- * Foreign Language Studies
- Chinese
- ESL
- Science & Mathematics
- Astronomy & Space Sciences
- Biology
- Study Aids & Test Prep
- Book Notes
- College Entrance Exams
- Teaching Methods & Materials
- Early Childhood Education
- Education Philosophy & Theory All categories
- * Business
- Business Analytics
- Human Resources & Personnel Management
- Career & Growth
- Careers
- Job Hunting
- Computers
- Applications & Software
- CAD-CAM
- Finance & Money Management
- Accounting & Bookkeeping
- Auditing
- Law
- Business & Financial
- Contracts & Agreements
- Politics
- American Government
- International Relations
- Technology & Engineering
- Automotive
- Aviation & Aeronautics All categories
- * Art
- Antiques & Collectibles
- Architecture
- Biography & Memoir
- Artists and Musicians
- Entertainers and the Rich & Famous
- Comics & Graphic Novels
- History
- Ancient
- Modern
- Philosophy
- Language Arts & Discipline
- Composition & Creative Writing
- Linguistics
- Literary Criticism
- Social Science
- Anthropology
- Archaeology
- True Crime All categories
- Hobbies & Crafts Documents
- Cooking, Food & Wine
- Beverages
- Courses & Dishes
- Games & Activities
- Card Games
- Fantasy Sports
- Home & Garden
- Crafts & Hobbies
- Gardening
- Sports & Recreation
- Baseball
- Basketball All categories
- Cooking, Food & Wine
- Personal Growth Documents
- Lifestyle
- Beauty & Grooming
- Fashion
- Religion & Spirituality
- Buddhism
- Christianity
- Self-Improvement
- Addiction
- Mental Health
- Wellness
- Body, Mind, & Spirit
- Diet & Nutrition All categories
- Lifestyle
67% found this document useful [3 votes]
2K views
6 pages
Copyright
© Attribution Non-Commercial [BY-NC]
Available Formats
PDF, TXT or read online from Scribd
Share this document
Did you find this document useful?
67% found this document useful [3 votes]
2K views6 pages
Tom Tat Cong Thuc Toan Cao Cap A1, 2
H
μ
m mét biÕn
1. C«ng thøc tÝnh ®¹o hµm
•
[u
α
]’ =
α
.u’.u
α
-1
[
α
: H»ng sè, U: Hµm sè]
•
[a
U
]’ = u’.
ln
a.a
U
[a: H»ng sè, U: Hµm sè]
•
[e
U
]’ = u’.e
U
•
[Sin u]’ = u’.cos u
•
Cos u]’ = - u’.sin u
•
[Tg u]’=
uCosu
2
'
;
•
[Cotg u]’=
uSinu
2
'
−
•
[Log
a
u]’ =
auu
ln.'
•
[arcsin u]’ =
2
1'
uu
−
;
•
[arccos u]’ =
2
1'
uu
−−
•
[arctg u]’ =
2
1'
uu
+
;
•
[arccotg u]’ =
2
1'
uu
+−
•
[u ± v]’=u’ ± v’
•
[u.v]’= u’v+v’u
•
[
vu
]’ =
2
''
vuvvu
−
2. Vi ph©n
du = u’.dx
3. Giíi h¹n
- V« cïng bÐ t
−
¬ng ®
−
¬ng
:
0][
\=
→
x Lim
a x
α
\=>
α
[x] ®
−
îcgäi lµ v« cïng bÐ khi x->a
1][][
\=
→
x x Lim
a x
β α
-->
α
[x] vµ
β
[x] lµ hai v« cïng bÐ t
−
¬ng ®
−
¬ng khi x->a Ký hiÖu
:
α
[x]
∼β
[x] khi x->a
§Þnh lý
: NÕu
α
[x]
∼α
1
[x] vµ
β
[x]
∼β
1
[x]khi x->a th×
][][][][
11
x x Lim x x Lim
a xa x
β α β α
→→
\=
Sin x
∼
x khi x->0
ArcSin x
∼
x khi x->0
Tg x
∼
x khi x->0
ArcTg x
∼
x khi x->0
e
x
-1
∼
x khi x->0
ln[1+x]
∼
x khi x->0 - C«ng thøc Lopital khö d¹ng
00
;
∞∞
:
1
][']['][][
x g x f Lim x g x f Lim
a xa x
→→
\=
4. TÝnh liªn tôc cña hµm sè
Hµm sè: y = f[x] liªn tôc t¹i x = x
0
nÕu
: + f[x
0
] x¸c ®Þnh vµ h÷u h¹n +
][][
0
0
x f x f Lim
x x
\=
→
[NÕu hµm sè kh«ng liªn tôc t¹i x
0
th× x
0
®c gäi lµ ®iÓm gi¸m ®o¹n]
Hµm sè s¬ cÊp y = f[x] sÏ liªn tôc t¹i mäi ®iÓm mµ hµm sè x¸c ®Þnh
5. TÝch ph©n
- C«ng thøc nguyªn hµm
•
C xdx x
++\=
+
∫
1
.]1[1
α α
α
[
α
\>0]
•
C aadxa
x x
+\=
∫
.ln1
•
C edxe
x x
+\=
∫
•
C xdx x
+\=
∫
cos.sin
•
∫
\=
dx x
.sin1
2
-
cotg
x
+ C
•
C xdx x
+−\=
∫
sin.cos
•
∫
\=
dx x
.cos1
2
tg
u
+ C
•
C a xdx xa
+\=−
∫
arcsin.1
22
•
∫
+
dx xa
.1
22
\=
a
1
.
arctg
a x
+C
•
C xdx x
+\=
∫
ln.1
- TÝch ph©n tõng phÇn:
∫ ∫
−\=
vduvudvu
..
H
μ
m nhiÒu biÕn
7. §¹o hµm riªng vµ vi ph©n toµn phÇn
•
x y x f y x x f Lim x y x f y x f
x x
Δ−Δ+\=∂∂\=
→Δ
],[],[],[ ],[
0000 00000'
•
y y x f y y x f Lim y y x f y x f
y y
Δ−Δ+\=∂∂\=
→Δ
],[],[],[ ],[
0000 00000'
•
Vi ph©n toµn phÇn cÊp 1:
dy y x f dx y x f y xdf
y x
],[],[],[
''
+\=
•
Vi ph©n toµn phÇn cÊp 2:
222222
],[],[2],[],[
dy y x f dxdy y x f dx y x f y x f d
yy xy xx
++\=
•
C«ng thøc tÝnh gÇn ®óng: f[x+
Δ
x, y+
Δ
- = f[x,y] + f
x
’[x,y].
Δ
x + f
y
’[x,y].
Δ
y
•
§¹o hµm cña hµm hîp: F[u,v], trong ®ã u =u[x,y]; v=v[x,y]
:
⎪⎪⎩⎪⎪⎨⎧∂∂∂∂+∂∂∂∂\=∂∂∂∂∂∂+∂∂∂∂\=∂∂
yvv F yuu F y F xvv F xuu F x F
•
§¹o hµm cña hµm Èn
: *NÕu F[x,y] = 0
; y= y[x]: \=>
],[],[]['
''
y x F y x F x y
y x
−\=
*NÕu F[x,y,z] = 0
; z= z[x,y]: \=>
],,[ ],,[ ]['
''
z y x F z y x F x z
x x
−\=
;
],,[ ],,[ ]['
''
z y x F z y x F y z
y x
−\=
. Cù trÞ hµm nhiÒu biÕn 8
B
−
íc1: T×m ®iÓm c¸c ®iÓm dõng M[x
i
,y
i
] lµ nghiÖm cña hÖ PT:
⎪⎩⎪⎨⎧\=\=
0],[ 0],[
''
y x f y x f
y x
B
−
íc2: KiÓm tra ®iÓm M[x
i
,y
i
] cã lµ cùc trÞ A=f
xx
”[x
i
,y
i
]; B=f
xy
”[x
i
,y
i
]; C=f
yy
”[x
i
,y
i
]; B
2
-AC < 0 A0: M[x
i
,y
i
]--- Cùc tiÓu B
2
-AC \> 0 M[x
i
,y
i
]--- kh«ng lµ cùc trÞ B
2
-AC \= 0 M[x
i
,y
i
]--- Ch
−
a kÕt luËn ®
−
îc
Cùc trÞ cã ®iÒu kiÖn:
T×m cùc trÞ hµm: u=f[x,y,z] víi ®k: g[x,y,z]=0 Gi¶i hÖ PT:
⎪⎩⎪⎨⎧\=\=\=
0],,[
''''''
z y x g g f g f g f
z z y y x x
\=> NghiÖm M[x,y,z]
9. TÝch ph©n kÐp
Trong hÖ täa ®é ®Ò c¸c: -
NÕu miÒn D lµ h×nh ch÷ nhËt x¸c ®Þnh bëi: a
≤
x
≤
b
vµ c
≤
y
≤
d
th×:
∫∫∫∫
\=
d cba D
dy y x f dxdxdy y x f
],[],[
-
NÕu miÒn D lµ h×nh ch÷ nhËt x¸c ®Þnh bëi: a
≤
x
≤
b
vµ y
1
[x]
≤
y
≤
y
2
[x]
th×:
∫∫∫∫
\=
][][
21
],[],[
x y x yba D
dy y x f dxdxdy y x f
2
§æi biÕn trong tÝch ph©n kÐp: x=x[u,v] ; y=y[u,v]
∫∫∫∫
\=
D D
dudvvu yvu x f J dxdy y x f
]],[],,[[.||],[
trong ®ã: J=
''''
],[],[
vuvu
y y x xvu D y x D
\=
Trong hÖ täa ®é cùc:
I\= [x\= r.cos
ϕ
; y= r.sin
ϕ
]
∫∫∫∫
\=
'
.].sin,cos[],[
D D
drd r r r f dxdy y x f
ϕ ϕ ϕ
Dxy
ϕ2ϕ1
r=g2[
ϕ]
r=g1[
ϕ]
D xy
ϕ2ϕ1
r=g[
ϕ]
xy0
0 0 Dr=g[
ϕ]
3
DL
10. TÝch ph©n ®
−
êng lo¹i 1
- NÕu: y=y[x], a
≤
x
≤
b
th×:
2
[ , ] [ , [ ]] 1 ' [ ].
ba AB
f x y ds f x y x y x dx
\= +
∫ ∫
∫ ∫
\=
21][2][1
.].sin,cos[
ϕ ϕ ϕ ϕ
ϕ ϕ ϕ
g g
dr r r r f d I
∫ ∫
\=
π ϕ
ϕ ϕ ϕ
20][0
.].sin,cos[
g
dr r r r f d I
∫ ∫
\=
21][0
.].sin,cos[
ϕ ϕ ϕ
ϕ ϕ ϕ
g
dr r r r f d I
- NÕu: x=x[t], y=y[x], t
1
≤
t
≤
t
2
th×:
21
2 2
[ , ] [ [ ], [ ]]. ' [ ] ' [ ].
t t AB
f x y ds f x t y t x t y t dt
\= +
∫ ∫
. TÝch ph©n ®
−
êng lo¹i 2 11
- NÕu
AB
®
−
îc cho bëi: y=y[x], a,b lµ hoµnh ®é cña A vµ B th×
[ , ] [ , ] [ [ , [ ]] [ , [ ]]. '[ ]]
ba AB
P x y dx Q x y dy P x y x Q x y x y x dx
+ \= +
∫ ∫
- NÕu
AB
cho bëi: x=x[t], y=y[t], t=t
A
[t¹i A], t=t
B
[t¹i B] th×
:
B
[ , ] [ , ] [ [ [ ], [ ]]. '[ ] [ [ ], [ ]]. '[ ]]
B A
t t AB
P x y dx Q x y dy P x t y t x t Q x t y t y t dt
+ \= +
∫ ∫
- C«ng thøc Green
:
[ , ] [ , ] [ ]
L D
P Q P x y dx Q x y dy dxdy x y
∂ ∂+ \= −∂ ∂
∫ ∫∫
[
L- lµ miÒn biªn cña D và lµ mét ®
−
êng khÐp kÝn
] HÖ qu¶: NÕu
Q P x y
∂ ∂\=∂ ∂
trong D th×:
[ , ] [ , ] 0
L
P x y dx Q x y dy
+ \=
∫
•
§Þnh lý 4 mÖnh ®Ò t
−
¬ng ®
−
¬ng: Cho P[x,y] vµ Q[x,y] liªn tôc, cã ®¹o hµm riªng cÊp 1 trong miÒn D. Khi ®ã, 4 mÖnh ®Ò sau lµ t
−
¬ng ®
−
¬ng: [1]
Q P y
∂ ∂\=∂ ∂
[2]
∃
u[x,y] sao cho: d
u[x,y]
\=
P[x,y]
dx+
Q[x,y]
dy [3] Mäi ®
−
êng cong kÝn L
⊂
D th×:
[ , ] [ , ] 0
L
P x y dx Q x y dy
+
+ \=
∫
[L
+
- ®Þnh h
−
íng d
−
¬ng, do c«ng thøc Green] [4]
TÝch ph©n kh«ng phô thuéc vµo ®
−
êng cong nèi 2 ®iÓm A,B
[ , ] [ , ]
AB
P x y dx Q x y dy
+
∫