hoi 6n rap 86

~~p ~

~~ %

Chuong III. Cac dinh Ii v~ gia Iri trung binh va lIng dung 99

1. Cdc dinh live gia lrj trung blnh 99

2. C6ng tlurc Taylor 102

3. Quy tile L'Hospital 107

4. Thg dung dao ham d6 khao sal ham 56 III

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5. DUOng cong eho boi phtrong trtnh tham s6 123

6. DuOng cong trong he tea 09 CLfC 128

Cau hoi on tap 133Bai tap 134

Dap so 137

Cfurcng IV. Dinh thee - Ma tran - H~ phtrcng trlnh tuyen tinh 141

1. Khai niem mo dau ve rna tran 141

2. Dinh thire 143

3. Ma tran 147

4. H~ plnrcng tnnh tuyen tfnh ]55

Cau hoi on rap 162

Bai qp 163Dap 56 168

Chuang v. Kh6ng gian vccto 1711. Khai niem ve khong gian vecto 17]

2. Khong gian con. H~ sinh 174

3. Hang cua mot ho vecto 183

4. Bai wan d6i cc sa 1845. Anh xa tuyen tfnh 188

Cau hoi on tap 198Bai t~p 199Dap s6 205

Cheong VI. Phep tinh ttch philo ella hamsomot bien s6 207I. Tfeh phan bat dinh 207

2. Tfch phan xac dinh 226

3. Mot s6 irng dung hinh hoc cua tfch phan xac dinh 237

4. Tieh phan suy rong 250

~~irn* ill

~* ~~~ ~

Tal Ii~u tham khao 271

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CHUONG I. TAp HOP vA ANH XA - 56THUC vA" . .56 PHUC

Ml,JC EliCH YEU CAU

Ourong I danh de on t(\P va b6 sung nhirng kien tlurc ve tap hcp va anh xa,

ve so thuc ail dtroc hQC a b(lC Trung hoc PhC) thong. trtnh bay nhimg kienthuc ca ban ve so plurc, cac phep tinh ve so plurc.

Sinh vien can hidu ki cac kien ttuic do, lam quell vai s6 phcc. lam tfnh thanh

thac d6i vdi cac s6 phtrc, bier sit dung dang hrong glee ella so plurc.

l. NHAc L~I V~ Mi):NHD~ TOAN HQC VAKI HIi):U LOGIC

1.1. Men h de roan hQCMenh de toan hQC la mot khang dinh wan hoc chi co the dung hoac sai,

khong th~ vita dung vira sai, vee khong dung vim khong sai.

Vi du 1: 2 < 4 la menh de toan hoc dung;

5 > 7 lit menh de toan hoc sai.

1.2. Ki hieu logic

Trang suy dien toan hoc, ngtroi ta dung cac kf hieu sau:

Gla su co hai menh de A va B.

Ki hieu A :::::::> B doc lil "tir rnenb de A suy fa rnenh de B".

Kf lueu A cc- B doc la "menh de A tucng dccng vel menh de B". Dieu do

co nghla 1" A => B, dong thci B => A.

Neu A => B thi ta nci A la dieu kien du d~ co B, con B la dieu kien din co

dircc tit A. Neu A B thi A lil dieu kien can va du cua B, d6ng thci B cling

la dieu kien can va du cua A.

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Vi d 2: Dieu kien can va du d~ plnrong trlnh bac hai: ax' + bx + C == 0 (a '1':- 0)

co hai nghiem tlurc phan bier la 6. == b2 - 4ac > O. Ta viet:

[phucng trtnh: ax? + bx + C == 0 (a '* 0) co hai nghiern thuc phan biet] b2 - 4ae > O.

Kf hieu : == doc la "dtroc dinb nghia Ia".

Kf hieu '

VidT:ll:A=o Ixlx2 - I =oO} d9Cla"Alat~ph9Pcac~6xsaochox2-I =00".

D6 chfnh la tap hop httu han (-I; 1I.Cac tap hop thuong gap trong toan h9C Ia:

N =0 {O, I, 2, } 13. tap hop cac s6 U! nhien.

N' = (I, 2, 3, I Iii tap hop eric so nguyen duong.

Z =0 10, 1, 2, ... ] Ia tap hop cac s6 nguyen.

iQJ = I1'.1 p, q E :2:, q " 0 I Iii tap hop cac sc hitu ti.q

R til tap hop eric s6 tlurc.

IW.' = I x E IW.] x " 0 I Iii tap hop cac s6 thtrc khac khong.

IW., = Ix E IW.I x " 0 I Iii tap hop cdc s6 tlurc khong am.

lFt_ =0 {x E lFtl x $ O} 13. tap hop cac s6 tlurc khong dtrcmg.

Tap hop vo han dUqc goi la de'rn duoc neu co the danh s6 cac phan tic cua no

theo tlur til til nhien. Trong tnrcng hop trai Iai, tap hop dtroc goi Ia kh6ng dem

duoc, Ole tap hop N, N',:2:, iQJ Iii nhirng tap hop dem dtrcc. Chang han, ta c6 th~

danh s6 cac phan ri'r cua Z (rap hop cac s6 nguyen) thea cac rmli ten nhu sau"

o -> 1 2.j. /' .j.

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/' .j.

-3

/' ...

Ole tap hop IW., IW.', 1W... 1W._lii nhitng rap hop khcng dcm dtrcc.

2,2, Tap hop con, Tap hop bang nbau

Cho hai tap hop A va B. Neu moi phan tll cua A deu Iii phan tic ella B, ta n6i

rang A 13. tap hop con ella B, hay A baa ham trong B, hay rap hop B chua tap

hop A, kf hieu A c B hay B :J A.

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Nhu v~y, ta cling co A C A.

Vai cdc tap hop df lict kc o rrcn, taco N* C N C:- c Q c R

Ta guy tree : Tap hop rong Ia tap hop con cue moi tap hop.

Hai ti~_p hop A va B goi la bang nhau neu A c B va Be A, kf hleu : A = B.

2.3. Cac phep loan ve lap hopDe de hinh dung tap hop va cac phan tu cua no,

nguci ra thtrong dung each bieu dien hlnh hoc, xem

moi ph.in nr ella tap hop la mot diem nam trong mothinh phang gioi han bel mor dtrong cong kfn, goi la

bieu d6 Ven (hlnh 1.1).

A

!nnh l.l

2.3.1. Phep lufp

Hop cua hai tap hcp A va B la tap hop g6m nhifng pharr til thuoc t;;tp hcp A,

hoac tap hop B, kf hieu A U B.

AU B = {x]x E A hoac A E B} (hlnh 1.2).

Phep hop cac tap hop co cac tfrth char sau:

AU (B U C) = (A U B) U C (tfnh chat kit hop);

AU B = B U A (tfnh char giao hoan). !nnh 1.2

2.3.2. Phep giao

Giao cua hai t~p hop A va B la tap hop gem nhimg phan tir vira thuoc A vita

thuoc B, kf hieu A n B.

An B = {xlx E A va x E B } (hinh 1.3).

Phep giao cue cac tap hop co cac tinh chat sau :

A n (B n C) = (A n B) n c;

AnB~BnA.

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A

!nnh 1.3

B

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Hai phep roan tren ducc lien he vci nhau bci Iuat phan ph6i :

A U (B n C) = (A U B) n (A U C);

A n (B U C) = (A n B) U (A n C).

2.3.3. Phip tru

Hleu cua tap hop A va tap hop B Iii tap hop g6m

nhirng phan tlr thuoc tap hop A nlurng kh6ng thuoctap h91' B, ki hieu A \ B,

A \ B= [x] x E A, x e B} (hlnh 1.4).

A

Hinh 1.4

B

2.3.4. Tdp hap bu (phdn bil)

Xet tap hop E, A la tap hop con cua E. Tap hop bu cua A rrong E Iii tap hop

E \ A, ki hieu A, A = E \ A (hlnh 1.5).

Vi du 2:

Khi do