B O Ä M O Â N T O A Ù N Ö Ù N G D U ÏN G - Ñ H B K

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T O A Ù N 1 H K 1 0 7 0 8

• B A Ø I 2 : H A Ø M S O Á ( S V )

• T S . NGUY E Ã N QUOÁ C L A Â N (0 9 / 2 0 0 7 )

N O Ä I D U N G------------------------------------------------------------------------------------------------

---------------------------------

1- K H A Ù I N IE Ä M

H A Ø M S O Á 2 - C A Ù C C A Ù C H X A Ù C Ñ Ò N H H A Ø M

S O Á 3 - N H A É C L A Ï I : H A Ø M C Ô

B A Û N ( P H O Å T H O Â N G ) 4 - H A Ø M S O Á

N G Ö Ô ÏC 5 - H A Ø M L Ö Ô ÏN G

G IA Ù C N G Ö Ô ÏC 6 - H A Ø M

H YP E R B O L IC 7 - A Ù P D U ÏN G

K YÕ T H U A Ä T

K H A Ù I N IE Ä M H A Ø M S O Á -------------------------------------------------------------------------------------------------

----------------------------------

V D : Ñ o à t h ò

V N IN D E X ( c h ö ù n g

k h o a ù n ) → H a ø m

s o á : g ia ù c h ö ù n g

k h o a ù n t h e o ? ? ?

( T h ô ø i g ia n ? G ia ù

v a ø n g ? B ie á n

ñ o ä n g c h ín h t r ò ? &

B ie å u t h ö ù c y = ? ? ?

Ñ a ï i lö ô ïn g A b ie á n t h ie â n p h u ï

t h u o ä c ñ a ï i lö ô ïn g B :• Ñ ô ø i s o á n g : T ie à n ñ ie ä n t h e o

s o á k w h t ie â u t h u ï , g ia ù v a ø n g

t r o n g n ö ô ù c t h e o t h e á g iô ù i …• K y õ t h u a ä t : T o ïa ñ o ä c h a á t

ñ ie å m t h e o t h ô ø i g ia n …

T ö ô n

g

q u a n

h a ø

m

s o á

L Ò C H S Ö Û -------------------------------------------------------------------------------------------------

----------------------------------

G iö õ a T K 18 , E u le r :

B ie å u d ie ã n h a ø m

s o á q u a k y ù t ö ï → y

= f ( x )

17 8 6 ,

S c o t la n d :

T h e

Co mme r c ia l

a n

P o l it ic a l

A t l a s ,

P la y f a ir .

Ñ o à t h ò s o

s a ù n h

x u a á t &

n h a ä p

k h a å u t ö ø

A n h s a n g

Ñ a n M a ïc h

+ N a U y

x :Vaøo

f :Haøm

tính Maùy y :Ra

Ñ Ò N H N G H Ó A T O A Ù N H O ÏC -------------------------------------------------------------------------------------------------

----------------------------------

M X Ñ D f = { x | f ( x )

c o ù n g h ó a }

RX ⊂RY ⊂H a ø m s o á y = f ( x ) : X ⊂ R

→ Y ⊂ R : Q u y lu a ä t t ö ô n g

ö ù n g x ∈ X → y ∈ Y .

B ie á n s o á x , g ia ù t r ò y .

T ö ô n g q u a n h a ø m s o á :

1 g ia ù t r ò x c h o r a 1

g ia ù t r ò y

M o ä t x → N h ie à u

y : K 0 p h a û i h a ø m

n g h ó a t h o â n g

t h ö ô ø n g ( N h ö n g

h a ø m ñ a t r ò ? )

M G T r ò Im f : {y

=f ( x ) , x ∈D f }y = s in x ⇒ D = R , Im f

= [ – 1, 1]

C A Ù C C A Ù C H X A Ù C Ñ Ò N H H A Ø M S O Á -------------------------------------------------------------------------------------------------

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B o á n c a ù c h c ô b a û n x a ù c ñ ò n h h a ø m

s o á : M o â t a û ( ñ ô n g ia û n ) - B ie å u t h ö ù c

( t h o â n g d u ïn g ) – B a û n g g ia ù t r ò ( t h ö ïc

t e á ) – Ñ o à t h ò ( k y õ t h u a ä t ) M o â t a û : Ñ ô n g ia û n , d e ã p h a ù t

h ie ä n t ö ô n g q u a n h a ø m s o á

T r o ïn g

lö ô ïn gG ia ù

t ie à n

≤ 2 0

g r18 . 0

0 0 ñ

2 0 –

4 0 g r3 0 . 0

0 0 ñ

V D : B a û n g c ö ô ù c p h í g ö û i t h ö

b a è n g b ö u ñ ie ä n ñ i c h a â u A â u

B a û n g g ia ù t r ò : T h ö ïc t e á , r o õ r a ø n g ,

t h íc h h ô ïp c a ù c h a ø m ít g ia ù t r ò

V D : P h í g ö û i t h ö b ö u ñ ie ä n ñ i n ö ô ù c

n g o a ø i p h u ï t h u o ä c t r o ïn g lö ô ïn g

4 0 –

6 0 g r4 2 . 0

0 0 ñ

X A Ù C Ñ Ò N H H A Ø M S O Á Q U A B IE Å U T H Ö Ù C ( H A Y G A Ë P N H A Á T )

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Q u e n t h u o ä c ( d a ïn g

h ie ä n ) : y = f ( x )V D : y = x 2 , y = e x , h a ø m s ô

c a á p c ô b a û n …

D a ïn g

t h a m

s o á

( )( )

==

tyy

txx

V D : x = 1 + t , y = 1 – t →

Ñ ö ô ø n g t h a ú n g

: 1 t → 1

( x , y )

V D : x = a c o s t , y = a s in t →

Ñ ö ô ø n g t r o ø n

D a ïn g a å n F ( x , y ) = 0 ⇒ y =

f ( x ) ( im p l ic i t )V D : Ñ t r o ø n x 2 + y 2

– 4 = 0 ,

01916

22

=−+ yx

B ie å u

t h ö ù c :

M A P L E : K H A I B A Ù O H A Ø M S O Á , V E Õ Ñ O À T H Ò

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( K h a i b a ù o h a ø m s o á )

p : = x ^ 3 + x ^ 2 + 1; ( T ín h g ia ù t r ò h a ø m

s o á ) s u b s ( x =1, p ) ; ( T ín h g iô ù i h a ïn h a ø m s o á )

l im it ( s in ( 2 *x ) /x , x = 0 ) ; ( T ín h ñ a ïo h a ø m ) d i f f ( p , x ) ; ( T ín h

ñ h a ø m c a á p 2 ) d i f f ( p , x $ 2 ) ( V e õ ñ o à t h ò ) p lo t ( s in ( x ) , x = 0 . . P i ) ;

( N h ie à u ñ o à t h ò ) p lo t ( [ s in ( x ) , c o s ( x ) ] , x

= 0 . . 2 *P i , c o lo r = [ r e d , b lu e ] ) ; ( Ñ o à t h ò t h a m s o á ly ù t h u ù )

p lo t ( [ 3 1*c o s ( t ) -7 *c o s ( 3 1* t /7 ) ,

3 1* s in ( t ) -7 * s in ( 3 1* t /7 ) , t = 0 . . 14 *P i ] ) ; p lo t ( [ 17 *c o s ( t ) +7 *c o s ( 17 * t /7 ) ,

17 * s in ( t ) - …, t = 0 . . 14 *P i ] ) ;

H A Ø M Q U E N T H U O Ä C ( P H O Å T H O Â N G ) -------------------------------------------------------------------------------------------------

----------------------------------

T ín h c h a á t h a ø m y = x α: M X Ñ , ñ ô n ñ ie ä u …

t u y ø t h u o ä c α > 0 & < 0 !

H a ø m h a è n g , t u y e á n t ín h ( b a ä c 1) : y

= a x + b → Ñ ö ô ø n g t h a ú n g H a ø m lu y õ t h ö ø a : y = x α → Ñ a t h ö ù c : y

= a 0 x n + a 1x n – 1 + … , h a ø m p h a â n t h ö ù c : y

= 1/x , y = P ( x ) /Q ( x ) , h a ø m c a ê n y =

...n x

H a ø m y = x α: α t ö ï n h ie â n ⇒ M X Ñ : R , α

n g u y e â n a â m : M X Ñ x ≠ 0 , α ∈ R : n o ù i

c h u n g x > 0 ( N e á u h a ø m c a ê n : t u y ø t ín h

c h a ü n le û ) T ín h ñ ô n ñ ie ä u y = x α, x > 0 : α > 0 →

T a ê n g , α < 0 → G ia û m G iô ù i h a ïn x → +∞ : α > 0 → l im x α = +∞ , α

< 0 → l im x α = 0

Ñ O À T H Ò H A Ø M L U YÕ T H Ö Ø A -------------------------------------------------------------------------------------------------

----------------------------------

leû nhieân,töï αα :xy = chaün nhieân,töï αα :xy =

1&1: ><<= ααα 0 xy 0: <= αα xy

H A Ø M M U Õ , L O G -------------------------------------------------------------------------------------------------

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H a ø m ñ a t h ö ù c : c o ù c ö ïc

t r ò , k h o â n g c o ù t ie ä m c a ä nH a ø m p h a â n t h ö ù c : t c a ä n

ñ ö ù n g , x ie â n ( n g a n g ) t u y ø b a ä c

S v ie

â n

t ö ï

x e m

H a ø m c a ê n : m ie à n x a ù c

ñ ò n h , t ie ä m c a ä n …

H a ø m lo g a r i t : y = ln x → T o å n g q u a ù t : y =

lo g a x ( a > 1 & 0 < a < 1)

+∞==<<

−∞=+∞=>

+→+∞→

+→+∞→

xxa

xxa

ax

ax

ax

ax

loglim&0loglim:10

loglim&loglim:1

0

0

R :MGTrò0x:MXÑ >

H a ø m m u õ : y = e x → y = a x ( a > 1 & 0 < a

< 1) . D = R ; M G T :Ñ ô n ñ ie ä u y = a x : a > 1 ⇒ H a ø m t a ê n g &

0 < a < 1: H a ø m g ia û m ∞==<<=∞=>−∞→∞→−∞→∞→

x

x

x

x

x

x

x

xaaaaaa lim&0lim:10;0lim&lim:1

*+R

Ñ O À T H Ò H A Ø M M U Õ , L O G A R IT : S O S A Ù N H V Ô Ù I L U YÕ T H Ö Ø A

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0,

10&1:

>=

<<>=

ααxy

aaay x Ñ ie å m ñ a ë c

b ie ä t : ≠ n h a uK h i a > 1 & α >

0 : C u ø n g ↑ , →

+∞ , n h ö n g m u õ

n h a n h h ô n lu y õ

t h ö ø a

0,

10&1:log

>=

<<>=

ααxy

aaxy a

Ñ ie å m ñ a ë c

b ie ä t : ≠ n h a uK h i a > 1 & α >

0 : C u ø n g ↑ , →

+∞ , n h ö n g lu y õ

t h ö ø a n h a n h

h ô n lo g

H A Ø M L Ö Ô ÏN G G IA Ù C : s in x , c o s x -------------------------------------------------------------------------------------------------

----------------------------------

y = s in x , y = c o s x ⇒ M X Ñ R , M G T r ò [ –

1, 1] , T u a à n h o a ø n …

xy

xy

cos

sin

==

H A Ø M L Ö Ô ÏN G G IA Ù C : t g x , c o t g x -------------------------------------------------------------------------------------------------

----------------------------------

y = t g x ( x ≠ π/2 + k π ) , y = c o t g x ( x ≠ k π ) :

M G T R , T C ñ ö ù n g

xy

xy

cotg

tg

==

H A Ø M H Ô ÏP . H A Ø M S Ô C A Á P -------------------------------------------------------------------------------------------------

----------------------------------

2 h a ø m y = f ( x ) , y = g ( x ) → H a ø m h ô ïp : f o

g = f ( g ) : y ( x ) = f ( g ( x ) )

x :Vaøo g :Haøm ( )xg :Ra f :Haøm ( )( )xgf :trò Giaù

V D : P h a â n b ie ä t f ( g ) & g ( f ) : f = x 2 & g =

c o s x ⇒ f ( g ) = … ≠ g ( f ) = … H a ø m s ô c a á p : T o å n g , h ie ä u , t íc h ,

t h ö ô n g , h ô ïp ( n g ö ô ïc ) … c u û a n h ö õ n g

h a ø m c ô b a û n → H a ø m s ô c a á p : D ie ã n

t a û q u a 1 c o â n g t h ö ù cV D : y = ( s in 2 ( x ) – ln ( t g x +2 ) ) /( e c o s x – 1) : s ô

c a á p → L t u ïc , ñ h a ø m … V D

:

ñhaøm! khoâng:caáp sô Khoâng thöùc coâng 2 →

<−≥

== :0,

0,

xx

xxxy

H A Ø M N G Ö Ô ÏC -------------------------------------------------------------------------------------------------

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f – s o n g a ù n h ⇔ P h ö ô n g t r ìn h f ( x ) = y ( * )

c o ù n g h ie ä m x d u y n h a á t( ) XYfYyyfxxfy →∈∀=⇔= −− ::)( 11 :ngöôïc haøm thöùc bieåu

T ìm h a ø m n g ö ô ïc : G ia û i ( * ) ( a å n x ) ⇒ B ie å u

t h ö ù c h a ø m n g ö ô ïc x = f −1( y )

H a ø m s o á y = f ( x ) : X → Y

t h o a û t c h a á t :

∀ y ∈ Y , ∃! x ∈ X s a o c h o

y = f ( x ) ⇒ f : s o n g a ù n h

( t ö ô n g ö ù n g m o ä t –

m o ä t )

V D : T ìm m ie à n x a ù c ñ ò n h v a ø m ie à n g ia ù

t r ò ñ e å t r e â n ñ o ù h a ø m s o á s a u c o ù

h a ø m n g ö ô ïc v a ø c h æ r a h a ø m n g ö ô ïc

ñ o ù y = x 2 + 1

C h u ù y ù : C a å n

t h a ä n c h o ïn X & Y

V D : y = f ( x ) = 2 x + 1

⇒ f – 1 = ?

H A Ø M L Ö Ô ÏN G G IA Ù C N G Ö Ô ÏC ----------------------------------------------------------------------------------------------------

----------------------------------

V D : α = a r c s in ( 1/2 ) =

s in -1 ( 1/2 ) :

D u ø n g p h ím s in -1

t r e â n M T B T u ù i

[ ] yxyxyx arcsinsin:1,1,2

,2

=⇔=−∈

−∈ Nghieäm ptrGiaûi ππ

y = a r c s in x : D = [ – 1,

1] , M G T

βαβαππsinsin&

2,

21 =⇔=

− −

y = s in x : s o n g a ù n h :

→ H a ø m n g ö ô ïc y =

a r c s in x : →

−

2,

2ππ [ ]1,1−

[ ]1,1−

−

2,

2ππ

( ) ( ) Cxx

dx

u

uu

xx +=

−−=

−= ∫ arcsin

1&

1

''arcsin&

1

1'arcsin

222

H a ø m a r c c o s , a r c t g , a r c c o t g : T o a ù n 1, Ñ C K , t r a n g 2 1 – 2 3

--------------------------------------------------------------------------------------------------------------------------------------

y = c o s x s o n g a ù n h : [ 0 , π ] → [ – 1, 1] ⇒ y

= a r c c o s x : [ – 1, 1] … [ ] [ ] ( )

2

1

1

1'arccos&

cos

,0,1,1cosarccos

xx

yx

yxxxy

−−=

=∈−∈

⇒== − π

−→=⇒→

−=

2,

2:arctg

2,

2:tg

ππππRxyRxy :aùnh song

[ ] [ ]ππ ,0:arccotg,0:cotg →=⇒→= RxyRxy :aùnh song

( ) ( )

( ) ( )2

222

11'arccotg

arctg1

&1

''arctg&

11

'arctg

xx

Cxxdx

uu

ux

x

+−=

+=++

=+

= ∫

H A Ø M H YP E R B O L IC ( T o a ù n 1, Ñ C K , t r a n g 2 3 – 2 4 )

--------------------------------------------------------------------------------------------------------------------------------

,2

shsinhxx ee

xx−−== RD

eexx

xx

=+==−

.2

chcosh

C o â n g t h ö ù c h a ø m h y p e r b o l ic : N h ö

c o â n g t h ö ù c lö ô ïn g g ia ù c & ñ o å i d a á u

r ie â n g v ô ù i t h ö ø a s o á t íc h c h ö ù a 2 s in

( h o a ë c t h a y c o s x → c h x , s in x → i s h x ( i :

s o á a û o , i 2 = – 1) !

M T B T u ù i : B a á m h y p + s in , h y p + c o s .

V D : T ín h s h ( 0 ) , c h ( 0 ) V D : C h ö ù n g m in h : a / c h ( x ) > 0 ∀ x ( T h a ä t

r a c h ( x ) ≥ 1 ∀ x )

b / s h x < c h x ∀ x c / c h ( x ) : h a ø m c h a ü n ,

s h ( x ) : h a ø m le û )V D : C h ö ù n g m in h c h 2 x – s h 2 x = 1 ∀ x ( S o

s a ù n h : c o s 2 x + s in 2 x = 1)

V D : G ia û i p h ö ô n g

t r ìn h : s h ( x ) = 1

( )21ln2 +=⇔=−⇔ − xee xx

B A Û N G C O Â N G T H Ö Ù C H A Ø M H YP E R B O L IC

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1cossin 22 =+ xx 1shch 22 =− xx

( ) yxyxyx sinsincoscoscos =± ( ) yxyxyx shshchchch ±=±( ) xyyxyx cossincossinsin ±=± ( ) xyyxyx chshchshsh ±=±( ) xxx 22 sin211cos22cos −=−= ( ) xxx 22 sh211ch22ch +=−=

( ) xxx cossin22sin = ( ) xxx chsh22sh =

2cos

2cos2coscos

yxyxyx

−+=+2

ch2

ch2chchyxyx

yx−+=+

2sin

2sin2coscos

yxyxyx

−+−=−2

sh2

sh2chchyxyx

yx−+=−

Coâng thöùc HyperbolicCoâng thöùc löôïng giaùc

Ñ h a ø m : ( s h x ) ’ = c h x , ( c h x ) ’ = s h x . Ñ N : t h x

= s h x /c h x ; c t h x = 1/t h x

A Ù P D U ÏN G H A Ø M M U Õ , L O G : P H A Â N R A Õ P H O Ù N G X A Ï

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T o á c ñ o ä p h a â n r a õ c u û a v a ä t l ie ä u

p h o ù n g x a ï t y û le ä t h u a ä n v ô ù i k h o á i

lö ô ïn g h ie ä n c o ù . H a õ y t ìm q u y lu a ä t

p h a â n r a õ c u û a v a ä t l ie ä u n a ø y ?G ia û i : G o ï i R ( t ) – k h o á i lö ô ïn g v a ä t t h ô ø i

ñ ie å m t ⇒ t o á c ñ o ä p h a â n r a õ : R ’ ( t ) =

d R /d t < 0 ( v ì R g ia û m ) . T h e o q u a n s a ù t :( )0 leätyû soá haèng >−= :kkRdtdR ( ) kteRtRkdt

RdR −=⇒−=⇒ ∫ ∫ 0

C a r b o n C – 14 : C h u k y ø b a ù n p h a â n

r a õ : 5 7 3 0 n a ê m ⇒ T ìm R ( t ) ?G ia û i : T – c h u k y ø b a ù n p h a â n r a õ ⇒

K h o á i lö ô ïn g : R 0 /2 t a ï i t h /ñ ie å m T : T

kkTeRR kT 2ln

2ln2 0

0 =⇒=⇒= − ( ) teRtRT 000121.005730 −=⇒=

T A Á M V A Û I L IE Ä M T H A Ø N H T U R IN ---------------------------------------------------------------------------------------------------

-----------------------------

N a ê m 13 5 6 , c a ù c n h a ø k h a û o c o å

p h a ù t h ie ä n t a ï i t h a ø n h T u r in ( YÙ ) t a á m

v a û i c o ù a û n h a â m b a û n h ie ä n h ìn h

n g ö ô ø i ñ ö ô ïc x e m la ø C h u ù a J e s u s →

T r u y e à n t h u y e á t : T a á m v a û i l ie ä m

t h a ø n h T u r in . N a ê m 19 8 8 , T o a ø t h a ù n h

V a t ic a n c h o p h e ù p V ie ä n B a û o t a ø n g

A n h x a ù c ñ ò n h n ie â n ñ a ï i t a á m v a û i

b a è n g p h ö ô n g p h a ù p ñ o à n g v ò p h o ù n g

x a ï C – 14 → S ô ï i v a û i c h ö ù a 9 2 % - 9 3 %

lö ô ïn g C – 14 b a n ñ a à u . K e á t lu a ä n ?

G ia û i : T ö ø c o â n g

t h ö ù c t r ö ô ù c :

( ) teRtR 000121.0

0

−= ( )

−=⇒0

ln000121.0

1RtR

t

R /R 0 : 0 . 9 2 →

0 . 9 3 ⇒

( ) ( ) 60093.0ln&68992.0ln 21 ≈=≈= tt

T h ö ïc n g h ie ä m : 19 8 8 ⇒ T u o å i t a á m v a û i

k h i ñ o ù : 6 0 0 – 6 8 8 ⇒ K lu a ä n ?