(书上一句话)设函数y=f(x)在x0点的某邻域内有定义.什么是有定义? 函数 y=f(x)在点Xo的某一领域内有定义的问题
\u201cy=F\uff08X\uff09\u5728\u70b9X\u96f6 \u7684\u67d0\u4e00\u90bb\u57df\u5185\u6709\u5b9a\u4e49 \u201d \u60f3\u8981\u8bf4\u660e\u4ec0\u4e48\u5728\u70b9X0\u7684\u67d0\u4e00\u90bb\u57df\u5185\u6709\u5b9a\u4e49\u8bf4\u660e\uff1a
1.f(x0)\u5b58\u5728
2.\u5982\u679c\u77e5\u9053f(x)\u7684\u5177\u4f53\u8868\u8fbe\u5f0f\u5219\u53ef\u4ee5\u7528\u5bfc\u6570\u7684\u5b9a\u4e49\u5224\u65ad\u5728\u70b9x0\u5904f(x)\u662f\u5426\u53ef\u5230\uff0c\u5982\u679c\u5bfc\u6570\u5b58\u5728\uff0c\u5bfc\u6570\u503c\u662f\u591a\u5c11
\u4e0d\u53ef\u4ee5\u7b80\u5355\u8ba4\u4e3a\u201c\u67d0\u90bb\u57df\u201d\u4e3a\u8be5\u51fd\u6570\u7684\u5b9a\u4e49\u57df\u3002\u90bb\u57df\u9996\u5148\u5c31\u662f\u4e00\u4e2a\u6781\u9650\u7684\u6982\u5ff5\uff0c\u7b80\u5355\u5730\u8bf4\u5c31\u662f \u4e00\u4e2a\u70b9\u53ca\u6b64\u70b9\u5de6\u53f3\u4e24\u4fa7\u65e0\u7a77\u5c0f\u7684\u8303\u56f4 \u90bb\u57df\u53ef\u4ee5\u4f5c\u4e3a\u5b9a\u4e49\u57df\u7684\u4e00\u90e8\u5206 \u4f46\u4ec5\u51ed\u5728\u70b9X0\u7684\u67d0\u4e00\u90bb\u57df\u5185\u6709\u5b9a\u4e49 \u662f\u65e0\u6cd5\u786e\u5b9af(x)\u7684\u5b9a\u4e49\u57df\u7684
\u53ea\u80fd\u8bf4 \u6b64\u90bb\u57df\u5305\u542b\u5728f(x)\u7684\u5b9a\u4e49\u57df\u5185 \u4f46\u65e0\u6cd5\u77e5\u9053f(x)\u5b9a\u4e49\u57df
\u51fd\u6570 y=f(x)\u5728\u70b9Xo\u7684\u67d0\u4e00\u9886\u57df\u5185\u6709\u5b9a\u4e49 ,\u5c31\u662f\u5f53x=Xo\u65f6\uff0c\u51fd\u6570 y=f(x)\u5177\u6709\u786e\u5b9a\u7684\u503c.
\u4ea6\u5373\u5728x=Xo\u65f6,\u51fd\u6570 y=f(x)\u6709\u610f\u4e49\u3002\u3002\u3002\u3002\u6211\u4eec\u8981\u5148\u7406\u89e3\u9886\u57df\u7684\u542b\u4e49\uff0c \u5728\u6570\u5b66\u5206\u6790\u91cc\u4e00\u7ef4\u7a7a\u95f4\u4e2d\u7684\u9886\u57df\u5176\u5b9e\u5c31\u662f\u6570\u8f74\u4e0a\u7684\u4e00\u4e2a\u5f00\u533a\u95f4\uff0c\uff0c\uff0c\uff0c\u5728\u67d0\u4e00\u9886\u57df\u5185\u662f\u5b83\u7684\u524d\u63d0\uff0c,\u5426\u5219\u65e0\u8bba\u25b3X\u53d6\u591a\u5c0f,\u90fd\u53ef\u80fd\u662f\u95f4\u65ad\u7684\u8303\u56f4\u3002\u800c\u4e14\u5f88\u5bb9\u6613\u5c31\u80fd\u4e3e\u51fa\u53cd\u4f8b\u7684\u51fd\u6570\uff0cy=x(x\u5b9a\u4e49\u57df\u662f\u5168\u4f53\u6709\u7406\u6570),\u8fd9\u4e2a\u51fd\u6570\u80af\u5b9a\u4e0d\u8fde\u7eed\uff0c\u800c\u4e14\u6709\u65e0\u7a77\u4e2a\u95f4\u65ad\u70b9\uff0c\u56e0\u4e3a\u6211\u4eec\u53ef\u4ee5\u8bbe\u25b3X=1/n\uff0cn\u662f\u6574\u6570\uff0c\u8fd9\u6837\u51fd\u6570\u6c38\u8fdc\u90fd\u6709\u610f\u4e49\uff0c\u5f53n\u8d8b\u4e8e\u65e0\u7a77\uff0c\u25b3X\u8d8b\u4e8e\u96f6\uff0c\u25b3Y\u4e5f\u8d8b\u4e8e\u96f6\u3002
\u5e0c\u671b\u80fd\u7406\u89e3\uff0c\uff0c\u671b\u91c7\u7eb3
什么是有定义?
在x0点有定义就是允许自变量取X0这个值。
在x0点的某邻域内有定义就是允许自变量取X0附近的值。
绛旓細浣犻渶瑕佹煡涓涓嬶紙涓嶆煡涔熷彲浠ワ紝浣嗚鑷繁绠楋級涓嶅畾绉垎琛,鍑芥暟f(x)=鈭氾紙1+4x^2锛夊搴旂殑涓嶅畾绉垎缁撴灉:F(x)=鈭氾紙x^2+0.25)+(ln|x+鈭氾紙x^2+0.25)|)/4,閭ｄ箞浣犺姹傜殑"鍑芥暟闀垮害"涓篎(1)-F(0).璁惧嚱鏁皔=f(x)鍦ㄥ尯闂碵a,b]涓婂彲瀵,鍏跺鍑芥暟涓篺'(x),閭ｄ箞f(x)鍦╗a,b]涓婄殑涓绔洸绾...
绛旓細鍥犱负鏉′欢1鍙槸婊¤冻f(x)鈮鎵浠ワ紝鍦1鏉′欢涓嬬殑f(x锛鐨勫彇鍊煎畬鍏ㄥ彲浠ヤ笉绛変簬M锛屾潯浠2鏄檺瀹氫簡鍦ㄨ寖鍥碔涓繀鏈変竴涓彇鍊间娇鍑芥暟鍊间负M
绛旓細F(XY)=F(X)+F(Y)浠诲彇0<x1<x2,鍒檟2/x1>1 鈭磃(x2)=f(x1*x2/x1)=f(x1)+f(x2/x1)鈭磃(x2)-f(x1)=f(x2/x1)鈭靛綋x>1鏃 F(X)<0 鈭碏(x2/x1)<0 鍗砯(x2)-f(x1)<0 鈭磃(x1)>f(x2)鈭碏(x)鍦(0,姝ｆ棤绌)鏄噺鍑芥暟 绗2闂В涓嶇瓑寮忕敤F(1/3)=1 ...
绛旓細fx琛ㄧず鍑芥暟銆備竴鑸殑锛岃鍦ㄦ煇涓彉鍖栬繃绋嬩腑锛屾湁2涓彉閲弜銆y锛屽鏋滃浜巟鍦ㄦ煇涓鑼冨洿鍐呯殑姣忎竴涓‘瀹氱殑鍊硷紝鎸夌収鏌愪釜瀵瑰簲娉曞垯f锛寉閮芥湁涓涓敮涓纭畾鐨勫间笌鍏跺搴旓紝閭ｄ箞灏辩Оx涓鸿嚜鍙橀噺锛寉鏄痻鐨勫嚱鏁般傝浣 锛涘湪璇ュ嚱鏁颁腑锛寈鐨勫彇鍊艰寖鍥存瀯鎴愮殑闆嗗悎绉颁负璇ュ嚱鏁扮殑瀹氫箟鍩燂紱y鐨勫彇鍊艰寖鍥存瀯鎴愮殑闆嗗悎绉颁负璇ュ嚱鏁扮殑...
绛旓細f(y)=f(xy/x)=f(xy)-f(x)閭ｄ箞f(x)+f(y)=f(xy)f(x)-f[1/(x-3)]鈮2 f[x(x-3)]鈮(2)+f(2)f(x²-3x)鈮(4)鍥犱负y=f(x)鍗曡皟閫掑 鎵浠²-3x鈮4 -1鈮鈮4 瑙ｉ泦涓篬-1,4]
绛旓細=f(1-x)鏄垚绔嬬殑銆傛墍浠=1鏄叾瀵圭О杞淬傜涓夐棶锛氫富瑕佹槸鐢ㄥ埌鍛ㄦ湡鍑芥暟浜嗐傚苟涓旂敱f(x+2)=-f(x),f(x+4)=f(x).鍋氫竴涓嬭浆鍖栧氨鍙互姹傚嚭瑙ｆ瀽寮忎簡銆傚拰涓婁竴闂殑杞寲宸笉澶...绗洓闂細鍙鎶婄涓夐棶鐨勮В鏋愬紡灏卞嚭鏉ワ紝杩欎釜闂灏卞緢瀹规槗浜...鍙璁╁皬浜庢眰鍑哄嚱鏁扮粷瀵瑰肩殑鏈灏忓煎氨OK浜......
绛旓細鍙嶅嚱鏁扮櫨搴︾櫨绉 璇峰弬鑰 http://baike.baidu.com/view/359.htm y=f-1(x)鎸囩殑鏄痻鍜寉鎹綅缃 -1鎸囩殑涓嶆槸娆℃柟 渚嬪瓙 f(x)涓 y=2x+1 y-1=2x 1/2y-1/2=x 閭ｄ箞 鍙鍑芥暟f-1(x)涓 y=1/2x-1/2
绛旓細鍑芥暟鐨勮繛缁у彲浠ラ氳繃瀹氫箟鏉ヨ璁恒傝繛缁х殑瀹氫箟锛璁惧嚱鏁f锛坸锛夊湪鐐x=x0鐨勬煇閭诲煙鍐呮湁瀹氫箟锛屽鏋滃綋鑷彉閲忕殑澧為噺螖x=x−x0瓒嬩簬闆舵椂锛屽搴旂殑鍑芥暟澧為噺螖y=f锛坸0+螖x锛−f锛坸0锛変篃瓒嬩簬闆讹紝鍗砽im螖x鈫0螖y=0锛屽垯绉鍑芥暟y=f锛坸锛鍦▁0鐐硅繛缁傛澶栵紝杩樻湁涓浜涘叧浜庤繛缁х殑鍏朵粬鎬ц川锛...
绛旓細x] f(t)dt+xf(x)-2xf(x)=鈭玔0,x] f(t)dt-xf(x)F''(x)=f(x)-f(x)-xf'(x)=-xf'(x) 鐢遍鎰忕煡f'(x)<0鎭掓垚绔 x>0鏃,F''(x)>0,x<0鏃,F''(x)<0,鎵浠'(x)鏈灏=F'(0),鎵浠'(x)鈮'(0)=0,褰撲笖浠呭綋x=0鏃跺彇鍒扮瓑鍙.鎵浠(x)鍦≧涓婂崟璋冮掑 ...
绛旓細绛旀鏄-1/e鏂广y = ƒ(x) x = 蠄(y)dx/dy = 蠄'(y) = 1/(dy/dx) = 1/ƒ'(x)d²x/dy² = 蠄''(y) = d(dx/dy)/dy = d[1/(dy/dx)]/dy = [1/ƒ'(x)]' = - ƒ''(x)/[ƒ'(x)]²ƒ(x) = 鈭(1鈫...
