1. 楂樹竴鏁板 楂樹腑鏁板
楂樹竴鏄鎴戜滑榪涘叆楂樹腑鏃舵湡鐨勭涓闃舵碉紝鎴戜滑搴旇ュ畬鍠勫繁韜錛屽ソ濂藉︿範銆傝屾暟瀛︿篃鏄鎴戜滑蹇呴』瀛︿範鐨勯噸瑕佽劇▼涔嬩竴錛屾垜涓哄悇浣嶅悓瀛︽暣鐞嗕簡楂 涓騫寸駭鏁板 蹇呬慨浜旂煡璇嗙偣 鎬葷粨 錛屽笇鏈涘逛綘鏈夋墍甯鍔! 楂樹竴鏁板 蹇呬慨浜旂煡璇嗙偣鎬葷粨1 銆愬樊鏁板垪鐨勫熀鏈鎬ц川銆 鈶村叕宸涓篸鐨勭瓑宸鏁板垪,鍚勯」鍚屽姞涓鏁版墍寰楁暟鍒椾粛鏄絳夊樊鏁板垪,鍏跺叕宸浠嶄負d. 鈶靛叕宸涓篸鐨勭瓑宸鏁板垪,鍚勯」鍚屼箻浠ュ父鏁発鎵寰楁暟鍒椾粛鏄絳夊樊鏁板垪,鍏跺叕宸涓簁d. 鈶惰嫢{a}銆亄b}涓虹瓑宸鏁板垪,鍒檣a鹵b}涓巤ka+b}(k銆乥涓洪潪闆跺父鏁)涔熸槸絳夊樊鏁板垪. 鈶峰逛換浣昺銆乶,鍦ㄧ瓑宸鏁板垪{a}涓鏈夛細a=a+(n-m)d,鐗瑰埆鍦,褰搈=1鏃,渚垮緱絳夊樊鏁板垪鐨勯氶」鍏寮,姝ゅ紡杈冪瓑宸鏁板垪鐨勯氶」鍏寮忔洿鍏鋒湁涓鑸鎬. 鈶搞佷竴鑸鍦,濡傛灉l,k,p,鈥,m,n,r,鈥︾殕涓鴻嚜鐒舵暟,涓攍+k+p+鈥=m+n+r+鈥(涓よ竟鐨勮嚜鐒舵暟涓鏁扮浉絳),閭ｄ箞褰搟a}涓虹瓑宸鏁板垪鏃,鏈夛細a+a+a+鈥=a+a+a+鈥. 鈶瑰叕宸涓篸鐨勭瓑宸鏁板垪,浠庝腑鍙栧嚭絳夎窛紱葷殑欏,鏋勬垚涓涓鏂版暟鍒,姝ゆ暟鍒椾粛鏄絳夊樊鏁板垪,鍏跺叕宸涓簁d(k涓哄彇鍑洪」鏁頒箣宸). 鈶哄傛灉{a}鏄絳夊樊鏁板垪,鍏宸涓篸,閭ｄ箞,a,a,鈥,a銆乤涔熸槸絳夊樊鏁板垪,鍏跺叕宸涓-d;鍦ㄧ瓑宸鏁板垪{a}涓,a-a=a-a=md.(鍏朵腑m銆乲銆) 鈶誨湪絳夊樊鏁板垪涓,浠庣涓欏硅搗,姣忎竴欏(鏈夌┓鏁板垪鏈欏歸櫎澶)閮芥槸瀹冨墠鍚庝袱欏圭殑絳夊樊涓欏. 鈶煎綋鍏宸甦>0鏃,絳夊樊鏁板垪涓鐨勬暟闅忛」鏁扮殑澧炲ぇ鑰屽炲ぇ;褰揹<0鏃,絳夊樊鏁板垪涓鐨勬暟闅忛」鏁扮殑鍑忓皯鑰屽噺灝;d=0鏃,絳夊樊鏁板垪涓鐨勬暟絳変簬涓涓甯告暟. 鈶借綼,a,a涓虹瓑宸鏁板垪涓鐨勪笁欏,涓攁涓巃,a涓巃鐨勯」璺濆樊涔嬫瘮=(鈮-1),鍒檃=. 鈶存暟鍒梴a}涓虹瓑宸鏁板垪鐨勫厖瑕佹潯浠舵槸錛氭暟鍒梴a}鐨勫墠n欏瑰拰S鍙浠ュ啓鎴怱=an+bn鐨勫艦寮(鍏朵腑a銆乥涓哄父鏁). 鈶靛湪絳夊樊鏁板垪{a}涓,褰撻」鏁頒負2n(nN)鏃,S-S=nd,=;褰撻」鏁頒負(2n-1)(n)鏃,S-S=a,=. 鈶惰嫢鏁板垪{a}涓虹瓑宸鏁板垪,鍒橲,S-S,S-S,鈥︿粛鐒舵垚絳夊樊鏁板垪,鍏宸涓. 鈶瘋嫢涓や釜絳夊樊鏁板垪{a}銆亄b}鐨勫墠n欏瑰拰鍒嗗埆鏄瘲銆乀(n涓哄囨暟),鍒=. 鈶稿湪絳夊樊鏁板垪{a}涓,S=a,S=b(n>m),鍒橲=(a-b). 鈶圭瓑宸鏁板垪{a}涓,鏄痭鐨勪竴嬈″嚱鏁,涓旂偣(n,)鍧囧湪鐩寸嚎y=x+(a-)涓. 鈶鴻扮瓑宸鏁板垪{a}鐨勫墠n欏瑰拰涓篠.鈶犺嫢a>0,鍏宸甦<0,鍒欏綋a鈮0涓攁鈮0鏃,S;鈶¤嫢a<0,鍏宸甦>0,鍒欏綋a鈮0涓攁鈮0鏃,S鏈灝. 銆愮瓑姣旀暟鍒楃殑鍩烘湰鎬ц川銆 鈶村叕姣斾負q鐨勭瓑姣旀暟鍒,浠庝腑鍙栧嚭絳夎窛紱葷殑欏,鏋勬垚涓涓鏂版暟鍒,姝ゆ暟鍒椾粛鏄絳夋瘮鏁板垪,鍏跺叕姣斾負q(m涓虹瓑璺濈葷殑欏規暟涔嬪樊). 鈶靛逛換浣昺銆乶,鍦ㄧ瓑姣旀暟鍒梴a}涓鏈夛細a=a路q,鐗瑰埆鍦,褰搈=1鏃,渚垮緱絳夋瘮鏁板垪鐨勯氶」鍏寮,姝ゅ紡杈冪瓑姣旀暟鍒楃殑閫氶」鍏寮忔洿鍏鋒湁鏅閬嶆. 鈶朵竴鑸鍦,濡傛灉t,k,p,鈥,m,n,r,鈥︾殕涓鴻嚜鐒舵暟,涓攖+k,p,鈥,m+鈥=m+n+r+鈥(涓よ竟鐨勮嚜鐒舵暟涓鏁扮浉絳),閭ｄ箞褰搟a}涓虹瓑姣旀暟鍒楁椂,鏈夛細a.a.a.鈥=a.a.a.鈥.. 鈶瘋嫢{a}鏄鍏姣斾負q鐨勭瓑姣旀暟鍒,鍒檣|a|}銆亄a}銆亄ka}銆亄}涔熸槸絳夋瘮鏁板垪,鍏跺叕姣斿垎鍒涓簗q|}銆亄q}銆亄q}銆亄}. 鈶稿傛灉{a}鏄絳夋瘮鏁板垪,鍏姣斾負q,閭ｄ箞,a,a,a,鈥,a,鈥︽槸浠q涓哄叕姣旂殑絳夋瘮鏁板垪. 鈶瑰傛灉{a}鏄絳夋瘮鏁板垪,閭ｄ箞瀵逛換鎰忓湪n,閮芥湁a路a=a路q>0. 鈶轟袱涓絳夋瘮鏁板垪鍚勫瑰簲欏圭殑縐緇勬垚鐨勬暟鍒椾粛鏄絳夋瘮鏁板垪,涓斿叕姣旂瓑浜庤繖涓や釜鏁板垪鐨勫叕姣旂殑縐. 鈶誨綋q>1涓攁>0鎴00涓01鏃,絳夋瘮鏁板垪涓洪掑噺鏁板垪;褰搎=1鏃,絳夋瘮鏁板垪涓哄父鏁板垪;褰搎<0鏃,絳夋瘮鏁板垪涓烘憜鍔ㄦ暟鍒. 楂樹腑鏁板﹀繀淇浜旓細絳夋瘮鏁板垪鍓峮欏瑰拰鍏寮廠鐨勫熀鏈鎬ц川 鈶村傛灉鏁板垪{a}鏄鍏姣斾負q鐨勭瓑姣旀暟鍒,閭ｄ箞,瀹冪殑鍓峮欏瑰拰鍏寮忔槸S= 涔熷氨鏄璇,鍏姣斾負q鐨勭瓑姣旀暟鍒楃殑鍓峮欏瑰拰鍏寮忔槸q鐨勫垎孌靛嚱鏁扮殑涓緋誨垪鍑芥暟鍊,鍒嗘電殑鐣岄檺鏄鍦╭=1澶.鍥犳,浣跨敤絳夋瘮鏁板垪鐨勫墠n欏瑰拰鍏寮,蹇呴』瑕佸紕娓呭叕姣攓鏄鍙鑳界瓑浜1榪樻槸蹇呬笉絳変簬1,濡傛灉q鍙鑳界瓑浜1,鍒欓渶鍒唓=1鍜宷鈮1榪涜岃ㄨ. 鈶靛綋宸茬煡a,q,n鏃,鐢ㄥ叕寮廠=;褰撳凡鐭a,q,a鏃,鐢ㄥ叕寮廠=. 鈶惰嫢S鏄浠q涓哄叕姣旂殑絳夋瘮鏁板垪,鍒欐湁S=S+qS.鈶 鈶瘋嫢鏁板垪{a}涓虹瓑姣旀暟鍒,鍒橲,S-S,S-S,鈥︿粛鐒舵垚絳夋瘮鏁板垪. 鈶歌嫢欏規暟涓3n鐨勭瓑姣旀暟鍒(q鈮-1)鍓峮欏瑰拰涓庡墠n欏圭Н鍒嗗埆涓篠涓嶵,嬈n欏瑰拰涓庢n欏圭Н鍒嗗埆涓篠涓嶵,鏈鍚巒欏瑰拰涓巒欏圭Н鍒嗗埆涓篠涓嶵,鍒橲,S,S鎴愮瓑姣旀暟鍒,T,T,T浜︽垚絳夋瘮鏁板垪 涓囪兘鍏寮忥細sin2偽=2tan偽/(1+tan^2偽)(娉錛歵an^2偽鏄鎸噒an騫蟲柟偽) cos2偽=(1-tan^2偽)/(1+tan^2偽)tan2偽=2tan偽/(1-tan^2偽) 鍗囧籙鍏寮忥細1+cos偽=2cos^2(偽/2)1-cos偽=2sin^2(偽/2)1鹵sin偽=(sin(偽/2)鹵cos(偽/2))^2 闄嶅籙鍏寮忥細cos^2偽=(1+cos2偽)/2sin^2偽=(1-cos2偽)/21)sin(2k蟺+偽)=sin偽,cos(2k蟺+偽)=cos偽,tan(2k蟺+偽)=tan偽,cot(2k蟺+偽)=cot偽,鍏朵腑k鈭圸; (2)sin(-偽)=-sin偽,cos(-偽)=cos偽,tan(-偽)=-tan偽,cot(-偽)=-cot偽 (3)sin(蟺+偽)=-sin偽,cos(蟺+偽)=-cos偽,tan(蟺+偽)=tan偽,cot(蟺+偽)=cot偽 (4)sin(蟺-偽)=sin偽,cos(蟺-偽)=-cos偽,tan(蟺-偽)=-tan偽,cot(蟺-偽)=-cot偽 (5)sin(蟺/2-偽)=cos偽,cos(蟺/2-偽)=sin偽,tan(蟺/2-偽)=cot偽,cot(蟺/2-偽)=tan偽 (6)sin(蟺/2+偽)=cos偽,cos(蟺/2+偽)=-sin偽, tan(蟺/2+偽)=-cot偽,cot(蟺/2+偽)=-tan偽 (7)sin(3蟺/2+偽)=-cos偽,cos(3蟺/2+偽)=sin偽, tan(3蟺/2+偽)=-cot偽,cot(3蟺/2+偽)=-tan偽 (8)sin(3蟺/2-偽)=-cos偽,cos(3蟺/2-偽)=-sin偽, tan(3蟺/2-偽)=cot偽,cot(3蟺/2-偽)=tan偽(k路蟺/2鹵偽),鍏朵腑k鈭圸 娉ㄦ剰錛氫負鏂逛究鍋氶,涔犳儻鎴戜滑鎶娢辯湅鎴愭槸涓涓浣嶄簬絎涓璞￠檺涓斿皬浜90擄鐨勮; 褰搆鏄濂囨暟鐨勬椂鍊,絳夊紡鍙寵竟鐨勪笁瑙掑嚱鏁板彂鐢熷彉鍖,濡俿in鍙樻垚cos.鍋舵暟鍒欎笉鍙; 鐢ㄨ(k路蟺/2鹵偽)鎵鍦ㄧ殑璞￠檺紜瀹氱瓑寮忓彸杈逛笁瑙掑嚱鏁扮殑姝ｈ礋.渚嬶細tan(3蟺/2+偽)=-cot偽 鈭靛湪榪欎釜寮忓瓙涓璳=3,鏄濂囨暟,鍥犳ょ瓑寮忓彸杈瑰簲鍙樹負cot 鍙,鈭佃(3蟺/2+偽)鍦ㄧ鍥涜薄闄,tan鍦ㄧ鍥涜薄闄愪負璐熷,鍥犳や負浣跨瓑寮忔垚絝,絳夊紡鍙寵竟搴斾負-cot偽.涓夎掑嚱鏁板湪鍚勮薄闄愪腑鐨勬ｈ礋鍒嗗竷 sin錛氱涓絎浜岃薄闄愪腑涓烘;絎涓夌鍥涜薄闄愪腑涓鴻礋cos錛氱涓絎鍥涜薄闄愪腑涓烘;絎浜岀涓夎薄闄愪腑涓鴻礋cot銆乼an:絎涓絎涓夎薄闄愪腑涓烘;絎浜岀鍥涜薄闄愪腑涓鴻礋銆 楂樹竴鏁板﹀繀淇浜旂煡璇嗙偣鎬葷粨2 (涓)銆佹槧灝勩佸嚱鏁般佸弽鍑芥暟 1銆佸瑰簲銆佹槧灝勩佸嚱鏁頒笁涓姒傚康鏃㈡湁鍏辨у張鏈夊尯鍒錛屾槧灝勬槸涓縐嶇壒孌婄殑瀵瑰簲錛岃屽嚱鏁板張鏄涓縐嶇壒孌婄殑鏄犲皠. 2銆佸逛簬鍑芥暟鐨勬傚康錛屽簲娉ㄦ剰濡備笅鍑犵偣錛 (1)鎺屾彙鏋勬垚鍑芥暟鐨勪笁瑕佺礌錛屼細鍒ゆ柇涓や釜鍑芥暟鏄鍚︿負鍚屼竴鍑芥暟. (2)鎺屾彙涓夌嶈〃紺烘硶鈥斺斿垪琛ㄦ硶銆佽В鏋愭硶銆佸浘璞℃硶錛岃兘鏍瑰疄闄呴棶棰樺繪眰鍙橀噺闂寸殑鍑芥暟鍏崇郴寮忥紝鐗瑰埆鏄浼氭眰鍒嗘靛嚱鏁扮殑瑙ｆ瀽寮. (3)濡傛灉y=f(u)錛寀=g(x)錛岄偅涔坹=f[g(x)]鍙鍋歠鍜実鐨勫嶅悎鍑芥暟錛屽叾涓璯(x)涓哄唴鍑芥暟錛宖(u)涓哄栧嚱鏁. 3銆佹眰鍑芥暟y=f(x)鐨勫弽鍑芥暟鐨勪竴鑸姝ラわ細 (1)紜瀹氬師鍑芥暟鐨勫煎煙錛屼篃灝辨槸鍙嶅嚱鏁扮殑瀹氫箟鍩; (2)鐢眣=f(x)鐨勮В鏋愬紡奼傚嚭x=f-1(y); (3)灝唜錛寉瀵規崲錛屽緱鍙嶅嚱鏁扮殑涔犳儻琛ㄨ揪寮弝=f-1(x)錛屽苟娉ㄦ槑瀹氫箟鍩. 娉ㄦ剰鈶狅細瀵逛簬鍒嗘靛嚱鏁扮殑鍙嶅嚱鏁幫紝鍏堝垎鍒奼傚嚭鍦ㄥ悇孌典笂鐨勫弽鍑芥暟錛岀劧鍚庡啀鍚堝苟鍒頒竴璧. 鈶＄啛鎮夌殑搴旂敤錛屾眰f-1(x0)鐨勫礆紝鍚堢悊鍒╃敤榪欎釜緇撹猴紝鍙浠ラ伩鍏嶆眰鍙嶅嚱鏁扮殑榪囩▼錛屼粠鑰岀畝鍖栬繍綆. (浜)銆佸嚱鏁扮殑瑙ｆ瀽寮忎笌瀹氫箟鍩 1銆佸嚱鏁板強鍏跺畾涔夊煙鏄涓嶅彲鍒嗗壊鐨勬暣浣擄紝娌℃湁瀹氫箟鍩熺殑鍑芥暟鏄涓嶅瓨鍦ㄧ殑錛屽洜姝わ紝瑕佹ｇ『鍦板啓鍑哄嚱鏁扮殑瑙ｆ瀽寮忥紝蹇呴』鏄鍦ㄦ眰鍑哄彉閲忛棿鐨勫瑰簲娉曞垯鐨勫悓鏃訛紝奼傚嚭鍑芥暟鐨勫畾涔夊煙.奼傚嚱鏁扮殑瀹氫箟鍩熶竴鑸鏈変笁縐嶇被鍨嬶細 (1)鏈夋椂涓涓鍑芥暟鏉ヨ嚜浜庝竴涓瀹為檯闂棰橈紝榪欐椂鑷鍙橀噺x鏈夊疄闄呮剰涔夛紝奼傚畾涔夊煙瑕佺粨鍚堝疄闄呮剰涔夎冭檻; (2)宸茬煡涓涓鍑芥暟鐨勮В鏋愬紡奼傚叾瀹氫箟鍩燂紝鍙瑕佷嬌瑙ｆ瀽寮忔湁鎰忎箟鍗沖彲.濡傦細 鈶犲垎寮忕殑鍒嗘瘝涓嶅緱涓洪浂; 鈶″伓嬈℃柟鏍圭殑琚寮鏂規暟涓嶅皬浜庨浂; 鈶㈠規暟鍑芥暟鐨勭湡鏁板繀欏誨ぇ浜庨浂; 鈶ｆ寚鏁板嚱鏁板拰瀵規暟鍑芥暟鐨勫簳鏁板繀欏誨ぇ浜庨浂涓斾笉絳変簬1; 鈶や笁瑙掑嚱鏁頒腑鐨勬ｅ垏鍑芥暟y=tanx(x鈭圧錛屼笖k鈭圸)錛屼綑鍒囧嚱鏁皔=cotx(x鈭圧錛寈鈮爇蟺錛宬鈭圸)絳. 搴旀敞鎰忥紝涓涓鍑芥暟鐨勮В鏋愬紡鐢卞嚑閮ㄥ垎緇勬垚鏃訛紝瀹氫箟鍩熶負鍚勯儴鍒嗘湁鎰忎箟鐨勮嚜鍙橀噺鍙栧肩殑鍏鍏遍儴鍒(鍗充氦闆). (3)宸茬煡涓涓鍑芥暟鐨勫畾涔夊煙錛屾眰鍙︿竴涓鍑芥暟鐨勫畾涔夊煙錛屼富瑕佽冭檻瀹氫箟鍩熺殑娣卞埢鍚涔夊嵆鍙. 宸茬煡f(x)鐨勫畾涔夊煙鏄痆a錛宐]錛屾眰f[g(x)]鐨勫畾涔夊煙鏄鎸囨弧瓚砤鈮g(x)鈮b鐨剎鐨勫彇鍊艱寖鍥達紝鑰屽凡鐭f[g(x)]鐨勫畾涔夊煙[a錛宐]鎸囩殑鏄痻鈭圼a錛宐]錛屾ゆ椂f(x)鐨勫畾涔夊煙錛屽嵆g(x)鐨勫煎煙. 2銆佹眰鍑芥暟鐨勮В鏋愬紡涓鑸鏈夊洓縐嶆儏鍐 (1)鏍規嵁鏌愬疄闄呴棶棰橀渶寤虹珛涓縐嶅嚱鏁板叧緋繪椂錛屽繀欏誨紩鍏ュ悎閫傜殑鍙橀噺錛屾牴鎹鏁板︾殑鏈夊叧鐭ヨ瘑瀵繪眰鍑芥暟鐨勮В鏋愬紡. (2)鏈夋椂棰樿劇粰鍑哄嚱鏁扮壒寰侊紝奼傚嚱鏁扮殑瑙ｆ瀽寮忥紝鍙閲囩敤寰呭畾緋繪暟娉.姣斿傚嚱鏁版槸涓嬈″嚱鏁幫紝鍙璁緁(x)=ax+b(a鈮0)錛屽叾涓璦錛宐涓哄緟瀹氱郴鏁幫紝鏍規嵁棰樿炬潯浠訛紝鍒楀嚭鏂圭▼緇勶紝奼傚嚭a錛宐鍗沖彲. (3)鑻ラ樿劇粰鍑哄嶅悎鍑芥暟f[g(x)]鐨勮〃杈懼紡鏃訛紝鍙鐢ㄦ崲鍏冩硶奼傚嚱鏁癴(x)鐨勮〃杈懼紡錛岃繖鏃跺繀欏繪眰鍑篻(x)鐨勫煎煙錛岃繖鐩稿綋浜庢眰鍑芥暟鐨勫畾涔夊煙. (4)鑻ュ凡鐭f(x)婊¤凍鏌愪釜絳夊紡錛岃繖涓絳夊紡闄f(x)鏄鏈鐭ラ噺澶栵紝榪樺嚭鐜板叾浠栨湭鐭ラ噺(濡俧(-x)錛岀瓑)錛屽繀欏繪牴鎹宸茬煡絳夊紡錛屽啀鏋勯犲叾浠栫瓑寮忕粍鎴愭柟紼嬬粍錛屽埄鐢ㄨВ鏂圭▼緇勬硶奼傚嚭f(x)鐨勮〃杈懼紡. (涓)銆佸嚱鏁扮殑鍊煎煙涓庢渶鍊 1銆佸嚱鏁扮殑鍊煎煙鍙栧喅浜庡畾涔夊煙鍜屽瑰簲娉曞垯錛屼笉璁洪噰鐢ㄤ綍縐 鏂規硶 奼傚嚱鏁板煎煙閮藉簲鍏堣冭檻鍏跺畾涔夊煙錛屾眰鍑芥暟鍊煎煙甯哥敤鏂規硶濡備笅錛 (1)鐩存帴娉曪細浜︾О瑙傚療娉曪紝瀵逛簬緇撴瀯杈冧負綆鍗曠殑鍑芥暟錛屽彲鐢卞嚱鏁扮殑瑙ｆ瀽寮忓簲鐢ㄤ笉絳夊紡鐨勬ц川錛岀洿鎺ヨ傚療寰楀嚭鍑芥暟鐨勫煎煙. (2)鎹㈠厓娉曪細榪愮敤浠ｆ暟寮忔垨涓夎掓崲鍏冨皢鎵緇欑殑澶嶆潅鍑芥暟杞鍖栨垚鍙︿竴縐嶇畝鍗曞嚱鏁板啀奼傚煎煙錛岃嫢鍑芥暟瑙ｆ瀽寮忎腑鍚鏈夋牴寮忥紝褰撴牴寮忛噷涓嬈″紡鏃剁敤浠ｆ暟鎹㈠厓錛屽綋鏍瑰紡閲屾槸浜屾″紡鏃訛紝鐢ㄤ笁瑙掓崲鍏. (3)鍙嶅嚱鏁版硶錛氬埄鐢ㄥ嚱鏁癴(x)涓庡叾鍙嶅嚱鏁癴-1(x)鐨勫畾涔夊煙鍜屽煎煙闂寸殑鍏崇郴錛岄氳繃奼傚弽鍑芥暟鐨勫畾涔夊煙鑰屽緱鍒板師鍑芥暟鐨勫煎煙錛屽艦濡(a鈮0)鐨勫嚱鏁板煎煙鍙閲囩敤姝ゆ硶奼傚緱. (4)閰嶆柟娉曪細瀵逛簬浜屾″嚱鏁版垨浜屾″嚱鏁版湁鍏崇殑鍑芥暟鐨勫煎煙闂棰樺彲鑰冭檻鐢ㄩ厤鏂規硶. (5)涓嶇瓑寮忔硶奼傚煎煙錛氬埄鐢ㄥ熀鏈涓嶇瓑寮廰+b鈮[a錛宐鈭(0錛+鈭)]鍙浠ユ眰鏌愪簺鍑芥暟鐨勫煎煙錛屼笉榪囧簲娉ㄦ剰鏉′歡鈥滀竴姝ｄ簩瀹氫笁鐩哥瓑鈥濇湁鏃墮渶鐢ㄥ埌騫蟲柟絳夋妧宸. (6)鍒ゅ埆寮忔硶錛氭妸y=f(x)鍙樺艦涓哄叧浜巟鐨勪竴鍏冧簩嬈℃柟紼嬶紝鍒╃敤鈥溾柍鈮0鈥濇眰鍊煎煙.鍏墮樺瀷鐗瑰緛鏄瑙ｆ瀽寮忎腑鍚鏈夋牴寮忔垨鍒嗗紡. (7)鍒╃敤鍑芥暟鐨勫崟璋冩ф眰鍊煎煙錛氬綋鑳界『瀹氬嚱鏁板湪鍏跺畾涔夊煙涓(鎴栨煇涓瀹氫箟鍩熺殑瀛愰泦涓)鐨勫崟璋冩э紝鍙閲囩敤鍗曡皟鎬ф硶奼傚嚭鍑芥暟鐨勫煎煙. (8)鏁板艦緇撳悎娉曟眰鍑芥暟鐨勫煎煙錛氬埄鐢ㄥ嚱鏁版墍琛ㄧず鐨勫嚑浣曟剰涔夛紝鍊熷姪浜庡嚑浣曟柟娉曟垨鍥捐薄錛屾眰鍑哄嚱鏁扮殑鍊煎煙錛屽嵆浠ユ暟褰㈢粨鍚堟眰鍑芥暟鐨勫煎煙. 2銆佹眰鍑芥暟鐨勬渶鍊間笌鍊煎煙鐨勫尯鍒鍜岃仈緋 奼傚嚱鏁版渶鍊肩殑甯哥敤鏂規硶鍜屾眰鍑芥暟鍊煎煙鐨勬柟娉曞熀鏈涓婃槸鐩稿悓鐨勶紝浜嬪疄涓婏紝濡傛灉鍦ㄥ嚱鏁扮殑鍊煎煙涓瀛樺湪涓涓鏈灝(澶)鏁幫紝榪欎釜鏁板氨鏄鍑芥暟鐨勬渶灝(澶)鍊.鍥犳ゆ眰鍑芥暟鐨勬渶鍊間笌鍊煎煙錛屽叾瀹炶川鏄鐩稿悓鐨勶紝鍙鏄鎻愰棶鐨勮掑害涓嶅悓錛屽洜鑰岀瓟棰樼殑鏂瑰紡灝辨湁鎵鐩稿紓. 濡傚嚱鏁扮殑鍊煎煙鏄(0錛16]錛屽兼槸16錛屾棤鏈灝忓.鍐嶅傚嚱鏁扮殑鍊煎煙鏄(-鈭烇紝-2]鈭猍2錛+鈭)錛屼絾姝ゅ嚱鏁版棤鍊煎拰鏈灝忓礆紝鍙鏈夊湪鏀瑰彉鍑芥暟瀹氫箟鍩熷悗錛屽倄>0鏃訛紝鍑芥暟鐨勬渶灝忓間負2.鍙瑙佸畾涔夊煙瀵瑰嚱鏁扮殑鍊煎煙鎴栨渶鍊肩殑褰卞搷. 3銆佸嚱鏁扮殑鏈鍊煎湪瀹為檯闂棰樹腑鐨勫簲鐢 鍑芥暟鐨勬渶鍊肩殑搴旂敤涓昏佷綋鐜板湪鐢ㄥ嚱鏁扮煡璇嗘眰瑙ｅ疄闄呴棶棰樹笂錛屼粠鏂囧瓧琛ㄨ堪涓婂父甯歌〃鐜頒負鈥滃伐紼嬮犱環鏈浣庘濓紝鈥滃埄娑︹濇垨鈥滈潰縐(浣撶Н)(鏈灝)鈥濈瓑璇稿氱幇瀹為棶棰樹笂錛屾眰瑙ｆ椂瑕佺壒鍒鍏蟲敞瀹為檯鎰忎箟瀵硅嚜鍙橀噺鐨勫埗綰︼紝浠ヤ究鑳芥ｇ『奼傚緱鏈鍊. (鍥)銆佸嚱鏁扮殑濂囧伓鎬 1銆佸嚱鏁扮殑濂囧伓鎬х殑瀹氫箟錛氬逛簬鍑芥暟f(x)錛屽傛灉瀵逛簬鍑芥暟瀹氫箟鍩熷唴鐨勪換鎰忎竴涓獂錛岄兘鏈塮(-x)=-f(x)(鎴杅(-x)=f(x))錛岄偅涔堝嚱鏁癴(x)灝卞彨鍋氬囧嚱鏁(鎴栧伓鍑芥暟). 姝ｇ『鐞嗚В濂囧嚱鏁板拰鍋跺嚱鏁扮殑瀹氫箟錛岃佹敞鎰忎袱鐐癸細(1)瀹氫箟鍩熷湪鏁拌醬涓婂叧浜庡師鐐瑰圭О鏄鍑芥暟f(x)涓哄囧嚱鏁版垨鍋跺嚱鏁扮殑蹇呰佷笉鍏呭垎鏉′歡;(2)f(x)=-f(x)鎴杅(-x)=f(x)鏄瀹氫箟鍩熶笂鐨勬亽絳夊紡.(濂囧伓鎬ф槸鍑芥暟瀹氫箟鍩熶笂鐨勬暣浣撴ц川). 2銆佸囧伓鍑芥暟鐨勫畾涔夋槸鍒ゆ柇鍑芥暟濂囧伓鎬х殑涓昏佷緷鎹銆備負浜嗕究浜庡垽鏂鍑芥暟鐨勫囧伓鎬э紝鏈夋椂闇瑕佸皢鍑芥暟鍖栫畝鎴栧簲鐢ㄥ畾涔夌殑絳変環褰㈠紡錛 娉ㄦ剰濡備笅緇撹虹殑榪愮敤錛 (1)涓嶈篺(x)鏄濂囧嚱鏁拌繕鏄鍋跺嚱鏁幫紝f(|x|)鎬繪槸鍋跺嚱鏁; (2)f(x)銆乬(x)鍒嗗埆鏄瀹氫箟鍩烡1銆丏2涓婄殑濂囧嚱鏁幫紝閭ｄ箞鍦―1鈭〥2涓婏紝f(x)+g(x)鏄濂囧嚱鏁幫紝f(x)路g(x)鏄鍋跺嚱鏁幫紝綾諱技鍦版湁鈥滃嚶卞=濂団濃滃嚸楀=鍋垛濓紝鈥滃伓鹵鍋=鍋垛濃滃伓脳鍋=鍋垛濃滃嚸楀伓=濂団; (3)濂囧伓鍑芥暟鐨勫嶅悎鍑芥暟鐨勫囧伓鎬ч氬父鏄鍋跺嚱鏁; (4)濂囧嚱鏁扮殑瀵煎嚱鏁版槸鍋跺嚱鏁幫紝鍋跺嚱鏁扮殑瀵煎嚱鏁版槸濂囧嚱鏁般 3銆佹湁鍏沖囧伓鎬х殑鍑犱釜鎬ц川鍙婄粨璁 (1)涓涓鍑芥暟涓哄囧嚱鏁扮殑鍏呰佹潯浠舵槸瀹冪殑鍥捐薄鍏充簬鍘熺偣瀵圭О;涓涓鍑芥暟涓哄伓鍑芥暟鐨勫厖瑕佹潯浠舵槸瀹冪殑鍥捐薄鍏充簬y杞村圭О. (2)濡傝佸嚱鏁扮殑瀹氫箟鍩熷叧浜庡師鐐瑰圭О涓斿嚱鏁板兼亽涓洪浂錛岄偅涔堝畠鏃㈡槸濂囧嚱鏁板張鏄鍋跺嚱鏁. (3)鑻ュ囧嚱鏁癴(x)鍦▁=0澶勬湁鎰忎箟錛屽垯f(0)=0鎴愮珛. (4)鑻f(x)鏄鍏鋒湁濂囧伓鎬х殑鍖洪棿鍗曡皟鍑芥暟錛屽垯濂(鍋)鍑芥暟鍦ㄦｈ礋瀵圭О鍖洪棿涓婄殑鍗曡皟鎬ф槸鐩稿悓(鍙)鐨勩 (5)鑻f(x)鐨勫畾涔夊煙鍏充簬鍘熺偣瀵圭О錛屽垯F(x)=f(x)+f(-x)鏄鍋跺嚱鏁幫紝G(x)=f(x)-f(-x)鏄濂囧嚱鏁. (6)濂囧伓鎬х殑鎺ㄥ箍 鍑芥暟y=f(x)瀵瑰畾涔夊煙鍐呯殑浠諱竴x閮芥湁f(a+x)=f(a-x)錛屽垯y=f(x)鐨勫浘璞″叧浜庣洿綰縳=a瀵圭О錛屽嵆y=f(a+x)涓哄伓鍑芥暟.鍑芥暟y=f(x)瀵瑰畾涔夊煙鍐呯殑浠-x閮芥湁f(a+x)=-f(a-x)錛屽垯y=f(x)鐨勫浘璞″叧浜庣偣(a錛0)鎴愪腑蹇冨圭О鍥懼艦錛屽嵆y=f(a+x)涓哄囧嚱鏁般 楂樹竴鏁板﹀繀淇浜旂煡璇嗙偣鎬葷粨3 1.鍑芥暟鐨勬傚康錛氳続銆丅鏄闈炵┖鐨勬暟闆嗭紝濡傛灉鎸夌収鏌愪釜紜瀹氱殑瀵瑰簲鍏崇郴f錛屼嬌瀵逛簬闆嗗悎A涓鐨勪換鎰忎竴涓鏁皒錛屽湪闆嗗悎B涓閮芥湁紜瀹氱殑鏁癴(x)鍜屽畠瀵瑰簲錛岄偅涔堝氨縐癴錛欰鈫払涓轟粠闆嗗悎A鍒伴泦鍚圔鐨勪竴涓鍑芥暟.璁頒綔錛歽=f(x)錛寈鈭圓.鍏朵腑錛寈鍙鍋氳嚜鍙橀噺錛寈鐨勫彇鍊艱寖鍥碅鍙鍋氬嚱鏁扮殑瀹氫箟鍩;涓巟鐨勫肩浉瀵瑰簲鐨剏鍊煎彨鍋氬嚱鏁板礆紝鍑芥暟鍊肩殑闆嗗悎{f(x)|x鈭圓}鍙鍋氬嚱鏁扮殑鍊煎煙. 娉ㄦ剰錛氬傛灉鍙緇欏嚭瑙ｆ瀽寮弝=f(x)錛岃屾病鏈夋寚鏄庡畠鐨勫畾涔夊煙錛屽垯鍑芥暟鐨勫畾涔夊煙鍗蟲槸鎸囪兘浣胯繖涓寮忓瓙鏈夋剰涔夌殑瀹炴暟鐨勯泦鍚;鍑芥暟鐨勫畾涔夊煙銆佸煎煙瑕佸啓鎴愰泦鍚堟垨鍖洪棿鐨勫艦寮. 瀹氫箟鍩熻ˉ鍏 鑳戒嬌鍑芥暟寮忔湁鎰忎箟鐨勫疄鏁皒鐨勯泦鍚堢О涓哄嚱鏁扮殑瀹氫箟鍩燂紝奼傚嚱鏁扮殑瀹氫箟鍩熸椂鍒椾笉絳夊紡緇勭殑涓昏佷緷鎹鏄錛 (1)鍒嗗紡鐨勫垎姣嶄笉絳変簬闆; (2)鍋舵℃柟鏍圭殑琚寮鏂規暟涓嶅皬浜庨浂; (3)瀵規暟寮忕殑鐪熸暟蹇呴』澶т簬闆; (4)鎸囨暟銆佸規暟寮忕殑搴曞繀欏誨ぇ浜庨浂涓斾笉絳変簬1. (5)濡傛灉鍑芥暟鏄鐢變竴浜涘熀鏈鍑芥暟閫氳繃鍥涘垯榪愮畻緇撳悎鑰屾垚鐨.閭ｄ箞錛屽畠鐨勫畾涔夊煙鏄浣垮悇閮ㄥ垎閮芥湁鎰忎箟鐨剎鐨勫肩粍鎴愮殑闆嗗悎. (6)鎸囨暟涓洪浂搴曚笉鍙浠ョ瓑浜庨浂 2.鏋勬垚鍑芥暟鐨勪笁瑕佺礌錛氬畾涔夊煙銆佸瑰簲鍏崇郴鍜屽煎煙 鍐嶆敞鎰忥細 (1)鏋勬垚鍑芥暟涓変釜瑕佺礌鏄瀹氫箟鍩熴佸瑰簲鍏崇郴鍜屽煎煙.鐢變簬鍊煎煙鏄鐢卞畾涔夊煙鍜屽瑰簲鍏崇郴鍐沖畾鐨勶紝鎵浠ワ紝濡傛灉涓や釜鍑芥暟鐨勫畾涔夊煙鍜屽瑰簲鍏崇郴瀹屽叏涓鑷達紝鍗崇О榪欎袱涓鍑芥暟鐩哥瓑(鎴栦負鍚屼竴鍑芥暟) (2)涓や釜鍑芥暟鐩哥瓑褰撲笖浠呭綋瀹冧滑鐨勫畾涔夊煙鍜屽瑰簲鍏崇郴瀹屽叏涓鑷達紝鑰屼笌琛ㄧず鑷鍙橀噺鍜屽嚱鏁板肩殑瀛楁瘝鏃犲叧銆傜浉鍚屽嚱鏁扮殑鍒ゆ柇鏂規硶錛氣憼琛ㄨ揪寮忕浉鍚;鈶″畾涔夊煙涓鑷(涓ょ偣蹇呴』鍚屾椂鍏峰) 鍊煎煙琛ュ厖 (1)銆佸嚱鏁扮殑鍊煎煙鍙栧喅浜庡畾涔夊煙鍜屽瑰簲娉曞垯錛屼笉璁洪噰鍙栦粈涔堟柟娉曟眰鍑芥暟鐨勫煎煙閮藉簲鍏堣冭檻鍏跺畾涔夊煙.(2).搴旂啛鎮夋帉鎻′竴嬈″嚱鏁般佷簩嬈″嚱鏁般佹寚鏁般佸規暟鍑芥暟鍙婂悇涓夎掑嚱鏁扮殑鍊煎煙錛屽畠鏄奼傝В澶嶆潅鍑芥暟鍊煎煙鐨勫熀紜.(3).奼傚嚱鏁板煎煙鐨勫父鐢ㄦ柟娉曟湁錛氱洿鎺ユ硶銆佸弽鍑芥暟娉曘佹崲鍏冩硶銆侀厤鏂規硶銆佸潎鍊間笉絳夊紡娉曘佸垽鍒寮忔硶銆佸崟璋冩ф硶絳. 3.鍑芥暟鍥捐薄鐭ヨ瘑褰掔撼 (1)瀹氫箟錛氬湪騫抽潰鐩磋掑潗鏍囩郴涓錛屼互鍑芥暟y=f(x),(x鈭圓)涓鐨剎涓烘í鍧愭爣錛屽嚱鏁板紋涓虹旱鍧愭爣鐨勭偣P(x錛寉)鐨勯泦鍚圕錛屽彨鍋氬嚱鏁皔=f(x),(x鈭圓)鐨勫浘璞. C涓婃瘡涓鐐圭殑鍧愭爣(x錛寉)鍧囨弧瓚沖嚱鏁板叧緋粂=f(x)錛屽弽榪囨潵錛屼互婊¤凍y=f(x)鐨勬瘡涓緇勬湁搴忓疄鏁板箈銆亂涓哄潗鏍囩殑鐐(x錛寉)錛屽潎鍦–涓.鍗寵頒負C={P(x,y)|y=f(x),x鈭圓} 鍥捐薄C涓鑸鐨勬槸涓鏉″厜婊戠殑榪炵畫鏇茬嚎(鎴栫洿綰),涔熷彲鑳芥槸鐢變笌浠繪剰騫寵屼笌Y杞寸殑鐩寸嚎鏈澶氬彧鏈変竴涓浜ょ偣鐨勮嫢騫叉潯鏇茬嚎鎴栫繪暎鐐圭粍鎴. (2)鐢繪硶 A銆佹弿鐐規硶錛氭牴鎹鍑芥暟瑙ｆ瀽寮忓拰瀹氫箟鍩燂紝奼傚嚭x,y鐨勪竴浜涘瑰簲鍊煎苟鍒楄〃錛屼互(x,y)涓哄潗鏍囧湪鍧愭爣緋誨唴鎻忓嚭鐩稿簲鐨勭偣P(x,y)錛屾渶鍚庣敤騫蟲粦鐨勬洸綰垮皢榪欎簺鐐硅繛鎺ヨ搗鏉. B銆佸浘璞″彉鎹㈡硶(璇峰弬鑰冨繀淇4涓夎掑嚱鏁) 甯哥敤鍙樻崲鏂規硶鏈変笁縐嶏紝鍗沖鉤縐誨彉鎹銆佷幾緙╁彉鎹㈠拰瀵圭О鍙樻崲 (3)浣滅敤錛 1銆佺洿瑙傜殑鐪嬪嚭鍑芥暟鐨勬ц川;2銆佸埄鐢ㄦ暟褰㈢粨鍚堢殑鏂規硶鍒嗘瀽瑙ｉ樼殑鎬濊礬銆傛彁楂樿В棰樼殑閫熷害銆 鍙戠幇瑙ｉ樹腑鐨勯敊璇銆 4.蹇鍘諱簡瑙ｅ尯闂寸殑姒傚康 (1)鍖洪棿鐨勫垎綾伙細寮鍖洪棿銆侀棴鍖洪棿銆佸崐寮鍗婇棴鍖洪棿;(2)鏃犵┓鍖洪棿;(3)鍖洪棿鐨勬暟杞磋〃紺. 5.浠涔堝彨鍋氭槧灝 涓鑸鍦幫紝璁続銆丅鏄涓や釜闈炵┖鐨勯泦鍚堬紝濡傛灉鎸夋煇涓涓紜瀹氱殑瀵瑰簲娉曞垯f錛屼嬌瀵逛簬闆嗗悎A涓鐨勪換鎰忎竴涓鍏冪礌x錛屽湪闆嗗悎B涓閮芥湁紜瀹氱殑鍏冪礌y涓庝箣瀵瑰簲錛岄偅涔堝氨縐板瑰簲f錛欰B涓轟粠闆嗗悎A鍒伴泦鍚圔鐨勪竴涓鏄犲皠銆傝頒綔鈥渇錛欰B鈥 緇欏畾涓涓闆嗗悎A鍒癇鐨勬槧灝勶紝濡傛灉a鈭圓,b鈭圔.涓斿厓緔燼鍜屽厓緔燽瀵瑰簲錛岄偅涔堬紝鎴戜滑鎶婂厓緔燽鍙鍋氬厓緔燼鐨勮薄錛屽厓緔燼鍙鍋氬厓緔燽鐨勫師璞 璇存槑錛氬嚱鏁版槸涓縐嶇壒孌婄殑鏄犲皠錛屾槧灝勬槸涓縐嶇壒孌婄殑瀵瑰簲錛屸憼闆嗗悎A銆丅鍙婂瑰簲娉曞垯f鏄紜瀹氱殑;鈶″瑰簲娉曞垯鏈夆滄柟鍚戞р濓紝鍗沖己璋冧粠闆嗗悎A鍒伴泦鍚圔鐨勫瑰簲錛屽畠涓庝粠B鍒癆鐨勫瑰簲鍏崇郴涓鑸鏄涓嶅悓鐨;鈶㈠逛簬鏄犲皠f錛欰鈫払鏉ヨ達紝鍒欏簲婊¤凍錛(鈪)闆嗗悎A涓鐨勬瘡涓涓鍏冪礌錛屽湪闆嗗悎B涓閮芥湁璞★紝騫朵笖璞℃槸鐨;(鈪)闆嗗悎A涓涓嶅悓鐨勫厓緔狅紝鍦ㄩ泦鍚圔涓瀵瑰簲鐨勮薄鍙浠ユ槸鍚屼竴涓;(鈪)涓嶈佹眰闆嗗悎B涓鐨勬瘡涓涓鍏冪礌鍦ㄩ泦鍚圓涓閮芥湁鍘熻薄銆 甯哥敤鐨勫嚱鏁拌〃紺烘硶鍙婂悇鑷鐨勪紭鐐癸細 鍑芥暟鍥捐薄鏃㈠彲浠ユ槸榪炵畫鐨勬洸綰匡紝涔熷彲浠ユ槸鐩寸嚎銆佹姌綰褲佺繪暎鐨勭偣絳夌瓑錛屾敞鎰忓垽鏂涓涓鍥懼艦鏄鍚︽槸鍑芥暟鍥捐薄鐨勪緷鎹;瑙ｆ瀽娉曪細蹇呴』娉ㄦ槑鍑芥暟鐨勫畾涔夊煙;鍥捐薄娉曪細鎻忕偣娉曚綔鍥捐佹敞鎰忥細紜瀹氬嚱鏁扮殑瀹氫箟鍩;鍖栫畝鍑芥暟鐨勮В鏋愬紡;瑙傚療鍑芥暟鐨勭壒寰;鍒楄〃娉曪細閫夊彇鐨勮嚜鍙橀噺瑕佹湁浠ｈ〃鎬э紝搴旇兘鍙嶆槧瀹氫箟鍩熺殑鐗瑰緛. 娉ㄦ剰鍟婏細瑙ｆ瀽娉曪細渚誇簬綆楀嚭鍑芥暟鍊箋傚垪琛ㄦ硶錛氫究浜庢煡鍑哄嚱鏁板箋傚浘璞℃硶錛氫究浜庨噺鍑哄嚱鏁板 琛ュ厖涓錛氬垎孌靛嚱鏁(鍙傝佽炬湰P24-25) 鍦ㄥ畾涔夊煙鐨勪笉鍚岄儴鍒嗕笂鏈変笉鍚岀殑瑙ｆ瀽琛ㄨ揪寮忕殑鍑芥暟銆傚湪涓嶅悓鐨勮寖鍥撮噷奼傚嚱鏁板兼椂蹇呴』鎶婅嚜鍙橀噺浠ｅ叆鐩稿簲鐨勮〃杈懼紡銆傚垎孌靛嚱鏁扮殑瑙ｆ瀽寮忎笉鑳藉啓鎴愬嚑涓涓嶅悓鐨勬柟紼嬶紝鑰屽氨鍐欏嚱鏁板煎嚑縐嶄笉鍚岀殑琛ㄨ揪寮忓苟鐢ㄤ竴涓宸﹀ぇ鎷鍙鋒嫭璧鋒潵錛屽苟鍒嗗埆娉ㄦ槑鍚勯儴鍒嗙殑鑷鍙橀噺鐨勫彇鍊兼儏鍐.(1)鍒嗘靛嚱鏁版槸涓涓鍑芥暟錛屼笉瑕佹妸瀹冭璁や負鏄鍑犱釜鍑芥暟;(2)鍒嗘靛嚱鏁扮殑瀹氫箟鍩熸槸鍚勬靛畾涔夊煙鐨勫苟闆嗭紝鍊煎煙鏄鍚勬靛煎煙鐨勫苟闆. 琛ュ厖浜岋細澶嶅悎鍑芥暟 濡傛灉y=f(u),(u鈭圡),u=g(x),(x鈭圓),鍒檡=f[g(x)]=F(x)錛(x鈭圓)縐頒負f銆乬鐨勫嶅悎鍑芥暟銆
楂樹竴鏁板﹀繀淇浜旂煡璇嗙偣鎬葷粨鐩稿叧 鏂囩珷 錛 鈽 楂樹腑鏁板﹀﹂湼鎻愬垎縐樼睄:蹇呬慨浜旂煡璇嗙偣鎬葷粨 鈽 楂樹腑鏁板﹀繀淇5鏁板垪鐭ヨ瘑鐐規葷粨 鈽 楂樹竴鏁板﹀繀淇浜旀暟鍒楃煡璇嗙偣 鈽 楂樹腑鏁板﹀繀淇5鍏寮忔葷粨 鈽 楂樹腑鏁板﹀繀淇5鍏ㄩ儴鍏寮 鈽 楂樹竴鏁板︾瓑姣旀暟鍒楃煡璇嗙偣鎬葷粨 鈽 楂樹竴鏁板﹀繀淇浜旂瓑姣斾腑欏瑰繀鑰冪煡璇嗙偣 鈽 楂樹竴鏁板﹀繀淇涓鐭ヨ瘑鐐規葷粨 鈽 楂樹腑鏁板﹀繀鑰冪煡璇嗙偣褰掔撼鏁寸悊 鈽 楂樹腑鏁板︽帹鐞嗙煡璇嗙偣鎬葷粨