鍙嬫儏鎻愮ず锛氭湰绔欐彁渚涘叏鍦�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�鈥� 涓嬭級
绗竴绔� 鍑芥暩(sh霉)鑸囨サ闄�
1銆佸嚱鏁�(sh霉)鐨勬湁鐣屾€у湪瀹氱京鍩熷収(n猫i)鏈塮(x)鈮1鍓囧嚱鏁�(sh霉)f(x)鍦ㄥ畾缇╁煙涓婃湁涓嬬晫锛孠1鐐轰笅鐣岋紱濡傛灉鏈塮(x)鈮2锛屽墖鏈変笂鐣岋紝K2绋辩偤涓婄晫銆傚嚱鏁�(sh霉)f(x)鍦ㄥ畾缇╁煙鍏�(n猫i)鏈夌晫鐨勫厖鍒嗗繀瑕佹浠舵槸鍦ㄥ畾缇╁煙鍏�(n猫i)鏃㈡湁涓婄晫鍙堟湁涓嬬晫銆�
2銆佹暩(sh霉)鍒楃殑妤甸檺瀹氱悊(妤甸檺鐨勫敮涓€鎬�)鏁�(sh霉)鍒梴xn}涓嶈兘鍚屾檪鏀舵杺浜庡叐鍊嬩笉鍚岀殑妤甸檺銆�
瀹氱悊(鏀舵杺鏁�(sh霉)鍒楃殑鏈夌晫鎬�)濡傛灉鏁�(sh霉)鍒梴xn}鏀舵杺锛岄偅涔堟暩(sh霉)鍒梴xn}涓€瀹氭湁鐣�銆�
濡傛灉鏁�(sh霉)鍒梴xn}鐒$晫锛岄偅涔堟暩(sh霉)鍒梴xn}涓€瀹氱櫦(f膩)鏁o紱浣嗗鏋滄暩(sh霉)鍒梴xn}鏈夌晫锛屽嵒涓嶈兘鏂峰畾鏁�(sh霉)鍒梴xn}涓€瀹氭敹鏂�锛屼緥濡傛暩(sh霉)鍒�1锛�-1锛�1锛�-1锛�(-1)n+1鈥﹁┎鏁�(sh霉)鍒楁湁鐣屼絾鏄櫦(f膩)鏁�锛屾墍浠ユ暩(sh霉)鍒楁湁鐣屾槸鏁�(sh霉)鍒楁敹鏂傜殑蹇呰姊濅欢鑰屼笉鏄厖鍒嗘浠�銆�
瀹氱悊(鏀舵杺鏁�(sh霉)鍒楄垏鍏跺瓙鏁�(sh霉)鍒楃殑闂�(gu膩n)绯�)濡傛灉鏁�(sh霉)鍒梴xn}鏀舵杺浜巃锛岄偅涔堝畠鐨勪换涓€瀛愭暩(sh霉)鍒椾篃鏀舵杺浜巃.濡傛灉鏁�(sh霉)鍒梴xn}鏈夊叐鍊嬪瓙鏁�(sh霉)鍒楁敹鏂備簬涓嶅悓鐨勬サ闄�锛岄偅涔堟暩(sh霉)鍒梴xn}鏄櫦(f膩)鏁g殑锛屽鏁�(sh霉)鍒�1锛�-1锛�1锛�-1锛�(-1)n+1鈥︿腑瀛愭暩(sh霉)鍒梴x2k-1}鏀舵杺浜�1锛寋xnk}鏀舵杺浜�-1锛寋xn}鍗绘槸鐧�(f膩)鏁g殑锛涘悓鏅備竴鍊嬬櫦(f膩)鏁g殑鏁�(sh霉)鍒楃殑瀛愭暩(sh霉)鍒椾篃鏈夊彲鑳芥槸鏀舵杺鐨�銆�
3銆佸嚱鏁�(sh霉)鐨勬サ闄愬嚱鏁�(sh霉)妤甸檺鐨勫畾缇╀腑0<|x-x0|琛ㄧずx鈮爔0锛屾墍浠鈫抶0鏅俧(x)鏈夋矑鏈夋サ闄愯垏f(x)鍦ㄩ粸(di菐n)x0鏈夋矑鏈夊畾缇╃劇闂�(gu膩n)銆�
瀹氱悊(妤甸檺鐨勫眬閮ㄤ繚铏熸€�)濡傛灉lim(x鈫抶0)鏅俧(x)=A锛岃€屼笖A>0(鎴朅<0)锛屽氨瀛樺湪钁楅粸(di菐n)閭d箞x0鐨勬煇涓€鍘诲績閯板煙锛岀暥(d膩ng)x鍦ㄨ┎閯板煙鍏�(n猫i)鏅傚氨鏈塮(x)>0(鎴杅(x)>0)锛屽弽涔嬩篃鎴愮珛銆�
鍑芥暩(sh霉)f(x)鐣�(d膩ng)x鈫抶0鏅傛サ闄愬瓨鍦ㄧ殑鍏呭垎蹇呰姊濅欢鏄乏妤甸檺鍙虫サ闄愬悇鑷瓨鍦ㄥ苟涓旂浉绛�锛屽嵆f(x0-0)=f(x0+0)锛岃嫢涓嶇浉绛夊墖limf(x)涓嶅瓨鍦�銆�
涓€鑸殑瑾紝濡傛灉lim(x鈫掆垶)f(x)=c锛屽墖鐩寸窔y=c鏄嚱鏁�(sh霉)y=f(x)鐨勫湒褰㈡按骞虫几杩戠窔銆傚鏋渓im(x鈫抶0)f(x)=鈭�锛屽墖鐩寸窔x=x0鏄嚱鏁�(sh霉)y=f(x)鍦栧舰鐨勯墰鐩存几杩戠窔銆�
4銆佹サ闄愰亱(y霉n)绠楁硶鍓囧畾鐞嗘湁闄愬€嬬劇绐皬涔嬪拰涔熸槸鐒$??锛涙湁鐣屽嚱鏁�(sh霉)鑸囩劇绐皬鐨勪箻绌嶆槸鐒$灏�锛涘父鏁�(sh霉)鑸囩劇绐皬鐨勪箻绌嶆槸鐒$??锛涙湁闄愬€嬬劇绐皬鐨勪箻绌嶄篃鏄劇绐皬锛涘畾鐞嗗鏋淔1(x)鈮2(x)锛岃€宭imF1(x)=a锛宭imF2(x)=b锛岄偅涔坅鈮.
5銆佹サ闄愬瓨鍦ㄦ簴(zh菙n)鍓囧叐鍊嬮噸瑕佹サ闄恖im(x鈫�0)(sinx/x)=1锛沴im(x鈫掆垶)(1+1/x)x=1.澶鹃€兼簴(zh菙n)鍓囧鏋滄暩(sh霉)鍒梴xn}銆亄yn}銆亄zn}婊胯冻涓嬪垪姊濅欢锛歽n鈮n鈮n涓攍imyn=a锛宭imzn=a锛岄偅涔坙imxn=a锛屽皪浜庡嚱鏁�(sh霉)瑭叉簴(zh菙n)鍓囦篃鎴愮珛銆�
鍠(di脿o)鏈夌晫鏁�(sh霉)鍒楀繀鏈夋サ闄�銆�
6銆佸嚱鏁�(sh霉)鐨勯€g簩(x霉)鎬цō(sh猫)鍑芥暩(sh霉)y=f(x)鍦ㄩ粸(di菐n)x0鐨勬煇涓€閯板煙鍏�(n猫i)鏈夊畾缇�锛屽鏋滃嚱鏁�(sh霉)f(x)鐣�(d膩ng)x鈫抶0鏅傜殑妤甸檺瀛樺湪锛屼笖绛変簬瀹冨湪榛�(di菐n)x0铏曠殑鍑芥暩(sh霉)鍊糵(x0)锛屽嵆lim(x鈫抶0)f(x)=f(x0)锛岄偅涔堝氨绋卞嚱鏁�(sh霉)f(x)鍦ㄩ粸(di菐n)x0铏曢€g簩(x霉)銆�
涓嶉€g簩(x霉)鎯呭舰锛�1銆佸湪榛�(di菐n)x=x0娌掓湁瀹氱京锛�2銆侀洊鍦▁=x0鏈夊畾缇╀絾lim(x鈫抶0)f(x)涓嶅瓨鍦紱3銆侀洊鍦▁=x0鏈夊畾缇╀笖lim(x鈫抶0)f(x)瀛樺湪锛屼絾lim(x鈫抶0)f(x)鈮爁(x0)鏅傚墖绋卞嚱鏁�(sh霉)鍦▁0铏曚笉閫g簩(x霉)鎴栭枔鏂�銆�
濡傛灉x0鏄嚱鏁�(sh霉)f(x)鐨勯枔鏂烽粸(di菐n)锛屼絾宸︽サ闄愬強鍙虫サ闄愰兘瀛樺湪锛屽墖绋眡0鐐哄嚱鏁�(sh霉)f(x)鐨勭涓€椤為枔鏂烽粸(di菐n)(宸﹀彸妤甸檺鐩哥瓑鑰呯ū鍙幓闁撴柗榛�(di菐n)锛屼笉鐩哥瓑鑰呯ū鐐鸿烦韬嶉枔鏂烽粸(di菐n))銆傞潪绗竴椤為枔鏂烽粸(di菐n)鐨勪换浣曢枔鏂烽粸(di菐n)閮界ū鐐虹浜岄闁撴柗榛�(di菐n)(鐒$闁撴柗榛�(di菐n)鍜岄渿钑╅枔鏂烽粸(di菐n))銆�
瀹氱悊鏈夐檺鍊嬪湪鏌愰粸(di菐n)閫g簩(x霉)鐨勫嚱鏁�(sh霉)鐨勫拰銆佺⿳銆佸晢(鍒嗘瘝涓嶇偤0)鏄€嬪湪瑭查粸(di菐n)閫g簩(x霉)鐨勫嚱鏁�(sh霉)銆�
瀹氱悊濡傛灉鍑芥暩(sh霉)f(x)鍦ㄥ崁(q奴)闁揑x涓婂柈瑾�(di脿o)澧炲姞鎴栨笡灏戜笖閫g簩(x霉)锛岄偅涔堝畠鐨勫弽鍑芥暩(sh霉)x=f(y)鍦ㄥ皪鎳�(y墨ng)鐨勫崁(q奴)闁揑y={y|y=f(x)锛寈鈭圛x}涓婂柈瑾�(di脿o)澧炲姞鎴栨笡灏戜笖閫g簩(x霉)銆傚弽涓夎鍑芥暩(sh霉)鍦ㄤ粬鍊戠殑瀹氱京鍩熷収(n猫i)閮芥槸閫g簩(x霉)鐨�銆�
瀹氱悊(鏈€澶у€兼渶灏忓€煎畾鐞�)鍦ㄩ枆鍗€(q奴)闁撲笂閫g簩(x霉)鐨勫嚱鏁�(sh霉)鍦ㄨ┎鍗€(q奴)闁撲笂涓€瀹氭湁鏈€澶у€煎拰鏈€灏忓€�銆傚鏋滃嚱鏁�(sh霉)鍦ㄩ枊鍗€(q奴)闁撳収(n猫i)閫g簩(x霉)鎴栧嚱鏁�(sh霉)鍦ㄩ枆鍗€(q奴)闁撲笂鏈夐枔鏂烽粸(di菐n)锛岄偅涔堝嚱鏁�(sh霉)鍦ㄨ┎鍗€(q奴)闁撲笂灏变笉涓€瀹氭湁鏈€澶у€煎拰鏈€灏忓€笺€�
瀹氱悊(鏈夌晫鎬у畾鐞�)鍦ㄩ枆鍗€(q奴)闁撲笂閫g簩(x霉)鐨勫嚱鏁�(sh霉)涓€瀹氬湪瑭插崁(q奴)闁撲笂鏈夌晫锛屽嵆m鈮(x)鈮.瀹氱悊(闆堕粸(di菐n)瀹氱悊)瑷�(sh猫)鍑芥暩(sh霉)f(x)鍦ㄩ枆鍗€(q奴)闁揫a锛宐]涓婇€g簩(x霉)锛屼笖f(a)鑸噁(b)鐣拌櫉(鍗砯(a)脳f(b)<0)锛岄偅涔堝湪闁嬪崁(q奴)闁�(a锛宐)鍏�(n猫i)鑷冲皯鏈夊嚱鏁�(sh霉)f(x)鐨勪竴鍊嬮浂榛�(di菐n)锛屽嵆鑷冲皯鏈変竴榛�(di菐n)尉(a<尉<b锛夈€�
鎺ㄨ珫鍦ㄩ枆鍗€(q奴)闁撲笂閫g簩(x霉)鐨勫嚱鏁�(sh霉)蹇呭彇寰椾粙浜庢渶澶у€糓鑸囨渶灏忓€糾涔嬮枔鐨勪换浣曞€�銆�
绗簩绔� 灏�(d菐o)鏁�(sh霉)鑸囧井鍒�
1銆佸皫(d菐o)鏁�(sh霉)瀛樺湪鐨勫厖鍒嗗繀瑕佹浠跺嚱鏁�(sh霉)f(x)鍦ㄩ粸(di菐n)x0铏曞彲灏�(d菐o)鐨勫厖鍒嗗繀瑕佹浠舵槸鍦ㄩ粸(di菐n)x0铏曠殑宸︽サ闄恖im(h鈫�-0)[f(x0+h)-f(x0)]/h鍙婂彸妤甸檺lim(h鈫�+0)[f(x0+h)-f(x0)]/h閮藉瓨鍦ㄤ笖鐩哥瓑锛屽嵆宸﹀皫(d菐o)鏁�(sh霉)f-鈥�(x0)鍙冲皫(d菐o)鏁�(sh霉)f+鈥�(x0)瀛樺湪鐩哥瓑銆�
2銆佸嚱鏁�(sh霉)f(x)鍦ㄩ粸(di菐n)x0铏曞彲灏�(d菐o)=>鍑芥暩(sh霉)鍦ㄨ┎榛�(di菐n)铏曢€g簩(x霉)锛涘嚱鏁�(sh霉)f(x)鍦ㄩ粸(di菐n)x0铏曢€g簩(x霉)鈮�>鍦ㄨ┎榛�(di菐n)鍙皫(d菐o)銆傚嵆鍑芥暩(sh霉)鍦ㄦ煇榛�(di菐n)閫g簩(x霉)鏄嚱鏁�(sh霉)鍦ㄨ┎榛�(di菐n)鍙皫(d菐o)鐨勫繀瑕佹浠惰€屼笉鏄厖鍒嗘浠�銆�
3銆佸師鍑芥暩(sh霉)鍙皫(d菐o)鍓囧弽鍑芥暩(sh霉)涔熷彲灏�(d菐o)锛屼笖鍙嶅嚱鏁�(sh霉)鐨勫皫(d菐o)鏁�(sh霉)鏄師鍑芥暩(sh霉)灏�(d菐o)鏁�(sh霉)鐨勫€掓暩(sh霉)銆�
4銆佸嚱鏁�(sh霉)f(x)鍦ㄩ粸(di菐n)x0铏曞彲寰�=>鍑芥暩(sh霉)鍦ㄨ┎榛�(di菐n)铏曞彲灏�(d菐o)锛涘嚱鏁�(sh霉)f(x)鍦ㄩ粸(di菐n)x0铏曞彲寰殑鍏呭垎蹇呰姊濅欢鏄嚱鏁�(sh霉)鍦ㄨ┎榛�(di菐n)铏曞彲灏�(d菐o)銆�
绗笁绔� 涓€煎畾鐞嗚垏灏�(d菐o)鏁�(sh霉)鐨勬噳(y墨ng)鐢�
1銆佸畾鐞�(缇呯埦瀹氱悊)濡傛灉鍑芥暩(sh霉)f(x)鍦ㄩ枆鍗€(q奴)闁揫a锛宐]涓婇€g簩(x霉)锛屽湪闁嬪崁(q奴)闁�(a锛宐)鍏�(n猫i)鍙皫(d菐o)锛屼笖鍦ㄥ崁(q奴)闁撶榛�(di菐n)鐨勫嚱鏁�(sh霉)鍊肩浉绛�锛屽嵆f(a)=f(b)锛岄偅涔堝湪闁嬪崁(q奴)闁�(a锛宐)鍏�(n猫i)鑷冲皯鏈変竴榛�(di菐n)尉(a<尉<b锛�锛屼娇鐨勫嚱鏁�(sh霉)f锛坸锛夊湪瑭查粸(di菐n)鐨勫皫(d菐o)鏁�(sh霉)绛変簬闆讹細f'锛埼撅級= 0.
2銆佸畾鐞�(鎷夋牸鏈楁棩涓€煎畾鐞�)濡傛灉鍑芥暩(sh霉)f(x)鍦ㄩ枆鍗€(q奴)闁揫a锛宐]涓婇€g簩(x霉)锛屽湪闁嬪崁(q奴)闁�(a锛宐)鍏�(n猫i)鍙皫(d菐o)锛岄偅涔堝湪闁嬪崁(q奴)闁�(a锛宐)鍏�(n猫i)鑷冲皯鏈変竴榛�(di菐n)尉(a<尉<b锛夛紝浣跨殑绛夊紡f锛坆锛�-f锛坅锛�= f'锛埼撅級锛坆-a锛夋垚绔嬪嵆f'锛埼撅級= [f锛坆锛�-f锛坅锛塢/锛坆-a锛�銆�
3銆佸畾鐞�(鏌タ涓€煎畾鐞�)濡傛灉鍑芥暩(sh霉)f(x)鍙奆(x)鍦ㄩ枆鍗€(q奴)闁揫a锛宐]涓婇€g簩(x霉)锛屽湪闁嬪崁(q奴)闁�(a锛宐)鍏�(n猫i)鍙皫(d菐o)锛屼笖F'(x)鍦�(a锛宐)鍏�(n猫i)鐨勬瘡涓€榛�(di菐n)铏曞潎涓嶇偤闆�锛岄偅涔堝湪闁嬪崁(q奴)闁�(a锛宐)鍏�(n猫i)鑷冲皯鏈変竴榛�(di菐n)尉锛屼娇鐨勭瓑寮廩f(b)-f(a)]/[F(b)-F(a)]=f'(尉)/F'(尉)鎴愮珛銆�
4銆佹礇蹇呴仈(d谩)娉曞墖鎳�(y墨ng)鐢ㄦ浠跺彧鑳界敤鑸囨湭瀹氬瀷璜稿0/0銆佲垶/鈭�銆�0脳鈭�銆佲垶-鈭炪€�00銆�1鈭炪€佲垶 0绛夊舰寮�銆�
5銆佸嚱鏁�(sh霉)鍠(di脿o)鎬х殑鍒ゅ畾娉曡ō(sh猫)鍑芥暩(sh霉)f(x)鍦ㄩ枆鍗€(q奴)闁揫a锛宐]涓婇€g簩(x霉)锛屽湪闁嬪崁(q奴)闁�(a锛宐)鍏�(n猫i)鍙皫(d菐o)锛岄偅涔堬細(1)濡傛灉鍦�(a锛宐)鍏�(n猫i)f'(x)>0锛岄偅涔堝嚱鏁�(sh霉)f(x)鍦╗a锛宐]涓婂柈瑾�(di脿o)澧炲姞锛�(2)濡傛灉鍦�(a锛宐)鍏�(n猫i)f鈥�(x)<0锛岄偅涔堝嚱鏁�(sh霉)f(x)鍦╗a锛宐]涓婂柈瑾�(di脿o)娓涘皯銆�
濡傛灉鍑芥暩(sh霉)鍦ㄥ畾缇╁崁(q奴)闁撲笂閫g簩(x霉)锛岄櫎鍘绘湁闄愬€嬪皫(d菐o)鏁�(sh霉)涓嶅瓨鍦ㄧ殑榛�(di菐n)澶栧皫(d菐o)鏁�(sh霉)瀛樺湪涓旈€g簩(x霉)锛岄偅涔堝彧瑕佺敤鏂圭▼f'(x)=0鐨勬牴鍙奻鈥�(x)涓嶅瓨鍦ㄧ殑榛�(di菐n)渚嗗妰鍒嗗嚱鏁�(sh霉)f(x)鐨勫畾缇╁崁(q奴)闁�锛屽氨鑳戒繚璀塮'(x)鍦ㄥ悇鍊嬮儴鍒嗗崁(q奴)闁撳収(n猫i)淇濇寔鍥哄畾绗﹁櫉锛屽洜鑰屽嚱鏁�(sh霉)f(x)鍦ㄦ瘡鍊嬮儴鍒嗗崁(q奴)闁撲笂鍠(di脿o)銆�
6銆佸嚱鏁�(sh霉)鐨勬サ鍊煎鏋滃嚱鏁�(sh霉)f(x)鍦ㄥ崁(q奴)闁�(a锛宐)鍏�(n猫i)鏈夊畾缇╋紝x0鏄�(a锛宐)鍏�(n猫i)鐨勪竴鍊嬮粸(di菐n)锛屽鏋滃瓨鍦ㄨ憲榛�(di菐n)x0鐨勪竴鍊嬪幓蹇冮劙鍩�锛屽皪浜庨€欏幓蹇冮劙鍩熷収(n猫i)鐨勪换浣曢粸(di菐n)x锛宖(x)f(x0)鍧囨垚绔嬶紝灏辩ūf(x0)鏄嚱鏁�(sh霉)f(x)鐨勪竴鍊嬫サ灏忓€�銆�
鍦ㄥ嚱鏁�(sh霉)鍙栧緱妤靛€艰檿锛屾洸绶氫笂鐨勫垏绶氭槸姘村钩鐨�锛屼絾鏇茬窔涓婃湁姘村钩鏇茬窔鐨勫湴鏂�锛屽嚱鏁�(sh霉)涓嶄竴瀹氬彇寰楁サ鍊�锛屽嵆鍙皫(d菐o)鍑芥暩(sh霉)鐨勬サ鍊奸粸(di菐n)蹇呭畾鏄畠鐨勯榛�(di菐n)(灏�(d菐o)鏁�(sh霉)鐐�0鐨勯粸(di菐n))锛屼絾鍑芥暩(sh霉)鐨勯榛�(di菐n)鍗讳笉涓€瀹氭槸妤靛€奸粸(di菐n)銆�
瀹氱悊(鍑芥暩(sh霉)鍙栧緱妤靛€肩殑蹇呰姊濅欢)瑷�(sh猫)鍑芥暩(sh霉)f(x)鍦▁0铏曞彲灏�(d菐o)锛屼笖鍦▁0铏曞彇寰楁サ鍊�锛岄偅涔堝嚱鏁�(sh霉)鍦▁0鐨勫皫(d菐o)鏁�(sh霉)鐐洪浂锛屽嵆f'(x0)=0.瀹氱悊(鍑芥暩(sh霉)鍙栧緱妤靛€肩殑绗竴绋厖鍒嗘浠�)瑷�(sh猫)鍑芥暩(sh霉)f(x)鍦▁0涓€鍊嬮劙鍩熷収(n猫i)鍙皫(d菐o)锛屼笖f鈥�(x0)=0锛岄偅涔堬細(1)濡傛灉鐣�(d膩ng)x鍙杧0宸﹀伌(c猫)鑷ㄨ繎鐨勫€兼檪锛宖'(x)鎭掔偤姝�锛涚暥(d膩ng)x鍘粁0鍙冲伌(c猫)鑷ㄨ繎鐨勫€兼檪锛宖鈥�(x)鎭掔偤璨�(f霉)锛岄偅涔堝嚱鏁�(sh霉)f(x)鍦▁0铏曞彇寰楁サ澶у€�锛�(2)濡傛灉鐣�(d膩ng)x鍙杧0宸﹀伌(c猫)鑷ㄨ繎鐨勫€兼檪锛宖'(x)鎭掔偤璨�(f霉)锛涚暥(d膩ng)x鍘粁0鍙冲伌(c猫)鑷ㄨ繎鐨勫€兼檪锛宖鈥�(x)鎭掔偤姝�锛岄偅涔堝嚱鏁�(sh霉)f(x)鍦▁0铏曞彇寰楁サ灏忓€�锛�(3)濡傛灉鐣�(d膩ng)x鍙杧0宸﹀彸鍏╁伌(c猫)鑷ㄨ繎鐨勫€兼檪锛宖'(x)鎭掔偤姝f垨鎭掔偤璨�(f霉)锛岄偅涔堝嚱鏁�(sh霉)f(x)鍦▁0铏曟矑鏈夋サ鍊笺€�
瀹氱悊(鍑芥暩(sh霉)鍙栧緱妤靛€肩殑绗簩绋厖鍒嗘浠�)瑷�(sh猫)鍑芥暩(sh霉)f(x)鍦▁0铏曞叿鏈変簩闅庡皫(d菐o)鏁�(sh霉)涓攆'(x0)=0锛宖''(x0)鈮�0閭d箞锛�(1)鐣�(d膩ng)f''(x0)<0鏅傦紝鍑芥暩(sh霉)f(x)鍦▁0铏曞彇寰楁サ澶у€�锛�(2)鐣�(d膩ng)f''(x0)>0鏅傦紝鍑芥暩(sh霉)f(x)鍦▁0铏曞彇寰楁サ灏忓€�锛涢榛�(di菐n)鏈夊彲鑳芥槸妤靛€奸粸(di菐n)锛屼笉鏄榛�(di菐n)涔熸湁鍙兘鏄サ鍊奸粸(di菐n)銆�
7銆佸嚱鏁�(sh霉)鐨勫嚬鍑告€у強鍏跺垽瀹氳ō(sh猫)f(x)鍦ㄥ崁(q奴)闁揑x涓婇€g簩(x霉)锛屽鏋滃皪浠绘剰鍏╅粸(di菐n)x1锛寈2鎭掓湁f[(x1+x2)/2]<[f(x1)+f(x1)]/2锛岄偅涔堢ūf(x)鍦ㄥ崁(q奴)闁揑x涓婂湒褰㈡槸鍑圭殑锛涘鏋滄亽鏈塮[(x1+x2)/2]>[f(x1)+f(x1)]/2锛岄偅涔堢ūf(x)鍦ㄥ崁(q奴)闁揑x涓婂湒褰㈡槸鍑哥殑銆�
瀹氱悊瑷�(sh猫)鍑芥暩(sh霉)f(x)鍦ㄩ枆鍗€(q奴)闁揫a锛宐]涓婇€g簩(x霉)锛屽湪闁嬪崁(q奴)闁�(a锛宐)鍏�(n猫i)鍏锋湁涓€闅庡拰浜岄殠灏�(d菐o)鏁�(sh霉)锛岄偅涔�(1)鑻ュ湪(a锛宐)鍏�(n猫i)f'鈥�(x)>0锛屽墖f(x)鍦ㄩ枆鍗€(q奴)闁揫a锛宐]涓婄殑鍦栧舰鏄嚬鐨�锛�(2)鑻ュ湪(a锛宐)鍏�(n猫i)f'鈥�(x)<0锛屽墖f(x)鍦ㄩ枆鍗€(q奴)闁揫a锛宐]涓婄殑鍦栧舰鏄嚫鐨�銆�
鍒ゆ柗鏇茬窔鎷愰粸(di菐n)(鍑瑰嚫鍒嗙晫榛�(di菐n))鐨勬椹�(1)姹傚嚭f'鈥�(x)锛�(2)浠'鈥�(x)=0锛岃В鍑洪€欐柟绋嬪湪鍗€(q奴)闁�(a锛宐)鍏�(n猫i)鐨勫(sh铆)鏍�锛�(3)灏嶄簬(2)涓В鍑虹殑姣忎竴鍊嬪(sh铆)鏍箈0锛屾鏌'鈥�(x)鍦▁0宸﹀彸鍏╁伌(c猫)閯拌繎鐨勭铏�锛屽鏋渇'鈥�(x)鍦▁0宸﹀彸鍏╁伌(c猫)閯拌繎鍒嗗垾淇濇寔涓€瀹氱殑绗﹁櫉锛岄偅涔堢暥(d膩ng)鍏╁伌(c猫)鐨勭铏熺浉鍙嶆檪锛岄粸(di菐n)(x0锛宖(x0))鏄嫄榛�(di菐n)锛岀暥(d膩ng)鍏╁伌(c猫)鐨勭铏熺浉鍚屾檪锛岄粸(di菐n)(x0锛宖(x0))涓嶆槸鎷愰粸(di菐n)銆�
鍦ㄥ仛鍑芥暩(sh霉)鍦栧舰鐨勬檪鍊�锛屽鏋滃嚱鏁�(sh霉)鏈夐枔鏂烽粸(di菐n)鎴�?q奴)顫�?sh霉)涓嶅瓨鍦ㄧ殑榛�(di菐n)锛岄€欎簺榛�(di菐n)涔熻浣滅偤鍒嗛粸(di菐n)銆�
绗洓绔� 涓嶅畾绌嶅垎
1銆佸師鍑芥暩(sh霉)瀛樺湪瀹氱悊瀹氱悊濡傛灉鍑芥暩(sh霉)f(x)鍦ㄥ崁(q奴)闁揑涓婇€g簩(x霉)锛岄偅涔堝湪鍗€(q奴)闁揑涓婂瓨鍦ㄥ彲灏�(d菐o)鍑芥暩(sh霉)F(x)锛屼娇灏嶄换涓€x鈭圛閮芥湁F'(x)=f(x)锛涚啊鍠殑瑾€g簩(x霉)鍑芥暩(sh霉)涓€瀹氭湁鍘熷嚱鏁�(sh霉)銆�
鍒嗛儴绌嶅垎鐧�(f膩)濡傛灉琚⿳鍑芥暩(sh霉)鏄啰鍑芥暩(sh霉)鍜屾浣欏鸡鎴栧啰鍑芥暩(sh霉)鍜屾寚鏁�(sh霉)鍑芥暩(sh霉)鐨勪箻绌�锛屽氨鍙互鑰冩叜鐢ㄥ垎閮ㄧ⿳鍒嗘硶锛屽苟瑷�(sh猫)鍐嚱鏁�(sh霉)鍜屾寚鏁�(sh霉)鍑芥暩(sh霉)鐐簎锛岄€欐ǎ鐢ㄤ竴娆″垎閮ㄧ⿳鍒嗘硶灏卞彲浠ヤ娇鍐嚱鏁�(sh霉)鐨勫啰闄嶄綆涓€娆�銆傚鏋滆绌嶅嚱鏁�(sh霉)鏄啰鍑芥暩(sh霉)鍜屽皪鏁�(sh霉)鍑芥暩(sh霉)鎴栧啰鍑芥暩(sh霉)鍜屽弽涓夎鍑芥暩(sh霉)鐨勪箻绌�锛屽氨鍙ō(sh猫)灏嶆暩(sh霉)鍜屽弽涓夎鍑芥暩(sh霉)鐐簎.
2銆佸皪浜庡垵绛夊嚱鏁�(sh霉)渚嗚锛屽湪鍏跺畾缇╁崁(q奴)闁撲笂锛屽畠鐨勫師鍑芥暩(sh霉)涓€瀹氬瓨鍦�锛屼絾鍘熷嚱鏁�(sh霉)涓嶄竴瀹氶兘鏄垵绛夊嚱鏁�(sh霉)銆�
绗簲绔� 瀹氱⿳鍒�
1銆佸畾绌嶅垎瑙f焙鐨勫吀鍨嬪晱椤�(1)鏇查倞姊舰鐨勯潰绌�(2)璁婇€熺洿绶氶亱(y霉n)鍕曠殑璺▼
2銆佸嚱鏁�(sh霉)鍙⿳鐨勫厖鍒嗘浠跺畾鐞嗚ō(sh猫)f(x)鍦ㄥ崁(q奴)闁揫a锛宐]涓婇€g簩(x霉)锛屽墖f(x)鍦ㄥ崁(q奴)闁揫a锛宐]涓婂彲绌�锛屽嵆閫g簩(x霉)=>鍙⿳銆�
瀹氱悊瑷�(sh猫)f(x)鍦ㄥ崁(q奴)闁揫a锛宐]涓婃湁鐣�锛屼笖鍙湁鏈夐檺鍊嬮枔鏂烽粸(di菐n)锛屽墖f(x)鍦ㄥ崁(q奴)闁揫a锛宐]涓婂彲绌�銆�
3銆佸畾绌嶅垎鐨勮嫢骞查噸瑕佹€ц唱(zh矛)鎬ц唱(zh矛)濡傛灉鍦ㄥ崁(q奴)闁揫a锛宐]涓奻(x)鈮�0鍓団埆abf(x)dx鈮�0.鎺ㄨ珫濡傛灉鍦ㄥ崁(q奴)闁揫a锛宐]涓奻(x)鈮(x)鍓団埆abf(x)dx鈮も埆abg(x)dx.鎺ㄨ珫|鈭玜bf(x)dx|鈮も埆ab|f(x)|dx.鎬ц唱(zh矛)瑷�(sh猫)M鍙妋鍒嗗垾鏄嚱鏁�(sh霉)f(x)鍦ㄥ崁(q奴)闁揫a锛宐]涓婄殑鏈€澶у€煎拰鏈€灏忓€硷紝鍓噈(b-a)鈮も埆abf(x)dx鈮(b-a)锛岃┎鎬ц唱(zh矛)瑾槑鐢辫绌嶅嚱鏁�(sh霉)鍦ㄧ⿳鍒嗗崁(q奴)闁撲笂鐨勬渶澶у€煎強鏈€灏忓€煎彲浠ヤ及瑷�(j矛)绌嶅垎鍊肩殑澶ц嚧鑼冨湇銆�
鎬ц唱(zh矛)(瀹氱⿳鍒嗕腑鍊煎畾鐞�)濡傛灉鍑芥暩(sh霉)f(x)鍦ㄥ崁(q奴)闁揫a锛宐]涓婇€g簩(x霉)锛屽墖鍦ㄧ⿳鍒嗗崁(q奴)闁揫a锛宐]涓婅嚦灏戝瓨鍦ㄤ竴鍊嬮粸(di菐n)尉锛屼娇涓嬪紡鎴愮珛锛氣埆abf(x)dx=f(尉)(b-a)銆�
4銆侀棞(gu膩n)浜庡唬缇╃⿳鍒嗚ō(sh猫)鍑芥暩(sh霉)f(x)鍦ㄥ崁(q奴)闁揫a锛宐]涓婇櫎榛�(di菐n)c(a<c<b锛夊閫g簩(x霉)锛岃€屽湪榛�(di菐n)c鐨勯劙鍩熷収(n猫i)鐒$晫锛屽鏋滃叐鍊嬪唬缇╃⿳鍒嗏埆acf锛坸锛塪x鑸団埆cbf锛坸锛塪x閮芥敹鏂傦紝鍓囧畾缇┾埆abf锛坸锛塪x=鈭玜cf锛坸锛塪x+鈭玞bf锛坸锛塪x锛屽惁鍓囷紙鍙鍏朵腑涓€鍊嬬櫦(f膩)鏁o級灏辩ū寤g京绌嶅垎鈭玜bf锛坸锛塪x鐧�(f膩)鏁�銆�
绗叚绔� 瀹氱⿳鍒嗙殑鎳�(y墨ng)鐢�
姹傚钩闈㈠湒褰㈢殑闈㈢⿳(鏇茬窔鍦嶆垚鐨勯潰绌�)
鐩磋鍧愭(bi膩o)绯讳笅(鍚弮鏁�(sh霉)鑸囦笉鍚弮鏁�(sh霉))
妤靛潗妯�(bi膩o)绯讳笅(r锛屛�锛寈=rcos胃锛寉=rsin胃)(鎵囧舰闈㈢⿳鍏紡S=R2胃/2)
鏃嬭綁(zhu菐n)楂旈珨绌�(鐢遍€g簩(x霉)鏇茬窔銆佺洿绶氬強鍧愭(bi膩o)杌告墍鍦嶆垚鐨勯潰绌嶇篂鍧愭(bi膩o)杌告棆杞�(zhu菐n)鑰屾垚)(涓旈珨绌峍=鈭玜b蟺[f(x)]2dx锛屽叾涓璮(x)鎸囨洸绶氱殑鏂圭▼)
骞宠鎴潰闈㈢⿳鐐哄凡鐭ョ殑绔嬮珨楂旂⿳(V=鈭玜bA(x)dx锛屽叾涓瑼(x)鐐烘埅闈㈤潰绌�)
鍔�銆佹按澹撳姏銆佸紩鍔�
鍑芥暩(sh霉)鐨勫钩鍧囧€�(骞冲潎鍊紋=1/(b-a)*鈭玜bf(x)dx)
绗竷绔� 澶氬厓鍑芥暩(sh霉)寰垎娉曞強鍏舵噳(y墨ng)鐢�
1銆佸鍏冨嚱鏁�(sh霉)妤甸檺瀛樺湪鐨勬浠舵サ闄愬瓨鍦ㄦ槸鎸嘝(x锛寉)浠ヤ换浣曟柟寮忚定浜嶱0(x0锛寉0)鏅�锛屽嚱鏁�(sh霉)閮界劇闄愭帴杩戜簬A锛屽鏋淧(x锛寉)浠ユ煇涓€鐗规畩鏂瑰紡锛屼緥濡傛部钁椾竴姊濆畾鐩寸窔鎴栧畾鏇茬窔瓒ㄤ簬P0(x0锛寉0)鏅傦紝鍗充娇鍑芥暩(sh霉)鐒¢檺鎺ヨ繎鏌愪竴纰哄畾鍊�锛屾垜鍊戦倓涓嶈兘鐢辨鏂峰畾鍑芥暩(sh霉)妤甸檺瀛樺湪銆傚弽閬庝締锛屽鏋滅暥(d膩ng)P(x锛寉)浠ヤ笉鍚屾柟寮忚定浜嶱0(x0锛寉0)鏅�锛屽嚱鏁�(sh霉)瓒ㄤ簬涓嶅悓鐨勫€硷紝閭d箞灏卞彲浠ユ柗瀹氶€欏嚱鏁�(sh霉)鐨勬サ闄愪笉瀛樺湪銆備緥濡傚嚱鏁�(sh霉)锛歠(x锛寉)={0(xy)/(x^2+y^2)x^2+y^2鈮�0
2銆佸鍏冨嚱鏁�(sh霉)鐨勯€g簩(x霉)鎬у畾缇╄ō(sh猫)鍑芥暩(sh霉)f(x锛寉)鍦ㄩ枊鍗€(q奴)鍩�(鎴栭枆鍗€(q奴)鍩�)D鍏�(n猫i)鏈夊畾缇�锛孭0(x0锛寉0)鏄疍鐨勫収(n猫i)榛�(di菐n)鎴栭倞鐣岄粸(di菐n)涓擯0鈭圖锛屽鏋渓im(x鈫抶0锛寉鈫抷0)f(x锛寉)=f(x0锛寉0)鍓囩ūf(x锛寉)鍦ㄩ粸(di菐n)P0(x0锛寉0)閫g簩(x霉)銆�
鎬ц唱(zh矛)(鏈€澶у€煎拰鏈€灏忓€煎畾鐞�)鍦ㄦ湁鐣岄枆鍗€(q奴)鍩烡涓婄殑澶氬厓閫g簩(x霉)鍑芥暩(sh霉)锛屽湪D涓婁竴瀹氭湁鏈€澶у€煎拰鏈€灏忓€�銆�
鎬ц唱(zh矛)(浠嬪€煎畾鐞�)鍦ㄦ湁鐣岄枆鍗€(q奴)鍩烡涓婄殑澶氬厓閫g簩(x霉)鍑芥暩(sh霉)锛屽鏋滃湪D涓婂彇寰楀叐鍊嬩笉鍚岀殑鍑芥暩(sh霉)鍊�锛屽墖瀹冨湪D涓婂彇寰椾粙浜庨€欏叐鍊嬪€间箣闁撶殑浠讳綍鍊艰嚦灏戜竴娆°€�
3銆佸鍏冨嚱鏁�(sh霉)鐨勯€g簩(x霉)鑸囧彲灏�(d菐o)濡傛灉涓€鍏冨嚱鏁�(sh霉)鍦ㄦ煇榛�(di菐n)鍏锋湁灏�(d菐o)鏁�(sh霉)锛屽墖瀹冨湪瑭查粸(di菐n)蹇呭畾閫g簩(x霉)锛屼絾灏嶄簬澶氬厓鍑芥暩(sh霉)渚嗚锛屽嵆浣垮悇鍋忓皫(d菐o)鏁�(sh霉)鍦ㄦ煇榛�(di菐n)閮藉瓨鍦�锛屼篃涓嶈兘淇濊瓑鍑芥暩(sh霉)鍦ㄨ┎榛�(di菐n)閫g簩(x霉)銆傞€欐槸鍥�?y脿n)妤﹂珜鐝�?d菐o)鏁�(sh霉)瀛樺湪鍙兘淇濊瓑榛�(di菐n)P娌胯憲骞宠浜庡潗妯�(bi膩o)杌哥殑鏂瑰悜瓒ㄤ簬P0鏅傦紝鍑芥暩(sh霉)鍊糵(P)瓒ㄤ簬f(P0)锛屼絾涓嶈兘淇濊瓑榛�(di菐n)P鎸変换浣曟柟寮忚定浜嶱0鏅�锛屽嚱鏁�(sh霉)鍊糵(P)閮借定浜巉(P0)銆�
4銆佸鍏冨嚱鏁�(sh霉)鍙井鐨勫繀瑕佹浠朵竴鍏冨嚱鏁�(sh霉)鍦ㄦ煇榛�(di菐n)鐨勫皫(d菐o)鏁�(sh霉)瀛樺湪鏄井鍒嗗瓨鍦ㄧ殑鍏呭垎蹇呰姊濅欢锛屼絾澶氬厓鍑芥暩(sh霉)鍚勫亸灏�(d菐o)鏁�(sh霉)瀛樺湪鍙槸鍏ㄥ井鍒嗗瓨鍦ㄧ殑蹇呰姊濅欢鑰屼笉鏄厖鍒嗘浠�锛屽嵆鍙井=>鍙亸灏�(d菐o)銆�
5銆佸鍏冨嚱鏁�(sh霉)鍙井鐨勫厖鍒嗘浠跺畾鐞�(鍏呭垎姊濅欢)濡傛灉鍑芥暩(sh霉)z=f(x锛寉)鐨勫亸灏�(d菐o)鏁�(sh霉)瀛樺湪涓斿湪榛�(di菐n)(x锛寉)閫g簩(x霉)锛屽墖鍑芥暩(sh霉)鍦ㄨ┎榛�(di菐n)鍙井鍒�銆�
6.澶氬厓鍑芥暩(sh霉)妤靛€煎瓨鍦ㄧ殑蹇呰銆佸厖鍒嗘浠跺畾鐞�(蹇呰姊濅欢)瑷�(sh猫)鍑芥暩(sh霉)z=f(x锛寉)鍦ㄩ粸(di菐n)(x0锛寉0)鍏锋湁鍋忓皫(d菐o)鏁�(sh霉)锛屼笖鍦ㄩ粸(di菐n)(x0锛寉0)铏曟湁妤靛€�锛屽墖瀹冨湪瑭查粸(di菐n)鐨勫亸灏�(d菐o)鏁�(sh霉)蹇呯偤闆躲€�
瀹氱悊(鍏呭垎姊濅欢)瑷�(sh猫)鍑芥暩(sh霉)z=f(x锛寉)鍦ㄩ粸(di菐n)(x0锛寉0)鐨勬煇閯板煙鍏�(n猫i)閫g簩(x霉)涓旀湁涓€闅庡強浜岄殠閫g簩(x霉)鍋忓皫(d菐o)鏁�(sh霉)锛屽張fx(x0锛寉0)=0锛宖y(x0锛寉0)=0锛屼护fxx(x0锛寉0)=0=A锛宖xy(x0锛寉0)=B锛宖yy(x0锛寉0)=C锛屽墖f(x锛寉)鍦ㄩ粸(di菐n)(x0锛寉0)铏曟槸鍚﹀彇寰楁サ鍊肩殑姊濅欢濡備笅锛�(1)AC-B2>0鏅傚叿鏈夋サ鍊�锛屼笖鐣�(d膩ng)A<0鏅傛湁妤靛ぇ鍊�锛岀暥(d膩ng)A>0鏅傛湁妤靛皬鍊�锛�(2)AC-B2<0鏅傛矑鏈夋サ鍊硷紱(3)AC-B2=0鏅傚彲鑳芥湁涔熷彲鑳芥矑鏈�銆�
7銆佸鍏冨嚱鏁�(sh霉)妤靛€煎瓨鍦ㄧ殑瑙f硶(1)瑙f柟绋嬬祫fx(x锛寉)=0锛宖y(x锛寉)=0姹傜殑涓€鍒囧(sh铆)鏁�(sh霉)瑙�锛屽嵆鍙眰寰椾竴鍒囬榛�(di菐n)銆�
(2)灏嶄簬姣忎竴鍊嬮榛�(di菐n)(x0锛寉0)锛屾眰鍑轰簩闅庡亸灏�(d菐o)鏁�(sh霉)鐨勫€糀(ch菙)銆丅銆丆.(3)瀹氬嚭AC-B2鐨勭铏燂紝鎸夊厖鍒嗘浠堕€�(j矛n)琛屽垽瀹歠(x0锛寉0)鏄惁鏄サ澶у€�銆佹サ灏忓€笺€�
娉ㄦ剰锛氬湪鑰冩叜鍑芥暩(sh霉)鐨勬サ鍊煎晱椤屾檪锛岄櫎浜嗚€冩叜鍑芥暩(sh霉)鐨勯榛�(di菐n)澶栵紝濡傛灉鏈夊亸灏�(d菐o)鏁�(sh霉)涓嶅瓨鍦ㄧ殑榛�(di菐n)锛岄偅涔堝皪閫欎簺榛�(di菐n)涔熸噳(y墨ng)鐣�(d膩ng)鑰冩叜鍦ㄥ収(n猫i)銆�
绗叓绔� 浜岄噸绌嶅垎
1銆佷簩閲嶇⿳鍒嗙殑涓€浜涙噳(y墨ng)鐢ㄦ洸闋傛煴楂旂殑楂旂⿳鏇查潰鐨勯潰绌�(A=鈭埆鈭歔1+f2x(x锛寉)+f2y(x锛寉)]d蟽)
骞抽潰钖勭墖鐨勮唱(zh矛)閲忓钩闈㈣杽鐗囩殑閲嶅績鍧愭(bi膩o)(x=1/A鈭埆xd蟽锛寉=1/A鈭埆yd蟽锛涘叾涓瑼=鈭埆d蟽鐐洪枆鍗€(q奴)鍩烡鐨勯潰绌嶃€�
骞抽潰钖勭墖鐨勮綁(zhu菐n)鍕曟叄閲�(Ix=鈭埆y2蟻(x锛寉)d蟽锛孖y=鈭埆x2蟻(x锛寉)d蟽锛涘叾涓�(x锛寉)鐐哄湪榛�(di菐n)(x锛寉)铏曠殑瀵嗗害銆�
骞抽潰钖勭墖灏嶈唱(zh矛)榛�(di菐n)鐨勫紩鍔�(FxFyFz)
2銆佷簩閲嶇⿳鍒嗗瓨鍦ㄧ殑姊濅欢鐣�(d膩ng)f(x锛寉)鍦ㄩ枆鍗€(q奴)鍩烡涓婇€g簩(x霉)鏅�锛屾サ闄愬瓨鍦紝鏁呭嚱鏁�(sh霉)f(x锛寉)鍦―涓婄殑浜岄噸绌嶅垎蹇呭畾瀛樺湪銆�
3銆佷簩閲嶇⿳鍒嗙殑涓€浜涢噸瑕佹€ц唱(zh矛)鎬ц唱(zh矛)濡傛灉鍦―涓婏紝f(x锛寉)鈮は�(x锛寉)锛屽墖鏈変笉绛夊紡鈭埆f(x锛寉)dxdy鈮も埆鈭�(x锛寉)dxdy锛岀壒娈婂湴鐢变簬-|f(x锛寉)|鈮(x锛寉)鈮f(x锛寉)|鍙堟湁涓嶇瓑寮弢鈭埆f(x锛寉)dxdy|鈮も埆鈭珅f(x锛寉)|dxdy.鎬ц唱(zh矛)瑷�(sh猫)M锛宮鍒嗗垾鏄痜(x锛寉)鍦ㄩ枆鍗€(q奴)鍩烡涓婄殑鏈€澶у€煎拰鏈€灏忓€�锛屜冩槸D鐨勯潰绌�锛屽墖鏈塵蟽鈮も埆鈭玣(x锛寉)d蟽鈮蟽銆�
鎬ц唱(zh矛)(浜岄噸绌嶅垎鐨勪腑鍊煎畾鐞�)瑷�(sh猫)鍑芥暩(sh霉)f(x锛寉)鍦ㄩ枆鍗€(q奴)鍩烡涓婇€g簩(x霉)锛屜冩槸D鐨勯潰绌�锛屽墖鍦―涓婅嚦灏戝瓨鍦ㄤ竴榛�(di菐n)(尉锛屛�)浣垮緱涓嬪紡鎴愮珛锛氣埆鈭玣(x锛寉)d蟽=f(尉锛屛�)*蟽4銆佷簩閲嶇⿳鍒嗕腑妯�(bi膩o)閲忓湪鐩磋鑸囨サ鍧愭(bi膩o)绯讳腑鐨勮綁(zhu菐n)鎻涙妸浜岄噸绌嶅垎寰炵洿瑙掑潗妯�(bi膩o)绯绘彌鐐烘サ鍧愭(bi膩o)绯�锛屽彧瑕佹妸琚⿳鍑芥暩(sh霉)涓殑x锛寉鍒嗗垾鎻涙垚ycos胃銆乺sin胃锛屽苟鎶婄洿瑙掑潗妯�(bi膩o)绯讳腑鐨勯潰绌嶅厓绱燿xd
鍏嶈铂(z茅)鑱叉槑锛氭湰鏂囩郴杞�(zhu菐n)杓夎嚜缍�(w菐ng)绲�(lu貌)锛屽鏈変镜鐘紝璜嬭伅(li谩n)绯绘垜鍊戠珛鍗冲埅闄�锛屽彟锛氭湰鏂囧儏浠h〃浣滆€呭€嬩汉瑙€榛�(di菐n)锛岃垏鏈恫(w菐ng)绔欑劇闂�(gu膩n)銆傚叾鍘熷壍(chu脿ng)鎬т互鍙婃枃涓櫝杩版枃瀛楀拰鍏�(n猫i)瀹规湭缍�(j墨ng)鏈珯璀夊(sh铆)锛屽皪鏈枃浠ュ強鍏朵腑鍏ㄩ儴鎴栬€呴儴鍒嗗収(n猫i)瀹�銆佹枃瀛楃殑鐪熷(sh铆)鎬с€佸畬鏁存€�銆佸強鏅傛€ф湰绔欎笉浣滀换浣曚繚璀夋垨鎵胯锛岃珛璁€鑰呭儏浣滃弮鑰�锛屽苟璜嬭嚜琛屾牳瀵�(sh铆)鐩搁棞(gu膩n)鍏�(n猫i)瀹�銆�