多元函數導數題目及答案

一、單項選擇題1.函數\(z=f(x,y)\)在點\((x_0,y_0)\)處可微的必要條件是（）A.\(f(x,y)\)在點\((x_0,y_0)\)處連續(xù)B.\(f_x(x_0,y_0)\)與\(f_y(x_0,y_0)\)都存在C.\(f(x,y)\)在點\((x_0,y_0)\)處沿任意方向的方向導數都存在D.\(f(x,y)\)在點\((x_0,y_0)\)處的偏導數連續(xù)答案：B2.設\(z=\ln(x+y^2)\)，則\(\frac{\partialz}{\partialx}\)在點\((1,1)\)處的值為（）A.\(\frac{1}{2}\)B.\(1\)C.\(2\)D.\(\frac{1}{3}\)答案：A3.已知\(z=x^2y+\sin(xy)\)，則\(\frac{\partialz}{\partialy}\)等于（）A.\(x^2+x\cos(xy)\)B.\(2xy+\cos(xy)\)C.\(x^2+\cos(xy)\)D.\(2xy+x\cos(xy)\)答案：A4.設\(u=e^{x+2y+3z}\)，\(x=t\)，\(y=t^2\)，\(z=t^3\)，則\(\frac{du}{dt}\)為（）A.\(e^{x+2y+3z}(1+4t+9t^2)\)B.\(e^{x+2y+3z}(1+2t+3t^2)\)C.\(e^{x+2y+3z}(4t+9t^2)\)D.\(e^{x+2y+3z}(1+4t)\)答案：A5.函數\(z=f(x,y)\)在點\((x_0,y_0)\)處的全微分\(dz\)等于（）A.\(f_x(x_0,y_0)\Deltax+f_y(x_0,y_0)\Deltay\)B.\(f_x(x_0,y_0)dx+f_y(x_0,y_0)dy\)C.\(\lim\limits_{\Deltax\to0}\frac{f(x_0+\Deltax,y_0)-f(x_0,y_0)}{\Deltax}\Deltax+\lim\limits_{\Deltay\to0}\frac{f(x_0,y_0+\Deltay)-f(x_0,y_0)}{\Deltay}\Deltay\)D.\(f(x_0+\Deltax,y_0+\Deltay)-f(x_0,y_0)\)答案：B6.設\(z=\arctan\frac{y}{x}\)，則\(\frac{\partial^2z}{\partialx^2}\)等于（）A.\(\frac{2xy}{(x^2+y^2)^2}\)B.\(-\frac{2xy}{(x^2+y^2)^2}\)C.\(\frac{y^2-x^2}{(x^2+y^2)^2}\)D.\(-\frac{y^2-x^2}{(x^2+y^2)^2}\)答案：B7.已知\(z=f(u,v)\)，\(u=x+y\)，\(v=xy\)，則\(\frac{\partialz}{\partialx}\)為（）A.\(f_u+f_v\)B.\(f_u+yf_v\)C.\(xf_u+yf_v\)D.\(xf_u+f_v\)答案：B8.函數\(z=x^3+y^3-3xy\)的駐點是（）A.\((0,0)\)和\((1,1)\)B.\((0,0)\)和\((-1,-1)\)C.\((1,1)\)和\((-1,-1)\)D.\((0,1)\)和\((1,0)\)答案：A9.設\(z=f(x^2-y^2,e^{xy})\)，則\(\frac{\partialz}{\partialy}\)為（）A.\(-2yf_1+xe^{xy}f_2\)B.\(2yf_1+xe^{xy}f_2\)C.\(-2yf_1-xe^{xy}f_2\)D.\(2yf_1-xe^{xy}f_2\)答案：A10.函數\(z=f(x,y)\)在區(qū)域\(D\)內具有二階連續(xù)偏導數，且\(A=f_{xx}(x_0,y_0)\)，\(B=f_{xy}(x_0,y_0)\)，\(C=f_{yy}(x_0,y_0)\)，若\(AC-B^2<0\)，則點\((x_0,y_0)\)（）A.是極大值點B.是極小值點C.不是極值點D.是否為極值點不能確定答案：C二、多項選擇題1.下列關于多元函數偏導數的說法正確的是（）A.若函數\(z=f(x,y)\)在點\((x_0,y_0)\)處偏導數存在，則函數在該點連續(xù)B.函數\(z=f(x,y)\)的偏導數\(f_x(x,y)\)是將\(y\)看作常數對\(x\)求導C.函數\(z=f(x,y)\)的偏導數\(f_y(x,y)\)是將\(x\)看作常數對\(y\)求導D.若函數\(z=f(x,y)\)在某區(qū)域內偏導數連續(xù)，則函數在該區(qū)域內可微答案：BCD2.設\(z=x^y\)（\(x>0\)），則（）A.\(\frac{\partialz}{\partialx}=yx^{y-1}\)B.\(\frac{\partialz}{\partialy}=x^y\lnx\)C.\(\frac{\partial^2z}{\partialx^2}=y(y-1)x^{y-2}\)D.\(\frac{\partial^2z}{\partialy^2}=x^y(\lnx)^2\)答案：ABCD3.對于函數\(z=f(x,y)\)在點\((x_0,y_0)\)處的全微分\(dz\)，以下說法正確的是（）A.\(dz\)是\(\Deltaz=f(x_0+\Deltax,y_0+\Deltay)-f(x_0,y_0)\)的線性主部B.\(dz=f_x(x_0,y_0)\Deltax+f_y(x_0,y_0)\Deltay\)C.當\(\vert\Deltax\vert\)，\(\vert\Deltay\vert\)很小時，\(\Deltaz\approxdz\)D.函數\(z=f(x,y)\)在點\((x_0,y_0)\)處可微則偏導數\(f_x(x_0,y_0)\)，\(f_y(x_0,y_0)\)一定存在答案：ABCD4.已知\(z=f(x,y)\)，\(x=r\cos\theta\)，\(y=r\sin\theta\)，則（）A.\(\frac{\partialz}{\partialr}=f_x\cos\theta+f_y\sin\theta\)B.\(\frac{\partialz}{\partial\theta}=-rf_x\sin\theta+rf_y\cos\theta\)C.\(\frac{\partial^2z}{\partialr^2}=f_{xx}\cos^2\theta+2f_{xy}\cos\theta\sin\theta+f_{yy}\sin^2\theta\)D.\(\frac{\partial^2z}{\partial\theta^2}=r^2(f_{xx}\sin^2\theta-2f_{xy}\sin\theta\cos\theta+f_{yy}\cos^2\theta)-rf_x\cos\theta-rf_y\sin\theta\)答案：ABCD5.函數\(z=f(x,y)\)在點\((x_0,y_0)\)處取得極值的充分條件有（）A.\(f_x(x_0,y_0)=0\)，\(f_y(x_0,y_0)=0\)且\(AC-B^2>0\)（其中\(zhòng)(A=f_{xx}(x_0,y_0)\)，\(B=f_{xy}(x_0,y_0)\)，\(C=f_{yy}(x_0,y_0)\)），當\(A>0\)時為極小值，當\(A<0\)時為極大值B.\(f(x,y)\)在點\((x_0,y_0)\)處的梯度\(\nablaf(x_0,y_0)=\vec{0}\)C.函數\(z=f(x,y)\)在點\((x_0,y_0)\)處可微且\(f_x(x_0,y_0)=0\)，\(f_y(x_0,y_0)=0\)D.\(f(x,y)\)在點\((x_0,y_0)\)處連續(xù)且在該點某去心鄰域內偏導數存在，且\(f_x(x,y)\)與\(f_y(x,y)\)在經過點\((x_0,y_0)\)時變號答案：AD6.設\(z=\sin(x+2y)\)，則（）A.\(\frac{\partialz}{\partialx}=\cos(x+2y)\)B.\(\frac{\partialz}{\partialy}=2\cos(x+2y)\)C.\(\frac{\partial^2z}{\partialx\partialy}=-2\sin(x+2y)\)D.\(\frac{\partial^2z}{\partialy^2}=-4\sin(x+2y)\)答案：ABCD7.已知函數\(u=f(x,y,z)\)，\(x=g(t)\)，\(y=h(t)\)，\(z=k(t)\)，則\(\frac{du}{dt}\)等于（）A.\(f_xg^\prime(t)+f_yh^\prime(t)+f_zk^\prime(t)\)B.\(\frac{\partialu}{\partialx}\frac{dx}{dt}+\frac{\partialu}{\partialy}\frac{dy}{dt}+\frac{\partialu}{\partialz}\frac{dz}{dt}\)C.\(f_1g^\prime(t)+f_2h^\prime(t)+f_3k^\prime(t)\)（這里\(f_1,f_2,f_3\)分別表示\(f\)對第一個、第二個、第三個變量的偏導數）D.\(f(x,y,z)\)對\(t\)的全導數答案：ABCD8.對于二元函數\(z=f(x,y)\)，以下結論正確的是（）A.若\(f(x,y)\)在點\((x_0,y_0)\)處的兩個偏導數都連續(xù)，則\(f(x,y)\)在點\((x_0,y_0)\)處可微B.若\(f(x,y)\)在點\((x_0,y_0)\)處可微，則\(f(x,y)\)在點\((x_0,y_0)\)處連續(xù)C.若\(f(x,y)\)在點\((x_0,y_0)\)處可微，則\(f(x,y)\)在點\((x_0,y_0)\)處的兩個偏導數都存在D.若\(f(x,y)\)在點\((x_0,y_0)\)處的兩個偏導數存在，則\(f(x,y)\)在點\((x_0,y_0)\)處可微答案：ABC9.設\(z=f(x^2+y^2)\)，則（）A.\(\frac{\partialz}{\partialx}=2xf^\prime(x^2+y^2)\)B.\(\frac{\partialz}{\partialy}=2yf^\prime(x^2+y^2)\)C.\(\frac{\partial^2z}{\partialx^2}=2f^\prime(x^2+y^2)+4x^2f^{\prime\prime}(x^2+y^2)\)D.\(\frac{\partial^2z}{\partialy^2}=2f^\prime(x^2+y^2)+4y^2f^{\prime\prime}(x^2+y^2)\)答案：ABCD10.函數\(z=f(x,y)\)在點\((x_0,y_0)\)處的方向導數\(\frac{\partialz}{\partiall}\)與偏導數的關系是（）A.\(\frac{\partialz}{\partiall}=f_x(x_0,y_0)\cos\alpha+f_y(x_0,y_0)\cos\beta\)，其中\(zhòng)(\cos\alpha,\cos\beta\)是方向\(l\)的方向余弦B.方向導數是偏導數在某一方向上的推廣C.若函數在點\((x_0,y_0)\)處可微，則沿任意方向\(l\)的方向導數都存在D.當方向\(l\)與\(x\)軸正向一致時，方向導數就是\(f_x(x_0,y_0)\)；當方向\(l\)與\(y\)軸正向一致時，方向導數就是\(f_y(x_0,y_0)\)答案：ABCD三、判斷題1.若函數\(z=f(x,y)\)在點\((x_0,y_0)\)處的兩個偏導數\(f_x(x_0,y_0)\)和\(f_y(x_0,y_0)\)都存在，則函數在該點一定連續(xù)。（）答案：錯誤2.函數\(z=f(x,y)\)的偏導數\(f_x(x,y)\)和\(f_y(x,y)\)在點\((x_0,y_0)\)處連續(xù)是函數在該點可微的充分條件。（）答案：正確3.對于函數\(z=f(x,y)\)，\(\frac{\partial^2z}{\partialx\partialy}\)和\(\frac{\partial^2z}{\partialy\partialx}\)一定相等