已知函数f（x）=ex，其图象在点P（2，f（2））处的切线为l．（1）求y=f（x）、直线x=2及两坐标轴围成的图 已知函数f（x）=ex（ax2-2x-2），a∈R且a≠0．...
\u5df2\u77e5\u51fd\u6570fx=e^x,\u5176\u56fe\u50cf\u5728\u70b9p(2,f(2))\u5904\u7684\u5207\u7ebf\u4e3al\uff0c\u6c42\u66f2\u7ebfy=fx\uff0c\u76f4\u7ebfl\u53cay\u8f74\u56f4\u6210f'(x)=e^x
f(2)=e²\uff0cf'(2)=e²
\u5207\u7ebfl\u7684\u65b9\u7a0b\u4e3ay=e²(x-2)+e²
\u5373y=e²x-e²
\u9762\u79ef=\u222b(0,2)[e^x-e²x+e²]dx
=[e^x-e²x²/2+e²x](0, 2)
=[e²-2e²+2e²]-[1]
=e²-1
\u7531\u9898\u610f\u5f97\uff1af'\uff08x\uff09=\uff08ex\uff09'?\uff08ax2-2x-2\uff09+ex?\uff08ax2-2x-2\uff09'=ex(ax2?2x?2)+ex(2ax?2)\uff1daex(x?2a)(x+2)\uff1b\uff083\u5206\uff09\uff081\uff09\u7531\u66f2\u7ebfy=f\uff08x\uff09\u5728\u70b9P\uff082\uff0cf\uff082\uff09\uff09\u5904\u7684\u5207\u7ebf\u5782\u76f4\u4e8ey\u8f74\uff0c\u7ed3\u5408\u5bfc\u6570\u7684\u51e0\u4f55\u610f\u4e49\u5f97f'\uff082\uff09=0\uff0c\u5373a?e2?(2?2a)(2+2)=4ae2?2a?2a\uff1d0\uff0c\u89e3\u5f97a=1\uff1b\uff086\u5206\uff09\uff082\uff09\u8bbe|sinx|=t\uff080\u2264t\u22641\uff09\uff0c\u5219\u53ea\u9700\u6c42\u5f53a\uff1e0\u65f6\uff0c\u51fd\u6570y=f\uff08t\uff09\uff080\u2264t\u22641\uff09\u7684\u6700\u5c0f\u503c\uff0e\u4ee4f'\uff08x\uff09=0\uff0c\u89e3\u5f97x\uff1d2a\u6216x=-2\uff0c\u800ca\uff1e0\uff0c\u53732a\uff1e?2\uff0e\u4ece\u800c\u51fd\u6570f\uff08x\uff09\u5728\uff08-\u221e\uff0c-2\uff09\u548c(2a\uff0c+\u221e)\u4e0a\u5355\u8c03\u9012\u589e\uff0c\u5728(?2\uff0c2a)\u4e0a\u5355\u8c03\u9012\u51cf\uff0e\u5f532a\u22651\u65f6\uff0c\u53730\uff1ca\u22642\u65f6\uff0c\u51fd\u6570f\uff08x\uff09\u5728[0\uff0c1]\u4e0a\u4e3a\u51cf\u51fd\u6570\uff0cymin=f\uff081\uff09=\uff08a-4\uff09e\uff1b\u5f530\uff1c2a\uff1c1\uff0c\u5373 a\uff1e2\u65f6\uff0c\u51fd\u6570f\uff08x\uff09\u7684\u6781\u5c0f\u503c\uff0c\u5373\u4e3a\u5176\u5728\u533a\u95f4[0\uff0c1]\u4e0a\u7684\u6700\u5c0f\u503c\uff0cymin\uff1df(2a)\uff1d?2e2a\uff0e\u7efc\u4e0a\u53ef\u77e5\uff0c\u5f530\uff1ca\u22642\u65f6\uff0c\u51fd\u6570f\uff08|sinx|\uff09\u7684\u6700\u5c0f\u503c\u4e3a\uff08a-4\uff09e\uff1b\u5f53a\uff1e2\u65f6\uff0c\u51fd\u6570f\uff08|sinx|\uff09\u7684\u6700\u5c0f\u503c\u4e3a?2e2a\uff0e\uff0812\u5206\uff09
（1）f（x）=ex、直线x=2及两坐标轴围成的图形绕x轴旋转一周所得几何体的体积是V＝
| ∫ | 2 绛旓細鏍规嵁棰樻剰锛屸埖y=f锛坸锛=ex锛鈭磝=lny锛涘張鈭祔=g锛坸锛=3ex锛屸埓x=lny3锛涒埓A銆丅涓ょ偣涔嬮棿鐨勮窛绂讳负lny-lny3=ln锛坹梅y3锛=ln3锛屾晠绛旀涓猴細ln3 绛旓細y=e^x灏辨槸涓涓櫘閫氱殑鎸囨暟鍑芥暟,缁忚繃(0,1)鐐箉=e^-x灏辨槸灏唝=e^x鐨勫浘鍍忓叧浜巠杞村仛杞村绉板悗鐨鍥惧儚,鍥犱负f(x)=e^x鐨勫浘鍍忎笌f(-x)=e^-x鍏充簬y杞村绉般倅=e^x/x y'=e^x/x-e^x/x=e^x(x-1)/x 浠'=0锛岃В寰梮=1 x<1 鏃讹紝y'<0 x>1 鏃讹紝y'>0 鏁呭嚱鏁 y=e^x/x 鍦 x... 绛旓細=- x2 4 锛屽嵆 x2 4 +etx+et(1-t)=0鏈変袱涓浉绛夌殑瀹炴暟鏍癸紝鈭粹柍=e2t-et(1-t)=0锛宔t+t-1=0锛岃蠁(t)=et+t-1锛屽垯蠁(0)=0锛屼笖蠁鈥(t)=et+1锛0锛屜(t)鍦≧涓婇掑锛屜(t)鍙湁涓涓浂鐐箃=0.鈭村瓨鍦ㄥ敮涓涓鏉＄洿绾縧涓庡嚱鍑芥暟f(x)涓巊(x)鐨鍥捐薄鍧囩浉鍒锛屽叾鏂圭▼涓簓=x+1. 绛旓細瑙ｏ細鈭礷锛坸锛=ex 鈭磃锛坸1+x2锛=ex1+x2=ex1•ex2=f锛坸1锛•f锛坸2锛夛紝鏁呪憼姝ｇ‘锛涒埖f锛坸1•x2锛=ex1•x2锛沠锛坸1锛+f锛坸2锛=ex1+ex2锛屾晠鈶￠敊璇紱鈭礶锛1锛屾晠鍑芥暟f锛坸锛=ex涓哄鍑芥暟锛屾晠f(x1)-f(x2)x1-x2锛0鎴愮珛锛屽嵆鈶㈡纭紱鑰鍑芥暟f锛坸锛=ex鍥捐薄涓... 绛旓細锛坅x2-2x-2锛夆=ex?锛坅x2-2x-2锛+ex?锛2ax-2锛=a?ex?锛坸-2a锛夛紙x+2锛夛紟鈭垫洸绾縴=f锛坸锛鍦ㄧ偣P锛2锛宖锛2锛夛級澶勭殑鍒囩嚎鍨傜洿浜巠杞达紝鐢卞鏁扮殑鍑犱綍鎰忎箟寰梖鈥诧紙2锛=0锛屸埓a=1锛庘埓瀹炴暟a鐨勫间负锛1锛庯紙2锛夌敱锛1锛夊彲鐭ヨ|sinx|=t锛岋紙0鈮鈮1锛夛紝鍒欒浆鍖栦负姹鍑芥暟y=f锛坱锛夛紝锛0鈮... 绛旓細瑙ｏ細濡傚浘锛岀洿绾縴=x-a涓鍑芥暟y=f锛坸锛=ex-1鐨鍥捐薄鍦▁鈮0澶勬湁涓涓垏鐐癸紝鍒囩偣鍧愭爣涓猴紙0锛0锛夛紝姝ゆ椂a=0锛涚洿绾縴=|x-a|涓庡嚱鏁皔=-x2-2x鐨勫浘璞″湪x锛0澶勬湁涓や釜鍒囩偣锛屽垏鐐瑰潗鏍囧垎鍒槸(?12锛34)鍜(?32锛34)锛屾鏃剁浉搴旂殑a锛14锛宎锛?94锛岃瀵熷浘璞″彲鐭ワ紝鏂圭▼f锛坸锛=|x-a|鏈変笁涓笉鍚岀殑瀹炴牴... 绛旓細锛1锛夎瘉鏄庯細鈭12[f(x1)+f(x2)]?f(x1+x22)=12(ex1+ex2)?ex1+x22=12(ex1+ex2?2ex1+x22)=12(ex12?ex22)2鈮0锛屸埓12[f(x1)+f(x2)]鈮(x1+x22)锛庯紙2锛夎В锛氳鐩寸嚎l涓鍑芥暟f锛坸锛鐨鍥捐薄鐩稿垏锛屽垏鐐逛负锛坱锛宔t锛夛紝鍒欑洿绾縧鐨勬柟绋嬩负y-et=et锛坸-t锛夛紝鍗硑=etx+et锛1-t锛夛紝鐩寸嚎... 绛旓細鍑芥暟y=ex鐨鍥捐薄涓庡嚱鏁皔=f锛坸锛夌殑鍥捐薄鍏充簬鐩寸嚎y=x瀵圭О锛屾墍浠锛坸锛夋槸y=ex鐨勫弽鍑芥暟锛鍗f锛坸锛=lnx锛屸埓f锛2x锛=ln2x=lnx+ln2锛坸锛0锛夛紝閫塂锛 绛旓細璁惧垏鐐筆锛坸锛寉锛夆埖f鈥锛坸锛=ex鐢卞鏁扮殑鍑犱綍鎰忎箟鍙緱鍒囩嚎鐨勬枩鐜噆=e=ex鈭磝=1锛寉=e鍗冲垏鐐瑰潗鏍囷紙1锛宔锛夋晠绛旀涓猴紙1锛宔锛 绛旓細ex绠浠嬶細鍏跺浘鍍鏄崟璋冮掑锛寈鈭圧锛寉>0锛屼笌y杞寸浉浜や簬锛0锛1锛夌偣锛屽浘鍍忎綅浜嶺杞翠笂鏂癸紝绗簩璞￠檺鏃犻檺鎺ヨ繎X杞淬 瑙ｏ細y=ex鏄簳鏁颁负鑷劧瀵规暟e锛屾寚鏁颁负x鐨勬寚鏁鍑芥暟锛宔绾︾瓑浜2.87>1鍗曡皟閫掑銆俥x濂囧伓鎬э細ex鏃笉鏄鍑芥暟锛屼篃涓嶆槸鍋跺嚱鏁般f(x)= ex 锛f(-x)= e-x 锛-f(x)=- ex 锛宖(x)鈮爁... 本站交流只代表网友个人观点，与本站立场无关 |
