Câu hỏi trắc nghiệm ôn thi giữa kỳ môn Giải tích 2 | Đại học Bách Khoa Hà Nội

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Môn: Giải tích 2

Trường: Đại học Bách Khoa Hà Nội

Thông tin:

16 trang 6 tháng trước

Tác giả:

Profile Thùy Dương
  f( x, y x) = ln (
2
+ y
2
)  D
f
     f( x, y E) ;
f
     f( x, y)
     
D
f
= IR E− {0 };
f
= IR D
f
= IR
2
− {( 0 , 0 ) }; E
f
= IR
2
! D
f
= IR
2
− {( 0 , 0 ) }; E
f
= IR " D
f
= IR
2
− {( 0 , 0 ) }; E
f
= ( 0 , +∞)
 # $% "& %  '( )  !) x
2
+ y
2
≤ 9 * x
2
+ y
2
≤ 2 y
8 π ! 4 π 1 0 π "   + 
 , $% "& %  '( )  !)
x
2
9
+
y
2
4
≤ 1 * y ≥ 0 , x ≤ 0
3 π
2
!
3 π
4
3 π "   + 
 - $. f
′
x
( 0 , 0 ) ; f
′
y
( 0 , 0 ) *) f ( x, y) =
( )x + y arctg(
x
y
)
2
, y = 0
π
2
x, y = 0
f
′
x
=
π
2
; ∃f
′
y
! f
′
x
= 0 ; f
′
y
= 0 f
′
x
=
π
2
; f
′
y
= 0 " f
′
x
=
π
2
; f
′
y
= 1
 / 01 2 2 3 % ' I =
2
0
dx
√
x
0
f( x, y) dy 4
4
2
dx
√
x
x−2
f( x, y) dy
I =
2
0
dy
y
y+2
f( x, y) dx I =
2
0
dy
y+2
y
f( x, y) dx
! 5  +  " I =
2
dy
4
f( x, y) dx
 6 $% I =
1
0
dy
1
√
y
c o s ( x
3
− 1 ) dx
I = −
1
2
s in 1 ! I = −
1
3
s in 1 I =
1
3
s in 1 " I =
1
2
s in 1
 7  f ( x, y e) = ( x + 2 y)
3 +x y
$% I =
∂

∂x

f( 1 , 0 )
I = 4 3.
9
e
3
! I = 3
9
e
3
I = 1 1 e
3
" I = 1 3 3.
9
e
3
 8  f ( x, y) =
sin x y( +
)
x
e
t
dt      
5  +  f
′
( x, y) = e
sin
( +x y
)
.cos y( x +
2
)
! f
′
x
( x, y) = e s y( x +
2
) − 2 xe
x
 9 $% I =
D
2 dxdy; D = {( x, y) ∈ IR
2
|0 ≤ x x;
2
≤ y; y ≤ x + 2 }
I =
20
3
! I =
10
3
I =
26
3
" I =
25
6
 : $% I = lim
x → 0
y → 0
( x
2
+ y
2
) c o s
1
x
2
+ y
2
I = 0 ! I = +∞ ∃I " I = 1
1
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   f( x, y) = g( x − 2 y, 2 x + y) ; < u( x, y x, y) = x − 2 y; v( ) = 2 x + y    
 
df( x, y g) =
′
u
dx g+
′
v
dy df( x, y g) = 3
′
u
dx g−
′
v
dy
! df( x, y g) = (
′
u
− 2 g
′
v
) dx + ( g
′
v
− 2 g
′
u
) dy " df ( x, y g) = (
′
u
+ 2 g
′
v
) dx + ( g
′
v
− 2 g
′
u
) dy
 # $.   ) 3 A =  f =    3 B =  f  f( x, y) = 1 − 3 x − 4 y
 > x
2
+ y
2
≤ 2 5
A = f( 3 , 2 ) = −1 6 ; B = f ( −3 , −2 ) = 1 8 A = f ( −3 , −4 ) = 2 6 ; B = f( 3 , 4 ) = −2 4
! 5  +  " A = f( 3 , ,−4 ) = 8 ; B = f( −3 4 ) = −6
 ,  f( x, y e) =
x y+
$. + 1 $  f 3 3' #   &  1 M
0
( 1 , 0 )
1 + ( x − 1 ) + +y
( 1)x−
2
+ y( x − 1 ) +
y
2
+ o( ρ
2
) ; =ρ
( x − 1 )
2
+ y
2
! e + e( x − 1 ) + ey e+
(x−1)
2
+ ey( x −1 ) + e
y
2
+ o( ρ
2
) ; =ρ
( x − 1 )
2
+ y
2
e − e( x − 1 ) + ey e+
(x−1)
2
− ey( x − 1 ) + e
y
2
+ o( ρ
2
) ; =ρ
( x − 1 )
2
+ y
2
" 5  + 
 - ?3 '2) . < '( 3' "&  < !&  z = 4 x
2
− y
2
+ 2 y  ( −1 , 2 , 4 )
8 x + 2 y + z = 0 ! 8 x + 2 y −z = 0 x + 2 y + z = 7 " 4 x+2 y −z +4 = 0
 / $.
df
dt
= !3 f( x, y t, y) = x ln ( x + 2 y) ; x = s in = c o s t
c o s t [ln ( x + 2 y) +
x
x y+2
] [ln ( x + 2 y) +
x
x+2y
] −
2x
x+2y
! c o s t [ln ( x
x 2x
"
x 2x
x+2y
s in t
 6 $. f
′
x
= !3 f( x, t e) =
sin (
)
−
t
x
2
e
sin (
)
! −
t
x
2
c o s (
t
x
) e
sin (
)
c o s (
t
x
2
) e
sin (
)
"
t
x
2
c o s (
t
x
) e
sin (
)
 7 $.
∂f
∂t
= !3 f( x, y) = e
x
s in y; x = st
2
, y s=
2
t
2 ste
st
s in ( s
2
t)   + 
! e
st
s in ( s
2
t e) +
st
c o s ( s
2
t) " 2 ste
st
s in ( s
2
t s) −
2
e
st
c o s ( s
2
t)
 8   f( x, y x) =
4
+ y
4
− 4 xy + 1 (   
f  2 1  ( 1 , 1 ) *  ( − −1 , 1 ) @ A  & 2 1 * & 2 
! f +  2   ( −1 , −1 ) " @ A  & 2 1
 9 $. f
′′′
xxy
= !3
2 y
3
e
xy
( 2 − xy
2
) ! 4 y
3
e
xy
2 y
3
e
xy
( 2 + xy
2
) "   + 
 #:  f( x, y) = ln ( x + y + 3 ) $. + 1 B   f 3 3' #
 &
ρ =
√
x
2
+ y
2
ln 3 +
x
3
+
y
3
−
x
2
1 8
−
xy
9
−
y
2
1 8
+ o( ρ
2
)   + 
! ln 3 +
x
3
+
y
3
−
x
2
9
−
xy
9
−
y
2
9
+ o( ρ
2
)
" ln 3 +
x
3
+
y
3
+
x
2
1 8
+
xy
9
+
y
2
1 8
+ o( ρ
2
)
 # $.
df( 0 , 1 ) = !3 f( x, y x) = ln ( x +
√
2
+ y
2
)
2 dx + dy ! 2 dx + 3 dy dx − dy " dx + dy
2
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 ##  < !&  x
2
+ x + 1 = z 0  < .
B<  ! B<  C!" ' "   + 
 #,  < !&  z + x
2
+ y
2
+ +x y = 3 0  < .
B<  ! C!" ' B<  " D'"
 #- $. 2  2 "  f( x, y x) = (
2
+ y) e
y/2
( 0 , −2  1 2  ( 0 , −2 ) +  1 "2
! ( 0 , 0  1 2  " ( 0 , −2 )  1 2 1
 #/ ?3 '2) . < '( 3' "&  <  z = e
x
−y
 ( 1 , −1 , 1 )
2 x+2 y−z +1 = 0 ! x + 2 y −z + 2 = 0 2 x −2 y +z −5 = 0 "   + 
 #6 $. z
′
y
= !3 z = z( x, y)   1   2 '2) . ln ( x + yz) = 1 + xy
2
z
3
2 xyz
3
( x + yz) − z
y + 3 xy
2
z
2
(
x + yz)
  + 
!
2 xyz
3
( x + yz) − z
y −3 xy
2
z
2
(
x + yz)
"
z −2 xyz
3
( x + yz)
y −3 xy
2
z
2
(
x + yz)
 #7 EF G .   f
′
x
( 1 , 2 )  H+ & & 3   3' 3  @I$$J
@I$$ *) 2)     y = 2 * f( x, y)  1   & !K
! @I$$ *) 2)     y = 1 * f ( x, y)  1   & !K #
@I$$ *) 2)     x = 1 * f( x, y)  1   & !K #
"   +
 #8 $.
df( 3 , 4 ) = !3 f( x, y) =
√
x
2
+ y
2
3 dx + 4 dy !
3
5
dx +
4
5
dy
3
1 0
dx +
4
1 0
dy "
7
5
 #9   3 f( x, y x) =
4
+ y
4
− x
2
− 2 xy − y
2
(     
  +  f +  2   ( − −1 , 1 )
! f  2 1  ( −1 , −1 ) " f  2   ( − −1 , 1 )
 ,:   f( x, y x) = 2
3
+ xy
2
+ 5 x
2
+ y
2
(   
f  2 1  = 2   ( 0 , 0 ) ( − −1 , 2 )
! f  , 1 "2
f  2   ( 0 , 0 ) = +  2   ( − −1 , 2 )
" f  2
 , $.   ) 3 H$LMJ=    3 H$MMJ  f( x, y x) =
2
+ 2 y
2
 
x
2
+ y
2
≤ 1
$LM = 1 = $MM = 0 $LM = 2 = $MM = −1
! $LM = 0 = $MM = −1 "   + 
 ,# $.    3'  z
′′
xx
( 1 , 0 )   # !3 z = ln ( x + y
2
+ 1 )
−1
4
!
1
2
1
4
"
1
3
3
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 ,, 
f( x, y) =
xy
x
+ y
$% df( 2 , −1 )
dx + 4 dy ! dx + dy   +  " 4 dx + dy
 ,- $% % ' I =
D
( )x + y dxdy x*) > )  !)  2)
2
+ y
2
= 1 , x
2
+ y
2
= 4 =, y
0 0, y = x 3 ' x ≥
  +  ! I =
2
3
I =
1
3
" I =
7
3
 ,/  < !& 
x +
√
3 y
2
+ z
2
− 1 = 0 0  < .
B<  ! C!" ' M2 <  " B<  & '%
 ,6 $% % ' I =
D
1
√
x
2
+ y
2
dxdy x*) > )  !)  2)
2
+y
2
= 4 3
, y = x, y = x
√
3 ' y ≥ x
I =
π
3
! I =
π
6
I =
2
9
"   + 
 ,7 $% % ' I =
D
2 ydxdy *) > )  !)  2) x = y
2
+ y − 1 , x = y + 3
I = −1 6 ! I = 0 I = 1 6 " I = 4
 ,8 $% % ' I =
D
( )x + y dxdy *) > )  !)  2) y = x
2
, y = x
I = 3 /2 0 ! I = 1 /3 I = 3 /1 0 "   + 
 ,9 $. z
′
x
= !3 z
e
x
= 0
e
y
+ ze
x
y + e
x
! −
e
+ ze
y + e
x
−
e
y + e
x
" −
y + e
x
e
y
+ ze
x
 -: 
f( x, y) =
2 c o s x
e
y
$. + 1 B   f 3 3' #
  +  2 − 2 y − x
2
+ y
2
+ o( ρ
2
)
! 1 + 2 y + x
2
− y
2
+ o( ρ
2
) " 2 x −2 y −x
2
+ y
2
+ o( ρ
2
)
 - $% % ' I =
D
xdxdy *) >    NO5= O( 0 , 0 ) , A , B( 1 , 1 ) ( 0 , 1 )
I =
1
9
!   +  I =
1
6
" I =
1
3
 -#  f( x, y) = x ln ( xy) $% f
′′′
0
1
xy
 -,  < !& 
√
4 − 2 x
2
− z
2
+ y − 1 = 0 0  < .
M2 < '" ! C!" ' B<  " B< 
 -- $% I =
xdxdydz *) )  !) y = x; y = 3 x; x = 1 ; z = 0 ; z = 4 − y
I =
2
5
! I =
1
3
I =
5
3
"   + 
4
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 -/   2    z = 5 − 4 8x − y *)  +& x
2
− 8 y
2
= 8  P ( 4 , −1 )  1
"2   L 2 *)
λ =
1
2
(     
P  1 2 1   +& P  1 2    +&
!   +   " P +  1 2    +&
 -6 $% I =
xdxdydz *) )  !) y = x; y = 2 x; x = 1 ; z = 0 ; z = 4 − x
I =
1
3
! I =
1 3
1 2
I =
2
1 3
"   + 
 -7 $. * ' 3' & dz   # !3 z = s in x + cosy + xy
  +  dz = ( c o s x − y) dx + ( x −s in y) dy
! dz = ( c o s x + y) dx + ( x −s in y) dy " dz = ( c o s x + y) dx + ( x + s in y) dy
 -8 $. y
′
( (x) = !3 y = y x)   1   2 '2) . y
5
+ x
2
y
3
= 1 + ye
x
2 xye
x
5 y
4
+ 3 x
2
y
2
− e
x
!
2 xye
x
+ 2 xy
3
5 y
4
+ 3 x
2
y
2
  +  "
2 xy
3
− 2 xye
x
5 y
4
+ 3 x
2
y
2
− e
x
 -9 @ f( t e) =
√
t
*) t = x
2
+ y
2
 '2) .   
  +  ! xf
′
x
+ yf
′
y
= 0 yf
′
x
+ xf
′
y
= 0 " yf
′
x
− xf
′
y
= 0
 /:   3 f ( x, y) = a r c t g (
x
y
) $% df( 1 , 1 )
1
5
dx +
2
5
dy !
1
2
dx −
1
2
dy 2 dx −
2
5
dy "
1
2
dx +
1
2
dy
 / $% % '
D
≤ 1 ; 0 ≤ ≤y 2
I = 3 ! I = 5   +  " I = 2
 /# $% % ' I =
D
1 − x
2
− y
2
dxdy *) >  .  ) *
I =
2 π
3
! I =
π
2
  +  " I = π
 /,  < !&  x
2
− y
2
− z
2
= 2 y + 1 0  < .
B<  ! C!" ' B<  " B<   '%
 /-   2  2 "   z = 3 ( x
2
+ y
2
) − x
3
+ 4 y  P ( 0 , −
2
3
) (   
 
P  1 2 1 P +  1 2 
! P  1 
 // 
f( x, y) =
8 e
x
2 +
y
$. + 1 B   f 3 3' #
  +  4 + 4 x −2 y + 2 x
2
− 2 xy + y
2
+ o( ρ
2
)
! 4 + 2 + 4x − 3 y x
2
− 2 xy + y
2
+ o( ρ
2
) " 4 x + 2 + 2y x
2
+ 2 xy + y
2
+ o( ρ
2
)
 /6  f( x, y) =
√
x
3
− y
3
$% f
′
x
( 0 , 0 ) , f
′
y
( 0 , 0 )
f
′
x
( 0 , 0 ) = 1 , f
′
y
( 0 , 0 ) = −1   + 
! f
′
x
( 0 , 0 ) = 1 , f
′
y
( 0 , 0 ) = 1 " +  
5
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 /7 $% % ' I =
D
1
√
x
2
+ y
2
dxdy x*) >  
2
+ y
2
≤
2 x; y ≤ ≥x
√
3 ; y x
I =
√
3 +
√
2 ! I =
√
3 −
√
2 I =
√
2 "   + 
 /8  < !&  x
2
+ z
2
+ y = 2 x + 1 0  < .
B<  ! C!" ' M & '% " B< 
 /9 $% % ' I =
D
( )xy + 2 y dxdy *) >    NO5= O( 0 ( 1 ( 2, 0 ) , A , 1 ) , B , 0 )
  +  ! I = 2 I = 1 " I = −1
 6: $. df( −6 , 4 ) = !3 f( x, y) = s in ( 2 x + 3 y)
2 dx + 3 dy ! 3 dx + dy   +  " 2 3dx − dy
 6  < !& 
√
4 − x
2
− z
2
+ 3 − y = 0 0  < .
M2 <  ! C!" ' B<  " B<  & '%
 6# 
f( x, y) =
1
√
x
2
+ y
2
$.    D
f
*    E
f
D
f
= IR
2
\{( 0 , 0 ) }; E
f
= ( 0 , +∞)   + 
! D
f
= IR E\{0 };
f
= [0 , +∞) " D
f
= IR
2
\{( 0 , 0 ) }; E
f
= [0 , +∞)
 6, $% I =
D
xdxdy x*) D  2 . 
2
+ ( y − 2 )
2
≤ 1 , x ≥ 0
  +  ! I =
3
I =
−1
" I =
2
3
 6-   z = z( x, y)   2 '2) . z
3
− 4 xz + y
2
− 4 = 0 $% z
′
y
( 1 , −2 ) 3
z( 1 , −2 ) = 2
−
1
2
!   + 
2
3
"
1
2
 6/ 01 2 2 3 % '  % ' +'
1
0
dy
1
−
√
y
f( x, y) dx
0
−1
dx
1
x
f( x, y) dy4
1
0
dx
x
0
f( x, y) dy   + 
!
1
−1
dx
1
x
f( x, y) dy "
0
−1
dx
1
x
f( x, y) dy4
1
0
dx
1
0
f( x, y) dy
 66  f( x, y) = y ln ( xy) $% f
′′
xx
−y
x
!   +  "
y
x
 67 
f( x, y) =
x + y
2
x + y
$% df( 1 , 1 )
2
3
dx −
1
3
dy !   + 
−1
9
dx +
1
9
dy "
−1
3
dx +
1
3
dy
 68  f = f( u, v e) =
uv
, u x, y x= u( ) =
3
y, v x, y x= v( ) =
2
$. df
ve
uv
( 3 x
2
ydx x+
3
dy ue) +
uv
2 xdx ve
uv
x
3
dy + ue
uv
2 xdx
! ve
uv
3 x
2
ydx + ue
uv
2 xdy "   + 
 69  < !& 
y +
√
4 x
2
+ z
2
+ 2 = 0 0  < .
B<  ! M2 <  C!" ' " B<  & '%
6
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 7: $.   ) 3=    3  z = x
2
+ xy − 1    O5 *)
A( 1 , 1 ) ; B( 2 , 2 ) ; C( 3 , 1 )
z
max
= 1 1 , z
min
= 7   + 
! z
max
= 1 1 , z
min
= −7 " z
max
= 1 1 , z
min
= 1
 7   ) 3 M *  3 m  f ( x, y) = 3 + 2 xy  D = {( x, y) ∈ IR
2
: x
2
+ y
2
≤ 1 }
M = 4 = 2, m ! M = 4 = 0, m   +  " M = 4 = 3, m
 7#  < !&  x
2
+ z
2
− y
2
= 2 x + 2 z − 2 0  < .
C!" ' ! B<  B<  # '% " B< 
 7,  f( x, y x) = 2
2
− 3 xy + y
3
$% d
2
f( 1 , 1 )
4 dx
2
− 3 dxdy + 6 dy
2
4 dx
2
− 6 dxdy + 6 dy
2
!   +  " 2 dx
2
+ 6 dxdy + 6 dy
2
 7- $% % ' I =
D
1 2 ydxdy *) > )  !)  2) x = y
2
, x = y
I =
3
2 0
! I = 1   +  " I = 4
 7/   # !3 z = ( x + y
2
) e
x/2
* 1 P ( −2 , 0 ) (     
P +  1 "2   + 
! P  1  2 1 " P  1  2 
 76 $% % ' I =
2 xdxdy *) > )  !)  2) y = 2 − x
2
, y = x
I =
3
2 0
! I =
2
I =
1 0
"   + 
 77 $% I =
D
ydxdy x*) D  2 . 
2
+ ( y − 1 )
2
≤ 1 , x ≤ 0
I =
1
2
! I =
π
3
I =
π
2
"   + 
 78 01 2 2 3 % '  % ' +'
2
−1
dy
y+1
y
−1
f( x, y) dx
3
−1
dx
√
x+1
x−1
f( x, y) dy
!
0
−1
dx
√
x+1
−
√
f( x, y) dy4
3
dx
√
x+1
f( x, y) dy
0
−1
dx
√
0
f( x, y) dy4
0
dx
x−1
f( x, y) dy
"   + 
 79 
f( x, y) =
x
1 +
x + 2 y
$. + 1 B   f 3 3' ,
x − x
2
− 2 xy + x
3
+ 4 x
2
y + 4 xy
2
+ o( ρ
3
)   + 
! x − x
2
− 2 xy + x
3
+ 2 xy
2
+ o( ρ
3
) " x + x
2
+ 2 xy − 4 x
2
y + 2 xy
2
+ o( ρ
3
)
 8: $% % ' I =
D
3 dxdy *) > )  !)  2) y = x
2
, y x= 4
2
, y = 4 ( x ≥ 0 )
  +  ! I = 2 I = 8 " I = 6
7
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 8   ) 3 M *  3 m   f( x, y) = xy + x − y  
D = {( x, y) ∈ IR
2
: x ≥ 0 , y ≥ 0 4, x + y ≤ } 
  +  ! M = 5 , m = −4 M = 4 , m = −1 " M = 4 , m = −4
 8#   )' f = f( u, v) = *) u = 2 x + 3 y, v = x
2
+ 2 y $. df( x, y)
( 2 f
′
u
+ 2 xf
′
v
) dx + ( 3 f
′
u
+ 2 f
′
v
) dy 2 f
′
u
dx f+ 2
′
v
dy
! ( 2 + 2 + 3x) dx dy "   +  
 8,  < !&  x
2
− z
2
+ y
2
= 2 x + 2 z 0  < .
B<  ! B< '" B<  # '% " B< 
 8-  f( x, y x) = ln (
2
+ y
2
) $.    D
f
*    E
f
D
f
= IR
2
\{( 0 , 0 ) }; E
f
= [0 , +∞) D
f
= IR
2
; E
f
= [1 , +∞)
!   +  " D
f
= IR
2
\{( 0 , 0 ) }; E
f
= IR
 8/ 
f( x, y) =
2 x − y
x
+ y
$% df( 1 , 1 )
1
3
dx −
2
3
dy !
3
4
dx −
3
4
dy   +  "
−3
2
dx +
1
2
dy
 86  < !&  x
2
+ y
2
+ 2 x − 4 y − 2 = 0 0  < .
B<  ! C!" ' B<  " B<  '
 87  < !& 
x +
√
1 − y
2
− z
2
− 2 = 0 0  < .
C!" ' ! B<  M2 <  " B<  & '%
 88  f( x, y) =
  +  D = {( x, y) ∈ IR
2
| }x = 0
! D = IR
2
\{ ( 0 , 0 ) } " D = IR
2
 89   z = z( x, y)   1 2)   2 '2) . z − x = y c o s ( z − x) $.
I = dz(
π
4
, 0 ) ; !3 z(
π
4
, 0 ) =
π
2
I = dx −
√
2
2
dy ! I = dx +
√
2
2
dy I = −dx +
√
2
2
dy "   + 
 9:  f( x, y x) =
3
− 3 xy + 2 y
2
$% d
2
f( 2 , 1 )
1 2 dx
2
− 6 dxdy + 4 dy
2
1 2 dx
2
− 3 dxdy + 4 dy
2
! 2 dx
2
− 6 dxdy + 4 dy
2
"   + 
 9  f( x, y) = a r c t a n (
x
) $% f
′′
( 1 , 1 )
−1
2
−2
 9#   # !3 z = ( x
2
− 2 y
2
) e
x−y
* 1 P ( 0 , 0 ) (     
z +  2   P P +  1 "2
!   +  " P  1  2 1
 9,   2  2 "   f ( x, y x) =
2
+ y
2
− 3 2 ln ( )xy
@  1 2 1  ( 4 , 4 ) * 1 2   ( −4 , −4 )
! 5  + 
@  1 2 1  ( − −4 , 4 ) * 1 2   ( 4 , 4 )
" @   1 2 1  ( 4 , 4 ) * ( −4 , −4 )
8
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 9- $. * ' dz   # !3 z = s in x + c o s y + xy
dz = ( c o s x − y) dx + ( x − s in y) dy dz = ( c o s x − y) dx + ( x + s in y) dy
! 5  +  " dz = ( c o s x + y) dx + ( x −s in y) dy
 9/ $. + 1 B 
f( x, y) =
x
x
+ y + 2
3 3' #= < ρ =
√
x
2
+ y
2
x
2
−
x
2
4
−
xy
4
+ 0 ( ρ
2
)
x
2
−
x
2
2
−
xy
4
+ 0 ( ρ
2
)
!
x
2
+
x
2
4
−
xy
4
+ 0 ( ρ
2
) " 5  + 
 96 $. 2    f ( x, y) = x + 2 y *)  +& x
2
+ y
2
= 5 (    

f  2 1  ( 1 , 2 ) f  2   ( − −1 , 2 )
! f  2   ( 1 , 2 ) " 5  + 
 97  < !&  x
2
+ y
2
= 2 x + 2 + 1y 0  < .
C!" '
! 5  +  B<  " B< 
 98   3 f ( x, y) = a r c t g (
x
y
) $% A = f
′′
xx
+ f
′′
yy
A = 1 ! A = 0 A = 2 xy " 5  + 
 99   3 z = x
2
y + cos xy( ) + y 0( 2    
z
′
y
= 2 xy + s in ( xy) + 1 5  + 
! z
′
y
= x
2
− xy) + 1
 :: $. I =
D
dxdy
!3  '( D )  !) y =
x
2
; y = 2 x; xy = 2 ' x ≥ 0
5  +  ! I = 2 I = ln 2 " I = 2 ln 2
 : $. * ' 3' #   # !3 z = xe
y
d
2
z = e
y
dxdy + xe
y
dy
2
d
2
z = e
y
dx
2
+ e
y
dxdy + xe
y
dy
2
! 5  +  " d
2
z = 2 e
y
dxdy + xe
y
dy
2
 :# $.   ) 3 H$LMJ *    3 H$MMJ  f( x, y) = 1 + x + 2 y  
 x ≥ 0 , y ≥ 0 , x + y ≤ 1
$LM  3 = $MM  2 5  + 
! $LM  3 = $MM  1 " $LM  2 = $MM  1
 $. xf
′
x
+ y
x
+ y
0 ! 1 −1 " 5  + 
 :, 
f( x, y) = a r c t g
y
x
$% df( 1 , 1 )
−
dx
2
+
dy
2
! 5  + 
dx
2
+
dy
4
" −
dx
2
−
dy
2
 :- $.    3' & z
′
x
  # !3 z = ln y( x +
2
+ 1 )  ( 0 , 1 )
5  +  ! z
′
x
= 1 z
′
x
=
2
3
" z
′
x
=
−1
3
9
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 :/ $% % '
D
xdxdy x*) D )  !) x ≥ 0 ; y ≤ 2 −
2
; y ≥ x
5 ! 5  + 
1 2
5
"
5
1 2
 :6 $. df ( −2 , 4 ) = !3 f( x, y) = s in ( 4 x + 2 y)
4 dx + 2 dy !   +  3 dx + 2 dy " 4 dx − 2 dy
 :7 $. 2   f ( x, y) = 2 − x − 2 y *)  +& ϕ( x, y x) =
2
+ y
2
= 5 0< 0$  1
2 1; 00  1 2 
 # 0$  ( 1 , 2 ) * ( −1 , −2 ) 00  ( 1 , 2 ) ; 0$  ( − −1 , 2 )
! 0$  ( 1 , 2 ) ; 00( − −1 , 2 ) " 5  + 
 :8 $% I =
D
1 0 ydxdy= D 2) )  !) y = x
2
* y = 1
I = 6 ! I = 4 I = 8 " I = 3
 :9 $. f
′
x
*) f ( u, v v) = u ln (
2
) ; u( x, y) = y
2
+ 3 x; v( x, y) = xy
5  +  f
′
x
= 3 ln ( v
2
) +
2 u
v
y
! f
′
x
= 3 ln ( v
2
) +
2 u
v
" f
′
x
=
−4 ln ( v) +
2 u
v
y
 :  < !&  x + y
2
+ z
2
+ 2 y = 3 0  < .
B<  ! D'" C!" ' " B< 
   
f( x
@  2   ( 0 , 0 ) @  2 1  ( 0 , 0 )
! @ f ( x, y) +  2  " 5  + 
 # 
f( x, y) =
1
2 +
x + 2 y
$. + 1 B   f 3 3' #
1
2
−
x
4
−
y
2
+
x
2
8
+
xy
2
+
y
2
2
+ R
2
5  + 
!
1
2
−
x
4
+
y
2
−
x
2
8
+
xy
2
−
y
2
2
+ R
2
"
1
2
+
x
4
+
y
2
−
x
2
8
−
xy
2
+
y
2
2
+ R
2
 , @ f ( x, y x) =
3
− −3 xy y
3
@  & 1 2   & 1 2 1= & 1 2 
! 5  +  " @  & 1 2 1
 -  < !&  x
2
= 2 x + y + 1 0  < .
M & '% ! B<   B<  '! " C!" '
 / $% I =
OABC
|y − x
2
|dxdy; *) OHP=:J; 5H=:J; H=J; >HP=J
I =
11
15
! I =
8
5
I =
11
30
" I =
1
5
 6 $. d
2
z( 1 , 2 )   z = y ln x
d
2
z = −dx
2
+ 2 dxdy + 2 dy
2
d
2
z = −2 dx
2
+ dxdy
! d
2
z = −2 dx
2
+ 2 dxdy " d
2
z = −2 dx
2
+ 2 dxdy + dy
2
1 0
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 7  < !& 
z = 2 +
√
1 − x
2
− y
2
0  < .
B<  ! B<  & '% C!" ' " M2 < 
 8 $% % '
D
2 xdxdy *) D    NO5 *) O( 0 ( 1 ( 2, 0 ) ; A , 1 ) ; B , 0 )
−1 ! 2 : " 1
 9 $. 2  2 "   f( x, y e) =
4y−x
−y
@  2   ( 1 , 2 )   + 
! @  2   ( 0 , 2 ) " @  2 1  ( 0 , 2 )
 #:  f( x, y) = x ln ( xy) $. f
′
x
( 1 , e)
1 ! 5  +  e " 2
 # 
f( x, y) =
8 e
y
2 +
x
$. + 1 B   f 3 3' #
  +  −4 + 2 x −4 y + x
2
− 2 xy + 2 y
2
+ o( ρ
2
)
! 4 − 2 x + 4 y + x
2
− 2 xy + 2 y
2
+ o( ρ
2
) " 4 + 2 + 4x y + x
2
+ 2 + 2xy y
2
+ o( ρ
2
)
 ##  z = z( x, y)   1   2 '2) . z
3
− −2 xz x
2
+ 4 yz = 0 $% z
′
y
( 0 , −1 ) =
!3 z( 0 , −1 ) = 2
1
2
! 1
−1
2
" 5  + 
 #, 
f( x, y) =
x + 1
$. + 1 B   f 3 3' #  & ρ =
√
x
2
+ y
2
1
2
+
x
4
−
4
−
8
+ o( ρ
2
)
2
+
4
−
4
−
8
+
8
+ o( ρ
2
)
!
1
2
+
x
2
+
y
4
−
x
2
8
+
y
2
8
+ o( ρ
2
)
"
1
2
−
x
4
−
y
4
−
x
2
8
−
y
2
8
+ o( ρ
2
)
 #- $.   ) 3 A =  f =    3 B = f  f ( x, y x) = 2
2
+ 3 y
2
− −4 x 5
  > x
2
+ y
2
≤ 1 6
A = 4 3 ; B = −1 1 ! A = 4 7 ; B = −7 A = 4 7 ; B = −1 1 " A = 4 3 ; B = −7
 #/ 0   3'  z
′′
xx
   !3 z = xe
y
+ y
2
+ y s in x 
e
y
− y s in x ! e
y
+ y c o s x −y s in x " y s in x
 #6 $.   ) 3 M =    3 m  f ( x, y x) =
2
y
2
  |x| ≤ 1 , |y| ≤ 1
m = −1 ; " m = 1 ; M = 2
 #7 ? ' 3'    z = y ln x 
d
2
z
=
2
y
dxdy +
x
y
2
dy
2
d
2
z
=
1
y
dxdy +
x
y
2
dy
2
! d
2
z
=
1
x
dxdy −
y
x
2
dx
2
" d
2
z
=
2
x
dxdy −
y
x
2
dx
2
 #8 $% I =
D
e
−x
−y
dxdy y
= D 2) )  !) x =
√
4 −
2
*  
I =
π
2
e
−4
! I =
π
2
( 1 − e
−4
)
I =
π
2
( 2 + e
−4
)
" I =
π
2
( 2 − e
−4
)
1 1
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 #9 5K   1 2 2 % % ' I =
1
0
dy
1
√
y
√
x
3
+ 1 dx
I =
4
√
2 − 2
9
! I =
2
√
2 − 2
9
I =
4
√
2 + 2
9
" I =
2
√
2 + 2
9
 ,:    3 m  f ( x, y x) =
2
− 2 y   0 ≤ x ≤ ≤1 , 0 ≤ y 1
m = −
1
2
! m = 1 m = −2 " m = −1
 ,  
f( x, y) =
√
2 x
2
+ 4 y
2
+ 5 (   
( 0 , 0 ) @NQM '  1 )    2   ( 0 , 0 )
! f  2   ( 0 , 0 ) " f  2 1  ( 0 , 0 )
 ,# $% I =
D
2 ydxdy= D 2) )  !) y = x
2
+ 1 * y = 2
I =
6 4
5
! I =
3 2
1 5
I =
6 4
1 5
" I =
3 2
5
 ,, $. f
′
x
( 1 , −1 ) *) f ( u, v) = u
2
t g v; u( x, y) = x
2
y y; v( x, y) = x +
f
′
x
( 1 , −1 ) = 2 ! f
′
x
( 1 , −1 ) = 1 f
′
x
( 1 , −1 ) = 0 " f
′
x
( 1 , −1 ) = −1
 ,- I2 "  & 2 % % ' I =
1
0
dx
√
1−x
0
e
x
+y
dy
I =
π
8
( e
π π
" I =
π
4
( e − 1 )
 ,/ $.    D
f
*    E
f
 f ( x, y) =
e
, ( x, y) = ( 0 , 0 )
1
, ( x, y) = ( 0 , 0 )
D
f
= IR
2
; E
f
= ( 1 , +∞} D
f
= IR
2
\ {( 0 , 0 ) }; E
f
= [1 , +∞}
! D
f
= IR
2
; E
f
= ( 0 , 1 ] " D
f
= IR
2
; E
f
= [1 , +∞}
 ,6 
f( x, y) = 3 +
√
x
2
+ y
3
$. A = f
′
x
( 0 , 0 )
A = 1 ! A = 3    A " A = 0
 ,7 5K   1 2 2 % % ' I =
8
0
dy
2
√
y
e
x
dx
I =
e
16
4 4 4
" I =
e
8
− 1
4
 ,8 ? ' 3' &   z = a r c t g ( y − x) 
dz =
dy −dx
1 + ( )x − y
2
! dz =
−dy −dx
1 + ( )x − y
2
dz =
dx − dy
1 + ( )x − y
2
" dz =
dy + dx
1 + ( )x − y
2
 ,9 
z = f( x − y) $. A =
∂z
∂x
+
∂z
∂y
  +  ! A = 1 A = 3 " A = −1
 -:  f( x, y) = xe
3x y+4
$% df( 1 , 0 )
4 e
3
( )dx + 2 dy ! 5  +  4 e
3
( dx + dy) " 8 e
3
1 2
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 - $. 2  2 "  z = x
2
− 2 xy + 2 y
2
− 2 x + 2 + 4y  P ( 1 , 0 ) (   
5  +  P  1 2 1
! P +  1 "2 " P  1 2 
 -#  f( x, y e) = ( x + y)
xy
$% df( 1 , 1 )
5  +  ! 3 e( )dx + dy 6 e " 2 e( )dx + dy
 -,   f( x, y e) =
4y−x
−y
 1 P ( 1 , 2 ) (   
P +  1 "2 @  2   P
!   +  " @  2 1  P
 --  < !&  x
2
+ z
2
+ 2 x = 0 0  < .
B<  & '% M2 < 
! C!" ' " B< 
 -/  f( x, y) = 3
y/x
$% df( 1 , 1 )
3 ln 3 ( −dx + dy) ! 3 ln 3 ( 2 dx − dy)   +  " 3 ln 3 ( −dx + 2 dy)
 -6  < !& 
√
4 − x
2
− y
2
+ 2 = z 0  < .
C!" ' M2 < 
! B<  & '% " B< 
 -7 $% I =
D
2 dxdy *) D  2 .  ( x − 1 )
2
+ y
2
≤ 1 , y ≥ 0
I =
π
2
" I = −
π
2
 -8  f( x, y e) =
−x/y
$% df( 1 , 1 )
e
−1
( )−dx + dy !   +  e
−1
( )−dx − 2 dy " e
−1
( 2 dx + dy)
 -9
D
f( x, y) dxdy x*) D   )  !)
2
+ y
2
≤ ≤4 ; x 0 ; y ≥ 0 $. &  ϕ * r
π/2 ≤ ϕ ≤ π; 0 ≤ r ≤ 2 5  + 
! π/2 ≤ ϕ ≤ π; 0 ≤ r ≤ 4 " 0 ≤ ≤ϕ π; 0 ≤ ≤r 2
 /: $.
df( 1 , 1 ) = !3 f( x, y) =
x + 2 y
2 x − y
0 ! −3 dx + 5 dy 5  +  " −5 dx + 5 dy
 /  < !&
√
B<  C!" '
! B<  & '% " M2 < 
 /#  f( x, y) = x ln ( xy) $% f
′′
yy
x
y
!   +  0 "
−x
y
 /, $%
D
1
√
x
2
+ y
2
dxdy x*) D   )  !)
2
+ y
2
≤ ≥ ≥1 ; y 0 , x 0
5  +  !
π
2
π
4
" π
1 3
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 /-
D
f( x, y) dxdy x*) D )  !)
2
+ y
2
≤ 2 y; y ≤ −x $. &  ϕ * r
5  +  3 π/4 ≤ ϕ ≤ π; 0 ≤ ≤r 2 s in ϕ
! π/ π/4 ≤ ϕ ≤ 3 4 ; 0 ≤ r ≤ 2 s in ϕ " π/4 ≤ ϕ ≤ π; 0 ≤ r ≤ 2 s in ϕ
 //   2    z = 6 − 5 x − 4 y *)  +& x
2
− y
2
= 9  P ( 5 , −4 ) (
    
5  +  P  1 2 1
! P +  1 2  " P  1 2 
 /6  f( x, y e) = 6 s in y
x
$. + 1 B   f 3 3' ,
1 + 2 + 3 + 3y xy x
2
y −xy
2
+ y
3
+ o( ρ
3
)   + 
! 6 y + 6 + 3xy x
2
y −y
3
+ o( ρ
3
) " 3 6y − xy x+ 3
2
y −xy
2
+ o( ρ
3
)
 /7 $. + 1 $ 3 3' #   f( x, y) = x ln y   &  M
0
( 1 , 1 )
( y −1 ) + ( x − 1 ) ( y − 1 ) −
1
2
( y −1 )
2
+ R
2
( x, y)
! ( y − 1 ) + ( x −1 ) ( y − 1 ) −
1
2
( y −1 )
2
−
1
2
( x − 1 ) ( y − 1 )
2
+ R
2
( x, y)
( y − 1 ) + ( x − 1 ) ( y − 1 ) −
1
2!
( y −1 )
2
+ R
2
( x, y)
" 1 + ( 1 ) + (x − y − 1 ) + ( x − 1 ) ( y − 1 ) −
1
2
( y −1 )
2
+ R
2
( x, y)
 /8    !3 f ( x, y) = xe
xy
+ y c o s x f $. * ) * l=    
′
l
( −1 , 2 ) 
  ) 3
l = ( −
4
√
41
,
5
√
41
) l = (
5
√
41
, −
4
√
41
)
! l = ( −4 ,
 /9  z = z( x, y)   2 '2) . z
3
− 4 xz + y
2
− 4 = 0 $% z
′
x
, z
′
y
 M
0
( 1 , −2 , 2 )
z
′
x
= 1 , z
′
y
=
1
2
z
′
x
= 0 , z
′
y
= −1
! z
′
x
=
1
2
, z
′
y
= 1 " z
′
x
= 0 , z
′
y
= 1
 6: $. f
′
x
= !3 f( u, v u) =
2
s in v, u = x
2
+ y
2
, v =
y
x
f
′
x
= 4 xu s in v −
yu
x
c o s v f
′
x
= xu s in +v
yu
x
c o s v
! f
′
x
= 4 xu s in +v
yu
x
c o s v " 5  + 
 6 $. f
′
y
( 0 , 0 )   3  f( x, y) =
y
−x
x
+2y
, x
2
+ y
2
= 0
0 , x
2
+ y
2
= 0
1
2
+       x 
! −1 " 0
 6# $.    3'  z
′′
xy
( 0 ,
π
2
)   z = c o s ( xy − c o s )y
z
′′
xy
( 0 ,
π
2
) = −
π
2
z
′′
xy
( 0 ,
π
2
) = 0
! z
′′
xy
( 0 ,
π
2
) =
π
2
" z
′′
xy
( 0 ,
π
2
) = 1
 6, $. * ' dz   # !3 z =
x
√
x
+y
dz x= y(
2
+ y
2
)
−
( ydx −xdy) dz x= y(
2
+ y
2
)
−
( y
2
dx − xdy)
! dz x= (
2
+ y
2
)
−
( ydx − xdy) " 5  + 
1 4
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 6- $. * ' 3' #   # !3 z = e
xy
 M
0
( 1 , 1 )
d
2
z( 1 , 1 ) = e
2
( 4 dx
2
+ 6 dxdy + dy
2
) d
2
z( 1 , 1 ) = e
2
( 4 dx
2
+ 6 + 4dxdy dy
2
)
! d
2
z( 1 , 1 ) = e
2
( 4 dx
2
+ 3 dxdy + dy
2
) " d
2
z( 1 , 1 ) = e
2
( 4 dx
2
+ 6 + 4dxdy dy
2
)
 6/ $. 2    z = xy *)  +& x + y − 1 = 0 . (     
z  2   M(
1
2
,
1
2
) z +  2 
! z  2 1 M (
1
2
,
1
2
) " 5  + 
 66   # !3 z = 3 x −2 y + 1 =    D )  !) y = x − 1 , y = −x + 3 , x = 1
(     
  ) 3  z  5    3  z  4
!   ) 3  z  7 "    3  z  −2
 67   # !3 z = x
3
−y
3
+5 =    D = [0 , 1 ] 2 ]×[1 , (     
   3  z  − 3    3  z  −2
!   ) 3  z  4 "   ) 3  z  6
 68   # !3 z = x
2
+ y
2
+ xy − 1 2 x −3 y (     
z  2 1  M( 7 , −2 ) z +  1 "2
! z  2   M ( 7 , −2 ) " z +  2 
 69 R  &  % '
D
f( x, y) dxdy
D = {( x, y) (| x − 1 )
2
+ ( y − 2 )
2
≤ ≤4 , y 1 }
I =
1+ 3
√
1 3
−
√
dx
2−
√
4 1)−(x−
f( x, y) dy I =
1
−
√
3
dx
0
f( x, y) dy
! I =
1+ 3
√
1 3
−
√
dx
1
2+
√
4 ( 1)− x−
f( x, y) dy " I =
1
−1
dx
1
2−
√
4 1)−(x−
f( x, y) dy
 7:  % ' I =
1
0
dy
0
−
√
2y−y
f( x, y) dx $ 1 2 2 3 % '
I =
0
−1
dx
1
1 1
√
x
f( x, y) dy I =
0
−1
dx
1
1+ 1
√
x
f( x, y) dy
! I =
1
0
dx
1 1
−
√
−x
f( x, y) dy " I =
−1
dx
0
f( x, y) dy
1 5
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 7 $ 1 2 2 3 % ' I =
1
0
dy
y
y
f( x, y) dx
I =
1
0
dx
x
√
x
f( x, y) dy I =
1
0
dx
1
0
f( x, y) dy
! I =
1
0
dx
√
x
x
f( x, y) dy " 5  + 
 7# 0< ST
D
f( x, y) dxdy A= >      A  ( 0 , 1 ) , B , C( 0 , 2 ) ( 1 , 1 ) (  
  
I =
1
0
dx
2−x
1
f( x, y) dy =
2
1
dy
2−y
0
f( x, y) dx
! I =
1
0
dy
2−x
0
f( x, y) dx =
2
1
dx
2−y
0
f( x, y) dy
I =
1
0
dy
1
2−x
f( x, y) dx =
2
1
dx
2−y
0
f( x, y) dy
" I =
1
0
dx
2−x
1
f( x, y) dy =
2
1
dy
0
2 y
f( x, y) dx
 7, $ 1 2 2 3 % ' I =
2
0
dy
0
f( x, y) dx
I =
1
0
dx
2
2x
f( x, y) dy I =
1
0
dx
2
0
f( x, y) dy
! I =
1
0
dx
2x
2
f( x, y) dy " I =
2
0
dx
2
0
f( x, y) dy
 7- $% "& %  '( )  !) 2 x ≤ x
2
+ y
2
≤ ≤ ≥
6 x * y x
√
3 ; y 0
8 π
3
+ 2
√
3 !
8 π
3
4 π
3
+ 2
√
3 "   + 
1 6
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    f( x, y) = ln ( x2 + y2)   D     
     f( x, y)  f
f( x, y) ; Ef
     
Df = IR − {0 }; Ef = IR
Df = IR2 − {( 0 , 0 ) }; Ef = IR2 !
Df = IR2 − {( 0 , 0 ) }; Ef = IR "
Df = IR2 − {( 0 , 0 ) }; Ef = ( 0 , +∞) 
 #  $% "& %  '( )  !) x2 + y2 ≤ 9 * x2 + y2 ≤ 2 y  8 π ! 4 π  1 0 π "
  +  x2
 ,  $% "& %  '( )  !) y2 +
≤ 1 * y ≥ 0 , x ≤ 0 9 4 3 π 3 π   !   3 π "
  +  2 4
 ( x + y) arctg( x) 2, y  -  $. = 0
f′ ( 0 , 0 ) ; f′ y x y( 0 , 0 )
*) f( x, y) = π x, y = 0 2 f ′ = π;  = 0 ; f ′ = π ; f′ = π; f′ x  ∃f′ ! f ′ 2 y x y = 0  f′x 2 y = 0  " f ′x 2 y = 1  √ √ 2  x  4  x
 /  01 2 2 3 % ' I = dx
f ( x, y) dy 4 dx
f ( x, y) dy 0 0 2 x−2 2  y  2  y+2  I = dy
f ( x, y) dxI = dy
f( x, y) dx 0 y+2 0 y 2  4  !
5  +  " I = dy
f( x, y) dx 1  1 
 6  $% I = dy
c o s ( x3 − 1 ) dx 0 √ y
I = − 1 s in 1  !
I = − 1 s in 1  I = 1 s in 1  " I = 1 s in 1  2 3 3 2
 7   f( x, y) = ( x + 2 y) e3x+y $% I = ∂ f( 1 , 0 ) ∂x
I = 4 .3 9e3 ! I = 3 9e3 I = 1 1 e3 "
I = 1 3 .3 9e 3
sin(x+y ) 
 8   f( x, y) =
etdt       x
5  + 
f′ ( x, y) = esin (x+y).cos( x + y2)  !
f ′ ( x, y) = e
s( x + y2) x − 2 xex 
 9  $% I = 2 dxdy; D = {( x, y) ∈ IR2|0 ≤ x; x2 ≤ y; y ≤ x + 2 } D I = 20  ! I = 10  I = 26  " I = 25 3 3 3 6  :  $% 1 I = lim ( x2 + y2) c o s  x → 0 x2 + y2 y → 0 I = 0  ! I = +∞ ∃I " I = 1  1 CuuDuongThanCong.com
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    f( x, y) = g( x − 2 y, 2 x + y) ; < u( x, y) = x − 2 y; v( x, y) = 2 x + y      
df( x, y) = g′ dx + g′ ′ dx − g′ u v dy
df( x, y) = 3 gu vdy !
df( x, y) = ( g′ ) dx + ( g′
′ + 2 g′ ) dx + ( g′ u − 2 g′v
v − 2 g′u) dy "
df( x, y) = ( gu v
v − 2 g′u) dy
 #  $.   ) 3 A =  f=    3 B = f  f( x, y) = 1 − 3 x − 4 y 
 > x2 + y2 ≤ 2 5 
A = f( 3 , 2 ) = −1 6 ; B = f( −3 , −2 ) = 1 8 
A = f ( −3 , −4 ) = 2 6 ; B = f( 3 , 4 ) = −2 4  !
5  +  "
A = f ( 3 , −4 ) = 8 ; B = f( −3 , 4 ) = −6 
 ,   f( x, y) = ex+y $. + 1 $  f 3 3' #   &  1 M0( 1 , 0 )   
1 + ( x − 1 ) + y + (x−1) + y( x − 1 ) + y + o( ρ2) ; ρ = ( x − 1 ) 2 + y2 2 2  !
e + e( x − 1 ) + ey + e(x−1) + ey( x − 1 ) + ey + o( ρ2) ; ρ =
( x − 1 ) 2 + y2 2 2 
e − e( x − 1 ) + ey + e(x−1)  − ey( x − 1 ) + ey + o( ρ2) ; ρ = ( x − 1 ) 2 + y2 2 2 "
5  + 
 -  ?3 '2) . < '( 3' "&  < !&  z = 4 x2 − y2 + 2 y  ( −1 , 2 , 4 ) 
8 x + 2 y + z = 0  !
8 x + 2 y − z = 0 
x + 2 y + z = 7  "
4 x+2 y −z +4 = 0  df  /  $.
= !3 f( x, y) = x ln ( x + 2 y) ; x = s in t, y = c o s tdt
c o s t  [ln ( x + 2 y) + x ]
[ln ( x + 2 y) + x ] − 2x x+2y x+2y x+2y !
c o s t  [ln ( x x 2x  " x 2x s in tx+2y
 6  $. f′ = !3 sin ( ) x
f( x, t) = e t t t t t t − esin () ) ) ) ! − c o s (
) esin (   c o s (
) esin (  " c o s (
) e sin (  x2 x2 x x2 x2 x ∂f  7  $.
= !3 f( x, y) = e x s in y; x = st2, y = s 2t∂t
2 stest s in ( s 2t) 
  +  !
est s in ( s2 t) + est c o s ( s2t)  "
2 stest s in ( s2t) − s2 est c o s ( s2t) 
 8    f( x, y) = x4 + y4 − 4 xy + 1  (   
f  2 1  ( 1 , 1 ) *  ( −1 , −1 ) 
@ A  & 2 1 * & 2  !
f +  2   ( −1 , −1 )  "
@ A  & 2 1
 9  $. f′′′ = !3 xxy
2 y3exy( 2 − xy 2)  ! 4 y3 exy
2 y3exy ( 2 + xy2)  "
  + 
 #:   f( x, y) = ln ( x + y + 3 )  $. + 1 B   f 3 3' # √
 & ρ = x2 + y2 x y x2 xy y 2  ln 3 + + − − − + o( ρ2 ) 
  +  3 3 1 8 9 1 8 x y x2 xy y 2 x y x2 xy y2 ! ln 3 + + − − − + o( ρ2)  " ln 3 + + + + + + o( ρ2)  3 3 9 9 9 3 3 1 8 9 1 8 √
 #  $. df( 0 , 1 ) = !3 f( x, y) = ln ( x + x2 + y2)  2 dx + dy ! 2 dx + 3 dydx − dy " dx + dy 2 CuuDuongThanCong.com
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 ##   < !&  x2 + x + 1 = z 0  < .  B<  ! B< 
C!" ' "
  + 
 #,   < !&  z + x2 + y2 + x + y = 3  0  < .  B<  !
C!" '  B<  " D'"
 #-  $. 2  2 "  f( x, y) = ( x2 + y) ey/2
( 0 , −2  1 2 
( 0 , −2 ) +  1 "2 !
( 0 , 0  1 2  "
( 0 , −2 )  1 2 1
 #/  ?3 '2) . < '( 3' "&  <  z = ex−y  ( 1 , −1 , 1 ) 
2 x+2 y −z +1 = 0  !
x + 2 y − z + 2 = 0 
2 x−2 y +z −5 = 0  "
  + 
 #6  $. z′ = !3 y
z = z( x, y)   1   2 '2) . ln ( x + yz) = 1 + xy2z3
2 xyz3 ( x + yz) − z  
  + 
y + 3 xy2z2 ( x + yz)
2 xyz3 ( x + yz) − z
z − 2 xyz3( x + yz) !  " 
y − 3 xy2z 2( x + yz)
y − 3 xy2z2 ( x + yz)
 #7  EF G .   f′x( 1 , 2 )  H+ & & 3   3' 3  @I$$J
@I$$ *) 2)     y = 2 * f( x, y)  1   & !K  !
@I$$ *) 2)     y = 1 * f( x, y)  1   & !K #
@I$$ *) 2)     x = 1 * f( x, y)  1   & !K # "   + √
 #8  $. df( 3 , 4 ) = !3 f( x, y) = x2 + y2 3 4 3 4 7  3 dx + 4 dy ! dx + dydx + dy "  5 5 1 0 1 0 5
 #9    3 f( x, y) = x4 + y4 − x2 − 2 xy − y2 (     
  + 
f +  2   ( −1 , −1 )  !
f  2 1  ( −1 , −1 )  "
f  2   ( −1 , −1 ) 
 ,:    f( x, y) = 2 x3 + xy 2 + 5 x2 + y2 (   
f  2 1  ( 0 , 0 ) = 2   ( −1 , −2 )  !
f  , 1 "2
f  2   ( 0 , 0 ) = +  2   ( −1 , −2 )  "
f  2
 ,  $.   ) 3 H$LMJ=    3 H$MMJ  f( x, y) = x2 + 2 y 2  
x2 + y2 ≤ 1   $LM = 1 = $MM = 0 
$LM = 2 = $MM = −1  !
$LM = 0 = $MM = −1  "
  + 
 ,#  $.    3'  z′′xx( 1 , 0 )   # !3 z = ln ( x + y 2 + 1 )  1 1 1 −1  !    "  4 2 4 3 3 CuuDuongThanCong.com
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 ,,   f( x, y) =
 $% df( 2 , −1 ) x + y dx + 4 dy ! dx + dy
  +  " 4 dx + dy 
 ,-  $% % ' I =
( x + y) dxdy *) > )  !)  2) x 2 + y2 = 1 , x2 + y 2 = 4 , y = D
0 , y = x 3 ' x ≥ 0  2 1 7
  +  ! I =  I =  " I =  3 3 3 √
 ,/   < !&  x + 3 y2 + z 2 − 1 = 0  0  < .  B<  !
C!" '
M2 <  "
B<  & '%  1 √
 ,6  $% % ' I = √
dxdy *) > )  !)  2) x2+y2 = 4 , y = x, y = x 3 D x2 + y2
3 ' y ≥ xπ π 2 I =  ! I =  I =  "
  +  3 6 9 
 ,7  $% % ' I =
2 ydxdy *) > )  !)  2) x = y 2 + y − 1 , x = y + 3  D I = −1 6  ! I = 0  I = 1 6  " I = 4  
 ,8  $% % ' I =
( x + y) dxdy *) > )  !)  2) y = x2, y = xD I = 3 /2 0  ! I = 1 /3  I = 3 /1 0  "
  + 
 ,9  $. z′ = !3 x z ex = 0  ey + zex e + ze e y + ex   ! −− " −y + ex y + ex y + ex ey + zex 2 c o s x
 -:   f( x, y) =
 $. + 1 B   f 3 3' # ey
  + 
2 − 2 y − x2 + y2 + o( ρ2)  !
1 + 2 y + x2 − y2 + o( ρ2)  "
2 x − 2 y − x2 + y2 + o( ρ2 )  
 -  $% % ' I =
xdxdy *) >    NO5= O( 0 , 0 ) , A( 1 , 1 ) , B( 0 , 1 )  D 1 1 1 I =  !
  +  I =  " I =  9 6 3
 -#   f( x, y) = x ln ( xy)  $% f′′′  0  1  xy √
 -,   < !& 
4 − 2 x2 − z 2 + y − 1 = 0  0  < .
M2 < '"  !
C!" '  B<  " B<  
 --  $% I =
xdxdydz *)  )  !) y = x; y = 3 x; x = 1 ; z = 0 ; z = 4 − y  2 1 5 I =  ! I =  I =  "
  +  5 3 3 4 CuuDuongThanCong.com
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 -/    2    z = 5 − 4 x − 8 y *)  +& x2 − 8 y2 = 8   P ( 4 , −1 )  1 1
"2   L 2 *) λ =  (      2
P  1 2 1   +&
P  1 2    +& !
  +   "
P +  1 2    +& 
 -6  $% I =
xdxdydz *)  )  !) y = x; y = 2 x; x = 1 ; z = 0 ; z = 4 − x  1 1 3 2 I =  ! I =  I =  "
  +  3 1 2 1 3
 -7  $. * ' 3' & dz   # !3 z = s in x + cosy + xy
  + 
dz = ( c o s x − y) dx + ( x − s in y) dy !
dz = ( c o s x + y) dx + ( x − s in y) dy "
dz = ( c o s x + y) dx + ( x + s in y) dy
 -8  $. y′( x) = !3 y = y( x)   1   2 '2) . y5 + x2 y3 = 1 + yex 2 xyex
2 xyex + 2 xy 3
2 xy3 − 2 xyex   ! 
  +  " 
5 y4 + 3 x2y 2 − ex
5 y 4 + 3 x2y2
5 y 4 + 3 x2y2 − ex√  -9  @ 
f( t) = e t *) t = x2 + y2  '2) .   
  +  ! xf′ + yf ′ + xf ′ x y = 0  yf ′x y = 0  "
yf ′x − xf′y = 0 
 /:    3 f( x, y) = a r c t g ( x)  $% df( 1 , 1 ) y  1 dx + 2dy ! 1dx − 1dy  2 dx − 2dy " 1 dx + 1dy 5 5 2 2 5 2 2
 /  $% % '
≤ 1 ; 0 ≤ y ≤ 2  D I = 3  ! I = 5 
  +  " I = 2   
 /#  $% % ' I =
1 − x2 − y2 dxdy *) >  .  ) * D 2 π π I =  ! I = 
  +  " I = π 3 2
 /,   < !&  x2 − y2 − z2 = 2 y + 1  0  < .  B<  !
C!" '  B<  "
B<   '%
 /-    2  2 "   z = 3 ( x2 + y2) − x3 + 4 y  P ( 0 , −2)  (    3  
P  1 2 1
P +  1 2  !
P  1  8 ex
 //   f( x, y) =
 $. + 1 B   f 3 3' # 2 + y
  + 
4 + 4 x − 2 y + 2 x2 − 2 xy + y2 + o( ρ2)  !
4 + 2 x − 3 y + 4 x2 − 2 xy + y2 + o( ρ2)  "
4 x + 2 y + 2 x 2+ 2 xy + y 2 + o( ρ2)  √
 /6   f( x, y) =  x3 − y3 $% f′ ( 0 , 0 ) , f′ x y( 0 , 0 ) 
f′( 0 , 0 ) = 1 , f ′ x
y( 0 , 0 ) = −1 
  +  !
f′ ( 0 , 0 ) = 1 , f′ x
y( 0 , 0 ) = 1  "
+   5 CuuDuongThanCong.com
https://fb.com/tailieudientucntt  1 √
 /7  $% % ' I = √
dxdy *) >   x2 + y2 ≤ 2 x; y ≤ x 3 ; y ≥ xD x2 + y2 √ √ √ √ √ I = 3 + 2  ! I = 3 − 2  I = 2  "
  + 
 /8   < !&  x2 + z2 + y = 2 x + 1  0  < .  B<  !
C!" '
M & '% " B<  
 /9  $% % ' I =
( xy + 2 y) dxdy *) >    NO5= O( 0 , 0 ) , A( 1 , 1 ) , B( 2 , 0 )  D
  +  ! I = 2  I = 1  " I = −1 
 6:  $. df( −6 , 4 ) = !3 f( x, y) = s in ( 2 x + 3 y)  2 dx + 3 dy ! 3 dx + dy
  +  " 2 dx − 3 dy√
 6   < !& 
4 − x2 − z2 + 3 − y = 0  0  < .
M2 <  !
C!" '  B<  "
B<  & '% 1
 6#   f( x, y) = √
 $.    D x2 + y2
f *    Ef
Df = IR2\{( 0 , 0 ) }; Ef = ( 0 , +∞) 
  +  !
Df = IR\{0 }; Ef = [0 , +∞)  "
Df = IR2\{( 0 , 0 ) }; Ef = [0 , +∞)  
 6,  $% I =
xdxdy *) D  2 .  x2 + ( y − 2 ) 2 ≤ 1 , x ≥ 0  D 3 −1 2
  +  ! I =  I =  " I =  3
 6-    z = z( x, y)   2 '2) . z3 − 4 xz + y2 − 4 = 0  $% z′y( 1 , −2 ) 3
z( 1 , −2 ) = 2  1 2 1 − !
  +    "  2 3 2  1  1
 6/  01 2 2 3 % '  % ' +' dy
f ( x, y) dx √ 0 − y  0  1  1  xdx
f ( x, y) dy4 dx
f( x, y) dy
  +  −1 x 0 0  1  1  0  1  1  1 ! dx
f ( x, y) dy " dx
f ( x, y) dy4 dx
f( x, y) dy−1 x−1 x 0 0
 66   f( x, y) = y ln ( xy)  $% f′′ xx −y  !
  +    " y xxx + y
 67   f( x, y) =
 $% df( 1 , 1 ) 2 x + y  2 dx − 1dy !
  + 
−1 dx + 1 dy "
−1 dx + 1 dy 3 3 9 9 3 3
 68   f = f( u, v) = euv, u = u( x, y) = x3y, v = v( x, y) = x2 $. df
veuv( 3 x2 ydx + x3dy) + ueuv 2 xdx
veuvx3 dy + ueuv2 xdx !
veuv3 x2ydx + ueuv2 xdy "
  +  √
 69   < !&  y + 4 x2 + z 2 + 2 = 0  0  < .  B<  !
M2 < 
C!" ' "
B<  & '% 6 CuuDuongThanCong.com
https://fb.com/tailieudientucntt
 7:  $.   ) 3=    3  z = x2 + xy − 1    O5 *)
A( 1 , 1 ) ; B( 2 , 2 ) ; C( 3 , 1 )
zmax = 1 1 , zmin = 7 
  +  !
zmax = 1 1 , zmin = −7  "
zmax = 1 1 , zmin = 1 
 7    ) 3 M *  3 m  f( x, y) = 3 + 2 xy  D = {( x, y) ∈ IR2 : x 2 + y 2 ≤ 1 }
M = 4 , m = 2  !
M = 4 , m = 0 
  +  "
M = 4 , m = 3 
 7#   < !&  x2 + z2 − y2 = 2 x + 2 z − 2  0  < .
C!" ' ! B< 
B<  # '% " B< 
 7,   f( x, y) = 2 x2 − 3 xy + y3 $% d2f( 1 , 1 ) 
4 dx2 − 3 dxdy + 6 dy2
4 dx2 − 6 dxdy + 6 dy2 !
  +  "
2 dx2 + 6 dxdy + 6 dy 2 
 7-  $% % ' I =
1 2 ydxdy *) > )  !)  2) x = y 2, x = yD 3 I =  ! I = 1 
  +  " I = 4  2 0
 7/    # !3 z = ( x + y2) ex/2 * 1 P ( −2 , 0 )  (     
P +  1 "2
  +  !
P  1  2 1 "
P  1  2  
 76  $% % ' I =
2 xdxdy *) > )  !)  2) y = 2 − x2, y = x 3 I =  ! I =  I =  "
  +  2 0 2 1 0 
 77  $% I =
ydxdy *) D  2 .  x2 + ( y − 1 ) 2 ≤ 1 , x ≤ 0  D 1 π π I =  ! I =  I =  "
  +  2 3 2  2  y+1
 78  01 2 2 3 % '  % ' +' dy
f ( x, y) dx −1 y−1  3  √ x+1 dx
f( x, y) dy−1 x−1  0  √ x+1  3  √ x+1 ! dx
f ( x, y) dy4 dx
f ( x, y) dy√ −1 −  0  √ dx
f ( x, y) dy4 dx
f ( x, y) dy−1 0 0 x−1 "
  +  x
 79   f( x, y) =
 $. + 1 B   f 3 3' , 1 + x + 2 y
x − x2 − 2 xy + x3 + 4 x2y + 4 xy2 + o( ρ3) 
  +  !
x − x2 − 2 xy + x3 + 2 xy2 + o( ρ3)  "
x + x2 + 2 xy − 4 x2 y + 2 xy2 + o( ρ3)  
 8:  $% % ' I =
3 dxdy *) > )  !)  2) y = x2, y = 4 x2, y = 4 ( x ≥ 0 )  D
  +  ! I = 2  I = 8  " I = 6  7 CuuDuongThanCong.com
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 8    ) 3 M *  3 m   f( x, y)
= xy + x − y  
D = {( x, y) ∈ IR2 : x ≥ 0 , y ≥ 0 , x + y ≤ 4 } 
  +  !
M = 5 , m = −4 
M = 4 , m = −1  "
M = 4 , m = −4 
 8#    )' f = f( u, v) = *) u = 2 x + 3 y, v = x2 + 2 y $. df( x, y)
( 2 f′ + 2 xf′ ) dx + ( 3 f′ + 2 f′ dx + 2 f ′ u v u v ) dy  2 f ′u v dy !
( 2 + 2 x) dx + 3 dy "
  +  
 8,   < !&  x2 − z2 + y2 = 2 x + 2 z 0  < .  B<  !
B< '"
B<  # '% " B< 
 8-   f( x, y) = ln ( x2 + y2)  $.    D *     f Ef
Df = IR2\{( 0 , 0 ) }; Ef = [0 , +∞) 
Df = IR2; Ef = [1 , +∞)  !
  +  "
Df = IR2\{( 0 , 0 ) }; Ef = IR 2 x  8/   − y f ( x, y) =
 $% df( 1 , 1 ) x + y  1 dx − 2dy ! 3dx − 3dy
  +  "
−3 dx + 1 dy 3 3 4 4 2 2
 86   < !&  x2 + y2 + 2 x − 4 y − 2 = 0  0  < .  B<  !
C!" '
B<  "
B<  ' √
 87   < !&  x + 1 − y2 − z2 − 2 = 0  0  < .
C!" ' ! B< 
M2 <  "
B<  & '%
 88   f( x, y) =
  + 
D = {( x, y) ∈ IR2|x = 0 } !
D = IR2\{( 0 , 0 ) } " D = IR2
 89    z = z( x, y)   1 2)   2 '2) . z − x = y c o s ( z − x)  $.
I = dz( π, 0 ) ; !3 z( π, 0 ) = π  4 4 2 √ √ √
I = dx − 2 dy !
I = dx + 2dy
I = −dx + 2 dy "
  +  2 2 2
 9:   f( x, y) = x3 − 3 xy + 2 y2 $% d2f( 2 , 1 ) 
1 2 dx2 − 6 dxdy + 4 dy2
1 2 dx 2 − 3 dxdy + 4 dy2 !
2 dx2 − 6 dxdy + 4 dy2 "
  + 
 9   f( x, y) = a r c t a n ( x)  $% f ′′ ( 1 , 1 )  −1  −2  2
 9#    # !3 z = ( x2 − 2 y2) ex−y * 1 P ( 0 , 0 )  (     
z +  2   P
P +  1 "2 !
  +  "
P  1  2 1
 9,    2  2 "   f( x, y) = x2 + y2 − 3 2 ln ( xy)
@   1 2 1  ( 4 , 4 ) *  1 2   ( −4 , −4 )  !
5  + 
@   1 2 1  ( −4 , −4 ) * 1 2   ( 4 , 4 )  "
@   1 2 1  ( 4 , 4 ) * ( −4 , −4 )  8 CuuDuongThanCong.com
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 9-  $. * ' dz   # !3 z = s in x + c o s y + xy
dz = ( c o s x − y) dx + ( x − s in y) dy
dz = ( c o s x − y) dx + ( x + s in y) dy !
5  +  "
dz = ( c o s x + y) dx + ( x − s in y) dy x √
 9/  $. + 1 B  f( x, y) =
3 3' #= < ρ = x2 + y 2 x + y + 2 x x2 xy x x2 xy − − + 0 ( ρ2)  − − + 0 ( ρ 2)  2 4 4 2 2 4 x x2 xy ! + − + 0 ( ρ2)  "
5  +  2 4 4
 96  $. 2    f( x, y) = x + 2 y *)  +& x2 + y2 = 5  (     
f  2 1  ( 1 , 2 ) 
f  2   ( −1 , −2 )  !
f  2   ( 1 , 2 )  "
5  + 
 97   < !&  x2 + y2 = 2 x + 2 y + 1  0  < .
C!" ' !
5  +   B<  " B<   
 98    3 f( x, y) = a r c t g ( x)  $% A = f ′′ + f′′ y xx yy A = 1  ! A = 0  A = 2 xy "
5  + 
 99    3 z = x2y + cos( xy) + y 0( 2     
z′y = 2 xy + s in ( xy) + 1 
5  +  ! z′ = x2 y − xy) + 1   x
 ::  $. I =
dxdy !3  '( D )  !) y = ; y = 2 x; xy = 2 ' x ≥ 0  D 2
5  +  ! I = 2  I = ln 2  " I = 2 ln 2 
 :  $. * ' 3' #   # !3 z = xey
d2z = ey dxdy + xeydy2 
d2z = ey dx2 + eydxdy + xe ydy2  !
5  +  "
d2 z = 2 eydxdy + xey dy2 
 :#  $.   ) 3 H$LMJ *    3 H$MMJ  f( x, y) = 1 + x + 2 y  
 x ≥ 0 , y ≥ 0 , x + y ≤ 1 
$LM  3 = $MM  2 
5  +  !
$LM  3 = $MM  1  "
$LM  2 = $MM  1 
 $. xf′x + y x + y  0  ! 1  −1  "
5  +  y
 :,   f( x, y) = a r c t g
 $% df( 1 , 1 ) x dx dy dx dy dx dy − +  !
5  +   +  " − −  2 2 2 4 2 2
 :-  $.    3' & z′   # !3 2 x
z = ln( x + y + 1 )  ( 0 , 1 )  2 −1
5  +  ! z′   x = 1  z′x = " z ′ 3 x = 3 9 CuuDuongThanCong.com
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 :/  $% % '
xdxdy *) D )  !) x ≥ 0 ; y ≤ 2 − x2; y ≥ xD 1 2 5  5  !
5  +    "  5 1 2
 :6  $. df( −2 , 4 ) = !3 f( x, y) = s in ( 4 x + 2 y)  4 dx + 2 dy !
  +   3 dx + 2 dy " 4 dx − 2 dy
 :7  $. 2   f( x, y) = 2 − x − 2 y *)  +& ϕ( x, y) = x2 + y2 = 5  0< 0$  1
2 1; 00  1 2 
 # 0$  ( 1 , 2 ) * ( −1 , −2 ) 
00  ( 1 , 2 ) ; 0$  ( −1 , −2 )  !
0$  ( 1 , 2 ) ; 00( −1 , −2 )  "
5  +  
 :8  $% I = 1 0 ydxdy= D 2) )  !) y = x2 * y = 1  D I = 6  ! I = 4  I = 8  " I = 3 
 :9  $. f ′ *)
2 ) ; u( x, y) = y2 x
f ( u, v) = u ln ( v
+ 3 x; v( x, y) = xy′ 2 u
5  + 
fx = 3 ln ( v2) + yv 2 u 2 u !
f ′ = 3 ln ( v2  yx ) + " f ′ v
x = −4 ln ( v) + v
 :   < !&  x + y2 + z2 + 2 y = 3  0  < .  B<  ! D'"
C!" ' " B< 
     f( x
@  2   ( 0 , 0 ) 
@  2 1  ( 0 , 0 )  !
@ f( x, y) +  2  "
5  +  1
 #   f( x, y) =
 $. + 1 B   f 3 3' # 2 + x + 2 y 1 x y x2 xy y2 − − + + + + R
5  +  2 4 2 8 2 2 2 1 x y x2 xy y2 1 x y x2 xy y2 ! − + − + − + R + + − − + + R  2 4 2 8 2 2 2 " 2 4 2 8 2 2 2
 ,  @ f( x, y) = x3 − 3 xy − y3
@  & 1 2 
 & 1 2 1= & 1 2  !
5  +  "
@  & 1 2 1
 -   < !&  x2 = 2 x + y + 1  0  < .
M & '% !
B<  
B<  '! "
C!" ' 
 /  $% I =
|y − x2|dxdy; *) OHP=:J; 5H=:J; H=J; >HP=J OABC I = 11  ! I = 8 I = 11  " I = 1  15 5 30 5
 6  $. d2z( 1 , 2 )   z = y ln x
d2z = −dx2 + 2 dxdy + 2 dy2
d2z = −2 dx 2+ dxdy !
d2z = −2 dx2 + 2 dxdy "
d2 z = −2 dx2 + 2 dxdy + dy2 1 0 CuuDuongThanCong.com
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 7   < !&  z = 2 + 1 − x2 − y 2 0  < .  B<  !
B<  & '%
C!" ' "
M2 <  
 8  $% % '
2 xdxdy *) D    NO5 *) O( 0 , 0 ) ; A( 1 , 1 ) ; B( 2 , 0 )  D −1  ! 2   : " 1 
 9  $. 2  2 "   f( x, y) = e4y−x−y 
@  2   ( 1 , 2 ) 
  +  !
@  2   ( 0 , 2 )  "
@  2 1  ( 0 , 2 ) 
 #:   f( x, y) = x ln ( xy)  $. f′x( 1 , e)   1  !
5  +  e " 2  8 ey
 #   f( x, y) =
 $. + 1 B   f 3 3' # 2 + x
  + 
−4 + 2 x − 4 y + x2 − 2 xy + 2 y2 + o( ρ2)  !
4 − 2 x + 4 y + x2 − 2 xy + 2 y2 + o( ρ2)  "
4 + 2 x + 4 y + x2 + 2 xy + 2 y 2 + o( ρ2) 
 ##   z = z( x, y)   1   2 '2) . z3 − 2 xz − x2 + 4 yz = 0  $% z′y ( 0 , −1 ) =
!3 z( 0 , −1 ) = 2  1   ! 1  −1  "
5  +  2 2 x + 1 √
 #,   f( x, y) =
 $. + 1 B   f 3 3' #  & ρ = x 2 + y2 1 x  + − − + o( ρ2)   + − − + + o( ρ 2)  2 4 4 8 2 4 4 8 8 1 x y x2 y2 1 x y x 2 y 2 ! + + − + + o( ρ2)  " − − − − + o( ρ2)  2 2 4 8 8 2 4 4 8 8
 #-  $.   ) 3 A =  f=    3 B = f  f( x, y) = 2 x2 + 3 y2 − 4 x − 5
  > x2 + y2 ≤ 1 6 
A = 4 3 ; B = −1 1  !
A = 4 7 ; B = −7 
A = 4 7 ; B = −1 1  "
A = 4 3 ; B = −7 
 #/  0   3'  z′′    !3 xx
z = xey + y2 + y s in x 
ey − y s in x !
ey + y c o s x−y s in x " y s in x
 #6  $.   ) 3 M=    3 m  f( x, y) = x2y2   |x| ≤ 1 , |y| ≤ 1  m = −1 ; "
m = 1 ; M = 2 
 #7  ? ' 3'    z = y ln x  2 x 1 x d2z = dxdy + dy2
d 2z = dxdy + dy2  y y2 y y 2 1 y 2 y ! d2z = dxdy − dx2 "
d2 z = dxdy − dx2 x x2 x x2  √
 #8  $% I = e−x−ydxdy= D 2) )  !) x = 4 − y2 *   Dπ π π π I = e−4 ! I = ( 1 − e−4) 
I = ( 2 + e −4)  " I = ( 2 − e−4)  2 2 2 2 1 1 CuuDuongThanCong.com
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 #9  5K   1 2 2 % % ' I = dy x3 + 1 dx 0 √ y √ √ √ √ 4 2 − 2 2 2 − 2 4 2 + 2 2 2 + 2 I =  ! I =  I =  " I =  9 9 9 9
 ,:     3 m  f( x, y) = x2 − 2 y   0 ≤ x ≤ 1 , 0 ≤ y ≤ 1  1 m = − ! m = 1  m = −2  " m = −1  2 √
 ,    f( x, y) = 2 x2 + 4 y2 + 5  (   
( 0 , 0 ) @NQM '  1 ) 
  2   ( 0 , 0 )  !
f  2   ( 0 , 0 )  "
f  2 1  ( 0 , 0 )  
 ,#  $% I = 2 ydxdy= D 2) )  !) y = x2 + 1 * y = 2  D6 4 3 2 6 4 3 2 I =  ! I =  I =  " I =  5 1 5 1 5 5
 ,,  $. f ′x( 1 , −1 )
*) f( u, v) = u2 t g v; u( x, y) = x2y; v( x, y) = x + y
f′x( 1 , −1 ) = 2  !
f ′x( 1 , −1 ) = 1 
f ′x( 1 , −1 ) = 0  "
f ′x( 1 , −1 ) = −1  √ 1  1−x 
 ,-  I2 "  & 2 % % ' I = dx
ex +ydy 0 0 π π π π I = ( e " I = ( e − 1 )  8 4  
 ,/  $.    D *    
e, ( x, y) = ( 0 , 0 )  f Ef f ( x, y) = 1 ,
( x, y) = ( 0 , 0 ) D  
f = IR 2; Ef = ( 1 , +∞}
 Df = IR2 \ {( 0 , 0 ) }; Ef = [1 , +∞} !
Df = IR2; Ef = ( 0 , 1 ] "
Df = IR2 ; Ef = [1 , +∞}√
 ,6   f( x, y) = 3 + x2 + y3 $. A = f ′ ( 0 , 0 ) x A = 1  ! A = 3 
   A " A = 0  8  2 
 ,7  5K   1 2 2 % % ' I = dy exdx 0 √y e16 e8 − 1 I = " I =  4 4 4 4
 ,8  ? ' 3' &   z = a r c t g ( y − x)  dy − dx −dy − dx dx − dy dy + dx dz =  ! dz =  dz =  " dz =  1 + ( x − y) 2 1 + ( x − y) 2 1 + ( x − y) 2 1 + ( x − y) 2 ∂z ∂z
 ,9   z = f( x − y)  $. A = + ∂x ∂y
  +  ! A = 1  A = 3  " A = −1 
 -:   f( x, y) = xe3x+4y $% df( 1 , 0 ) 
4 e3( dx + 2 dy)  !
5  + 
4 e3( dx + dy)  " 8 e3 1 2 CuuDuongThanCong.com
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 -  $. 2  2 "  z = x2 − 2 xy + 2 y2 − 2 x + 2 y + 4   P ( 1 , 0 )  (   
5  + 
P  1 2 1 !
P +  1 "2 "
P  1 2 
 -#   f( x, y) = ( x + y) exy $% df( 1 , 1 )
5  +  !
3 e( dx + dy)   6 e "
2 e( dx + dy) 
 -,    f( x, y) = e4y−x−y   1 P ( 1 , 2 )  (   
P +  1 "2
@  2   P !
  +  "
@  2 1  P
 --   < !&  x2 + z 2 + 2 x = 0  0  < .
B<  & '%
M2 <  !
C!" ' " B< 
 -/   f( x, y) = 3 y/x $% df( 1 , 1 ) 
3 ln 3 ( −dx + dy)  !
3 ln 3 ( 2 dx − dy) 
  +  "
3 ln 3 ( −dx + 2 dy)  √
 -6   < !& 
4 − x2 − y2 + 2 = z 0  < .
C!" '
M2 <  !
B<  & '% " B<  
 -7  $% I =
2 dxdy *) D  2 .  ( x − 1 ) 2 + y2 ≤ 1 , y ≥ 0  D π π I =  " I = −  2 2
 -8   f( x, y) = e−x/y $% df( 1 , 1 ) 
e−1( −dx + dy)  !
  + 
e−1( −dx − 2 dy)  "
e−1( 2 dx + dy)    -9 
f( x, y) dxdy *) D   )  !) x2 + y2 ≤ 4 ; x ≤ 0 ; y ≥ 0  $. &  ϕ * r D
π/2 ≤ ϕ ≤ π; 0 ≤ r ≤ 2 
5  +  !
π/2 ≤ ϕ ≤ π; 0 ≤ r ≤ 4  "
0 ≤ ϕ ≤ π; 0 ≤ r ≤ 2  x + 2 y
 /:  $. df( 1 , 1 ) = !3 f( x, y) = 2 x − y  0  !
−3 dx + 5 dy
5  +  "
−5 dx + 5 dy√
 /   < !&  B< 
C!" ' !
B<  & '% "
M2 < 
 /#   f( x, y) = x ln ( xy)  $% f′′ yy x !
  +   0  " −x yy  1
 /,  $% √
dxdy *) D   )  !) x2 + y2 ≤ 1 ; y ≥ 0 , x ≥ 0 D x2 + y2 π π
5  +  !    " π 2 4 1 3 CuuDuongThanCong.com
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f( x, y) dxdy *) D )  !) x2 + y2 ≤ 2 y; y ≤ −x $. &  ϕ * r D
5  + 
3 π/4 ≤ ϕ ≤ π; 0 ≤ r ≤ 2 s in ϕ !
π/4 ≤ ϕ ≤ 3 π/4 ; 0 ≤ r ≤ 2 s in ϕ "
π/4 ≤ ϕ ≤ π; 0 ≤ r ≤ 2 s in ϕ
 //    2    z = 6 − 5 x − 4 y *)  +& x2 − y2 = 9   P ( 5 , −4 )  (
    
5  + 
P  1 2 1 !
P +  1 2  "
P  1 2 
 /6   f( x, y) = 6 s in y  ex $. + 1 B   f 3 3' ,
1 + 2 y + 3 xy + 3 x2y − xy2 + y 3 + o( ρ3) 
  +  !
6 y + 6 xy + 3 x2y − y3 + o( ρ3)  "
3 y − 6 xy + 3 x2y − xy2 + o( ρ3) 
 /7  $. + 1 $ 3 3' #   f( x, y) = x ln y   &  M0( 1 , 1 )
( y − 1 ) + ( x − 1 ) ( y − 1 ) − 1 ( y − 1 ) 2 + R 2 2( x, y)  !
( y − 1 ) + ( x − 1 ) ( y − 1 ) − 1 ( y − 1 ) 2 − 1( x − 1 ) ( y − 1 ) 2 + R 2 2 2( x, y) 
( y − 1 ) + ( x − 1 ) ( y − 1 ) − 1( y − 1 ) 2 + R 2! 2( x, y)  "
1 + ( x − 1 ) + ( y − 1 ) + ( x − 1 ) ( y − 1 ) − 1 ( y − 1 ) 2 + R 2 2 ( x, y)
 /8     !3 f( x, y) = xexy + y c o s x $. * ) * l=     f′l ( −1 , 2 ) 
  ) 3  l = ( − 4 √ , 5 √ )  l = ( 5 √ , − 4 √ )  41 41 41 41 !
l = ( −4 ,
 /9   z = z( x, y)   2 '2) . z3 − 4 xz + y2 − 4 = 0  $% z′ , z′  x y
M0( 1 , −2 , 2 )
z′ = 1 , z′ = 1  = 0 , z′ x y z′ 2 x y = −1  ! z′ = 1 , z′ = 0 , z ′ x 2 y = 1  " z′x y = 1 
 6:  $. f ′= !3
2 s in v, u = x2 + y2, v = y x
f ( u, v) = u x
f′ = 4 xu s in v
= xu s in v + yux
− yu c o s vf′ c o s vxx x !
f ′ = 4 xu s in v + yu x c o s v "
5  +  x  y−x  6  $.
, x2 + y 2 = 0 f ′ x+2y y( 0 , 0 )
  3  f( x, y) = 0 , x2 + y 2 = 0  1
+       x  2 ! −1  " 0
 6#  $.    3'  z′′ ( 0 ,π xy
)   z = c o s ( xy − c o s y) 2
z′′ ( 0 , π ) =  ( 0 , π xy −π z′′ ) = 0  2 2 xy 2 !
z′′ ( 0 , π) = π  ( 0 , π xy " z′′ ) = 1  2 2 xy 2
 6,  $. * ' dz   # !3 z = x √x+y
dz = y( x2 + y2) − 2  ( ydx − xdy) 
dz = y( x + y2) −  ( y2 dx − xdy)  !
dz = ( x2 + y2) − ( ydx − xdy)  "
5  +  1 4 CuuDuongThanCong.com
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 6-  $. * ' 3' #   # !3 z = exy  M0( 1 , 1 ) 
d2z( 1 , 1 ) = e2( 4 dx2 + 6 dxdy + dy2) 
d2z( 1 , 1 ) = e2( 4 dx2 + 6 dxdy + 4 dy2)  !
d2z( 1 , 1 ) = e2( 4 dx2 + 3 dxdy + dy2)  "
d2 z( 1 , 1 ) = e2( 4 dx2 + 6 dxdy + 4 dy2) 
 6/  $. 2    z = xy *)  +& x + y − 1 = 0 . (     
z  2   M ( 1 , 1)
z +  2  2 2 !
z  2 1 M ( 1 , 1) "
5  +  2 2
 66    # !3 z = 3 x − 2 y + 1 =    D )  !) y = x − 1 , y = −x + 3 , x = 1 
(     
  ) 3  z  5
   3  z  4 !
  ) 3  z  7 "
   3  z  −2
 67    # !3 z = x3 −y3 +5 =    D = [0 , 1 ]×[1 , 2 ] (     
   3  z  −3
   3  z  −2 !
  ) 3  z  4 "
  ) 3  z  6
 68    # !3 z = x2 + y2 + xy − 1 2 x − 3 y (     
z  2 1  M( 7 , −2 )
z +  1 "2 !
z  2   M ( 7 , −2 ) "
z +  2   
 69  R  &  % '
f( x, y) dxdy D
D = {( x, y) | ( x − 1 ) 2 + ( y − 2 ) 2 ≤ 4 , y ≤ 1 } √ 1+ 3  I = dx
f( x, y) dy I = dx
f ( x, y) dy √ √ √ 1− 3 2 0 − 4−(x−1)  1− 3 √ 1+ 3  1  1  1  ! I = dx
f ( x, y) dy " I = dx
f( x, y) dy √ √ √ 1− 3 2+ 4−(x−1) −1 2− 4−(x−1) 1  0 
 7:   % ' I = dy
f ( x, y) dx $ 1 2 2 3 % ' √ 0 − 2y−y  0  1  0  1  I = dx
f ( x, y) dy I = dx
f ( x, y) dy√ √ −1 1 1 x−1 1+ 1 x 1  ! I = dx
f ( x, y) dy " I = dx
f( x, y) dy 0 √ 1− 1−x−1 0 1 5 CuuDuongThanCong.com
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 7  $ 1 2 2 3 % ' I =
dy f ( x, y) dx 0 y 1  x  1  1  I = dx
f ( x, y) dyI = dx
f ( x, y) dy 0 √ x 0 0 √ 1  x ! I = dx
f ( x, y) dy "
5  +  0 x  
 7#  0< ST
f ( x, y) dxdy= >      A  A( 0 , 1 ) , B( 0 , 2 ) , C( 1 , 1 )  (   D
   1  2−x  2  2−yI = dx
f( x, y) dy = dy
f( x, y) dx 0 1 1 0 1  2−x  2  2−y ! I = dy
f ( x, y) dx = dx
f( x, y) dy 0 0 1 0 1  1  2  2−yI = dy
f ( x, y) dx = dx
f( x, y) dy 0 2−x 1 0 1  2−x  2  0  " I = dx
f ( x, y) dy = dy
f ( x, y) dx 0 1 1 2 y 2   
 7,  $ 1 2 2 3 % ' I =
dy f ( x, y) dx 0 0 1  2  1  2  I = dx
f( x, y) dy I = dx
f ( x, y) dy 0 2x 0 0 1  2x  2  2  ! I = dx
f ( x, y) dy " I = dx
f ( x, y) dy 0 2 0 0 √
 7-  $% "& %  '( )  !) 2 x ≤ x2 + y2 ≤ 6 x * y ≤ x 3 ; y ≥ 0 8 π √ 8 π 4 π √  + 2 3  !   + 2 3  "
  +  3 3 3 1 6 CuuDuongThanCong.com
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