常用的麦克劳林级数展开式
e^x=1+x+x^2/2!+x^3/3!+...+x^n/n!+...1/(1-x)=1+x+x^2+...+x^n+...
sinx=x-x^3/3!+x^5/5!+...+(-1)^n*x^(2n+1)/(2n+1)!+...
用kx代替上式中的x即可。
绛旓細楹﹀厠鍔虫灄鍏紡鏄娉板嫆鍏紡鐨勪竴绉嶇壒娈婂舰寮忋
绛旓細绉垎褰㈠紡灞曞紑锛氳嫢鍑芥暟f(z)鍦ㄥ骞抽潰鍐呭叿鏈楹﹀厠鍔虫灄绾ф暟灞曞紑寮锛屽垯璇ュ嚱鏁板湪骞抽潰涓婁换涓鐐瑰鍧囦竴鑷村湴浠ラ害鍏嬪姵鏋楃骇鏁版敹鏁涳紝涓斿叾鏀舵暃鍗婂緞涓烘棤绌峰ぇ銆備竴鑷存敹鏁涘畾鐞嗭細鑻ュ嚱鏁癴(z)鍦ㄤ互鍘熺偣涓轰腑蹇冪殑鏌愪竴鍖哄煙鍐呬竴鑷存敹鏁涳紝鍒欏叾楹﹀厠鍔虫灄绾ф暟灞曞紑寮忓湪璇ュ尯鍩熷唴涓鑷存敹鏁涖備互涓婂氨鏄嘲鍕掑睍寮寮忕殑涓浜甯哥敤鍏紡锛屽笇鏈涘浣犳湁鎵...
绛旓細绾ф暟灞曞紑鍏紡鏄埆cosxdx=sinx+C銆佲埆sinxdx=-cosx+C銆佲埆(secx)^2dx=tanx+C銆楹﹀厠鍔虫灄绾ф暟(Maclaurin's series)鏄娉板嫆绾ф暟(Taylor's series)鐨勭壒娈婃儏鍐碉紝鍗冲綋a=0鏃讹紝f(x)鐨勫睍寮寮忋傝繖绫诲叕寮忎笉闇瑕佺壒鎰忓幓鑳岃锛屽畠寰堥暱锛屼篃寰堝鏄撹娣枫傛渶濂界殑鍔炴硶灏辨槸鑷繁灏濊瘯鎺ㄥ銆傛湁绌锋暟鍒楃殑绾ф暟涓鑸氳繃鍒濈瓑浠ｆ暟...
绛旓細4.2锛夋暟瀛︾悊璁哄垎鏋愬拰璁＄畻 娉板嫆绾ф暟灞曞紑寮灏嗙畝鍗曠殑鍑芥暟寮忓瓙鍖栦负鏃犵┓澶氶」骞傚嚱鏁帮紝鐪嬩技鍖栫畝涓虹箒銆備絾浜嬪疄涓婃嘲鍕掔骇鏁板彲浠ヨВ鍐冲緢澶氭暟瀛﹂棶棰樸傚锛氣憼姹傛瀬闄愭椂鍙互鐢ㄥ嚱鏁鐨勯害鍏嬪姵鏋鍏紡(娉板嫆灞曞紑寮忕殑鐗规畩褰㈠紡)锛涒憽涓浜涢毦浠ョН鍒嗙殑鍑芥暟锛屽皢鍑芥暟娉板嫆灞曞紑鍙樹负骞傜骇鏁帮紝浣垮叾瀹规槗绉垎锛涒憿澶嶆潅绂绘暎鍑芥暟鐨勫椤瑰紡鎷熷悎锛岀敤浜庣粺璁...
绛旓細鐢楹﹀厠鍔虫灄灞曞紡锛屽垎瀛愬墠涓椤癸紝鍒嗘瘝鍓嶄笁椤瑰嵆鍙傚師寮=lim锛坸鈫0锛墈x^3/[x-(x-x^3/6)]}=6 xy'-y=0 => y'=y/x =>dy/y=dx/x =>ln|y|=ln|x|+lnC =>y=Cx锛屽幓缁濆鍊煎悗鐨勭鍙峰寘鍚湪C褰撲腑銆傛垨 xy'-y=0锛屽嵆 (xy'-y)/y^2=0,鍗 (y/x)'=0 y/x=C y=cx ...
绛旓細鑻ュ嚱鏁癴(x)鍦ㄥ寘鍚玿0鐨勬煇涓棴鍖洪棿[a,b]涓婂叿鏈塶闃跺鏁帮紝涓斿湪寮鍖洪棿(a,b)涓婂叿鏈(n+1)闃跺鏁帮紝鍒欏闂尯闂碵a,b]涓婁换鎰忎竴鐐箈锛屾垚绔嬩笅寮忥細鍏朵腑
绛旓細1銆乤rctanx鐨勯害鍏嬪姵鏋楃骇鏁板睍寮寮锛屽繀椤诲垎涓夋鑰冭檻锛氾紞鈭炩墹 x 鈮-1銆侊紞1 < x < +1銆1 < x < +鈭 2銆佸垎鎴愪笁娈电殑鍘熷洜鏄細(1)銆佸湪灞曞紑杩囩▼涓紝蹇呴』鍏堟眰瀵硷紝鍐嶇Н鍒嗭紱(2)銆佸湪姹傚璺熺Н鍒嗕箣闂达紝蹇呴』杩愮敤鍏瘮灏忎簬1鐨勬棤绌风瓑姣旀暟鍒楁眰鍜屽叕寮忥紱(3)銆佽繍鐢ㄧ瓑姣旀眰鍜屽叕寮忔椂锛屽繀椤昏冭檻鏀舵暃涓庡惁锛屽洜姝ゅ繀椤诲垎鎴...
绛旓細纭畾灞曞紑鐐癸細閫夋嫨涓涓睍寮鐐癸紝閫氬父鏄嚱鏁扮殑鏌愪釜鐗瑰畾鍊笺甯歌鐨閫夋嫨鏄浂鐐癸紝鍗冲睍寮鐐逛负x = 0锛岃繖鏃跺欐嘲鍕掔骇鏁颁篃琚О涓楹﹀厠鍔虫灄绾ф暟銆傝绠楀嚱鏁板湪灞曞紑鐐圭殑鍚勯樁瀵兼暟锛氳绠楀嚱鏁板湪灞曞紑鐐圭殑0闃跺埌n闃跺鏁帮紝鍏朵腑n鏄綘甯屾湜灞曞紑鐨绾ф暟鐨勯樁鏁般傝绠楃骇鏁颁腑鐨勭郴鏁帮細灏嗚绠楀緱鍒扮殑瀵兼暟浠ｅ叆娉板嫆绾ф暟鐨勫叕寮涓紝绯绘暟涓哄嚱鏁...
绛旓細8涓甯哥敤娉板嫆鍏紡锛屽涓嬪浘鎵绀猴細鍦ㄦ暟瀛︿腑锛娉板嫆绾ф暟鐢ㄦ棤闄愰」杩炲姞寮忊斺旂骇鏁版潵琛ㄧず涓涓嚱鏁帮紝杩欎簺鐩稿姞鐨勯」鐢卞嚱鏁板湪鏌愪竴鐐圭殑瀵兼暟姹傚緱銆傜浉鍏充俊鎭細娉板嫆鍏紡寰楀悕浜庤嫳鍥芥暟瀛﹀甯冮瞾鍏嬄锋嘲鍕掋備粬鍦1712骞寸殑涓灏佷俊閲岄娆″彊杩颁簡杩欎釜鍏紡锛屽敖绠1671骞磋┕濮嗘柉路鏍奸浄楂橀噷宸茬粡鍙戠幇浜嗗畠鐨勭壒渚嬨傛媺鏍兼湕鏃ュ湪1797骞翠箣鍓嶏紝鏈鍏...
绛旓細sinx鐢娉板嫆鍏紡灞曞紑鏄痵inx=x-1/3锛亁^3+1/5锛亁^5+o锛坸 ^5锛夈甯哥敤鐨勬嘲鍕鍏紡灞曞紑寮涓猴細Fx=fx0/0锛+f锛坸0锛/1锛侊紙x-x0锛+f锛坸0锛/2锛侊紙x-x0锛+...+f锛坸0锛/n锛侊紙x-x0锛塶娆℃柟+Rn锛坸锛夈傞珮绛夋暟瀛︿腑鐨勫簲鐢 鍦ㄩ珮绛夋暟瀛︾殑鐞嗚鐮旂┒鍙婂簲鐢ㄥ疄璺典腑,娉板嫆鍏紡鏈夌潃鍗佸垎閲嶈鐨勫簲鐢紝...
