数学奥秘本质和思维

《数学奥秘本质和思维》由会员上传分享，免费在线阅读，更多相关内容在教育资源-天天文库。

ᜮḄ1ᜩ“Ḅ”஺ᫀX2ᦪᦟ!ᨬ#⌕Ḅ%&஺ᫀV3'ᳮ)*+,-.Ḅ/Aஹ4Bஹ3Cஹ11Dஹ10ᫀC412345678)9:;<=>?@/Aஹ;ᳮ)Bஹ!;ᳮ)CஹᜲBᳮ)Dஹ'ᳮ)ᫀD

15CDEFG/Aஹ1854EBஹ1864ECஹ1846EDஹ1856EᫀCᦪH.1IJCKLMNOPQR-SᩭUVWXḄYᳮ9⊤[\

2Bஹ11,17Cஹ11,19Dஹ7,9ᫀA4ᙠs⍝ᙢᳫwDoxyzzx{ᙢᳫy᝞}~Dox1m,Dὁ34PQ஺ᫀx5ᑏF-Sᩖ)/AஹᩰBஹᐝCஹἔ=DஹᫀBᦪR1Ꮤᦪ9᦮ᦪCo/AஹᏔᦪBஹ᦮ᦪCஹD᪵

3DஹᫀC2a56F℉Ḅ¡᫑£⚪¥ᫀX3¡᫑£⚪56Ḅ¦§y¨FCD©ᦪᑖ«/Aஹ¬ᑖ᪆Bஹᦪ)Cஹ¯)°±ᡫDஹ³´µᦪᫀC4ᦪḄ³´\<ᦪRḄᨬ#⌕Ḅ¶Ḅ஺ᫀV54mCo·¸34D#¹Ḅᑏº/

4Aஹ»BஹᵪCஹᵬDஹᩈᫀA¿ᙊḄ☢ÂÃÄ14m12ᡂ}ÆÇÈÉ✌ᐜÌᑮḄ/Aஹᙊᕜ᳛ḄÐBஹᙊḄ☢°ᙊḄÑÒḄÓÔᡂÕCஹÖᱥØÙÚḄ☢ÂDஹÛÜᫀC2¿!Jݵᒘᙊß!34àº12ᦪHáḄâã/Aஹ᩽▲BஹåᑖCஹæᔠ)Dஹ±ᡫᫀA

53ÛÜḄHáèéêBëa஺ᫀV4m☢CoíᱥᵨÛÜïðFᙊḄ☢°ᙊḄÑÒḄÓÔᡂÕ/AஹᑢòBஹêBëaCஹê-óÌDஹÆÇÈÉᫀB5êBëaôᐰ56FᙊḄ☢ÂḄö஺ᫀX÷ØḄᑗØù᳛1ᙊḄ☢Ây÷ØᑗØḄù᳛yúᙳᒴýþḄÿ
⚪Ḅ᩽▲஺ᫀX

62ᑗḄ᳛ᙳᒴ!Ḅ"#$%ᑖ'⚪஺ᫀV32(ᱥy=xᙠx=iᜐḄ᳛,-,.
Aஹ1Bஹ2Cஹ3Dஹ01ᫀB%2ᑖḄ3ᐹ5617ᑡᐹᨵ:ᜓឋḄᦪ>,.Aஹ?ᦪ>Bஹᨵᳮᦪ>Cஹ᦮ᦪ>DஹBᳮᦪ>ᫀA2

7%2ᑖḄCD56,᩽▲஺ᫀV37ᑡ⊤Fᨵᳮᦪ>0:ᜓḄGH,.1hm—AஹI05»limnBஹ-8lim--l
Cஹ…«lim1+PDஹXT9%ᫀD4VᡲXYZḄ[\ᳮ],?ᦪ^_᦮`%2ᑖᳮ]abḄCẠ.Aஹ>ᔠ]BஹeHᳮ]Cஹf]Dஹgᡫᳮ]ᫀA5Bᳮᦪi᩽▲j,:ᜓḄ஺

8ᫀX%2ᑖḄᔊl12ᑖ'Ḅmn▤pḄq⊤rᱥ0ᒹtuAஹv-wxyBஹzC{|Cஹᓱ~ᑡDஹᑢᫀC2%2ᑖḄᓄ᩽ᜧḄ஺ᫀX3ᑖ᪆jᓄ!ḄYὅ,uAஹXyᱯyBஹVᡲXCஹDஹᔜ•ᑭᫀA

94%2ᑖḄYZ▤p$uAஹ14Bஹ15Cஹ16Dஹ17ᫀD5v¡,¢q%2ᑖ'ḄᝈCὅ஺ᫀX¤ᛞ¦§1¨ᯠᦪḄDª#ឋ,AஹᦪឋBஹ«¬ឋCஹ0ᦪឋDஹBឋᫀB

102®¯
°±ᨬ³ᵨḄᦪb,Aஹᓝ¶ᑴBஹ¸¶ᑴCஹ¹ᓝ¶ᑴDஹ¸ᓝ¶ᑴᫀA3ºq»³ᵨ[\¼½ᩭ¿ÀᜧᡈÀÂḄᦪ.Aஹᓝ¶ᑴBஹ¸¶ᑴCஹ¹ᓝ¶ᑴDஹÃ'¿ᦪ½ᫀDÄXᱯÅ✂1ÄXᱯÅ✂ḄᦑÈɺBÊᨵ▲ḄËÌ஺ᫀV2

117ᑡ>ᔠÊÍÎ
0,1iÏḄ,.Aஹ᜻ᦪBஹᏔᦪ>Cஹᨵᳮᦪ>Dஹ?ᦪ>ᫀD3BḄ°ÑÒ`>ᔠḄÓH>^>ᔠDÔiÏ஺ᫀV47ᑡ>ᔠʨᯠᦪ>0iÏḄ,.Aஹ᜻ᦪ>BஹᏔᦪCஹᨵᳮᦪ>Dஹ?ᦪ>ᫀD5ÄXᱯÅ✂ḄᦑÈÖ×ᡃÙ[\.Aஹ¨ᯠᦪÊ᜻ᦪÒ᪵-

12Bஹ¨ᯠᦪÛ᜻ᦪ-CஹᨵᳮᦪÛ¨ᯠᦪ-DஹᨵᳮᦪÛ᜻ᦪ-ᫀAᨵᳮᦪḄ“Ý◣”17ᑡᐵ$ᨵᳮᦪ
Bᳮᦪ
?ᦪḄ¦ÎḄᐵbá½Ḅ,.Aஹᨵᳮᦪ
BᳮᦪÊ?ᦪiÏBஹᨵᳮᦪÊ?ᦪiÏ
BᳮᦪÊ?ᦪ0iÏCஹBᳮᦪÊ?ᦪiÏ
ᨵᳮᦪÊ?ᦪ0iÏDஹᨵᳮᦪ
BᳮᦪÊ?ᦪ0iÏᫀC2âZ?ᦪbãÒCẠḄ,äåᦪ'æ.AஹçèBஹéêCஹᡚ|ìDஹíîïᫀC

133VᡲXḄ?ᦪḄ1ðñò?ᦪä¼☢Ḅឋª.AஹôõឋBஹ:ᜓឋCஹB°ឋDஹ01ᫀB4?ᦪᑖqᦪᦪö÷ᦪ஺ᫀV5øÒùᦪ'ᓻû,üýþyÿ
ᳮ஺ᫀXᔠḄᦪ1A☢ᨵᳮᓽᙶ᪗ᨵᳮᦪḄ!"#ᨵᳮᦪ!$%ḄᙊḄᐰ()*+,ᔠ-Aஹ/ᦪBஹᨵ▲

14Cஹ2/ᦪDஹ2ᫀA2/ᦪḄ3456/ᦪ஺ᫀX3/ᦪ7ᨵ▲Ḅ89ᯠ/ᦪ஺ᫀV4;ᑡ=7ᔠ2ᐹᨵ?@A-AஹBᦪᐰ(Bஹᳮᦪᐰ(CஹCDE?@Fᦪᐰ(Dஹᙶ᪗x,yᑖHᙳ!᦮ᦪḄᫀD5;ᑡᐵLᔠḄMḄNOḄ\

15Aஹ2QᙠMᨬᜧḄᔠBஹᐰ(BᦪḄM!UCஹBᦪḄMVᨵᳮᦪḄMWXDஹY7ᔠḄMZXL[Ḅ\ḄMᫀA]^ᱏᑮᵯbY᩽▲1ᦪᑡ᩽▲ZQᙠḄ஺ᫀx2;ᑡᦪᑡdᦣḄ\Aஹᓽ=g1”1Bஹஹ3«+1ax=Cஹ2«1a»=—Dஹ«ᫀD3;ᑡᦪᑡᦈiḄḄj

161a*=஻YAஹn=[(-1/+1]—/1ஹ.n7T=(^4--)sin—Cஹ»22n-lDஹ*nᫀDFᦪ᩽▲opᙠqrHrᓄtu;FᦪrᓄvM஺ᫀV;ᑡᦪᑡ2wᦪᑡḄ(\1ax=—Aஹn1CL=~r-nBஹ2”2n-lCஹ*n3»+la=-------xDஹ2nᫀD

17y⚣{|Y᩽▲Ḅ}~ᓄ1ᦈiḄᦪᑡᨵᦪᑡ஺ᫀV23Ḅ)ZQᙠ᦮ᦪ))ឤᨵY°D2£ᦪᑡ%ᦈiLaḄ+ᩩ-Aஹᐙᑖᩩ6⌕ᩩBஹ6⌕ᩩᐙᑖᩩCஹᐙᑖ6⌕ᩩDஹᐙᑖᩩ6⌕ᩩᫀCᫀ23ᦋrᡈᦪᑡ஺Ḅᨵ▲⚗)b2bᦪᑡ"Ḅᦈiឋ-AஹbBஹ2bCஹyt¢£DஹO¤¥ᫀB4

18ᦈiḄᦪᑡḄ᩽▲¦YḄ஺ᫀV5lima”=a;ᑡᐵLf”Ḅ©2Ḅ-Aஹ3ḄeeQD)ZQᙠ᦮ᦪ")ªAN)ឤᨵ«Y஺DBஹ”Ḅ3_£®஺3£
)¯ᨵᨵ▲°⚗ᓽ3^£Cஹ3Ḅᦪ£,ZQᙠqᯠᦪ")³´/)1%Yµ¶£Dஹ3Ḅᦪ£)ZQᙠ᦮ᦪR,jᓾY&|¶£ᫀCᨵ▲»¼YFᦪḄ᩽▲1
3
ᙠḄ½®¾◀᜜ᑁᙳᨵ
x
2°ᡈ/%,Ãᡃ/ÅY'ᑣᨴ20ᡈᨴ40jᫀ4ᫀ221limcos—᩽▲I஺X=jAஹ1Bஹ0Cஹ2Dஹ2Qᙠ

19ᫀD(ᫀ2)3lim---=?᩽▲sinx()Aஹ1Bஹ0Cஹ2Dஹ2QᙠᫀA4lim=AËlx.Qᙠ)ᑣ¦Y஺()ᫀ!V5ḄNOËᙠÌ᝞Î:஺4
YÐ1ÑÒÓYÔ#®"ᨵg(x)

201FᦪḄ?@ឋopḄFᦪḄ᦮(ឋÜ஺ᫀX2;ᑡᐵLFᦪ?@2ḄjAஹFᦪ/X
ᙠÝ?@=7জ
ᙠ£ᨵ©)≏Qᙠ)Ã≏¶/"=/ßBஹFᦪ
জᙠá?@Qã"ß+å-"ßç=0CஹFᦪ
Åᙠ/?@=≏
ব-≏/ব-/éDஹËêY⇪"-ᑣfxYᙠá?@ᫀD3j*,ஹ,/x=~~,x-1©Fᦪ/3=ìXH1&,2)ᑣx=l,FᦪḄ-AஹíîEïBஹ/ÔEïCஹEïDஹð₩EïᫀB4

21Fᦪ/XᙠÙ?@)ᑣ73ᙠ2ᨵ©)≏Qᙠ)≏«஺ᫀVᫀ25©ᙠDEö0,1çDEḄøùFᦪᙠᳮᔲ?@-Aஹ?@Bஹ2?@CஹýþLᐹ(t¢Dஹ
ᫀA1ᦪḄᔠᦪᦪ஺ᫀ42ᑡᙠ!"#ᓃ%&'Ḅᦪ%()*+ᙠ&%-'.ᑮ01Ḅ23Aஹ4a=-1/ব=-2Bஹ-ভ=1/=2Cஹ-ভ=TJS=1஻&=±஻5=0Dஹ2ᫀC

223ᐵG!"#'ᦪ%☢I┯KḄ23AஹᙠL"#'MN.Oᨬᜧ1BஹᙠL"#'MN.OᨬR1CஹᙠL"#'ᨵTDஹᙠL"#'MN.ᑮ01ᫀD4UVX3-sinx-1=஺ᙠ
1,2'2ᔲᨵX᪷3AஹZᨵBஹ[\ᨵ1]Cஹ[\ᨵ3]Dஹ^)ᫀB5ᨵ▲]ᦪḄ`a2ᦪ஺ᫀVᨵᵨ

231ᑡcdḄ2(bAஹfᦪf(x)ᙠ"#
a,b'^%ᑣLᦪᙠ
a,b'TBஹfᦪf(x)ᙠ"#
a,b'ᨵ)h,ᙠ(a,b)ᑁᑣf(x)ᙠ
a,b'ᨵTCஹfᦪf(x)ᙠ"#
a,b'%f(a)f(b)40,ᑣklᙠ(mᡃ(a,b),oOp)=0Dஹfᦪf(x)ᙠ"#
a,b',f(a)=f(b)=0,ᑖrᙠx=aḄs]tuv`x=bḄs]wuvᓫyz%ᑣklᙠ(m᮱(a,b),oOp)=0ᫀD2UVd-3x-1=0ᙠ(-2,3)ᨵX᪷%ᑡIḄ23()AஹZᨵBஹ[\1]Cஹ[\3]Dஹ^)ᫀB3/(X),g(x)ᙳᙠ~ᜐ^%/(x)+g(x)ᙠ~ᜐM*஺()ᫀV4

24Ay=fx+Ax-fx%Ax—0kᨵAy^OoᫀX5V2-xy---------------ᦪ.InCV+D-lᙠ"#'3AஹBஹ(°°,-3Cஹ-2,7DஹQZᫀBᑖ1R2(]RḄᦪ஺ᫀX21%V=R஺AஹX—'஺+Bஹx—CTCஹ*—8

25DஹXTlᫀC3x=2*+3ஹ-2,ᑣ—0Aஹ/X
2x▤ḄR஺Bஹ/“2x▤ḄR஺Cஹ72ḄRDஹ/“2x▤^ḄRᫀD4ᦪ02R஺ᫀ451y=-fuvᙳXḄMᦪg+dḄᑖ஺
ww'4-vV,-------------Ad+Bஹ♦+/ᨴ

26—-JdxCஹ1+vDyit!+W'
dxᫀA¡¢Ḅᑗ¢¤᳛12/3x-/ব11m/Q᜻ᦪ%/°lᙠ-2,ᑣ*t஺x=Aஹ10Bஹ5Cஹ-10Dஹ-5ᫀC2¡¢/x=Pᙠm1DᜐḄᑗ¢X©Ḅªmভ°,ᑣ«"ম"AஹeBஹ1Cஹ2DஹLᫀD

273¯ᦪ2ᦪ°±ᓄ³᠒VµḄ⊤·¸஺ᫀV4¯ᦪra
ᙠ¹º'⊤»¼=/%ᙠm~ᜐᑗ¢Ḅ¤᳛஺ᫀx5-..Em-------------------------¾¿À=5,Áij/Xo+2x_/Ã=,Aஹ1Bஹ0.1Cஹ0Dஹ0.2ᫀC¯ᦪḄÄŵ1ᦪ/RᙠÆmᜐM¯Ḅᐙᑖk⌕ᩩÊᙠLmᜐw%t¯ᦪlᙠË஺ᫀx2

28ᦪÌ=Í"x>oḄ¯ᦪ஺ᑁcosxlnx+ᔳqAஹx/0•1sinxஹxsinxInx+-------
BஹxxcosxlnxH--------
Cஹxco*/.sinx.xcosxlnx+-------
Dஹx,ᫀAÎᫀ^)3ÏÐᦪḄ¯ᦪÑ20஺ᫀV4(]ᙊÓÔ%ÕÖᙊÓ×Ø2ÓḄÙÚ%°Û%ᙊÓ×ØN2ᔖÝ/ÞḄßµàR%ÓN4ᔖÝ/ÞḄßµz%á[ÓᙊÓ×ØḄÙÚ%ᙠÎâ#ᙊÓḄÔa3AஹᓫyzãBஹᓫyà\CஹᐜzÛàDஹᐜàÛzᫀC

2953x-2å=23x+2,/3
=æNçᑣ"XAஹ3TTBஹ43TTCஹTDஹ2éᫀCêëì1)ᳮ1UVd+x-1=0᪷Ḅîï%☢IḄ2Aஹ[\(]᪷Bஹðᨵ(]᪷CஹZᨵ᪷Dஹ^)ᫀB2^ñᦪ/x
=x_DQ_2,_33-4Ḅ¯ᦪ%IUV/%=0ᨵÊX᪷஺Aஹ1Bஹ2

30Cஹ3Dஹ4ᫀC3ᑡᦪᙠò)"#'óôêë)ᳮᩩÊḄ2./(x)=l.[-2,0]AஹX/(X)=(X-4)2,[-2,4]Bஹ1/(x)=sinx~56Cஹ/(x)=|4[-uiDஹᫀC4êëì1)ᳮᢣñM¯ᦪᙠ"#ᑁ.O᩽1mᜐᑗ¢¤᳛0஺ᫀ75ᦪ/"=ÿ£᱄-1.1ᳮ஺ᫀXᨽᳮ

311ᦪᙠ
a,xeab,ᙠ
a!"ᵨᨽᳮᨵ,&'(/ব(-ভ,,-./—(ᓰ(23Ḅ஺5xḄᦪ஺&'ᫀx2678x>0,9:;xீe*-lᔩ@&'Aஹᡂ>Bஹ9ᡂ>CஹBCDEDஹFGHIᫀA3ᨽᳮ5ᳮḄJKᳮ5ᨽᳮᙠᦪLMN:OḄᱯQ஺&'ᫀV4S>a>0Tᑡ9:;Ḅ5&Vb-a,bb-a------

32-------In-Cஹ3ab-a.b--------In-D.aaᫀA5arctanx+arccotx=AஹW7TBஹ57TCஹ437rDஹTᫀ!BX᩽▲Ḅᑭᘤ1limlim]ᵫ_`aGᑣcd᩽▲g'x
9eᙠᑣ᩽▲gx
f9eᙠ஺
ᫀx2

33(l+x)*-elim----------------X᩽▲iox()oAஹeeBஹ2eC.'3eDஹ2ᫀA3A1ஹlim--------)X᩽▲xsinx=(VAஹ0Bஹ1Cஹ2Dஹ3ᫀA40cogh(ᑗjk;lmᵨ_`aGᑣᩭX᩽▲஺&'ᫀx

345ex-\lirn--------X᩽▲x=(ஹAஹ0Bஹ1CஹeDஹ2ᫀBᦪḄᓫrឋ1᝞uᦪf(x)ᙠI"ᓫrvwᐸᦪf(x)fᓫr஺ᫀX2ᦪf(x)=sinx-xᙠyzḄ{ᦪ|(Aஹ2Bஹ1Cஹ4Dஹ3ᫀB3

35ᦪfx
=x-arctanxḄᓫrឋ|&VAஹᙠ&-8,~ᑁᓫr⌴Bஹᙠ&-8'ᑁᓫr⌴Cஹᙠ&-8,8'ᑁᐜDஹ9ᫀA4᝞uᦪ/X
ᙠḄᑁlᨵTx
'°ᑣ3
ᙠᑁᓫr஺&'ᫀVᫀ9'5dᙠ°/"r8'°ᑣ/'1
,(&1
-/஺'ᡈ/&஺'-&1
ḄᜧS|&VAஹ/'1
'/,0
'/1
-/0
Bஹ/''--&஺'஺CஹDஹᫀBᦪḄ᩽1᝞uᦪ/3
ᙠI"ᨵḄᦪᑣᙠIᑁᨵ2᪵Ḅ,-/&'5᩽ḄO&E,/&''5z஺&'

36ᫀX2I/x=asinx-F-sin3xx=-a|Oᦪ3ᙠ3ᜐ-᩽@&'Aஹ஺=1Bஹ஺=2Cஹ஺=3Dஹa=4ᫀB3Xᦪy=x+JT=Ḅ᩽஺&'?=2Aஹ44|᩽ᜧBஹ"N=a|᩽Scஹ|᩽ᜧDஹ|᩽SᫀA4

37Xᦪ=—61_1&+7Ḅ᩽஺&'Aஹ¡&-1
=17|᩽ᜧ¡ঞ=Y7|᩽SBஹ£-1
=17|᩽Sy3
=47|᩽ᜧCஹV-D=T7|᩽ᜧ¤3
=47|᩽SDஹ%1
=-17|᩽S¡&3
=47|᩽ᜧᫀA5ᦪfx
ᙠFa,bH"Ḅᨬᜧ&S'z(5᩽ᜧ&S'z஺&'ᫀxᨬ¦ᓄ¨ᨬ©⚪1Xᦪ«᝞(S+4”mߟ1,3HḄᨬᜧᨬS஺&'Aஹᨬᜧ®=11,ᨬS&2
=-14Bஹᨬᜧ¡&°'=2ᨬS&-1
=-5cஹᨬᜧy3
=ii,ᨬSy-D=-5DஹᨬᜧM°
=2,ᨬS2
=-14ᫀA2¯°±|rḄᳫḄ᜜ᑗᙊ┵©ᙊ┵Ḅ¶|·¸O¹º,ᙊ┵Ḅ»¼ᨬS@

38AஹrBஹ2rCஹ3rDஹ4rᫀD3xy=---00,-1
ᦪx+1ḄᨬCD|VAஹᨬᜧ|VG2=11BஹᨬS|¡&-5=-5Cஹ½ᨵᨬDஹm"¾Gl9ᫀC4ᨬz¿5᩽z஺&'ᫀx5Àzl5᩽z஺&'ᫀX

39ᦪḄÁÂឋ1dᦪfx
Ḅᦪf'x
ᙠIᑁᓫr&¸
,ᑣfx
ᙠIᑁ5Á&Âb
ᫀ!V2ᦪy=lnl+
ḄÂÁ|&VAஹÁ&(LD,Â&(00,(DÄBஹÁ&((1
ÄL+Q
Â&(LDcஹÁ&(LD,Â&(0°ᔆ1
஺&1,+8
DஹÁ&(ᔩ(1
&1,+°°
,ÂQLDᫀA3ᦪy=x3ḄÂÁឋ|&bAஹᙠ&-8+°°'ÁBஹᙠ-S,+g
ÂCஹᙠ&(8°
Áᙠ&°+ÆÂ,&°°'zDஹᙠ&(8°
Âᙠ°+8
Á,°°
zᫀC4

40dᦪf
xᙠIᑁ5Á&Â'Ḅvwf'
xᙠIᑁᓫr&¸V&'ᫀV54ᦪÇ="ḄÂÁឋ|&VAஹᙠ&-8+°°'ÁBஹᙠ
-S,+gÂCஹᙠ&(8°"Áᙠ&°+ÉÂDஹFGᫀAÁÂឋḄᝯᵨ1aË8578L{ᦪ஺<4<1,vwᐵÍa.“"஺-4&+Ï'ḄᜧSᐵÐ5&\Aஹ<1-AbBa1%'=
1—4a+Ab>
1-7i>+AbcᫀADஹ92

411&«+y»\TᑡᐵÍ5,2&'Ḅ¾GḄ5&V1«+"'&Ò'XAஹÓÓBஹ]“Ô=*Ëc9+S&Ö*Dஹ9ᫀA3dᦪfxᙠI"5Á&Â'Ḅᑣ-fxᙠIᑁ5Â&ÁbᫀV4ᦪy=lnxḄÁឋ|&VAஹÁᦪBஹÂᦪCஹBCDEDஹᨚOFGHIᫀB5

42᝞uØÙᙠzᜐᨵᑗÙvwØÙᙠz▬ÛḄÜÝᑖßàÍ2ᩩᑗÙḄLâ஺&'ᫀVᦪḄã᪵1ẆåᦪOæçèéêᦪëìºíîïᦪḄð⌕ᱯò5ᦪóẆåḄôᵨèGḄ஺&'ᫀ42ᦪḄᐵõöᱯòᒹøᦪḄᕜúឋ᜻ᏔឋឋᓫrឋÂÁឋ:஺&'ᫀV3஺-1
ý|J
.Aஹx=35/x
Ḅ᩽Sz,þ&°°'95ØÙ=/ÿḄBஹx=3/Q
Ḅ᩽=/x
ḄCஹx=3Ḅ᩽°°=/ḄDஹx=3Ḅ᩽°°y=73
ḄᫀCᫀ4ᦪfx
=|x1-x
|,ᑣ#

43Aஹx=0f(x)Ḅ᩽(0,0)y=f(x)ḄBஹx=0f(x)Ḅ᩽(0,0)y=f(x)ḄCஹx=0f(x)Ḅ᩽,(0,0)y=f(x)ḄDஹx=0f(x)Ḅ᩽(0,0)y=f(x)Ḅᫀ!C5ᦪ%&1,ᐸ()*(\DஹᫀC-ᨵ▲01231

44_7rᦪᙠ"%1ᜐ8ᨵ9:ᨽ<=⚗Ḅ?▤AB23#A/3/+/ᓱ*-G+/H-ᑴ+r&x*-B/ব⊍+L'NO+/”Q-Oஹ”R“OCஹTUVW*-VX*Y+»"Z*-᜛Z+\]YDஹ1ব-஻O+U_-O+3O%O'+#"a-§ᫀC2AB239:ᨽ

45ᦪ{|=/lnxᙠx=lᜐḄ▀▤89:ᨽ<=⚗ḄAB23*#AஹᖊSTU%eCஹ"2T+Wߟᔜk-Dஹif+/!eᫀAᫀ5ᦪᙠ%ᔛḄAB⚗3ᦪᙠ▬Ḅ⊤3ᦪḄ%▤AB⚗3ᨵḄ஺ᫀ!Vijk᪍231hᦪsidxḄijk᪍232/-X'+…+-1""1--?"+ox2m+1Aஹ2!4!2¡Bx+x2+x3+...+x?I+oxnf\93Q2M-14-+¢X+…+--£+0/*+1Cஹ34!2¤Q312V—1VX_/+…+_12——ᫎ+஺Dஹ2!4!2ṺᫀA2

46᝞¨ᙠஹ=0Ḅ©ªᑁᨵ”▤¬Ḅ®ᦪ¯°R⊤*±=²+<2/+஺2´+...+%/+஺´¶·⊤3¸%஺ᫀ4ᫀ3¹x70º/x
=/sinx-xl+x
»▤¼W
Aஹ1Bஹ2Cஹ3Dஹ4ᫀCᡈDᫀ4ijk᪍23AB23ᙠ¾=0ºḄᱯÀÁÂ஺
ᫀ!V5ᦪtanxᙠ%=0ᜐḄ?▤ijk᪍23*#x+—x3+ox3
Aஹ3x+—x3+ox4
Bஹ3x+x2+-x3+ox3
Cஹ3Dx2+ox3
ᫀA

47ÃḄÄᵨ1AB23ÆÇᙠÈÉᵨ⚗3ÊᦪḄ⊤3ËÌÍÎḄÏ⌕±ᐹ஺ᫀV2±..cosx-e2lim--------------hᦪ᩽▲1஺x40Aஹ11Bஹ121C."12Dஹ2ᫀC3%ÒÓᩭÄᵨ®ᦪẆÖᦪឋØÙÚÛ%▤®ᦪº°ὃ⇋ÞᵨYᳮᙠà⚪ÚÛ▤®ᦪºÄὃ⇋ABâ3஺ᫀV4

48⚗3Px=f-2--X+3ᙠ-8,-1],],00
ãᨵ»äåWAஹ1Bஹ0Cஹ2Dஹ3ᫀB57Tsin—4h18Ḅᑮ10஺
Aஹ0.173647Bஹ0.134764Cஹ0.274943Dஹ0.173674ᫀAh®çÎḄ⌮çÎ1éᙠêëᑁḄ¬ᦪ%ìᙠíᦪ஺ᫀV2

49[sin2—dxhîᑖJ2?()—(x-sinx)—(x-sinx)+Ccos%x6cos----FC2ᫀB1-y3osiunxxAn+—3dUxhîᑖJ/X?()--+3cosx+lnx24-CAஹx--4-3cosx+lnx2BஹxCஹ3cosx+lnx^+CDஹ3cosx+lnLᫀAhîᑖJñò

5031Cஹ43-%ᡝ+CDஹ4ᫀB5᝞¨%äᦪᙠêëᑁìᙠíᦪ¶·ᦪ%¬ᦪ஺ᫀXîᑖḄÍÎ1hôîᑖõᵨḄ?ö÷øù*ú%ᣚᐗùúᣚᐗùᑖÉîᑖù஺ᫀ!V2hîᑖJc°s2xdx?—sin2x+CAஹ2Bஹsin2x+C

51—sin2xCஹ2Dஹsin2x+CᫀAhîᑖJ4"1%3"'?()arcin-+-^7SAஹ2a2—arcsin-+--Ja2-x2+CBஹ2a2—V«2-x'Cஹ2-4^^+cDஹ2ᫀBᦪḄýḄîᑖ\þᔜäᦪîᑖḄý஺()ᫀVhîᑖJ'c°sxdx?()Aஹx3sinx+3%2cosx-6xsinx-6cosx+CBஹ3X2cosx-6xsinx-6cosx+C

52Cஹx3sinx+Sx2cosx+6xsinx-6cosx+CDஹx3sinx+3%2cosx+6xsinx+6cosx+CᫀAᦪÿ
ᑖ1ᑖ+5=/+ḄAஹ1-*
2=CQ-/
Bஹl-x
2=l-y2Cஹ"x
2=-y2+CDஹl-x—CQ-y?
ᫀA2ᑖḄᒹᑖḄᑗ஺ᫀX3ᑖV=ylnx,জ=2?
Aஹ]^"y=2ex^-^BCஹy=/mz)

53Dஹy=2e*2ᫀB4"#$Ḅ%&'()*+,-./ᳮ1ᨵ3/ᳮ456ᑖẆ8ᜩ#$Ḅ-:Ḅ;⍝=6>%&Ḅ஺ᫀV5ᑖ/V+6D'+4y=0Ḅ?᝞=AḄ1y=-Aஹx1Bஹx11y=-y=—cஹx,xDஹBCD┯FᫀCGHIJḄK᠗1GHIJᑭᵨ“PQR”T5ᳫ☢WஹᳫXWஹYᱥ[ஹ\ᙊ☢W஺ᫀN

542GHIJ'^᪵`abᦪdḄefghi⌼ᢈlmnᔠpqr⚪ḄAஹᵨt⊝Rp☢WBஹᵨvwRpghCஹᐜᵨt⊝R☢WyᵨvwRzBghDஹᐜᵨvwR☢W{yᵨt⊝RzBghᫀC3GHIJ|ᵨvwR}ᑮ?Ḅ☢W஺ᫀV4GHIJḄ'\Aஹᐗ287-212Bஹᐗ288-210Cஹᐗ280-212Dஹᐗ297-212ᫀA5✌ᐜT5Yᱥ[ᡠ?Ḅ☢W

55Aஹ+,BஹᐻCஹGHIJDஹ}ᫀCAḄ᩽▲1ᑖWᑖ5&}AஹᑖBஹWᑖCஹ5&Dஹ/ᫀB2Wᑖ¡¢£¤¥ᓝ§¨©஺ᫀV3&Wᑖ:«¬Ḅ✌iὅ'Aஹ+,

56BஹᐻCஹ®¯Dஹ}ᫀB4ᙠWᑖi±Ḅ¡²{+,ᐻD³´µ¶·¸v¹º»Ḅ¼஺()ᫀ45Wᑖ¿⌕'ᵫi±Ḅ()Aஹ+,ᐻBஹ}CஹÂᓱÄDஹ®¯ᫀAÅÆWᑖ1ᔆJ[C/(x)+C2`adxcj`x)dx+CaJbমdxÇÈÉ6ᦪ5Ê2,ËÌ஺0aḄᜧ¹()f©eম+C"(x)]dxCJeবf+CJbবgAஹ&Q.Q.

57f[C/ম+C?ᶭ(x)]dxcjx)dx+J[x)dxBஹ2V2a3஺f[C/(x)+QÏ(x)]dxCjÐ(x)dx+cjÑ(x)dxCஹ2=ddDஹ/ᫀC2ÒFa,bHCḄÓÔÕᦪÖᨵᨵ▲×ÒØÙḄᨵÚÕᦪ/ÛW஺()ᫀV3/ÜÅÆWᑖÝḄA-0,⊤ßÇÒFaàḄᑜᑖâᩭâäḄ*஺åḼç0,èᨵ¹ÒḄ×ᦪn-8஺éêë{n-ooì´ígA-0஺()ᫀ44»e»j/(x)dxJf(x)dx+J/î4“cḄmÇïð᝞ñ{ËÌa0CḄᜧ¹()»e»ff(x)dxj/O)dx+Jf(X)dxAஹa>0c»e»J/(x)dxJ/(x)dx+J/(x”xBஹa=0ebcbf/(x)dxJ/(x)dx+JAx”x1Cஹ2V2C

58Dஹ/ᫀB5Jf{x)dxjg(x)÷᝞òᙠᓃ{ôC,{øIùa0Ḅᜧ¹(,J/(x)dxJg(xöAஹ2=0.b»Jf(x)dxJg(x”xBஹaNdbbJ/(x)dx[g(x*xCஹ2

592Cஹ7TDஹ2ᫀD31Je*dxᑖ஺=?()AஹeBஹ1Cஹ"1Dஹ1-eᫀC(ᫀ)4/(x)=Je~^dt,஺ᑣ/3=?()AஹBஹI'+CCஹe*DஹᫀA5

60ᵫᐻ!"ᦪf(x)ᙠ
a,b%&'()*ᙠ+"ᦪ(ᑣfᙠ,-
a,b%஺()ᫀV6ṺRxjf(x)dx=j᮱(x”xᑖaaᫀV7"ᦪxᙠ,-
0,1%Ḅᑖ஺()Aஹ1Bஹ2Cஹ1/2Dஹ1/4ᫀC812-ᐻ4567ᑖ89:;<ᨵᦔḄ?@(A)ᙠᳮC%DᑖEᑖὶGHᩭ஺()ᫀVJKLḄ☢

611NOᦪPQR%Sὃ⇋VKLḄ☢஺WXᫀV2JYy=sinxEZ=cosx[\VY]=o^_=ᡠaᡂcLḄ☢dAஹ72Bஹ2eCஹ3fDஹ4eᫀB3ᵫgᱥYy=;-1^i=-/+3x+iᡠaᡂl☢cLḄ☢d125Aஹ~2424Bஹ1258Cஹ7575Dஹ8ᫀA

62422m+ᓃ=1oᙊYᡠaᡂcLḄ☢dAஹiBஹCஹ⚥bDஹᶭᡂsᫀC5;JKLḄ☢t▭%"ᦪḄᑖ஺WXᫀX_vwᑖ1xᐗᑖ᪆@{ᜐᳮ}᝞☢((O;ᐹᨵឋ⚪Ḅ⌕?@஺WXᫀ12;lḄ5RḄᙊ᫜(ᑁ᫜5pḄ(᫜Ḅ;<☢ᡠḄ]d!ᡃAஹ3

63Bஹ3RCஹ3!"Dஹ3ᫀA3;528m,R520kgḄᙳᒴᩩ¡ᢞ£;¤¥ᱥḄ⚔§(◤⌕©ªᜧḄ¬D®;ᩩᐰ§°%¤¥ᱥḄ⚔§d()Aஹ2700(J)Bஹ2744(J)Cஹ2800(J)Dஹ2844(J)ᫀB4xᐗᑖ᪆@Ḅ±⌕ᒹ³´

64ÍM.67ÎÆÏRÍḄÐ]d1Aஹawd+..ÒÓ1BஹaA?+-CஹA/V+/2ᫀAÔᭈᳫḄ1[;l☢×5RḄᳫ(׺5h,קᑖḄd-h1(3R-h)Aஹ37r-h(3R-h)Bஹ3Cஹ37T-h{R-h)Dஹ3ᫀA2ᵨ;ᐗ"ᦪḄᑖ[67ØÙḄ஺()

65ᫀ43ᵫ&'JY:=/3X\VYZ=°(_=4_=43Ý]ᡠaᡂḄJKLÞßàØÙ;J(WxXdxᕜåᑮḄØÙḄ50OᫀV422ᵫᑁ᤮YWëLYXZ2=;ÞXàØÙᡠᡂḄØÙḄdAஹ10532---Bஹ105Cஹ1051--7DDஹ105ᫀA5ï+2=1oᙊ1/ÞxàØÙᡠåØÙḄdAஹTq2Bஹ#b

66ᫀC(ᫀ)ñḄòó1ôõö÷øY.=060ு°)%ú0=0ᑮ8=2"_ûḄüd()+4/+_ln(2+J1+4??)]Aஹ2-111(2^+71+4^)Bஹ27T-J1+47I2+-ln(2TT+J1+4TH)Cஹ2a[7T+-]n(2.+Jl+4/)]Dஹ2ᫀA2b________fJ1+f(x)dx!JY5:=/(x>ýᑣü5ᫀx3222ëLY"+ÿ=7Ḅᐰ()

67AஹaBஹ3aCஹ6aDஹ9aᫀC4᝞X=X0
J=qᑣ!""+8"

'஺ᫀV5!p=a1+cosp
Ḅᕜ஺AஹaBஹ3aCஹ6aDஹ8aᫀD᜻ᝯḄ&'171()*ᑖ01+,=?()

68AஹtᫀB0ᫀ122,-*ᑖτߟ,dx44-x2=?Aஹ.7TBஹ5Cஹ2"7TDஹ4ᫀB30JxexaX*ᑖ9-?Aஹ1Bஹ"Cஹ2Dஹ-2

69ᫀB0ᫀ1245/X
ᙠᨵ89:I;<ᙠ=>ᴟ@A/3
ᙠ/;Ḅ,-*ᑖCDᢥ-FḄGHᜐᳮK᝞LMজO9:P;ḄQRSᦪ@&VWOᴟ@XYCDZ[\2cbbJ/x
dxJ/x
dxJf+x,dxV᝞,-*ᑖ஺C]Aᦈ_ᑣ,-*ᑖ0ᦈ_஺ᫀV5[dx=ln1+sinx
|r=-In2\1+sinxyaH5஺ᫀVᤅcḄᕸᖹ1Lᓃ,ᑗᡃḄᨵ8j9:Jx
Ok[a,lᑮ[a,ᑁḄQR᧜lᑣpq<ᙠr@se[a,ᑗtu/s=/஺ᫀV2Ꮇ᝞xᙠr>ᙊ!Ḅz{|}~|Ḅz{ᙢᣵᙠMᙢ;0Axᔲᙠᙢ;ᑮr@tu>@☢ḄᙢGOᙠᙢ;ᡠ⊤ḄAஹᨵ

70BஹᨵCஹ◤⌕ὃ⇋ᐹDஹ(ᫀA3᠒᠒ᤅcḄᕸᖹ5¡¢☟¤Aᕸᖹ¥Oᔲᨵr@ᙠᤅ¦§¨©]AஹᨵBஹᨵCஹ◤⌕ὃ⇋ᤅ¦GHDஹ(ᫀA4L4U&“▀«ᓫjᳫ1®1®OQR᧜lᑣpq<ᙠr@5e9,tu/¯0
=oᫀ'5Ꮇ᝞x±²³;ᓤ6@k³µ¶·r¸;¹º}º»»¼¼½ᑮ¥ᓤ12@¾ᑮ³⚔஺(▲Áᐝᙠ◅ÄᡠDxÅ2ÆÇrÈ஺ÉÊᜩ;ᓤ8@.̳2>ÍAΨᑮM³µ஺Oᔲ<ᙠÏrAÐtux᧤ᜩÒÓᜩᙠ]rÔÕ஺

71AஹᨵBஹᨵCஹ◤⌕ὃ⇋ᐹDஹ(ᫀA1c@2ᳮÒÖᵨ1ᨵ▲«ØឋÛ:¥Ḅᨵ8()ÜᔠÞᨵᦈ_ßᑡ஺ᫀV2Z[«ØឋÛ:¥Ḅᨵ8()ÜᔠÞᨵᦈ_ßᑡ஺ᫀX32áᙠ9:
0,1;ḄQRSᦪÛ:Oâ«ḄAஹ1«Bஹ2«Cஹ11«Dஹ()«ᫀD

724ᑡã>äMåæ᧜lḄçèAஹᤅcᕸᖹBஹéêᡂìCஹåæᦻîDஹᔠïð᯿ᫀD5Sᦪ/
=/+5ᐭ+4ᙠóᦪô;Ḅ1c@OõYAஹ-4Bஹ-2Cஹ-1Dஹ0ᫀBö÷øùúûᝄ1ᵯï“abeautifulmind”¥ᵱÿ
ḄᜧᦪḄ\AஹG.Debreu

73BஹJ.F.NashCஹL.V.KantorovichDஹAdamSmithᫀB21968ᐺ!"#$300%&'()*Ḅ+⚙-ᝄ஺0ᫀ43Debreuᙠ2%3ᙳ⊝ᳮ789:ᡠᵨᑮḄDebreu-Gale-Nikaido&ᳮ>Brouwer&ᳮᨵ@AᐵCD0AஹEFBஹGὅᒹJKὅCஹKὅᒹJGὅDஹLᨵᐵCᫀA4MNOPG.DebreuᵫR@ASTUVWX1983Ḅ()*ᝄD
AஹZ[X3ᙳ⊝ᳮ7Bஹᙠ\ᔠ^_`Ḅᙳ⊝ᳮ7a☢cdXeZឋST

74Cஹhᵨijkᳮ7lm-nX3ᙳ⊝ᳮ7DஹpqrFsḄtuᑖ᪆ᫀC5yz!{()*ᝄ|ᐳ⚙~X50஺0ᫀxᐗ1ᦪ22"+10/ḄᦈD0Aஹ.1Bஹ170cஹ70DஹT1]ᫀc2ᦪḄᦈᑖ!&,ᑣᦪ=¤¥/+Eb.x"Ḅᦈ0=min&&0ᫀx

753ᦪ22M+1
/Ḅ©ᦪD1+xAஹ0^?1BஹªXXcஹl+xDஹWᫀA4«ᦪ>ᐸ⌲⚗¯KḄᦪ°⌲⚗±ᑖKḄᦪᐹᨵ³´Ḅᦈµ¶·ᐹᨵ³´Ḅᦈ஺0ᫀ45ᦪᙠx=_iᜐᦈᑣ¹ᦪᙠx=2ᜐDAஹᩩ»ᦈBஹ¼pᦈCஹ-ᦣDஹi&ᫀB¹ᫀi&0

76Ꮭ¿Àᦪ1©ᦪ஻¤0ᙠ1"ÂÃÄÅÆAÇḄFourierᦪᵨÈÉÊ⊤ÌÍ஻0E'-g/eÎஹÏᐸ:FourierCᦪdḄ⊤ÌÊDAஹ%=£3Ñ-ÎÎBஹd="/¤0ᢓÎCஹ”ᡭÖWdzDஹd=2×Jre,ndtᫀA2ÊØ/ÖUÙÚ1Ù°ᐸ:"?/„0ḄÜ@ADAஹ1Bஹ0CஹTDஹ-1ᫀB3Ýᑡßà℉^âã!äᑖ᪆Ḅ-åD0AஹæçèéBஹæêᯠìḄᦪᳮé

77CஹæíᦪçèᳮéDஹæîḄ2᪆ᳮ7éᫀD41822Fourier-⊤XḄ℉æîḄ2᪆ᳮ7é஺0ᫀ75FourierḄ¤^ïðp©ᦪᭆò^óᦋᓽ©ᦪâ'ᑖö⊤÷஺0ᫀV᰿឴úû1ᐵRᦪᓻýÝᑡþO┯ḄAஹᦪᓻᳮᦪḄ℉Ḅ័▲ឋឋᭆ"ᡠ$⍗Ḅ&'()័*+ᩭ஺Bஹ.ᦪᓻ/0ᑖ223456789:;ᐻᙠ/0ᑖ>Ḅ?@ABᜐDᐸᐵGHIJᔲ0ḄL⚪NOP஺CஹQᦪᓻᙠ1902RSRS័NOᦪTḄUPV᤯XᔠḄZẠ஺Dஹ\)]Qᦪᓻᦪ^\_`aᗐ?cdᓻ஺efgᫀiD

782kᑡ?m4័᪷oḄAஹ᪀ᡂ័Ḅr⚪ᡈὅuv>◚xḼᑭᵨូឋ}~Ḅᭆ"Bஹ᝞ᑭᵨᡲᩎSḄXᔠḄᭆᑣ᪀ᡂXᔠCஹ▲ᭆ"ḄDஹ56Ḅ▲ឋefgᫀiD3?aᐰឋᳮᵫḄAஹᱯBஹCஹDஹ¡9efgᫀiC4ᱯ¢£័¤ᯠu¦⊤¨uᑁª⌼ᡂḄ஺¢¬័B®¯°±²³´µ¶·ᙠᐸ>¸³ᦪ¹ᳮº±»¼ᡂ³ᳮ½)ᩭ¸³ᳮº±»¼ᡂ³ᦪ¹ᳮ஺]᪵Ḅ¶·¿aᐰḄ஺efgᫀiN

795ᡲᨬᜧZᦪ័ÂRS័ºᨵ³Ä⌕ḄᱯÆi¤ᢣឋ஺efgᫀiV