Cao Ho Thi 1

Chng I

O TH

1. H TA : (Coordinates System)

1.1 H to vung gc: (Cartersian Coordinates System)

H to vung gc trong mt mt phng c cu to bi hai trc s thc vung gc vi nhau. Trc nm ngang (Horizontal axes) gi l trc honh xox, trc thng ng (Vertical axes) gi l trc tung yoy. Giao im ca hai trc gi l gc ta (Origin) O. H ta vung gc chia mt phng lm 4 vng I, II, III v IV.

x

y

M(x,y)

x

y

x'

y'

1.2 Ta ca mt im trong mt phng:

V tr ca mt M trong mt phng c xc nh bng honh x (Abscisga) v tung (Ordinade) y.

(x,y) c gi l ta ca im M v c k hiu M(x,y).

1.3 Khong cch gia hai im:

Cho hai im M1(x1,y1) v M2(x2,y2) trong mt phng khong cch gia hai im M1,M2 c tnh theo cng thc sau:

Cao Ho Thi 2

x

y

M2(x2,y2)

x2

y2

x'

y'0

M1(x1,y1)

x1

y1

x = x2-x1

y = y2-y1

( ) ( )d M M x x y y= = + 1 2 2 1 2 2 1 2 ( ) ( ) ( ) ( )d x x y y= + = + 2 1 2 2 1 2 2 26 2 4 1

d = 5

1.4 Gia s: Gia s ca x l x = x2 -x1 y l y = y2 -y1

2. NG THNG

2.1 Phng trnh ca ng thng

- Dng tng qut (Dng chun):

CByAx =+ L phng trnh bc nht theo x v y. A, B, C l cc hng s

- Dng thng dng:

bmxy += m: dc (slope)

b: tung gc (intercept): x = 0 y = b

Cao Ho Thi 3

2.2 dc:

Gi m l dc ca ng thng (D)

myx

y yx x

tg= = =

2 1

2 1

ngha ca c dc: Khi thay i 1 n v th y thay i m n v.

x

y M2(x2, y2)

M1(x1, y1) y = y2 - y1

x = x2 - x1

(D)

y2

y1

b

0 x1 x2

Cao Ho Thi 4

Nhn xt:

ng thng (D) Dng th dc m

+ i ln (ng bin)

m>0

+ i xung (Nghch bin)

m

Cao Ho Thi 5

b. y = 2/3 x - 2

Xc nh phng trnh ng thng i qua mt im M(x1, y1) v bit trc dc m (Point-Slope Form).

Ta c: m y yx x

= 1

1

Phng trnh ng thng c dng: y - y1 = m(x - x1)

V d: Vit phng trnh ng thng c dc l 1/2 v i qua im (-4,3).

Gii: Phng trnh ng thng c dng: y -y1 = m(x - x1)

m= 12

, x1 = -4, y1 = 3

y -3 = 12

(x + 4) = 12

x +2

Vy: y = 12

x + 5

Xc nh phng trnh ng thng i qua 2 im M1(x1,y1) v M2 (x2,y2).

dc ca ng thng l:

m y yx x

= 2 1

2 1

Phng trnh ng thng c dng:

y - y1 = ( )y yx x x x2 12 1 1

hay y yy y

x xx x

=

1

2 1

1

2 1

(D) M(x,y)

M1(x1,y1)

y

y1

0 xx1

x

y = y2 - y1

M2 (x2,y2)

M1(x1,y1)

yy2

y1

x2 x10 x

x = x2 - x1

Cao Ho Thi 6

V d: Vit phng trnh ng thng i qua 2 im c ta (-3,2) v (-4,5).

Gii: Phng trnh ng thng c dng:

y yy y

x xx x

=

1

2 1

1

2 1

y x x =

=+

25 2

34 3

31

( )( )

-y + 2 = 3x + 9 y = -3x - 7

ng thng nm ngang v ng thng thng ng.

+ Phng trnh ng thng nm ngang: y = b

+ Phng trnh ng thng thng ng: x = b

V d: Phng trnh ng thng ng v ng nm ngang i qua im c ta (-2,3).

Gii: + Phng trnh ng thng nm ngang y=3

+ Phng trnh ng thng thng ng x= -2

ng thng song song v thng gc

Cho 2 ng thng (D1) v (D2) c dc tng ng l m1 v m2 + Nu (D1) // (D2) th m1 = m2 + Nu (D1) (D2) th m1*m2 = -1

V d: Cho ng thng (D) c phng trnh y = 12

x - 2 v im A(2,-3). Vit phng

trnh ca ng thng

a. (D1) i qua dim A v song song vi ng thng (D)

b. (D2) i qua im A v thng gc vi ng thng (D)

Gii:

a. Gi m1 l dc ca ng thng (D1)

y

x=-2

3 y=3

x -2 0

Cao Ho Thi 7

(D1) // (D) m1 = m = 12 (D1): y - y1 = m1 (x-xA)

y-(-3) = 12

(x-2)

y + 3 = 12

x 1 y = 12

x - 4

b. Gi m2 l dc ca ng thng (D2)

(D2) (D) m2* m = -1 m2 = = = 1 112

2m

(D2): y - yA = m2 (x - xA)

y + 3 = -2(x - 2)

y = -2x +1

3. HM S V TH

3.1 Hm s;

a. nh ngha hm s:

Mt hm s f t tp hp X n tp hp Y l mt qui tc sao cho vi mi phn t xX c tng ng vi nhiu nht mt phn t yY.

X: tp hp ngun f: X Y Y: tp hp ch x y = f(x) x: bin s (to nh) y: hm s (nh)

b. Min xc nh v min ga tr ca hm s:

Min xc nh D (Domain)

D = { }x X y f x =/ ( )

f YX

= f(x) yD

V

Cao Ho Thi 8

D l tp hp gm nhng phn t x c tng ng vi phn t y.

Min gi tr V V={ }y Y y f x =/ ( ) V l tp hp gm nhng phn t y c tng ng vi phn t x.

Ghi ch: Hm s f t tp hp X n tp hp Y chnh l mt nh x t D n Y, c ngha f l mt qui tc sao cho mi phn t x D u tng ng vi mt v ch mt phn t yY. c. Min xc nh ca mt s hm s c bn

Hm a thc: y = Pn(x) = anxn + an-1xn-1 + ........... + a2x2 + a1x1 + a0

Min xc nh D = R c ngha l y = f(x) c xc nh vi mi xR V d: Hm s y = f(x) = 3x2 - 2x + 1

D = R = (- , ) Hm hu t:

y P xQ x

n

m

= ( )( )

y = c xc nh khi Qm(x) 0 Hm v t

y P xn= ( ) y = f(x) c xc nh khi Pn(x) 0

V d: Tm min xc nh ca cc hm s sau:

y xx

= +2 1

1 y x= 3

Gii

a. y c xc nh khi x-1 0 hay x 1: D = R\ { }1 b. y c xc nh khi x-3 0 hay x 3: D = [ )3, 3.2 th

a. nh ngha: nghin cu hm s f(x) ta thng biu din cp s (x, f(x)) ln mt phng ta .

Tp hp cc im biu din cc cp s ny gi l th hm s f.

b. S bin thin ca hm s f.

Hm s ng bin.

Hm f ng bin trn khong (a,b)

Cao Ho Thi 9

[ x1, x2 (a,b), x1 < x2 f(x1) < f(x2)]

>

x x a bf x f x

x x1 22 1

2 1

0, ( , ),( ) ( )

x

y

y

y = f(x)

f(x2)

x2

f(x1)

x10 b Hm s nghch bin:

Hm s f nghch bin trn (a,b) [ ] x x a b x x f x f x1 2 1 2 1 2, ( , ), ( ) ( )p f

x x a bf x f x

x x1 22 1

2 1

0, ( , ),( ) ( ) p

x

y

y

y = f(x)

f(x2)

x2

f(x1)

x10 ba

c. S dch chuyn th ca hm s:

Dch chuyn theo phngng Gi (C) l th ca hm s y = f(x)

(C) dch ln trn: y = f(x) + k, k > 0 (C) dch xung di: y = f(x) - k, k > 0

Cao Ho Thi 10

Dch chuyn theo phng ngang

Dch qua phi: y = f(x - h), h > 0

Dch qua tri: y = f(x + h), h>0

x

y

y = f(x)

y = f(x+k)

y = f(x)+h

i xng qua trc X:

y = -f(x)

x

y

y = f(x)

y = -f(x)

Gin v co th:

Gin th: y = C f(x), C > 1

Co th: y = C f(x), 0 < C < 1

Cao Ho Thi 11

0 1 2 3 4

10

20

30

x

y

y = k*f(x)

y = f(x)/k

y = f(x)

Vn : Chuyn H Trc Ta .

i xng oxy: M(x,y), O(a,b)

i xng OXY: M(X,Y)

x a Xy b Y= += +

hay

X x aY y b= =

ng dng: V th

+ Cho th y = f(x)

+ V th y = f(x-K) v y = f(x+K) X x k

Y yx X K

y K= =

= +=

Kt Lun:

+ th y = f(x-K) l th y = f(x) dch qua phi K n v

+ th y = f(x+K) l th y = f(x) dch qua tri K n v

Vn : Chi Ph

Chi Ph

Tng Chi Ph (Total Cost - TC): Chi ph c nh (Fixed Cost - FC) Chi ph bin i (Variable Cost - VC)

TC = f(Q), Q: Sn lng FC l chi ph m mt x nghip nht thit phi chi tr d khng sn xut g c. VC l chi ph tng ln cng vi mc tng ca sn lng.

Yy

Yy M

0b

0 a x x

x X

Cao Ho Thi 12

Chi ph cn bin (Marginal Cost - MC): l chi ph gia tng sn xut thm mt n v sn phm.

Chi ph Bnh Qun (Average Cost - AC) AC TC

Q=

AFC FCQ

=

AVC VCQ

= V d: Mt cng ty sn xut giy nhn thy rng chi ph c nh l 300 USD mi ngy v tng chi ph l 4300 USD mi ngy ng vi tng sn lng mi ngy l 100 i giy. Gi s rng tng chi ph TC (USD) c quan h tuyn tnh vi sn lng x (i giy).

a. Xc nh phng trnh ca chi ph c nh FC theo sn lng x.

b. Xc nh phng trnh ca chi ph bin i VC theo sn lng x.

c. Xc nh phng trnh ca tng chi ph TC theo sn lng x Gii:

a. Xc nh phng trnh ca chi ph c nh TC:

FC = 300

b. Xc nh phng trnh ca chi ph bin i VC theo sn lng x: VC = f(x).

VC = f(x) = mx + b

x = 0 VC = 0 b = 0 x = 100 VC = TC - FC = 4300 - 300 = 4000 m = =

4000 0100 0

40

Vy: VC = 40x

c. Xc nh phng trnh ca tng chi ph TC theo sn lng x:

TC = FC + VC = 300 + 40x

TC

VC

VC FC

Q FC

C

Cao Ho Thi 13

x

C

FC

VC

TC

300

Vn : Thu Nhp B phn nghin cu th trng ca cng ty sn xut linh kin in t cho my vi tnh xc nh phng trnh ng cu ca linh kin in t l:

x = 10000 - 50p

Trong x l s lng linh kin c bn mc gi $p mi linh kin .

a. Trnh by thu nhp R di dng hm ca x

b. Tm min xc nh ca hm R.

Gii:

a. x = 10000 - 50p p = 200 - 150

x

R = px = (200 - 150

x) = 200x - 150

x2

b. Tm min xc nh ca R iu kin x 0 p 0 200 - 1

50x 0 x 10000

0 x 10000 hay D = [0, 10000]

Vn : im Ha Vn (Break - Even point Analysis).

Cao Ho Thi 14

x

$

FC

TC

300

TR=TC

QBE

TR

TC = FC + VC = FC + v*x

TR = p*x

im ha vn x = xBE Ta c TC = TR:

FC + v*xBE = xBE*p

x FCp vBE

=

V d: Mt cng ty sn xut bu thip nhn thy rng chi ph c nh sn xut bu thip l 9000$ v chi ph bin i l 3,5$ mi bu thip v gi mi bu thip l 5$. Tm sn lng ha vn ca cng ty.

Gii:

TC = 9000 + 3,5x v R = 5x

TC = R 9000 +3,5 x = 5x xBE = 90005 35 6000 =. bu thip

Cao Ho Thi 15

Lu :

Vn : M Hnh Khu Hao Tuyn Tnh (Deppreciation) (M Hnh Khu Hao u) (M Hnh Sn Lng - Straight Line)

Theo m hnh hu hao u. Chi ph khu hao D cho mi nm s l:

D P SVn

= Trong :

P: gi tr ca ti sn. SV: gi tr cn li n: thi k hu hao P-SV: Gi tr ti sn u t b gim

Gi tr bt ton ca ti sn cui nm t l Bvt

BVt

t21 nt nm

SV

P

Gi tr ti sn

Cao Ho Thi 16

BV P D t P P SVn

tt = = V d: Mt ti sn c gi tr ban u l 50 triu ng. Gi tr cn li sau 5 nm l 10 triu ng. Tnh chi ph khu hao hng nm v chi ph bt ton vo cui nm th hai.

Gii:

D = =50 105

8 tr/nm

BV2 = 50-8*2 = 34 tr/nm.

Vn : ng Cung v Cu .

1. ng Cu: (Demand Curve) - ng cu l s tng quan gia gi v lng cu ca mt mt hng (khi cc gi

tr khc c gi khng i).

- Phn bit gia Cu v lng cu: (Demand and Quantity Demanded).

+ Cu m t hnh vi ca ngi mua tt c mc gi.

+ Lng cu ch c ngha trong mi quan h vi mt mc gi c th.

Ni cch khc, danh t Cu ch ton b ng cu trong khi danh t lng cu ch mt im c th no trn ng cu.

- iu kin mt ng cu: tng quan gia gi v lng cu l nghch bin ngha l gi tng lng cu gim.

- dc ca ng cu phn nh mc p ng ca lng cu i vi cc thay i v gi.

2. ng Cung: (Supply Curve)

- ng cung l s tng quan gia gi v lng cu ca mt mt hng (khi cc gi tr khc c gia nguyn)

- Phn bit gia Cung v lng cu. (Supply and Quantity Supplied)

+ Cung m t ton din v s lng m ngi bn mun bn mi mc gi.

+ Lng cung c ngha trong mi quan h vi mc gi c th.

Gi (P)

Lng cu (Q)

n gin

Gi (P)

Q D1QD0 Lng cu (Q)

Po

P1

Cao Ho Thi 17

Ni cch khc, danh t Cung ch ton b ng cung trong khi danh t lng cung ch mt im c th no trn ng cung.

- iu kin mt ng cung, tng quan gia gi v lng cung l ng bin, ngha l gi tng lng cung gim.

- dc ng cung phn nh mc p ng ca lng cung i vi cc thay i v gi.

V d: Cung v cu v go.

Gi P (1000/kg)

Lng Cu QD (triu kg/thng)

Lng Cung QS (triu kg/thng)

2,5 2,4 2,3 2,2 2,1

9 10 12 15 20

18 16 12 7 2

Q

2.1

2.5

2.42.3

2.2

P0

SD

0 Q0=12

P

10 20

3. S cn bng gia Cung v Cu:

- Mt th trng tn ti khi c s giao tip gia cc ngi mua v ngi bn trong nic bn mt mt hng hay dch v.

PP1

P

QS1 QS0

Q

P0

Q

Cao Ho Thi 18

- Th trng cn bng khi ng cung gp ng cu. Giao im ca ng cung v ng cu l im cn bng. im cn bng ta c gi cn bng v lng cn bng.

Trn thc t, Cung v Cu khng phi lc no cng trong trng thi cn bng, nhng xu hng cc th trng u tin ti cn bng.

4. S dch chuyn ca ng Cung v Cu:

P D SD Cung

P0

D Cu

Q Q0

P

QS1 QS1

S

S D

P0

P1

Q project

Q0 Q

P SD D

P0

P1

Q project

Q0 Q QS

1 Q D1