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1
AO HAM, VI PHN
HAM MT BIN
Lecture 4
Nguyen Van Thuy
Review
inh ly (Kp). Nu () () () khi gn
va
thi
inh ly
lim ( ) lim ( ) lim ( )x a x a x a
f x L f x L f x
Giai tich 1 4-2 Nguyen Van Thuy-University of Science
lim ( ) lim ( )x a x a
f x h x L
lim ( )x a
g x L
Review
inh nghia. Ham f c goi la lin tuc tai a nu
f gian oan tai a nu f khng lin tuc tai a
f lin tuc trn khoang (a, b) nu f lin tuc tai moi
im thuc khoang o
Cu 65. Tim a ham s sau
lin tuc tai = 1
lim ( ) ( )x a
f x f a
2
2
2
1arctan , 1
( 1)( )
3, 1
1
xx
f xx x a
xx
Giai tich 1 4-3 Nguyen Van Thuy-University of Science
Review
inh ly. Tt ca nhng ham sau lin tuc trn min
xac inh
Ham a thc
Ham phn thc hu ty
Ham cn thc
Ham mu
Ham logarithm
Ham lng giac
Ham lng giac ngc
Giai tich 1 4-4 Nguyen Van Thuy-University of Science
Review
7 dang v inh
Cac gii han c ban
Vi du. Tinh
0 0.00
, , , ,1 ,0
0,
1/
0 0
sin 1lim 1, lim 1 , lim(1 )
u
u
u u u
ue u e
u u
0
tan 2) lim
x
xa
x
1) lim 1
2
x
xb
x
Giai tich 1 4-5 Nguyen Van Thuy-University of Science
H s gc ca ng thng
Giai tich 1 Nguyen Van Thuy-University of Science 4-6
?
2
H s gc ca ng thng
Giai tich 1 Nguyen Van Thuy-University of Science 4-7
= =
H s gc ca tip tuyn
Tinh
Tinh
Nhn xt
Giai tich 1 Nguyen Van Thuy-University of Science 4-8
0lim ABh
k
H s gc ca tip tuyn
Giai tich 1 4-9 Nguyen Van Thuy-University of Science
0
( ) ( )limtth
f a h f ak
h
Vn tc tc thi
Vn tc trung binh
Vn tc tc thi tai thi im =
( ) ( )sa h sav
h
0
( ) ()() lim
h
sah sava
h
Giai tich 1 4-10 Nguyen Van Thuy-University of Science
ao ham
inh nghia. ao ham cua ham s tai
Phng trinh tip tuyn tai im (, ())
= ()( ) + ()
0
( ) ( )'( ) lim
h
f a h f af a
h
Giai tich 1 4-11 Nguyen Van Thuy-University of Science
ao ham
Vi du. Tinh ao ham bng inh nghia
1) () = 2 + , tinh (3)
2) . Tinh (2) ( )f x x
2
0 0
2
0 0
(3 ) (3) (3 ) (3 ) 12'(3) lim lim
7lim lim( 7) 7
h h
h h
f h f h hf
h h
h hh
h
Giai tich 1 4-12 Nguyen Van Thuy-University of Science
3
ao ham
Ky hiu ao ham cua ham s = ()
Chu y. () la gia tri tai = cua ham ()
Vi du. () = , phat biu (0) =
0 bi vi (0) = 0 la hng s, va ao ham
cua hng s la zero ung hay sai?
'( ) ' ( ) ( ) ( )xdy df d
f x y f x Df x D f xdx dx dx
Giai tich 1 4-13 Nguyen Van Thuy-University of Science
ao ham
Cac cng thc ao ham c ban
1
2 2
2 2
2 2
'( )' ', ( )' ', (ln )'
( )' 'ln , (sin )' 'cos , (cos )' 'sin
(tan )' '(1 tan ),(
' '(arcsin )' ,(arccos )'
1 1
' '(arctan )' ,(arcc
cot )' '(1 cot )
ot )'1 1
u u
u u
uu u u e e u u
u
a a u a u u u u u u
u u u
u uu u
u u
u uu u
u
u u
u
u
Giai tich 1 4-14 Nguyen Van Thuy-University of Science
ao ham
Cac tinh cht cua ao ham
Vi du
'
2
( ) ' ' ', ( . ) ' . '
' '( ) ' ' ',
u v u v c u c u
u u v uvuv u v uv
v v
1 cos 1 cos 1 cos( ) .(1 cos ) ' .sinx x xd
e e x e xdx
ln ln cos ?d
xdx
Giai tich 1 4-15 Nguyen Van Thuy-University of Science
Khi nao ao ham tn tai?
Gii han nay co th khng tn tai
Nu () tn tai hu han, c goi la kha
vi tai
Nu kha vi tai a thi lin tuc tai
0
( ) ( )'( ) lim
h
f a h f af a
h
Giai tich 1 4-16 Nguyen Van Thuy-University of Science
ao ham
Vi du
() = || co va khng co ao ham
tai = 0
1, 0'( )
1, 0
xf x
x
Giai tich 1 4-17 Nguyen Van Thuy-University of Science
ao ham cp cao
= , . . . , () = ( 1 )
+ = () + ()
()()= ()
Vi du. Tinh cua ham s
= arctan + 1 + 2
Vi du. Tinh cua ham s
= 2 + 1 arctan + 1 ln (2 + 2 + 2)
Giai tich 1 4-18 Nguyen Van Thuy-University of Science
4
ao ham cp cao
Cng thc
( )
1
1 ( 1) !
( )
n n
n
n
x a x a
( )(sin ) sin2
nx x n
( )(cos ) cos2
nx x n
( )( )ax n n axe a e
( )(sin ) sin2
n nax a ax n
( )(cos ) cos2
n nax a ax n
Giai tich 1 4-19 Nguyen Van Thuy-University of Science
ao ham cp cao
Cng thc Leibniz
vi
Vi du. a) Tinh b) Tinh
(0) !,!( )!
k
n
nf fC
knk
2 (100)( )xx e( )
2
2 1
5 6
nx
x x
Giai tich 1 4-20 Nguyen Van Thuy-University of Science
() () ( )
0
0 (0) () 1 (1) ( 1) () (0)
( )n
n k k nk
n
k
n n n n
n n n
fg Cf g
Cf g Cf g Cf g
Vi phn ca ham s
Tai x=a
=
Tai x
=
Giai tich 1 Nguyen Van Thuy-University of Science 4-21
Vi phn ca ham s
Cng thc
=
Vi du. Tim vi phn cp 1 cua ham s
Vi du. Tim vi phn cp 1 cua ham s
Giai tich 1 Nguyen Van Thuy-University of Science 4-22
lnarctan
3
xy
(3)xy x
V phn cp cao
Vi phn cp n
= ()()
Vi du. Tim vi phn cp 2 cua ham s
Vi du. Tim vi phn cp 2 cua ham s
Giai tich 1 Nguyen Van Thuy-University of Science 4-23
2ln(12)y x
2cot( )yarc x
Quy tc LHospital
inh ly. Nu ()
() co dang
0
0,
khi va
tn tai lim
()
()= thi
lim
()
()= lim
()
()=
Chu y: co th hu han hoc v han
Giai tich 1 4-24 Nguyen Van Thuy-University of Science
5
Quy tc LHospital
Chu y. Qua trinh co th thay bi
+,
, ,
Vi du
3 20 0
0 0
sin 1 coslim lim
3
sin cos 1lim lim
6 6 6
0 0
0 0
0
0
x x
x x
x x x
x x
x x
x
Giai tich 1 4-25 Nguyen Van Thuy-University of Science
Quy tc LHospital
Vi du. Tinh
) = 0 ) =1
3 ) = 2 ) =
1
3
Vi du. Tinh
) = ) = 0 ) = 1 ) = 2
Giai tich 1 Nguyen Van Thuy-University of Science 4-26
30
arctanlim
0
0x
x xL
x
0
0.limlnx
L xx
Quy tc LHospital
Vi du. Tinh
) = 1 ) =1
2 ) =
1
4 ) =
1
8
Vi du. Tinh
) = 0 ) = ) = 2 ) u sai
Giai tich 1 Nguyen Van Thuy-University of Science 4-27
2
0( 2)lim(2) 0xx
L x
1
1lim
1lnx
xL
x x
ao ham ca ham n
inh nghia. Ham s = () cho bi
phng trinh (, ) = 0 c goi la ham
n
Vi du. Cho ham s = () xac inh bi
phng trinh 2 + 2 = 2
Phng trinh trn xac inh hai ham n
2 22 , 2y x y x
Giai tich 1 4-28 Nguyen Van Thuy-University of Science
ao ham ca ham n
tinh ao ham cua ham n, chu y rng
Chu y. la ham s theo , con la bin s
Vi du. Tinh () bit 2 + 2 = 2
Ly ao ham theo ca hai v, ta c
'
( , ) 0 ( , ) 0x
F x y F x y
2 2 ' 0 'x
x yy yy
Giai tich 1 4-29 Nguyen Van Thuy-University of Science
ao ham ca ham n
Vi du. Tim ao ham (0) cua ham n
= () c cho bi phng trinh
=
) 0 = ) 0 =
) (0) =1
) (0) =
1
Giai tich 1 Nguyen Van Thuy-University of Science 4-30
6
ao ham ca ham n
Vi du. Vit phng trinh tip tuyn cua
ng cong cardioid
tai (0, 1/2)
2 2 2 2 2(2 2 )x y x y x
Giai tich 1 4-31 Nguyen Van Thuy-University of Science
ao ham ca ham n
Vi du. Vit phng trinh tip tuyn cua
ng cong lemniscate
tai (3, 1)
2 2 2 2 22( ) 25( )x y x y
Giai tich 1 4-32 Nguyen Van Thuy-University of Science
ao ham ca ham s dang tham s
inh nghia. Ham s = () cho di
dang = (), = () c goi la ham
s cho di dang tham s
Vi du. Ham s = () cho bi =
, = , /2 /2
o la ham s
21 , 1 1y x x
Giai tich 1 4-33 Nguyen Van Thuy-University of Science
1 -1 0
x
y
ao ham ca ham s dang tham s
ao ham cua ham s cho di dang tham
s
Vi du. Cho ham s = () xac inh bi
'( )
'( )
'( )'( )
'( )
dy y t dt
dx x
y ty x
x tt dt
'( ) sin , '( ) cos
cos , s
'( ) '( ) / '( ) / cot
in
x t a t y t b t
x a t y b t
y x y t x t b a t
Giai tich 1 4-34 Nguyen Van Thuy-University of Science
ao ham ca ham s dang tham s
Vi du. Tim () tai 0 = 2 cua ham s
= () cho bi phng trin