全微分考試題及答案

單項(xiàng)選擇題（每題2分，共10題）1.函數(shù)\(z=xy\)在點(diǎn)\((1,1)\)處的全微分\(dz\)為（）A.\(dx+dy\)B.\(2dx+2dy\)C.\(dx-dy\)D.\(dy-dx\)2.若\(z=e^{x+y}\)，則\(dz\)等于（）A.\(e^{x+y}dx\)B.\(e^{x+y}dy\)C.\(e^{x+y}(dx+dy)\)D.\(e^{x+y}(dx-dy)\)3.函數(shù)\(z=\ln(x+y)\)的全微分\(dz\)是（）A.\(\frac{1}{x+y}(dx+dy)\)B.\(\frac{1}{x+y}(dx-dy)\)C.\(\frac{1}{x+y}dx\)D.\(\frac{1}{x+y}dy\)4.設(shè)\(z=x^2+y^2\)，則在點(diǎn)\((1,2)\)處\(dz\)為（）A.\(2dx+4dy\)B.\(4dx+2dy\)C.\(2dx-4dy\)D.\(4dx-2dy\)5.函數(shù)\(z=\sin(xy)\)的全微分\(dz\)為（）A.\(\cos(xy)(xdx+ydy)\)B.\(\cos(xy)(ydx+xdy)\)C.\(\cos(xy)(dx+dy)\)D.\(\cos(xy)(dx-dy)\)6.設(shè)\(z=x/y\)，則\(dz\)等于（）A.\(\frac{1}{y}dx-\frac{x}{y^2}dy\)B.\(\frac{1}{y}dx+\frac{x}{y^2}dy\)C.\(\frac{x}{y^2}dx-\frac{1}{y}dy\)D.\(\frac{x}{y^2}dx+\frac{1}{y}dy\)7.函數(shù)\(z=e^{xy}\)在點(diǎn)\((0,1)\)處的全微分\(dz\)是（）A.\(dx\)B.\(dy\)C.\(dx+dy\)D.\(dx-dy\)8.若\(z=x^3+y^3\)，則\(dz\)為（）A.\(3x^2dx+3y^2dy\)B.\(3x^2dx-3y^2dy\)C.\(3x^2dy+3y^2dx\)D.\(3x^2dy-3y^2dx\)9.函數(shù)\(z=\arctan(xy)\)的全微分\(dz\)為（）A.\(\frac{y}{1+x^{2}y^{2}}dx+\frac{x}{1+x^{2}y^{2}}dy\)B.\(\frac{y}{1+x^{2}y^{2}}dx-\frac{x}{1+x^{2}y^{2}}dy\)C.\(\frac{x}{1+x^{2}y^{2}}dx+\frac{y}{1+x^{2}y^{2}}dy\)D.\(\frac{x}{1+x^{2}y^{2}}dx-\frac{y}{1+x^{2}y^{2}}dy\)10.設(shè)\(z=\sqrt{x^{2}+y^{2}}\)，則\(dz\)為（）A.\(\frac{x}{\sqrt{x^{2}+y^{2}}}dx+\frac{y}{\sqrt{x^{2}+y^{2}}}dy\)B.\(\frac{x}{\sqrt{x^{2}+y^{2}}}dx-\frac{y}{\sqrt{x^{2}+y^{2}}}dy\)C.\(\frac{y}{\sqrt{x^{2}+y^{2}}}dx+\frac{x}{\sqrt{x^{2}+y^{2}}}dy\)D.\(\frac{y}{\sqrt{x^{2}+y^{2}}}dx-\frac{x}{\sqrt{x^{2}+y^{2}}}dy\)多項(xiàng)選擇題（每題2分，共10題）1.以下哪些函數(shù)可求全微分（）A.\(z=x+y\)B.\(z=\frac{1}{x+y}\)C.\(z=\sinx\cosy\)D.\(z=x^y\)2.若\(z=f(x,y)\)在點(diǎn)\((x_0,y_0)\)處可微，則（）A.\(z=f(x,y)\)在點(diǎn)\((x_0,y_0)\)處連續(xù)B.\(z=f(x,y)\)在點(diǎn)\((x_0,y_0)\)處偏導(dǎo)數(shù)存在C.全微分\(dz=\frac{\partialz}{\partialx}dx+\frac{\partialz}{\partialy}dy\)D.函數(shù)的增量\(\Deltaz=dz+o(\sqrt{\Deltax^{2}+\Deltay^{2}})\)3.對(duì)于函數(shù)\(z=x^2-y^2\)，下列說(shuō)法正確的是（）A.\(\frac{\partialz}{\partialx}=2x\)B.\(\frac{\partialz}{\partialy}=-2y\)C.\(dz=2xdx-2ydy\)D.在點(diǎn)\((1,1)\)處\(dz=2dx-2dy\)4.設(shè)\(z=e^{x}\siny\)，則（）A.\(\frac{\partialz}{\partialx}=e^{x}\siny\)B.\(\frac{\partialz}{\partialy}=e^{x}\cosy\)C.\(dz=e^{x}\sinydx+e^{x}\cosydy\)D.在點(diǎn)\((0,\frac{\pi}{2})\)處\(dz=dx\)5.函數(shù)\(z=xy+x^2\)的全微分性質(zhì)有（）A.\(dz=(y+2x)dx+xdy\)B.是\(\Deltax,\Deltay\)的線(xiàn)性函數(shù)C.與函數(shù)增量之差是\(\sqrt{\Deltax^{2}+\Deltay^{2}}\)的高階無(wú)窮小D.當(dāng)\(\Deltax,\Deltay\)很小時(shí)可近似代替函數(shù)增量6.全微分\(dz\)與偏導(dǎo)數(shù)關(guān)系正確的是（）A.\(dz\)存在則偏導(dǎo)數(shù)一定存在B.偏導(dǎo)數(shù)存在則\(dz\)一定存在C.偏導(dǎo)數(shù)連續(xù)則\(dz\)一定存在D.\(dz\)存在偏導(dǎo)數(shù)不一定連續(xù)7.若\(z=\ln(x^{2}+y^{2})\)，則（）A.\(\frac{\partialz}{\partialx}=\frac{2x}{x^{2}+y^{2}}\)B.\(\frac{\partialz}{\partialy}=\frac{2y}{x^{2}+y^{2}}\)C.\(dz=\frac{2x}{x^{2}+y^{2}}dx+\frac{2y}{x^{2}+y^{2}}dy\)D.在點(diǎn)\((1,1)\)處\(dz=dx+dy\)8.對(duì)于函數(shù)\(z=\frac{x}{y}\)，以下正確的是（）A.\(\frac{\partialz}{\partialx}=\frac{1}{y}\)B.\(\frac{\partialz}{\partialy}=-\frac{x}{y^{2}}\)C.\(dz=\frac{1}{y}dx-\frac{x}{y^{2}}dy\)D.在點(diǎn)\((2,1)\)處\(dz=dx-2dy\)9.設(shè)\(z=x\cosy\)，則（）A.\(\frac{\partialz}{\partialx}=\cosy\)B.\(\frac{\partialz}{\partialy}=-x\siny\)C.\(dz=\cosydx-x\sinydy\)D.在點(diǎn)\((0,\pi)\)處\(dz=dx\)10.函數(shù)\(z=e^{x+y^2}\)的全微分相關(guān)結(jié)論有（）A.\(\frac{\partialz}{\partialx}=e^{x+y^2}\)B.\(\frac{\partialz}{\partialy}=2ye^{x+y^2}\)C.\(dz=e^{x+y^2}(dx+2ydy)\)D.在點(diǎn)\((0,0)\)處\(dz=dx+dy\)判斷題（每題2分，共10題）1.函數(shù)\(z=f(x,y)\)在某點(diǎn)偏導(dǎo)數(shù)存在，則該點(diǎn)全微分一定存在。（）2.若\(z=x+y\)，則\(dz=dx+dy\)。（）3.全微分\(dz\)是\(\Deltax,\Deltay\)的線(xiàn)性函數(shù)。（）4.函數(shù)\(z=x^2y\)在點(diǎn)\((1,1)\)處的全微分\(dz=2dx+dy\)。（）5.函數(shù)可微必連續(xù)。（）6.函數(shù)\(z=\frac{1}{x^{2}+y^{2}}\)的全微分\(dz=-\frac{2x}{(x^{2}+y^{2})^{2}}dx-\frac{2y}{(x^{2}+y^{2})^{2}}dy\)。（）7.偏導(dǎo)數(shù)連續(xù)是函數(shù)可微的必要條件。（）8.若函數(shù)\(z=f(x,y)\)全微分存在，則偏導(dǎo)數(shù)一定存在且唯一。（）9.對(duì)于\(z=\sin(x+y)\)，\(dz=\cos(x+y)(dx+dy)\)。（）10.函數(shù)\(z=e^{xy}\)的全微分\(dz=e^{xy}(ydx-xdy)\)。（）簡(jiǎn)答題（每題5分，共4題）1.簡(jiǎn)述函數(shù)\(z=f(x,y)\)在點(diǎn)\((x,y)\)可微的定義。2.全微分有什么幾何意義？3.說(shuō)明函數(shù)可微、偏導(dǎo)數(shù)存在與連續(xù)之間的關(guān)系。4.求函數(shù)\(z=x^2y\)的全微分。討論題（每題5分，共4題）1.討論函數(shù)全微分在實(shí)際問(wèn)題中的應(yīng)用。2.若函數(shù)在某點(diǎn)的偏導(dǎo)數(shù)存在但不連續(xù)，能否判斷函數(shù)在該點(diǎn)可微？3.結(jié)合實(shí)例說(shuō)明全微分近似代替函數(shù)增量的好處。4.當(dāng)函數(shù)\(z=f(x,y)\)的全微分存在時(shí)，如何利用它來(lái)分析函數(shù)的變化情況？答案單項(xiàng)選擇題1-5：ACAAB6-10：AAAAD多項(xiàng)選擇題1.ABCD2.ABCD3.ABCD4.ABCD5.ABCD6.ACD7.ABC8.ABCD9.ABC10.ABC判斷題1.×2.√3.√4.√5.√6.√7.×8.√9.√10.×簡(jiǎn)答題1.若函數(shù)\(z=f(x,y)\)在點(diǎn)\((x,y)\)的全增量\(\Deltaz=f(x+\Deltax,y+\Deltay)-f(x,y)\)可表示為\(\Deltaz=A\Deltax+B\Deltay+o(\sqrt{\Deltax^{2}+\Deltay^{2}})\)，其中\(zhòng)(A,B\)不依賴(lài)于\(\Deltax,\Deltay\)，則稱(chēng)函數(shù)在該點(diǎn)可微。2.全微分的幾何意義是：函數(shù)\(z=f(x,y)\)在點(diǎn)\((x_0,y_0)\)的全微分\(dz\)，表示過(guò)曲面\(z=f(x,y)\)上點(diǎn)\((x_0,y_0,f(x_0,y_0))\)的切平面上點(diǎn)的豎坐標(biāo)的增量。3.可微能推出偏導(dǎo)數(shù)存在和函數(shù)連續(xù)；偏導(dǎo)數(shù)存在推不出函數(shù)可微和連續(xù)；函數(shù)連續(xù)推不出偏導(dǎo)數(shù)存在和可微；偏導(dǎo)數(shù)連續(xù)可推出函數(shù)可微。4.先求偏導(dǎo)數(shù)\(\frac{\partialz}{\partialx}=2xy\)，\(\frac{\partialz}{\partialy}=x^{2}\)，則\(dz=2xydx+x^{2}dy\)。討論題1.在物理學(xué)中可用于計(jì)算