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鍙嬫儏鎻愮ず锛氭湰绔欐彁渚涘叏鍦�(gu贸)400澶氭墍楂樼瓑闄㈡牎鎷涙敹纰╁＋銆佸崥澹爺绌剁敓鍏ュ(xu茅)鑰冭│姝峰勾鑰冪爺鐪熼銆佽€冨崥鐪熼銆佺瓟妗�锛岄儴鍒嗗(xu茅)鏍℃洿鏂拌嚦2012骞�锛�2013骞�锛涘潎鎻愪緵鏀惰不(f猫i)涓嬭級銆� 涓嬭級娴佺▼锛� 鑰冪爺鐪熼 榛�(di菐n)鎿娾€�鑰冪爺瑭﹀嵎鈥濃€濅笅杓�; 鑰冨崥鐪熼 榛�(di菐n)鎿娾€�鑰冨崥瑭﹀嵎搴�(k霉)鈥� 涓嬭級
闆诲瓙绉戞妧澶у(xu茅) target=_blank>鎴愰兘闆诲瓙绉戞妧澶у(xu茅) 2018 骞寸ⅸ澹爺绌剁敓鍏ュ(xu茅)鑰冭│鍒濊│鑷懡椤岀鐩€冭│澶х侗
鑰冭│绉戠洰 601 鏁�(sh霉)瀛�(xu茅)鍒嗘瀽 鑰冭│褰㈠紡 绛嗚│锛堥枆鍗凤級
鑰冭│鏅�(sh铆)闁� 180 鍒嗛悩 鑰冭│绺藉垎 150 鍒�
涓€銆佺附楂旇姹�
涓昏鑰冨療瀛�(xu茅)鐢熷皪(du矛)銆婃暩(sh霉)瀛�(xu茅)鍒嗘瀽銆嬬殑鍩烘湰鐭ヨ瓨(sh铆)銆佸熀鏈悊璜栧拰鍩烘湰鎶€鑳界殑鎺屾彙鎯呮硜浠ュ強鐢ㄦ暩(sh霉)瀛�(xu茅)鍒嗘瀽
鐨勭悊璜栬垏鏂规硶鍒嗘瀽鍟忛銆佽В姹哄晱椤岀殑鑳藉姏.
浜屻€佸収(n猫i)瀹�
1. 闆嗗悎鑸囧嚱鏁�(sh霉)
1锛� 瀵�(sh铆)鏁�(sh霉)闆� R 銆佹湁鐞嗘暩(sh霉)鑸囩劇鐞嗘暩(sh霉)鐨勭瀵嗘€�锛屽(sh铆)鏁�(sh霉)闆嗙殑鐣岃垏纰虹晫銆佺⒑鐣屽瓨鍦ㄦ€у畾鐞�銆佸柈瑾�(di脿o)鏈夌晫鎬�
瀹氱悊銆侀枆鍗€(q奴)闁撳瀹氱悊銆丅olzano-Weierstrass 瀹氱悊銆丆auchy 鏀舵杺鍘熺悊.
2锛� R
2
涓婄殑璺濋洟銆侀劙鍩�銆佽仛榛�(di菐n)銆佺晫榛�(di菐n)銆侀倞鐣�銆侀枊闆嗐€侀枆闆�銆佹湁鐣岋紙鐒＄晫锛夐泦銆� R
n
涓婄殑闁夌煩褰�
濂楀畾鐞�銆丠eine-Borel 瀹氱悊(鏈夐檺瑕嗚搵瀹氱悊)浠ュ強涓婅堪姒傚康鍜屽畾鐞嗗湪 R
n
涓婄殑鎺ㄥ唬.
3锛� 鍑芥暩(sh霉)銆佹槧灏�銆佽畩鎻涚瓑姒傚康鍙婂叾骞句綍鎰忕京锛岄毐鍑芥暩(sh霉)姒傚康锛屽弽鍑芥暩(sh霉)鑸囬€嗚畩鎻�锛屽弽鍑芥暩(sh霉)瀛樺湪鎬�
瀹氱悊锛屽垵绛夊嚱鏁�(sh霉)浠ュ強鑸囦箣鐩搁棞(gu膩n)鐨勬€ц唱(zh矛).
2. 妤甸檺鑸囬€ｇ簩(x霉)
1锛� 鏁�(sh霉)鍒楁サ闄�銆佹敹鏂傛暩(sh霉)鍒楃殑鍩烘湰鎬ц唱(zh矛)锛堟サ闄愬敮涓€鎬с€佹湁鐣屾€�銆佷繚铏�(h脿o)鎬с€佷笉绛夊紡鎬ц唱(zh矛)锛�.
2锛� 鏁�(sh霉)鍒楁敹鏂傜殑姊濅欢锛圕auchy 婧�(zh菙n)鍓�銆佽揩鏂傛€с€佸柈瑾�(di脿o)鏈夌晫鍘熺悊銆佹暩(sh霉)鍒楁敹鏂傝垏鍏跺瓙鍒楁敹鏂傜殑闂�(gu膩n)绯伙級锛�
妤甸檺 lim(1 ?
1
)
n
?e 鍙婂叾鎳�(y墨ng)鐢�.
n?? n
3锛変竴鍏冨嚱鏁�(sh霉)妤甸檺鐨勫畾缇�銆佸嚱鏁�(sh霉)妤甸檺鐨勫熀鏈€ц唱(zh矛)锛堝敮涓€鎬�銆佸眬閮ㄦ湁鐣屾€с€佷繚铏�(h脿o)鎬�銆佷笉绛夊紡鎬ц唱(zh矛)銆�
杩杺鎬э級锛孒eine 姝哥祼(ji茅)鍘熷墖鍜� Cauchy 鏀舵杺婧�(zh菙n)鍓囷紝鍏╁€�(g猫)閲嶈妤甸檺 lim sin x ? 1, lim(1 ? 1 )
x
?e
x ?0 x x?? x
鍙婂叾鎳�(y墨ng)鐢�锛岃▓(j矛)绠椾竴鍏冨嚱鏁�(sh霉)妤甸檺鐨勫悇绋柟娉�锛岀劇绐皬閲忚垏鐒＄澶ч噺銆侀殠鐨勬瘮杓冿紝瑷樿櫉(h脿o) O 鑸�
o 鐨勬剰缇�锛屽鍏冨嚱鏁�(sh霉)閲嶆サ闄愯垏绱妤甸檺姒傚康銆佸熀鏈€ц唱(zh矛)锛屼簩鍏冨嚱鏁�(sh霉)鐨勪簩閲嶆サ闄愯垏绱妤甸檺
鐨勯棞(gu膩n)绯�.
4锛� 鍑芥暩(sh霉)閫ｇ簩(x霉)鑸囬枔鏂枫€佷竴鑷撮€ｇ簩(x霉)鎬�銆侀€ｇ簩(x霉)鍑芥暩(sh霉)鐨勫眬閮ㄦ€ц唱(zh矛)锛堝眬閮ㄦ湁鐣屾€с€佷繚铏�(h脿o)鎬э級锛屾湁鐣岄枆
闆嗕笂閫ｇ簩(x霉)鍑芥暩(sh霉)鐨勬€ц唱(zh矛)锛堟湁鐣屾€с€佹渶澶у€兼渶灏忓€煎畾鐞�銆佷粙鍊煎畾鐞�銆佷竴鑷撮€ｇ簩(x霉)鎬э級.
3. 涓€鍏冨嚱鏁�(sh霉)寰垎瀛�(xu茅)
1锛夊皫(d菐o)鏁�(sh霉)鍙婂叾骞句綍鎰忕京銆佸彲灏�(d菐o)鑸囬€ｇ簩(x霉)鐨勯棞(gu膩n)绯�銆佸皫(d菐o)鏁�(sh霉)鐨勫悇绋▓(j矛)绠楁柟娉�锛屽井鍒嗗強鍏跺咕浣曟剰缇┿€佸彲寰�
鑸囧彲灏�(d菐o)鐨勯棞(gu膩n)绯�銆佷竴闅庡井鍒嗗舰寮忎笉璁婃€�.
2锛夊井鍒嗗(xu茅)鍩烘湰瀹氱悊锛欶ermat 瀹氱悊锛孯olle 瀹氱悊锛孡agrange 瀹氱悊锛孋auchy 瀹氱悊锛孴aylor 鍏紡
(Peano 浣欓爡(xi脿ng)鑸� Lagrange 浣欓爡(xi脿ng)).
3锛変竴鍏冨井鍒嗗(xu茅)鐨勬噳(y墨ng)鐢細鍑芥暩(sh霉)鍠(di脿o)鎬х殑鍒ゅ垾銆佹サ鍊�銆佹渶澶у€煎拰鏈€灏忓€�銆佸嚫鍑芥暩(sh霉)鍙婂叾鎳�(y墨ng)鐢�銆佹洸绶�
鐨勫嚬鍑告€�銆佹嫄榛�(di菐n)銆佹几杩戠窔銆佸嚱鏁�(sh霉)鍦栬薄鐨勮◣璜�銆佹礇蹇呴仈(d谩)锛圠'Hospital锛夋硶鍓囥€佽繎浼艰▓(j矛)绠�.
4. 澶氬厓鍑芥暩(sh霉)寰垎瀛�(xu茅)
1锛� 鍋忓皫(d菐o)鏁�(sh霉)銆佸叏寰垎鍙婂叾骞句綍鎰忕京锛屽彲寰垏鍋忓皫(d菐o)瀛樺湪銆侀€ｇ簩(x霉)涔嬮枔鐨勯棞(gu膩n)绯伙紝寰�(f霉)鍚堝嚱鏁�(sh霉)鐨勫亸灏�(d菐o)鏁�(sh霉)
鑸囧叏寰垎锛屼竴闅庡井鍒嗗舰寮忎笉璁婃€э紝鏂瑰悜?q奴)顫�?sh霉)鑸囨搴�锛岄珮闅庡亸灏�(d菐o)鏁�(sh霉)锛屾贩鍚堝亸灏�(d菐o)鏁�(sh霉)鑸囬爢搴忕劇
闂�(gu膩n)鎬�锛屼簩鍏冨嚱鏁�(sh霉)涓€煎畾鐞嗚垏 Taylor 鍏紡.
2锛� 闅卞嚱鏁�(sh霉)瀛樺湪瀹氱悊銆侀毐鍑芥暩(sh霉)绲勫瓨鍦ㄥ畾鐞�銆侀毐鍑芥暩(sh霉)锛堢祫锛夋眰灏�(d菐o)鏂规硶銆佸弽鍑芥暩(sh霉)绲勮垏鍧愭(bi膩o)璁婃彌.
3锛� 骞句綍鎳�(y墨ng)鐢紙骞抽潰鏇茬窔鐨勫垏绶氳垏娉曠窔銆佺┖闁撴洸绶氱殑鍒囩窔鑸囨硶骞抽潰銆佹洸闈㈢殑鍒囧钩闈㈣垏娉曠窔锛�.
4锛� 妤靛€煎晱椤岋紙蹇呰姊濅欢鑸囧厖鍒嗘浠讹級锛屾浠舵サ鍊艰垏 Lagrange 涔樻暩(sh霉)娉�銆�
5. 涓€鍏冨嚱鏁�(sh霉)绌嶅垎瀛�(xu茅)
1锛夊師鍑芥暩(sh霉)鑸囦笉瀹氱⿳鍒�銆佷笉瀹氱⿳鍒嗙殑鍩烘湰瑷�(j矛)绠楁柟娉曪紙鐩存帴绌嶅垎娉�銆佹彌鍏冩硶銆佸垎閮ㄧ⿳鍒嗘硶锛�銆佹湁
鐞�
鍑芥暩(sh霉)绌嶅垎锛�? R (cos x , sin x ) dx 鍨嬶紝? R ( x, ax
2
?bx? c )dx 鍨�.
2锛夊畾绌嶅垎鍙婂叾骞句綍鎰忕京銆佸彲绌嶆浠讹紙蹇呰姊濅欢銆佸厖瑕佹浠讹細??i?x i??锛�銆佸彲绌嶅嚱鏁�(sh霉)椤�.
3锛夊畾绌嶅垎鐨勬€ц唱(zh矛)锛堥棞(gu膩n)浜庡崁(q奴)闁撳彲鍔犳€с€佷笉绛夊紡鎬ц唱(zh矛)銆佺禃灏�(du矛)鍙⿳鎬�銆佸畾绌嶅垎绗竴涓€煎畾鐞嗭級銆佽畩
涓婇檺绌嶅垎鍑芥暩(sh霉)銆佸井绌嶅垎鍩烘湰瀹氱悊銆丯-L 鍏紡鍙婂畾绌嶅垎瑷�(j矛)绠�銆佸畾绌嶅垎绗簩涓€煎畾鐞�.
??
4锛� 鐒￠檺鍗€(q奴)闁撲笂鐨勫唬缇╃⿳鍒嗐€丆anchy 鏀舵杺婧�(zh菙n)鍓�銆佺禃灏�(du矛)鏀舵杺鑸囨浠舵敹鏂�銆乫 ( x)闈炶矤(f霉)鏅�(sh铆)?a f ( x )
dx
鐨勬敹鏂傛€у垽鍒ユ硶锛堟瘮杓冨師鍓�銆佹煰瑗垮垽鍒ユ硶锛�銆丄bel 鍒ゅ垾娉�銆丏irichlet 鍒ゅ垾娉曘€佺劇鐣屽嚱
鏁�(sh霉)寤ｇ京绌嶅垎姒傚康鍙婂叾鏀舵杺鎬у垽鍒ユ硶.
5锛夊井鍏冩硶銆佸咕浣曟噳(y墨ng)鐢紙骞抽潰鍦栧舰闈㈢⿳銆佸凡鐭ユ埅闈㈤潰绌嶅嚱鏁�(sh霉)鐨勯珨绌�銆佹洸绶氬姬闀�(zh菐ng)鑸囧姬寰垎銆佹棆
杞�(zhu菐n)楂旈珨绌嶏級锛屽叾浠栨噳(y墨ng)鐢�銆�
6. 澶氬厓鍑芥暩(sh霉)绌嶅垎瀛�(xu茅)
1锛変簩閲嶇⿳鍒嗗強鍏跺咕浣曟剰缇�銆佷簩閲嶇⿳鍒嗙殑瑷�(j矛)绠楋紙鍖栫偤绱绌嶅垎銆佹サ鍧愭(bi膩o)璁婃彌銆佷竴鑸潗妯�(bi膩o)璁婃彌锛�.
2锛変笁閲嶇⿳鍒�銆佷笁閲嶇⿳鍒嗚▓(j矛)绠楋紙鍖栫偤绱绌嶅垎銆佹煴鍧愭(bi膩o)銆佺悆鍧愭(bi膩o)璁婃彌锛�.
3锛夐噸绌嶅垎鐨勬噳(y墨ng)鐢紙楂旂⿳銆佹洸闈㈤潰绌嶃€侀噸蹇�銆佽綁(zhu菐n)鍕�(d貌ng)鎱ｉ噺绛夛級.
4锛夌涓€鍨嬫洸绶氱⿳鍒嗐€佹洸闈㈢⿳鍒嗙殑姒傚康銆佸熀鏈€ц唱(zh矛)銆佽▓(j矛)绠�.
5锛夌浜屽瀷鏇茬窔绌嶅垎姒傚康銆佹€ц唱(zh矛)銆佽▓(j矛)绠�锛汫reen 鍏紡锛屽钩闈㈡洸绶氱⿳鍒嗚垏璺緫鐒￠棞(gu膩n)鐨勬浠�.
6锛夋洸闈㈢殑鍋�(c猫)銆佺浜屽瀷鏇查潰绌嶅垎鐨勬蹇点€佹€ц唱(zh矛)銆佽▓(j矛)绠�锛孏auss 鍏紡銆丼tokes 鍏紡锛屽叐椤炵窔绌嶅垎銆�
鍏╅闈㈢⿳鍒嗕箣闁撶殑闂�(gu膩n)绯�.
7锛夊惈鍙冮噺姝ｅ父绌嶅垎鍙婂叾閫ｇ簩(x霉)鎬с€佸彲寰€�銆佸彲绌嶆€�锛岄亱(y霉n)绠楅爢搴忕殑鍙氦鎻涙€�.鍚弮閲忓唬缇╃⿳鍒嗙殑
涓€鑷存敹鏂傛€у強鍏跺垽鍒ユ硶锛屽惈鍙冮噺寤ｇ京绌嶅垎鐨勯€ｇ簩(x霉)鎬�銆佸彲寰€�銆佸彲绌嶆€э紝閬�(y霉n)绠楅爢搴忕殑鍙氦
鎻涙€�.
7. 鐒＄绱�(j铆)鏁�(sh霉)
1锛夋暩(sh霉)闋�(xi脿ng)绱�(j铆)鏁�(sh霉)
绱�(j铆)鏁�(sh霉)鍙婂叾鏂傛暎鎬�锛岀礆(j铆)鏁�(sh霉)鐨勫拰锛孋auchy 婧�(zh菙n)鍓囷紝鏀舵杺鐨勫繀瑕佹浠�锛屾敹鏂傜礆(j铆)鏁�(sh霉)鍩烘湰鎬ц唱(zh矛)锛涙闋�(xi脿ng)
绱�(j铆)鏁�(sh霉)鏀舵杺鐨勫厖鍒嗗繀瑕佹浠�锛屾瘮杓冨師鍓囥€佹瘮寮忓垽鍒ユ硶銆佹牴寮忓垽鍒ユ硶浠ュ強瀹冨€戠殑妤甸檺褰㈠紡锛涗氦
閷�(cu貌)绱�(j铆)鏁�(sh霉)鐨� Leibniz 鍒ゅ垾娉�锛涗竴鑸爡(xi脿ng)绱�(j铆)鏁�(sh霉)鐨勭禃灏�(du矛)鏀舵杺銆佹浠舵敹鏂傛€с€丄bel 鍒ゅ垾娉曘€�
Dirichlet 鍒ゅ垾娉�.
2锛夊嚱鏁�(sh霉)闋�(xi脿ng)绱�(j铆)鏁�(sh霉)
鍑芥暩(sh霉)鍒楄垏鍑芥暩(sh霉)闋�(xi脿ng)绱�(j铆)鏁�(sh霉)鐨勪竴鑷存敹鏂傛€�銆丆auchy 婧�(zh菙n)鍓�銆佷竴鑷存敹鏂傛€у垽鍒ユ硶锛圡-鍒ゅ垾娉曘€丄bel
鍒ゅ垾娉�銆丏irichlet 鍒ゅ垾娉曪級銆佷竴鑷存敹鏂傚嚱鏁�(sh霉)鍒�銆佸嚱鏁�(sh霉)闋�(xi脿ng)绱�(j铆)鏁�(sh霉)鐨勬€ц唱(zh矛)鍙婂叾鎳�(y墨ng)鐢�.
3锛夊啰绱�(j铆)鏁�(sh霉)
鍐礆(j铆)鏁�(sh霉)姒傚康銆丄bel 瀹氱悊銆佹敹鏂傚崐寰戣垏鍗€(q奴)闁�锛屽啰绱�(j铆)鏁�(sh霉)鐨勪竴鑷存敹鏂傛€�锛屽啰绱�(j铆)鏁�(sh霉)鐨勯€愰爡(xi脿ng)鍙⿳鎬�銆�
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鎮ㄧ従(xi脿n)鍦ㄧ殑浣嶇疆锛� 
