chapter 9 differentiation 2
TRANSCRIPT
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8/20/2019 Chapter 9 Differentiation 2
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105
9
9.1
)2(lim2
+→
x x
= 2 + 2
= 4
4
2
0 2
1. 2 1.9, 1.99 1.999, 4
4.
2. 2.1, 2.01, 2.001, 4
4.
3. , 2+ x 4 2 4)2(lim2
=+→
x x
4. H, 2+= x y 4 2→ x .
2
4lim
2
2 −
−
→ x
x
x
)2(
)2)(2(lim
2 −
−+
→ x x x
x
)2(lim2
+→
x x
22 +=
4=
2+= x y
F , .
.
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8/20/2019 Chapter 9 Differentiation 2
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106
)1
(lim x
x
x
+
∞→
)1
(lim x x
x
x+=
∞→
)1
1(lim x x
+=∞→
01+=
1=
9.1
F:
()2
2 23lim
x
x
x
+
∞→
() )32(lim1
+→
x x
()2
52lim x
x→
()3
9lim
2
4 −
−
→ x
x
x
9.2
9.21
1. .
2.
.
(1.1,2.4)
(1,2)
1. ,
.
2. .
. F :
∞=
0
1.
01 =∞
1 .
F ,
.
.
=∞ x
1
∞
1.
0
1=
∞.
o
o
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8/20/2019 Chapter 9 Differentiation 2
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107
2.
.
3. yδ
xδ .
(+ xδ , + yδ )
yδ
(, ) xδ + xδ
4. F , x
ym
PQδ
δ = .
5. B , x
ym
xP
δ
δ
δ 0lim→
=
6. x
y
x δ
δ
δ 0lim→
.
7. x
y
x δ
δ
δ 0lim→
, , dx
dy.
8.dx
dy .
9.22
( + xδ , + yδ )
(, )
2 x y =
22
2
2
)(
x x x x y y
x x y y
δ δ δ
δ δ
++=+
+=+
2 x y =
1
2
,
.
o
o
o
o
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8/20/2019 Chapter 9 Differentiation 2
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108
,
x x x
y
x x x
y
x
x x x
x
y x x x y
x x x x x y y y
x xδ
δ
δ
δ δ
δ
δ
δ δ
δ
δ
δ δ δ
δ δ δ
δ δ
+=
+=
+=
+=
−++=−+
→→
2limlim
2
22
2
00
2
2
222
xdx
dy2=
9.20
Fdx
dy .
()2
3 = () 52 2+= x y
()
2= ()
x y
35 −=
()2
31 x y −= () 53 2 −=
dx
dy
nax y =
G = (),
dx
dy )(
' x f ,
nax x f =)( ,
,
)()( '
x f dx
xdf =
2 1
1.. −
= n xna
dx
dy
xδ x
y
δ
δ .
.
F ,
x
y
x δ
δ
δ 0lim→
dx
dy. F
, xδ
.
,
xδ 0.
1..)('
−
= n xna x f
2 x y = , x
dx
dy 2= .
2)( x x f = , x x f 2)(' =
x y 2=
2
)2()(
=
=
dx
dy
xdx
d y
dx
d
x x f 2)( =
2)('
2)(
)2()]([
=
=
=
x f
dx
xdf
xdx
d x f
dx
d
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8/20/2019 Chapter 9 Differentiation 2
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109
2 x y =
)( 2 x
dx
d
dx
dy=
12.2.1 −= xdxdy
xdx
dy2=
x x f
x x x f
x x f
6)('
.0.6.2.3)('
63)(
1012
2
=
+=
+=
−−
9.21
1. Fdx
dy . H,
dx
dy 2= x .
() 42 4 −= x y
() )5( 2+= x x y
()2
22
x
x x y
+=
2. F )(' x f . H, )3(' f .
() x x x f 3)( 2 +=
()2)12()( −= x x f
() )3)(22()( −−= x x x f
9.3.1 D
D 42
1 4− x .
)42
1(
04 x xdx
d −
10144.0
2
1.4
−−
−= x x
32 x=
0
,
. 0 = 1.
,
.
C
.
.
.
)42
1(
4− x
dx
d , 4
2
1 4− x .
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8/20/2019 Chapter 9 Differentiation 2
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110
D )32)(32( +− x x .
x
x x
x xdx
d
x xdx
d
8
.9.0.4.2
)94(
)]32)(32[(
1012
02
=
−=
−
+−
−−
9.22
D .
() 53 2− x
() x x
22−
() )43)(43( +− x x
9.3
Fdx
dy
2)12( += x y
1
2)12( += x y
144 2 ++= x x y
2
2)12( += x y
12 += xu
2=dx
du
2u y =
F , .
.
0
,
. 0 = 1.
48 += xdx
dy
dudy
dxdu ,
dx
dy.
.
F , .
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8/20/2019 Chapter 9 Differentiation 2
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111
udu
dy2=
22 ×= udx
dy
u4=
)12(4 += x
48 += x
F , :
1. C . .2u y =
udu
dy2=
2. , .
12 += xu
2=dx
du
3. dx
dy.
4. 2 1
2.
Fdx
dy
5)43( += x y
43 += xu
3=dx
du
5u y =
45udu
dy=
dx
du
du
dy
dx
dy×=
dx
du
du
dy
dx
dy×=
,
dx
dy.
.
1
2. B,
4, 5 ,
1
2.
1.
5.
2.
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8/20/2019 Chapter 9 Differentiation 2
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112
35
4×=
udx
dy
415u= 4)43(15 += x
:
3.)43(5 15−+= x
dx
dy
4)43(15 += x
9.3
Fdx
dy .
()4)31( x y −=
()3
)1
1( x
y +=
()4
)53(
1
−
=
x y
9.4
9.41
)1)(12( ++= x x y
uv y =
vuuvvuuv y yvvuu y y
δ δ δ δ δ
δ δ δ
+++=+
++=+
))((
43 += xu 415u
dx
du
du
dy
dx
dy×=
2
1H ?
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8/20/2019 Chapter 9 Differentiation 2
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113
)(limlim00 x
vu
x
uv
x
vu
x
y
x
vu
x
uv
x
vu
x
y
x
vuuvvu
x
y
vuuvvu y
uvvuuvvuuv y y y
dxdx δ
δ δ
δ
δ
δ
δ
δ
δ
δ
δ δ
δ
δ
δ
δ
δ
δ
δ
δ δ δ δ
δ
δ
δ δ δ δ δ
δ δ δ δ δ
++=
++=
++=
++=
−+++=−+
→→
x
vu
x
uv
x
vu
x
y
dxdxdxdxdxdxdx δ
δ δ
δ
δ
δ
δ
δ
δ
0000000limlimlimlimlimlimlim→→→→→→→
×××××=
dx
dv
dx
duv
dx
dvu
dx
dy.0++=
Fdx
dy )1)(12( ++= x x y
)1)(12( ++= x x y
1
1
2
12
=
+=
=
+=
dx
dv
xv
dx
du
xu
dx
duvdx
dvudx
dy+= ,
dx
dy
)2)(1()1)(12( +++= x xdx
dy
34
2212
+=
+++=
x
x x
2 1
dx
duv
dx
dvu
dx
dy+=
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8/20/2019 Chapter 9 Differentiation 2
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114
9.40
1. Fdx
dy . H,
dx
dy 1−= x .
() )2)(31( +−= x x y
()
2
)3)(2( −+=
x x y ()
32 )2()3( ++= x x y
2. F )(' x f . H, )1(' f 2 )2(' − f
() )3)(52()( ++= x x x f
()5)4)(12()( +−= x x x f
()24)3()22()( −−= x x x f
9.4.2
D42 )52(3 − x x
42 )52(3 − x x
2.)52(4
)52(
6
3
3
4
2
−=
−=
=
=
xdx
dv
xv
xdx
du xu
3)52(8 −= x
)56()52(6
)]52(4[)52(6
)52(6)52(24
6.)52()52(8.3
])52(3[
3
3
432
432
42
−−=
−+−=
−+−=
−+−=
−
x x x
x x x x
x x x x
x x x x
x xdx
d
,
dx
duv
dx
dvu
dx
dy+= .
,
dx
duv
dx
dvu +
.
F
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8/20/2019 Chapter 9 Differentiation 2
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115
9.41
D .
()3)42)(53( x x −−
()53
)65(4 − x x
()43
)4()98( +− x x
9.42
1
12
+
+=
x
x y
v
u y =
vv
uu y y
δ
δ δ
+
+=+
, ,
+
−
=
+
−
=
×
+
−=
+
−=
+
−−+=
+
+−
+
+=
−
+
+=−+
→→ )(limlim
)(
1
)(
)(
)(
)(
)(
)(
)(
00 vvv
x
vu
x
uv
x
y
vvv
x
vu
x
uv
x
y
xvvv
vuuv
x
y
vvv
vuuv y
vvv
vuuvuvuv y
vvv
vvu
vvv
uuv y
v
u
vv
uu y y y
dxdx δ
δ
δ
δ
δ
δ
δ
δ
δ
δ
δ
δ
δ
δ
δ δ
δ δ
δ
δ
δ
δ δ δ
δ
δ δ δ
δ
δ
δ
δ δ
δ
δ δ
2
2 1
1 H ?
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8/20/2019 Chapter 9 Differentiation 2
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116
)(limlim
limlimlimlim
)(lim
00
0000
0
vvv
x
v
u x
u
v
vvv
x
vu
x
uv
dxdx
dxdxdxdx
dx
δ
δ
δ
δ
δ
δ
δ
δ
δ
δ
+×
×−×
=
+
−
=
→→
→→→→
→
vv
dx
dvu
dx
duv
dx
dy
.
−
=
Fdx
dy
1
12
+
+=
x
x y
2v
dx
dvu
dx
duv
dx
dy −
= .
1
1
2
12
=
+=
=
+=
dx
dv
xvdx
du
xu
dx
dy,
2)1(
)1)(12()2)(1(
+
+−+=
x
x x
dx
dy
2
2
2
)1(
3
)1(
1222
)1()12()1(2
+
=
+
+−+=
+
+−+=
x
x
x x
x x x
2v
dx
dvu
dx
duv
dx
dy −
=
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8/20/2019 Chapter 9 Differentiation 2
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117
9.42
1. Fdx
dy . H,
dx
dy 1= x .
()( )
52
23 2
−
+=
x
x y
()( )
( )2
3
1
12
+
+=
x
x y
()( )
( )3
2
74
65
−
−=
x
x y
2. F )(' x f . H, )1(' f 2 )5(' f
()52
2)(
2
−
+=
x
x x x f
()( )
52
23 3
−
+=
x
x y
()( )
x x
x y
52
232
2
−
+=
9.4.3
D 1
23 2
+
+
x
x x
1
1
26
23 2
=
+=
+=
+=
dx
dv
xv
xdx
du
x xu
+
+
1
23 2
x
x x
dx
d
2
2
)1(
)1)(23()26)(1(
+
+−++=
x
x x x x
2
22
)1(
23286
+
+−++=
x
x x x x
,
2v
dx
dvu
dx
duv
dx
dy −
= .
,
2v
dx
dv
udx
du
v −
.
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8/20/2019 Chapter 9 Differentiation 2
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118
2
2
)1(
2103
+
++=
x
x x
9.43
D .
()23
32
−
+
x
x
()2
34
x
x x +
()12
5 2
+ x
x
9.5
1. .
2.
. H, ,
1tan −=× normalgent mm
3. dx
dy . ,
, .
4. , .
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8/20/2019 Chapter 9 Differentiation 2
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119
F 592 2
−−= x x y
(3, 4)
94
592 2
−=
−−=
xdx
dy
x x y
A (1, 3),
9)3(4 −=dx
dy
3
912
=
−=
G 3.
3
1
13
1tan
−=
−=×
−=×
normal
normal
normalgent
m
m
mm
(3, 4)
63
933
3
3
3
−=
−=−
=
−
−
x y
x y x
y
(3, 4)
0123
393
3
1
3
3
=−+
−=−
−=
−
−
y x
x y
x
y
,
12
12
2−
−
x
y y. (, )
(3, 4).
(1, 3)
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8/20/2019 Chapter 9 Differentiation 2
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120
F 243 2
+−= x x y 52 =− x y .
46
243
2
−=
+−=
xdx
dy x x y
2
5
2
1
52
+=
=−
x y
x y
=2
1
normalm , 2
1=normalm .
2
12
1
1
tan
tan
tan
−=
−=×
−=×
gent
gent
normalgent
m
m
mm
46 −= xdx
dy 2tan −=gent m ,
3
1
246
=
−=−
x
x
2tan −=gent m 2
1=normalm
3
1= x .
243 2
+−= x x y
3
1= x ,
23
14
3
13
2
+
−
= y
1=
.
G
.
.
.
,
.
A ,
46 −= xdx
dy . F
,
.
2 21 .
243 2
+−= x x y .
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8/20/2019 Chapter 9 Differentiation 2
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121
H
1,
3
1.
1,
3
1
0563
1366
3
122
21
3
11
=+−
−=−
−=−
=
−
−
y x
x y
x y
x
y
9.5
F
:
( ) x x x y 46 23 +−= ; 3= x
() 2
)3)(2( −+= x x y ; 2−= x
()23
32
−
+=
x
x y ; 1= x
9.6
1. F
dx
dy.
2. 2
2
dx
yd .
G x x x y 46 23 +−= ,2
2
dx
yd .
126
4123
46
2
2
2
23
−=
+−=
+−=
xdx yd
x xdx
dy
x x x y
3
D
dx
d
dx
dy=
2
2
dx
yd
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8/20/2019 Chapter 9 Differentiation 2
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122
G x
x x x f 1
4)( 3 ++= , )('' x f .
3
''
3''
22'
13
3
26)(
26)(
143)(
4)(
14)(
x x x f
x x x f
x x x f
x x x x f
x x x x f
+=
+=
−+=
++=
++=
−
−
−
9.6
1. F2
2
dx
yd . H,
2
2
dx
yd 1= x .
() x x y 52 4
−=
() )5(2 2 x x x y +=
()2
2 32
x
x x y
+=
2. F )('' x f .
() x x x f 3)( 2 +=
()2)23()( −= x x f
()3)3)(24()( −−= x x x f
9.7
D
A
B
1. B :
() A C
() B D
() A, B, C D
)()( ' x f
dx
xdf = ,
)()( ''
2
2
x f dx
x f d =
D
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8/20/2019 Chapter 9 Differentiation 2
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123
2. A , 0=dx
dy
3.
0.dx
dy , 0=
dx
dy.
9.6.1
1 Fdx
dy
2 D ,
3 D (, ) .
() :
1
dxdy 0
() :
1
dx
dy 0
F 593 23 +−−= x x x y . H,
.
963
593
2
23
−−=
+−−=
x xdx
dy
x x x y
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8/20/2019 Chapter 9 Differentiation 2
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124
A , 0=dx
dy.
0)3)(1(
032
0963
2
2
=−+
=−−
=−−
x x
x x
x x
1−= x 3= x
() 1−= x
5)1(9)1(3)1( 23 +−−−−−= y (1, 10)
10=
() 3= x
5)3(9)3(3)3( 23
+−−= y (3, 22)
2= y
(1 , 10) (3,22)
()
() 2−= x
15
9)2(6)2(3 2
=
−−−−=
dx
dy
dx
dy
() 0= x
, (1, 10) .
()
() 2= x
9
9)2(6)2(3 2
=
−−=
dx
dy
dx
dy
() 4= x
, (3, 22) .
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8/20/2019 Chapter 9 Differentiation 2
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125
963
593
2
23
−−=
+−−=
x xdx
dy
x x x y
A , 0=dx
dy.
0)3)(1(
032
0963
2
2
=−+
=−−
=−−
x x
x x
x x
1−= x 3= x
() 1−= x
5)1(9)1(3)1( 23
+−−−−−= y (1, 10)10=
() 3= x
5)3(9)3(3)3( 23
+−−= y (3, 22)
2= y
(1 , 10) (3,22).
963 2
−−= x xdx
dy
662
2
−= xdx
yd
A (1, 10),
6)1(62
2
−−=
dx
yd
12−=
02
2
<
dx
yd
(1, 10)
A (3, 22),
6)3(62
2
−=
dx
yd
12=
.
593 23 +−−= x x x y
.
F
D
02
2
<
dx
yd , . 0
2
2
>
dx
yd ,
.
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8/20/2019 Chapter 9 Differentiation 2
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126
02
2
>
dx
yd
(3, 22) .
9.71. F . H,
.
( ) x
x y 1+=
()2
2 16)(
x x x f +=
2. F
( ) 28 x x y −=
( ) 22 96
2)( x x x f +=
9.8
1.
.
2.
dt
d
4.
A B .
1.41
. F
5 .
, A () B ().
G 14.1
−
= cmsdt
ds, cms 5=
3sV = .
dt
dV .
2
3
3sds
dV
sV
=
=
dt
dB
dB
dA
dt
dA×=
3sV =
. D .
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8/20/2019 Chapter 9 Differentiation 2
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127
dt
ds
ds
dV
dt
dV ×=
23s
ds
dV = .
dt
dss
dt
dV ×= )3(
2
1
4.1 −
= cmsdt
ds, 5=s ,
4.1)5(3 2
×=
dt
dV
13105
4.175
−
=
×=
scm
13
4 −
scm . F
3.
, A () B ().
G 13
4 −
−= scmdt
dV , cmr 3=
3
3
4r V π = .
dt
dr .
2
3
43
4
r dr
dV
r V
π
π
=
=
dt
dr
dr
dV
dt
dV ×=
2
4 r dr
dV π = ,
dt
dr
r dt
dV ×=
24π
5 32
dt
ds .
3
3
4
r V π =
. D
dt
dV
. ,
.
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128
13
4 −
−= scmdt
dV , cmr 3= ,
1
2
36
4
)3(44
−−
=
×=−
cmsdt
dr
dt
dr
π
π
1
9
1 −
−= cmsπ
9.8
1. 12
6 −
scm 5 . H,
.
2. G
45 += . 3 ,
= 4.
9.9
1. ( xδ ) ( yδ ) .
2. x
y
x δ
δ
δ 0lim→
=dx
dy.
2.
.
G 4 4.01.. F .
, A (A) B ().
G cmr 01.0=δ , cmr 4= 2
r A π = . Aδ .
r dr
dA
r A
π
π
2
2
=
=
dx
dy
x
y≈
δ
δ
3
24 r π
dt
dV
.
2r A π = .
D .
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129
dr
dA
r
A≈
δ
δ
r dr
dAπ 2= ,
r r
Aπ
δ
δ 2≈
r r A δ π δ ×≈ 2
cmr 01.0=δ , cmr 4= ,
)01.0()4(2 ×≈ π δ A
)01.0(8 ×≈ π
208.0 cmπ ≈
G 43 2 += x y . F 2 2.03.
, A B .
G 03.0= xδ , 2= x 43 2+= x y . yδ .
xdx
dy
x y
6
43 2
=
+=
dx
dy
x
y≈
δ
δ
xdx
dy6= ,
x x y
x x
y
δ δ
δ
δ
.6
6
≈
≈
03.0= xδ , 2= x ,
03.0.)2(6 ×≈ yδ
36.0≈
9.9
1. G 572 23
+−= x x y , dx
dy (3, 4). H, ,
2 1.97.
2. F 5 5.02.
r δ .
r δ .
Fdx
dy
xδ .
xδ .
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130
1. E :
()35
64lim
−
+
∞→ x
x
x
() )2
4(lim
2
2 −
−
→ x
x
x
() x
x
x −
+
→ 1
1lim
3
2. G 143 2
+−= x x y . Fdx
dy.
3. D2
35
x
x − .
4. G )1)(2()( 2 −+= x x x f . F )3('' f
5. G 5
)23( += . F = 1.
6. G 432
+−= xkx y = 2 13
1− . F .
7. F 2
3 x y −= (2, 1)
8. G xyP = 30=+ y x , .
9.
. G 60 .
() 2
4
430 x x A
+−=
π
() 142.3=π ,
.
10. G .572 23
+−= x x y F , = 3
5 .
11. G 23
2
9)1( t t p +−= . F
dt
dp 9=
dt
dp.
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12. G 2
3)( x
q px x f +=
3
2 1926)( x
x x f −=
,
()
() .
13. k x y =+4 5)32( 2
−−= x y E. F
()
()
14. D .
()32 )51( x+
()2
434+
−
x
x
15. G x
x x f 15)( 3 −= , )('' x f .
16. G x x y 32 2 −= x p −= 5 .
() Fdp
dy = 2,
() F 4 4.05.