多元函数导数题目及答案

多元函数导数题目及答案

一、单项选择题1.函数\(z=f(x,y)\)在点\((x_0,y_0)\)处可微的必要条件是（）A.\(f(x,y)\)在点\((x_0,y_0)\)处连续B.\(f_x(x_0,y_0)\)与\(f_y(x_0,y_0)\)都存在C.\(f(x,y)\)在点\((x_0,y_0)\)处沿任意方向的方向导数都存在D.\(f(x,y)\)在点\((x_0,y_0)\)处的偏导数连续答案：B2.设\(z=\ln(x+y^2)\)，则\(\frac{\partialz}{\partialx}\)在点\((1,1)\)处的值为（）A.\(\frac{1}{2}\)B.\(1\)C.\(2\)D.\(\frac{1}{3}\)答案：A3.已知\(z=x^2y+\sin(xy)\)，则\(\frac{\partialz}{\partialy}\)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.设\(u=e^{x+2y+3z}\)，\(x=t\)，\(y=t^2\)，\(z=t^3\)，则\(\frac{du}{dt}\)为（）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.函数\(z=f(x,y)\)在点\((x_0,y_0)\)处的全微分\(dz\)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.设\(z=\arctan\frac{y}{x}\)，则\(\frac{\partial^2z}{\partialx^2}\)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\(z=f(u,v)\)，\(u=x+y\)，\(v=xy\)，则\(\frac{\partialz}{\partialx}\)为（）A.\(f_u+f_v\)B.\(f_u+yf_v\)C.\(xf_u+yf_v\)D.\(xf_u+f_v\)答案：B8.函数\(z=x^3+y^3-3xy\)的驻点是（）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.设\(z=f(x^2-y^2,e^{xy})\)，则\(\frac{\partialz}{\partialy}\)为（）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.函数\(z=f(x,y)\)在区域\(D\)内具有二阶连续偏导数，且\(A=f_{xx}(x_0,y_0)\)，\(B=f_{xy}(x_0,y_0)\)，\(C=f_{yy}(x_0,y_0)\)，若\(AC-B^2<0\)，则点\((x_0,y_0)\)（）A.是极大值点B.是极小值点C.不是极值点D.是否为极值点不能确定答案：C二、多项选择题1.下列关于多元函数偏导数的说法正确的是（）A.若函数\(z=f(x,y)\)在点\((x_0,y_0)\)处偏导数存在，则函数在该点连续B.函数\(z=f(x,y)\)的偏导数\(f_x(x,y)\)是将\(y\)看作常数对\(x\)求导C.函数\(z=f(x,y)\)的偏导数\(f_y(x,y)\)是将\(x\)看作常数对\(y\)求导D.若函数\(z=f(x,y)\)在某区域内偏导数连续，则函数在该区域内可微答案：BCD2.设\(z=x^y\)（\(x>0\)），则（）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.对于函数\(z=f(x,y)\)在点\((x_0,y_0)\)处的全微分\(dz\)，以下说法正确的是（）A.\(dz\)是\(\Deltaz=f(x_0+\Deltax,y_0+\Deltay)-f(x_0,y_0)\)的线性主部B.\(dz=f_x(x_0,y_0)\Deltax+f_y(x_0,y_0)\Deltay\)C.当\(\vert\Deltax\vert\)，\(\vert\Deltay\vert\)很小时，\(\Deltaz\approxdz\)D.函数\(z=f(x,y)\)在点\((x_0,y_0)\)处可微则偏导数\(f_x(x_0,y_0)\)，\(f_y(x_0,y_0)\)一定存在答案：ABCD4.已知\(z=f(x,y)\)，\(x=r\cos\theta\)，\(y=r\sin\theta\)，则（）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.函数\(z=f(x,y)\)在点\((x_0,y_0)\)处取得极值的充分条件有（）A.\(f_x(x_0,y_0)=0\)，\(f_y(x_0,y_0)=0\)且\(AC-B^2>0\)（其中\(A=f_{xx}(x_0,y_0)\)，\(B=f_{xy}(x_0,y_0)\)，\(C=f_{yy}(x_0,y_0)\)），当\(A>0\)时为极小值，当\(A<0\)时为极大值B.\(f(x,y)\)在点\((x_0,y_0)\)处的梯度\(\nablaf(x_0,y_0)=\vec{0}\)C.函数\(z=f(x,y)\)在点\((x_0,y_0)\)处可微且\(f_x(x_0,y_0)=0\)，\(f_y(x_0,y_0)=0\)D.\(f(x,y)\)在点\((x_0,y_0)\)处连续且在该点某去心邻域内偏导数存在，且\(f_x(x,y)\)与\(f_y(x,y)\)在经过点\((x_0,y_0)\)时变号答案：AD6.设\(z=\sin(x+2y)\)，则（）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.已知函数\(u=f(x,y,z)\)，\(x=g(t)\)，\(y=h(t)\)，\(z=k(t)\)，则\(\frac{du}{dt}\)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.\(f_1g^\prime(t)+f_2h^\prime(t)+f_3k^\prime(t)\)（这里\(f_1,f_2,f_3\)分别表示\(f\)对第一个、第二个、第三个变量的偏导数）D.\(f(x,y,z)\)对\(t\)的全导数答案：ABCD8.对于二元函数\(z=f(x,y)\)，以下结论正确的是（）A.若\(f(x,y)\)在点\((x_0,y_0)\)处的两个偏导数都连续，则\(f(x,y)\)在点\((x_0,y_0)\)处可微B.若\(f(x,y)\)在点\((x_0,y_0)\)处可微，则\(f(x,y)\)在点\((x_0,y_0)\)处连续C.若\(f(x,y)\)在点\((x_0,y_0)\)处可微，则\(f(x,y)\)在点\((x_0,y_0)\)处的两个偏导数都存在D.若\(f(x,y)\)在点\((x_0,y_0)\)处的两个偏导数存在，则\(f(x,y)\)在点\((x_0,y_0)\)处可微答案：ABC9.设\(z=f(x^2+y^2)\)，则（）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.函数\(z=f(x,y)\)在点\((x_0,y_0)\)处的方向导数\(\frac{\partialz}{\partiall}\)与偏导数的关系是（）A.\(\frac{\partialz}{\partiall}=f_x(x_0,y_0)\cos\alpha+f_y(x_0,y_0)\cos\beta\)，其中\(\cos\alpha,\cos\beta\)是方向\(l\)的方向余弦B.方向导数是偏导数在某一方向上的推广C.若函数在点\((x_0,y_0)\)处可微，则沿任意方向\(l\)的方向导数都存在D.当方向\(l\)与\(x\)轴正向一致时，方向导数就是\(f_x(x_0,y_0)\)；当方向\(l\)与\(y\)轴正向一致时，方向导数就是\(f_y(x_0,y_0)\)答案：ABCD三、判断题1.若函数\(z=f(x,y)\)在点\((x_0,y_0)\)处的两个偏导数\(f_x(x_0,y_0)\)和\(f_y(x_0,y_0)\)都存在，则函数在该点一定连续。（）答案：错误2.函数\(z=f(x,y)\)的偏导数\(f_x(x,y)\)和\(f_y(x,y)\)在点\((x_0,y_0)\)处连续是函数在该点可微的充分条件。（）答案：正确3.对于函数\(z=f(x,y)\)，\(\frac{\partial^2z}{\partialx\partialy}\)和\(\frac{\partial^2z}{\partialy\partialx}\)一定相等