复旦大学：数学分析（习题全角、教学教案）：ex5-3



5



5BZMPS T??′[ T
1
? Lagrange?′? ?? xx x x )(1 )1ln( θ+ =+
 1)(0 << xθ
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 2/1)(lim 0 = → x x θ b ￡ ? )1ln( )1ln( )( xx xx x + +? =θ
|K'¤? 22 00 0 0 ln(1 ) ln(1 ) lim ( ) lim lim lim ln(1 ) ln(1 ) xx x x xxx xx x x x xx θ →→ → → ? +?+ =?=? + + 00 0 1 1 11 1 lim lim (lim ) 1 1 22 1 xx x x x →→ → ? + =?= ? + + 2 =b 2
 ! 2() 11 ( ) () '() "() ( ) 2! ! nn f xh fx fxh f xh f x hh n θ+= + + ++ +"
 )10( <<θ
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0)( )1( ≠ + xf n 1 1 lim 0 + = → n h θ b ￡ nn hhxf n hxfhxfxfhxf )( ! 1 )(" !2 1 )(')()( )(2 θ+++++=+ " )()( )!1( 1 )( ! 1 )(" !2 1 )(')( 11)1()(2 +++ + + +++++= nnnnn hhxf n hxf n hxfhxfxf D"
 ? ^ )1()( 1 1)()( )1( )()( D+ + = ?+ ? + xf nh xfhxf n nn θ θ θ b 7
¤? 0h → (1) (1) 0 1 lim ( ) ( ) 1 nn h f xf n θ ++ → ?= x +
 ?
 Hh?0)( )1( ≠ + xf n (1) () n f x +
'¤? 1 1 lim 0 + = → n h θ b 3
 ! fx x()= 3
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p ￥=Q?′ [ T #?[￥Vr T
i9 ?
x =1 1728 2 744aa.. fx() px 2 () p 2 2() 2 12599210 3 = . "
 b 116 3
? Lagrange?′ T (1) 1, (1.728) 1.2, (2.744) 1.4ff f== = 2 () () ( 1.728)( 2.744) ( 1)( 2.744) ( 1)( 1.728) 11.2 1.4 (1 1.728)(1 2.744) (1.728 1)(1.728 2.744) (2.744 1)(2.744 1.728) 0.7876( 1.728)( 2.744) 1.6224( 1)( 2.744) 0.7901( 1)( 1.728) fx p x x x xx xx xx xx xx ≈ ? ? ?? ?? =? + ? + ? ?? ?? ?? ≈?????+ =? 2 0.04465 +0.3965 +0.6481x x b 8 3 10 () 27 f xx ? ′′′ = , ?[ 2 8 3 5 ( ) ( 1)( 1.728)( 2.744) 81 rx x x x ξ =???b 2 (2) 1.2626p ≈ b 4
 !
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 p ￥=Q?′[ T fx x ()= 2 x =?101aa fx() px 2 () #?[￥Vr T
i9 ? ? ? ? ? ? ? 3 1 2 p bhD 5￥9 ?2TM1?is á 3μs￥eyb 3
? Lagrange?′ T (1) 0.5, (0) 1, (1) 2fff?= = = 2 2 () () ( 0)( 1) ( 1)( 1) ( 1)( 0) 0.5 1 2 ( 1 0)( 1 1) (0 1)(0 1) (1 1)(1 0) 0.25 ( 1) ( 1)( 1) ( 1) 0.25 +0.75 +1 fx p x xx xx xx xx x x x x x x ≈ ?? +? +? =? +? +? ?? ?? + ? + ? =???+++ = b 3 () ln22 x fx′′′ =?, ?[ 3 ln 2 () 2( 1)( 1) 6 n rx x xx ξ = +?b 2 1 ( ) 1.2778 3 p ≈ b D 5M1
'5μ?v￥ey ^ 2 ? ?|￥?? -W
7 5 2  ?|￥?? -W
 y 7μ?lb x =?101aa x =1 1728 2 744aa.. 5
 ! ???? ?)￥f ?′?/
 fx() x 1.4 1.7 2.3 3.1 117 fx() 65 58 44 36 kp ￥í ?′b f (.)28 3 ? Lagrange?′ T 3 () () ( 1.7)( 2.3)( 3.1) ( 1.4)( 2.3)( 3.1) 65 58 (1.4 1.7)(1.4 2.3)(1.4 3.1) (1.7 1.4)(1.7 2.3)(1.7 3.1) ( 1.4)( 1.7)( 3.1) ( 1.4)( 1.7)( 2.3) 44 3.6 (2.3 1.4)(2.3 1.7)(2.3 3.1) (3.1 1.4 fx px xxx xxx xxx xxx ≈ ??? ??? =? +? ??? ??? ??? ??? +? + ? ??? ? , )(3.1 1.7)(3.1 2.3)?? 3 (2.8) (2.8) 36.647fp≈≈b ) 6
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?? P¤ h abcaa af x h bf x cf x h()()(++ + ?D ′′fx()í ?￥¨Kú
$ 3 22 22 ()()() 11 [ () '() '() ] () [ () '() '() ] ( ) 1 ( ) ( ) ( ) '( ) ( ) ''( ) ( ) 2 af x h bf x cf x h a f x f xh f xh bf x c f x f xh f xh oh a b cfx a cf xh a cf xh oh ++ + ? =++++?++ =++ +? + + +
 2 ¤?Z?F 0 0 2 abc ac ac + +=? ? ?= ? ? += ?
3-¤? 1, 2ac b= ==?b n 7
|?′Hq|1 ?2? ￥f ?′?B¨? ?′
' n +1 px() ?@
 px fx px fx ni i ni i () () () () = ′ = ′ ? ? ? 0,1, 2, ,in= " ￥?′[ T?1 Hermite ?′[ T
 ±sZ? ?′p3?ù? 5×? μ×1T¨b ? V[|1 (0) (1) 0 () ( ) () ( ) () n nkkk k p xfxqxfxq = ′ x? ?=+ ? ? ∑ , ? ú
 ^ ?@Hq (0) (1) 0 {(),()} n kk qxqx =k 118 (0) (0) () , [ ]'() 0, ki ik k i qx q xδ==, 0,1, 2, ,ik n= " ? (1) (1) () 0, [ ]'() , ki k i ik qx q x δ== , 0,1, 2, ,ik n= " ￥f ?b k_v Lagrange?′[ T￥f ?/ b (0) (1) 0 {(),()} n kk qxqx =k 3 A?? H
ik≠ (0) (0) () [ ]() 0, ki k i qx q x ′ = = (0) (0) ()1,[ ]'()0 kk k k qx q x= =
 ! (0) 2 0 () ( ) [1 ( )] n i kk i ki ik xx qx cxx xx = ≠ ?? ? ?? =? ? ?? ?? ∏ ?
? (0) 0 2 []'() n kk i ki ik qx c xx = ≠ 0= ?= ? ∑ 3
¤ ? c (0) 2 00 2 ( ) ( ) [1 ( )( )], 0,1, 2, , n n i kk ii ki ki ikik xx qx xx k n xx xx == ≠≠ ?? ? ?? =??= ?? ?? ?? ∑∏ "
] ? V¤? (1) 2 0 ( ) ( ) ( ) 0,1, 2, , n i kk i ki ik xx qx xx k n xx = ≠ ?? ? ?? =?= ? ?? ?? ∏ " b 119