复旦大学：数学分析（习题全角、教学教案）：ex12-6

5



íHq′ 1
 ) ?/ f ?￥′

1

61222),( 2244 +??+= yxyxyxf
2

2244 2),( yxyxyxyxf ???+=
3

222 ),,( zyxzyxf ?+=
4

))((),( 42 xyxyyxf ??=
5
 y b x a xyyxf 33 ),( ++=
?è ?
0,0 >> ba
6
 zy z x y xzyxf 2 ),,( +++=

b 0,, >zyx 3 (1) 5p
?b? 3 3 440 824 x y fxx fy y ? =?= ? ? =?= ? ? 0
 3¤ 0, 1; 0, 3xy=± =±
 'f ?μ 9?
?b? 2 4(3 1) xx fx= ?
 0 xy f =
 2 24( 1) yy fy= ?
 V? 22 96(3 1)( 1)Hxy=??b ?¨? ? 12.6.2b
?
)0,0( )3,1(
 )3,1( ?
 )3,1(?
 )3,1( ?? ? @
 ? [ ^′?
 7?
?? ^′?b ? 0H > xx f ￥?|
 V?f ? ?|v′
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 )3,1( ?
 )3,1(?
 )3,1( ?? 1 ?|l′ b 13? ? '5 V P¨￥ZE¤? 22 2 2 ( , ) ( 1) 2( 3) 13fxy x y=?+ ??
 ?N^? )3,1(
 )3,1( ?
 )3,1(?
 )3,1( ?? 1?1f ?￥Kl′?
 Kl′1
f ? í K v ′
 ?1f ?￥v′?
 v′1 b 13? )0,0( 6
2
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?b? 145 3 3 422 422 x y fxxy fyxy ? =??= ? ? =??= ? ? 0 0
  TMh
 V3¤
 '
?1

0, 1xy==± )0,0( )1,1( )1,1( ?? ??b  ?

 2 12 2 xx fx=? 2 xy f =? 2 12 2 yy fy= ?
 V? 22 4(6 1)(6 1) 4Hx y=???b ?¨? ? 12.6.2b
?
)1,1( )1,1( ?? ?@
 ? [ ^′?
  ?  0H > xx f ￥?|
 V?f ?
)1,1( )1,1( ?? ?|l′ b 2?  ?
μ
O)0,0( 0H = (0,0) 0f = b??
 22 (,) 2 ( 2)fxx x x=? 4 (, ) 2f xx x?=
 V?f ? ??íM|
 ?[ ? ^′?b )0,0( )0,0(
3
5p
?b? 20 20 20 x y z fx fy fz ? = = ? = = ? ? =?= ?
 3¤ ^·B￥
?b?(0,0,0) (0,0,0) 0f =
 2 (, ,0) 2 f xy x y=+
 2 (0,0, )f z=?z 0 )
 V?f ? ??íM|
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 ?[f ?í′?b (0,0,0) 0,0,0) ? ?=Q[ T

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á ìμ?/2 ?
 ()f x n ∈Rx ! 1 ￥
?
5? V? 0 x ()f x 00 ()()()( T ff H?=? ?x x xx xx
a
 1Kl′￥ sA1Hq ^ H1??? ?
0 ()f x
b
 1Kv′￥ sA1Hq ^ H1?μ? ?
0 ()f x
c
 ? ^′￥ sA1Hq ^ H1?? ?b 0 ()f x '5??f ? ￥ Hesse ?1?? ?
 ?[ ? ^ ￥′?b (, ,)fxyz (0,0,0) (, ,)fxyz 146
4
5p
?b? 42 24 2(3 2 ) 0 20 x y fxxyxy fyxx ? =?? ? ? =??= ? ? =
 3¤ ;0xy== 1, 1x y=±=; 1 , 8 2 xy=± = 3
'
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 3 24 xy f xx=? ?

 V? 2 yy f = 42 2(30 12 2 )Hxyxy 32 (2 4 )=??x x?+ b ?¨? ? 12.6.2b
? ) 8 3 , 2 2 (
 ) 8 3 , 2 2 (? ?@
 ?[ ^′?
 ?  0H > xx f ￥?|
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 ) 8 3 , 2 2 (? |l′ 1 64 ? b 
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^?f ?  ??íM|
 ?[ ? ^′?b )0,0( )0,0(
5
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?b? 3 2 3 2 0 0 x y a fy x b fx y ? = ?= ? ? ? ? = ?= ? ?
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?b? 3 3 2 xx a f x =
 1 xy f =
 3 3 2 yy b f y =
 V? 33 33 4 1 ab H xy = ? b ?¨? ? 12.6.2b??
? 22 , ab ba ?? ?? ?? μ
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 147 V?f ? ),( 22 a b b a ?|l′ 3 b ab
6
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 3¤·B￥
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 ?[f ? )2,2,2( 4 3 2 1 4 1 |l′ 1 4 42? b 2
 !
￡ üf ? ￥Kl′1 0b xzxyzyxzyxf 2223),,( 222 +?++= f ￡ 5p
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 3¤·B
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??f ? ( ?￥ Hesse ? ^ ??￥
 ?[f ? ?|l′ (0,0,0) 0,0,0) 222 260 204 ? ?? ?? ? ?? (0,0,0) (0,0,0) 0f = b ? '5 V P¨￥ZE¤? 22 11 (, ,) ( 2) ( 2) 22 2 1 f xyz x y x z y=?+++
 ?N V?f ? ?|Kl′(0,0,0) (0,0,0) 0f = b 3. ￡ üf ? μík?v′?
 ?íl′ ?b yy yxyxf ecos)e1(),( ?+= ￡ ? (, ) (1 e)sin 0 (, ) ecos (1 )e 0 y x yy y fxy x fxy x y ? =? + = ? ? = ?+ = ? ?
 148 3¤ xkπ= , cos 1ykπ=?
 ?[
?1 (,cos 1)kkπ π ?
 0, 1, 2,k = ±±null b ?
(1 ) cos y xx fex=? + sin y xy f ex=?
 ecos (2 )e yy yy f xy=?+
 V? 
? (,cos 1)kkπ π ? )
 cos (1 ) yy Hkeπ=+e
 ?[? k 1 ? H
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? k 1} ? H
?0H > 0 xx f <
 V? (,cos 1kk)π π ? ^v′?b ?[f ?μík ?v′?
?íl′?b 4
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 ¤? cos cos cos( )x yx==+yb { }(, )|0 , 2xy xy x y π=<<+<D null  I n
 ¤? 2x yxyπ== ??
' 22 , 33 π π ? ? ?? ? ? ^f ?u×=?·B￥
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 '? 0x= 0y =  2xy π+ = H
 μ
7  u×=?·B￥
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 V?f ?￥Kv′1 2 33 max =f
Kl′1 b 0 min =f 5
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 ?[μK l ′
n? 15
3
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 ?[A? ^Kl′?by NKD°L1 6 1 ?= xξ b 6
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 ?? 4 5? 12 2 3 3 π α αα= == H ?Kv
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 ¤? 5 2 R h=
 | 5 2 R h= D 2 1 3 VRH Rππ=+ 2 h}?? 150 =?Z?
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 ?[? 21 5 hHR == H
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?N V??f ? 0=+ yx 122 22 =++ yxyx 1 2 =y )(xyy = ￥
?1
O?1x=± 1x=± Hμ 1y = ? b ??
?μ 2 1' ( ) 1 '' (1 2 ') 2(2) yxy yy x yxy y ++ =? + + =? ++
 ?  ￥?| V? "( 1)y ± )(xyy = 1x=? |v′ 1
 1x = |l′ b 1? ? '59 V? 222 22( )xxyyxyy++=++= 2 1 1
 ¤?
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 |l′ b 1x = 1? 9
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?1 D (0, 2)? 16 (0, ) 7 b 151 ? 2 2 4 12 8 z x zy ? = ???
 2 0 z xy ? = ??
 2 2 4 12 8 z yz ? = y? ??
 V?
? (0, 2)? D 16 (0, ) 7 μ b 0H >  ?

yN (0, 2)? 1z = 2 2 4 0 15 z x ? = > ?
 ?[ (0, 2)? 1l′?
 l′1
1z = 16 (0, ) 7 ?
 8 7 z =?
y N 2 2 4 0 15 z x ? =?< ?
 ? [ 16 (0, ) 7 1 v′?
v′1 8 7 z =? b ? 1 eZ? V[?1 22 22(2)(1)(78xyzzz++ =? +)
 ?PHdμ V¤
'(1)(78)0zz?+≥ 8 7 z ≤? ? b 1z ≥ ? 2 ?? bW?
Z?￥m^ ^ ? = ?w ?
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 Kl′ 1z =
 6B? 7 g_/
 K v′ 8 7 z =? b 10
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 P ???°L 0=x
 0=y
? ￥  ?￥üZ?Klb 0162 =?+ yx 3 ü ? ? (, )x y ??°L￥  ?üZ?1 22 216 (, ) ( ) 5 xy Dxyxy 2 + ? =++ b  pê? ?
 ,xy 2 2(216) 5 4 2 ( 2 16) 0, 5 x y Dxxy Dyxy ? =+ +?= ? ? ? ? =+ +?= ? ? 0,
 ¤? 81 , 55 xy== 6
 ?[f ?oμB?
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 V?f ? 
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 ¤? 2 π α βα==??β
 ?[ 6 π βα ==
 '?M????￥ ?1Klb 12
￡ ü
?￥ ?μ=¤ H??
[? H?￥ ?1Kvb n n ￡ !???1 1
?=¤ H?￥òH ?￥?? ?1n k α ),,2,1( nk null=
5 H?￥ ?1 n )]sin(sinsin[sin 2 1 121121 ?? +++?+++= nn S αααααα nullnull b ? 112 1 1 [cos cos( )] 0 2 n k S αααα α ? ? =?+++ ? null =
 )1,,2,1( ?= nk null
 w 12 1 12 1 2( nn )α ααπααα ? ? === =?+++nullnull
 ?[ 153 2 k n π α =
 (1,2,,kn)= null
 '=¤? H?￥ ?1Kvb n 13
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? +∞<<<< yx 0,10 H
? ??? T 1 e)1( ? <? xyx y b ￡ 7
)1(),( xyxyxf y ?= ypê?
 (1 )(1 ln ) 0 y f xxyx y ? =?+ = ?
 3¤ x y ln 1? = b%?￥
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 V?
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￥v′?1 x y ln 1? =
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 5

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? ^'( ) ln 0gx x=< (0 ) 1, (1 ) 0gg+= ?= () 0gx> )(x? ?ì?? 9Fb? 1 1 lim ( ) x x e? ? →? =
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 ,xy 223 244 x y Pxy Pxy αβ βα =? ? + =? ? ? =? ? + = ? ?
 3¤ 154 22 2 23 βα βα ? ? =x
 22 24 34 βα βα ? ? =y b y1 ^=Q[ T
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 2 xx P α=?<
 V? Hesse ? ^μ?￥
 ?[f ?μKv′
'? 22 2 23 βα βα ? ? =x
 22 24 34 βα βα ? ? =y Háè9 Kvb 155 12.6 9 ? L5
? =￥·?/
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 1
 ? ,q￥Fy3ü ?y?
 V U ,q?B?y? ?C￥F? ?
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 V U?=? y??C￥F? ?b°?¤ 8? ,q￥ D ?/
 x y x y x 0 1 3 6 8 5 4 2 y 1 2 2 4 4 3 3 2 ??t ? ￥??m
 s ? ì-W1"￥ü T
 i?E?wLb baxy += 3
??} ?1 hold off x=[0,1,3,6,8,5,4,2]; y=[1,2,2,4,4,3,3,2]; plot(x,y,'b*') hold on A=[x',ones(size(x'))]; B=y'; x1=A\B; a=x1(1);b=x1(2);y=a*x+b; plot(x,y,'r') string=['E?°L y=',num2str(a),'*x+',num2str(b)]; text(0.5,3.5,str,'FontSize',16) ?a
¤E?wL bm?1 = 0.37845 +1.2531yx 2
??v?￥?[c  D?aéc  ￥??2T?/V ? U
??} ?1 (%)x (%)y x 16
517
5 18
5 19
520
521
52
523
524
5 y 43
542
6 41
8 40
640
338
737
236
034
0 156 ??t ? ￥??m
s ? ì-W1"￥ü T
i? E?wLb 3
??} ?1 hold off x=[16.5,17.5,18.5,19.5,20.5,21.5,22.5,23.5,24.5]; y=[43.5,42.6,41.8,40.6,40.3,38.7,37.2,36.0,34.0]; plot(x,y,'b*') hold on A=[x',ones(size(x'))]; B=y'; x1=A\B; a=x1(1);b=x1(2);y=a*x+b; plot(x,y,'r') str=['E?°L y= ',num2str(a),'*x+',num2str(b)]; text(18,44,str,'FontSize',16) ?a
¤E?wL = 1.1483 +62.9519yx? bm?1 3
?á?Fy-￥c ￡ q
%
DFyac ￡ q
%
￥? k2 T?/V
 ? kI| i 1 2 3 4 5 6 7 8 9 10 Fy-￥ c ￡ q i x 16.7 18.2 18.0 17.9 17.4 16.6 17.2 17.7 15.7 17.1 Fya￥ c ￡ q i y 17.5 18.7 18.6 18.5 18.2 17.5 18.0 18.2 16.9 17.8 k??Fya￥c ￡ q DFy-c ￡ q ￥1"b y x 3
??} ?1 hold off x=[16.7,18.2,18.0,17.9,17.4,16.6,17.2,17.7,15.7,17.1]; y=[17.5,18.7,18.6,18.5,18.2,17.5,18.0,18.2,16.9,17.8]; plot(x,y,'b*') hold on 157 A=[x',ones(size(x'))]; B=y'; x1=A\B; a=x1(1);b=x1(2);y=a*x+b; plot(x,y,'r') str=['E?°L y=',num2str(a),'*x+',num2str(b)]; text(15.6,18.5,str,'FontSize',16) ?a
¤E?wL bm?1 = 0.73979 +5.2286yx 4
 9? ￡￥?
 P¨V???? ? ￡0? ￥? K
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 ?  P¨Q ?D??9v- W￥[/ 16 F ? b??t ? ￥??m
s P¨Q ? D ??9v -W￥1"
i?E?wLb x y x 2 3 4 5 6 7 8 9 y 6.42 8.20 9.58 9.50 9.70 10.00 9.93 9.99 x 10 11 12 13 14 15 16 17 y 10.50 10.59 10.60 10.63 10.60 10.90 10.76 10.80
4 U
 L ! b
 cbxaxy ++= 2 3
??} ?1 hold off x=2:17; y=[6.42,8.20,9.58,9.50,9.70,10.00,9.93,9.99,10.50,10.59,10.60,10.63,10.6 0,10.90,10.76,10.80]; plot(x,y,'b*') A=[x.^2',x',ones(size(x))']; a=A\y' hold on y=a(1)*x.^2+a(2)*x+a(3) plot(x,y,'r') string=['E?°L y=', 158 num2str(a(1)),'*x^2+',num2str(a(2)),'*x+',num2str(a(3))]; text(5,8,string) ?a
¤E?wL
 m?1 2 0.025746 0.68829 6.2507yxx=? + + 5
 ù??DQ? ? H
 ¤ ?/  ? b s L 7 Sa￥ HW t DQ?t￥  -W￥1"
i?E?wLb m t 3 6 9 12 15 18 21 24 m 57.6 41.5 31.2 22.9 15.4 12.1 8.9 6.4
4 U
 D ￥1"1 b
 m t bt am e= 3
??} ?1 hold off x=3:3:24; y=[57.6,41.5,31.2,22.9,15.4,12.1,8.9,6.4]; plot(x,y,'b*') hold on y1=log(y); A=[x',ones(size(x))']; a=A\y1'; y=exp(a(2))*exp(a(1)*x)
plot(x,y,'r') string=['E?°L m=',num2str(exp(a(2))),'*e^{',num2str(a(1)),'*t}']; text(8,28,string ,'FontSize',16) ?a
¤E?wL
 m?1 -0.10443t m=78.448e 159 160