高數(shù)下第9章測試題及答案VIP

高數(shù)下第9章測試題及答案

一、單項選擇題（每題2分，共10題）1.二元函數(shù)\(z=\ln(x+y)\)的定義域是（）A.\(x+y\gt0\)B.\(x+y\geq0\)C.\(x\gt0,y\gt0\)D.\(x\neq0,y\neq0\)2.設(shè)\(z=x^2y\)，則\(\frac{\partialz}{\partialx}\)等于（）A.\(2xy\)B.\(x^2\)C.\(y\)D.\(2x\)3.函數(shù)\(z=f(x,y)\)在點\((x_0,y_0)\)處可微是函數(shù)在該點連續(xù)的（）A.充分條件B.必要條件C.充要條件D.既不充分也不必要條件4.設(shè)\(D\)是由\(x=0,y=0,x+y=1\)圍成的區(qū)域，則\(\iint_Ddxdy\)等于（）A.\(\frac{1}{2}\)B.\(1\)C.\(\frac{1}{3}\)D.\(2\)5.二次積分\(\int_0^1dx\int_0^xf(x,y)dy\)交換積分次序后為（）A.\(\int_0^1dy\int_0^yf(x,y)dx\)B.\(\int_0^1dy\int_y^1f(x,y)dx\)C.\(\int_0^xdy\int_0^1f(x,y)dx\)D.\(\int_0^xdy\int_y^1f(x,y)dx\)6.設(shè)\(z=e^{xy}\)，則\(dz\)等于（）A.\(e^{xy}dx\)B.\(e^{xy}dy\)C.\(ye^{xy}dx+xe^{xy}dy\)D.\(e^{xy}(dx+dy)\)7.函數(shù)\(f(x,y)=x^2+y^2\)在點\((0,0)\)處（）A.有極大值B.有極小值C.無極值D.不是駐點8.設(shè)\(D\)是圓域\(x^2+y^2\leq1\)，則\(\iint_D(x^2+y^2)dxdy\)等于（）A.\(\frac{\pi}{4}\)B.\(\frac{\pi}{2}\)C.\(\pi\)D.\(2\pi\)9.已知\(z=f(x^2-y^2)\)，則\(\frac{\partialz}{\partialy}\)等于（）A.\(f^\prime(x^2-y^2)\)B.\(-2yf^\prime(x^2-y^2)\)C.\(2yf^\prime(x^2-y^2)\)D.\(-f^\prime(x^2-y^2)\)10.設(shè)\(z=\frac{y}{x}\)，則\(\frac{\partial^2z}{\partialx\partialy}\)等于（）A.\(\frac{1}{x^2}\)B.\(-\frac{1}{x^2}\)C.\(\frac{1}{x}\)D.\(-\frac{1}{x}\)二、多項選擇題（每題2分，共10題）1.下列哪些是多元函數(shù)的概念（）A.二元函數(shù)B.三元函數(shù)C.\(n\)元函數(shù)D.復(fù)合函數(shù)2.二元函數(shù)\(z=f(x,y)\)在點\((x_0,y_0)\)處偏導(dǎo)數(shù)存在，則（）A.函數(shù)在該點連續(xù)B.函數(shù)在該點可微C.函數(shù)沿\(x\)軸方向可導(dǎo)D.函數(shù)沿\(y\)軸方向可導(dǎo)3.計算二重積分\(\iint_Df(x,y)dxdy\)時，常用的方法有（）A.直角坐標(biāo)法B.極坐標(biāo)法C.換元法D.分部積分法4.設(shè)\(z=f(u,v)\)，\(u=\varphi(x,y)\)，\(v=\psi(x,y)\)，則復(fù)合函數(shù)\(z=f(\varphi(x,y),\psi(x,y))\)的偏導(dǎo)數(shù)\(\frac{\partialz}{\partialx}\)為（）A.\(\frac{\partialf}{\partialu}\frac{\partial\varphi}{\partialx}\)B.\(\frac{\partialf}{\partialv}\frac{\partial\psi}{\partialx}\)C.\(\frac{\partialf}{\partialu}\frac{\partial\varphi}{\partialx}+\frac{\partialf}{\partialv}\frac{\partial\psi}{\partialx}\)D.\(\frac{\partialf}{\partialu}\frac{\partial\varphi}{\partialy}+\frac{\partialf}{\partialv}\frac{\partial\psi}{\partialy}\)5.函數(shù)\(z=f(x,y)\)的駐點滿足（）A.\(\frac{\partialz}{\partialx}=0\)B.\(\frac{\partialz}{\partialy}=0\)C.\(\frac{\partial^2z}{\partialx^2}=0\)D.\(\frac{\partial^2z}{\partialy^2}=0\)6.下列區(qū)域中，是有界閉區(qū)域的有（）A.圓\(x^2+y^2\leq1\)B.矩形\(0\leqx\leq1,0\leqy\leq1\)C.半平面\(x\geq0\)D.區(qū)域\(x^2+y^2\lt1\)7.關(guān)于二元函數(shù)的全微分，下列說法正確的是（）A.全微分存在則偏導(dǎo)數(shù)存在B.偏導(dǎo)數(shù)存在則全微分存在C.全微分\(dz=\frac{\partialz}{\partialx}dx+\frac{\partialz}{\partialy}dy\)D.全微分與偏導(dǎo)數(shù)無關(guān)8.計算\(\iint_Dxydxdy\)，其中\(zhòng)(D\)由\(y=x\)，\(y=0\)，\(x=1\)圍成，下列計算正確的是（）A.\(\int_0^1dx\int_0^xxydy\)B.\(\int_0^1dy\int_y^1xydx\)C.\(\int_0^1xdx\int_0^xydy\)D.\(\int_0^1ydy\int_y^1xdx\)9.設(shè)\(z=x^3+y^3-3xy\)，則（）A.駐點為\((0,0)\)B.駐點為\((1,1)\)C.在\((0,0)\)處有極值D.在\((1,1)\)處有極值10.多元函數(shù)的極值點可能是（）A.駐點B.偏導(dǎo)數(shù)不存在的點C.邊界點D.連續(xù)點三、判斷題（每題2分，共10題）1.二元函數(shù)\(z=f(x,y)\)在點\((x_0,y_0)\)處連續(xù)，則偏導(dǎo)數(shù)一定存在。（）2.二重積分\(\iint_Df(x,y)dxdy\)的值與積分區(qū)域\(D\)的劃分方式無關(guān)。（）3.若函數(shù)\(z=f(x,y)\)在點\((x_0,y_0)\)處可微，則函數(shù)在該點沿任何方向的方向?qū)?shù)都存在。（）4.函數(shù)\(z=f(x,y)\)的駐點一定是極值點。（）5.交換二次積分\(\int_0^1dx\int_0^xf(x,y)dy\)的積分次序后為\(\int_0^1dy\int_0^yf(x,y)dx\)。（）6.設(shè)\(z=f(x,y)\)，則\(\frac{\partial^2z}{\partialx\partialy}=\frac{\partial^2z}{\partialy\partialx}\)一定成立。（）7.若\(D\)是由\(x=0,y=0,x+y=1\)圍成的區(qū)域，則\(\iint_D1dxdy=\frac{1}{2}\)。（）8.函數(shù)\(z=x^2+y^2\)在點\((0,0)\)處的梯度為\((0,0)\)。（）9.計算二重積分時，極坐標(biāo)下\(dxdy=rdrd\theta\)。（）10.多元函數(shù)\(z=f(x,y)\)的全微分\(dz\)是\(\Deltaz\)的線性主部。（）四、簡答題（每題5分，共4題）1.簡述二元函數(shù)\(z=f(x,y)\)在點\((x_0,y_0)\)處可微的定義。答案：若函數(shù)\(z=f(x,y)\)在點\((x_0,y_0)\)處的全增量\(\Deltaz=f(x_0+\Deltax,y_0+\Deltay)-f(x_0,y_0)\)可表示為\(\Deltaz=A\Deltax+B\Deltay+o(\rho)\)，其中\(zhòng)(A,B\)與\(\Deltax,\Deltay\)無關(guān)，\(\rho=\sqrt{(\Deltax)^2+(\Deltay)^2}\)，則稱函數(shù)在點\((x_0,y_0)\)處可微。2.寫出計算二重積分\(\iint_Df(x,y)dxdy\)的直角坐標(biāo)計算公式。答案：若\(D\)是\(X-\)型區(qū)域：\(a\leqx\leqb,\varphi_1(x)\leqy\leq\varphi_2(x)\)，則\(\iint_Df(x,y)dxdy=\int_a^bdx\int_{\varphi_1(x)}^{\varphi_2(x)}f(x,y)dy\)；若\(D\)是\(Y-\)型區(qū)域：\(c\leqy\leqd,\psi_1(y)\leqx\leq\psi_2(y)\)，則\(\iint_Df(x,y)dxdy=\int_c^ddy\int_{\psi_1(y)}^{\psi_2(y)}f(x,y)dx\)。3.求函數(shù)\(z=x^2+3xy+y^2\)的偏導(dǎo)數(shù)\(\frac{\partialz}{\partialx}\)和\(\frac{\partialz}{\partialy}\)。答案：\(\frac{\partialz}{\partialx}=2x+3y\)，\(\frac{\partialz}{\partialy}=3x+2y\)。4.簡述多元函數(shù)極值的必要條件。答案：若函數(shù)\(z=f(x_1,x_2,\cdots,x_n)\)在點\(P_0(x_1^0,x_2^0,\cdots,x_n^0)\)處取得極值，且函數(shù)在該點的偏導(dǎo)數(shù)存在，則\(\frac{\partialf}{\partialx_i}|_{P_0}=0\)，\(i=1,2,\cdots,n\)，即駐點是極值點的必要條件。五、討論題（每題5分，共4題）1.討論二元函數(shù)\(z=f(x,y)\)在點\((x_0,y_0)\)處連續(xù)、偏導(dǎo)數(shù)存在、可微之間的關(guān)系。答案：可微能推出連續(xù)且偏導(dǎo)數(shù)存在；偏導(dǎo)數(shù)存在且連續(xù)能推出可微；但連續(xù)推不出偏導(dǎo)數(shù)存在，偏導(dǎo)數(shù)存在也推不出連續(xù)，偏導(dǎo)數(shù)存在也推不出可微。2.在計算二重積分時，如何選擇合適的坐標(biāo)系（直角坐標(biāo)或極坐標(biāo)）？答案：若積分區(qū)域是矩形、三角形等邊界由直線圍成，或被積函數(shù)關(guān)于\(x,y\)形式簡單，用直角坐標(biāo)；若積分區(qū)域是圓、扇形等，或被積函數(shù)含\(x^2+y^2\)，用極坐標(biāo)可簡化計算。3.討論函數(shù)\(z=x^3-3x+y^2\)的極值情況。答案：先求偏導(dǎo)數(shù)\(\frac{\partialz}{\partialx}=3x^2-3\)，\(\frac{\partialz}{\partialy}=2y\)，得駐點\((1,0)\)和\((-1,0)\)。再求二階偏導(dǎo)數(shù)判斷，在\((1,0)\)處有極小值\(-2\)，在\((-1,0)\)處無極值。4.舉例說明多元函數(shù)在實際問題中的應(yīng)用。答案：如在經(jīng)濟學(xué)中，生產(chǎn)函數(shù)\(Q=f(K,L)\)（\(Q\)為產(chǎn)量，\(K\)為資本，\(L\)