2007級高等數學（下）期中考試試題及答案一、選擇題（每小題3分，共15分）C(,)(x,y)fxy和fxy(x,y)(x,y)(,)、f））xyB（yxln(xy)）arctanxy））ex）22xyxAy1xydxdyx22（））D223D3676532））））42322D（yz4x33）16）32）48）640:xyzR0y002x:xyzRz，，z，）22222221C（ydv4ydv））xdv4xdv））11zdv4zdvxyzdv4xyzdv11二、填空題（每小題3分，共15分）yzzxz33xyz1z2xy。x1y1z2x2y41,0)z441224102xyzxyzP(0,1,2)0u23(x,y)12(,)22fxy(,)ffxy12121yy12dxx(,)dfxyy1x16(x,y)f6limf(x,y)dxdyf20x2y22三、計算題（每小題7分，共56分）esin(xy)z。dzxy22zzeycos(xy)2xexcos(xy)2y2222xyxy解：，xydzzxzdxdy[2xcos(xy)]dx＋[xe2ycos(xy)]dy2222xyyyzf(xy,)zf。xyxzyyfyf()yff解：xxx1212222z11y11f[f][]xfffxfxyxx2xx111122212221yfff1x211x2223(x,y)(xy1)4xyf222221)280xyxxf222解：由2(01)(2)8fxyyxy222y(解得駐點：448xy44fffx，2y222，xxxyA4,B0,C4,ACB1602在點（，0所以點（，）不是極值點；A8,B0,C8,ACB64012在點（±，，且A0f(1,0)01所以點（±，0）是極小值點，極小值為y2dxdyD:xy3。x2222D解：原式xy2xy22222xy2222223xy(2xy)(xy2)2222x222yxy2223(dd2dd322000215222zxyz122xy122解：原式＝dddzd1d1dz1122200)10102dd112211101022[[11112222112[ln(1)arctan][ln21]210224解：設長方體容器的長、寬、高分別為xyz則它們滿足條件：，、、30xy10(2xz2yzxy)3602xyxzyz18，即Vxyz長方體的容器為Lxyz(2xyxzyz18)令，解下列方程組)0Lyzx)0Lxzy()03,23Lxyxyz得：z2y2axxyz4ax222222Vdv4d2d422dzaa200012a324d24d4()(4)a2ada2222230000144()aaa8d8d33333223300asinsindasinsind32322233200323212acos)d(cos)acoscos3233323232002a33321xy0zz22解：立體在旋轉拋物面z1xy上這部分的面積為22A1(2x)(2y)dxdyd2114d222100xy1133114d4]2222600A立體底面面積，所以立體的表面積為23AAA1)526612四、證明題（每小題7分，共14分）u)uxyzf(xy)f2222zz22(xyzxy)2222xy2211u)ffu。f44證明：由()得：zfxy22zxzf2x2xff(2y)2yf，yz2z2f4xf22fyf242x2y2，zz22(xy)(fxy)2222代入得xy222f4xf2f4yf(xyfxy))(2222224(xy)f(xy)(fu)2222即：，所以11fu4f(fu)f44，(x)[a,b]是f(x)fba)2D={(x,y)|axb,ay}f(y)D解：由輪換對稱性得：f(x)f(y)f(y)f(x)DD(x)f(x)f(y)f(y)f(x)()()所以：2ffxfydxdydxdydxdydxdyf(y)f(y)