偏导数题库及答案解析图

偏导数题库及答案解析图

一、单项选择题(每题2分,共10题)1.函数\(z=x^2+3xy+y^2\),\(\frac{\partialz}{\partialx}\)等于（）A.\(2x+3y\)B.\(3x+2y\)C.\(2x+y\)D.\(x+2y\)2.设\(z=e^{xy}\),则\(\frac{\partialz}{\partialy}\)为()A.\(xe^{xy}\)B.\(ye^{xy}\)C.\(e^{xy}\)D.\(xye^{xy}\)3.函数\(f(x,y)=x^3-3x+y^2\)在点\((1,1)\)处关于\(x\)的偏导数\(f_x(1,1)\)是()A.0B.1C.2D.34.已知\(z=\ln(x^2+y^2)\),\(\frac{\partialz}{\partialx}\)在点\((1,1)\)的值为()A.1B.\(\frac{1}{2}\)C.\(\frac{1}{\sqrt{2}}\)D.25.设\(z=x^y\),则\(\frac{\partialz}{\partialx}\)为()A.\(yx^{y-1}\)B.\(x^y\lnx\)C.\(yx^y\)D.\(x^y\lny\)6.函数\(z=\sin(xy)\),\(\frac{\partialz}{\partialy}\)等于()A.\(x\cos(xy)\)B.\(y\cos(xy)\)C.\(\cos(xy)\)D.\(xy\cos(xy)\)7.已知\(f(x,y)=x^2y+xy^2\),\(f_y(1,1)\)的值为()A.1B.2C.3D.48.函数\(z=\sqrt{x^2+y^2}\),\(\frac{\partialz}{\partialx}\)在点\((1,0)\)处的值为()A.0B.1C.\(\frac{1}{2}\)D.\(\frac{1}{\sqrt{2}}\)9.设\(z=\arctan(\frac{y}{x})\),则\(\frac{\partialz}{\partialy}\)为()A.\(\frac{-y}{x^2+y^2}\)B.\(\frac{x}{x^2+y^2}\)C.\(\frac{-x}{x^2+y^2}\)D.\(\frac{y}{x^2+y^2}\)10.函数\(z=e^{x+2y}\),\(\frac{\partialz}{\partialy}\)等于()A.\(e^{x+2y}\)B.\(2e^{x+2y}\)C.\(e^{x+y}\)D.\(2e^{x+y}\)二、多项选择题(每题2分,共10题)1.对于函数\(z=x^2+y^2\),以下说法正确的是（）A.\(\frac{\partialz}{\partialx}=2x\)B.\(\frac{\partialz}{\partialy}=2y\)C.\(z\)关于\(x\)的偏导数与\(y\)有关D.\(z\)关于\(y\)的偏导数与\(x\)有关2.设\(z=\sin(x+y)\),则()A.\(\frac{\partialz}{\partialx}=\cos(x+y)\)B.\(\frac{\partialz}{\partialy}=\cos(x+y)\)C.\(\frac{\partial^2z}{\partialx\partialy}=-\sin(x+y)\)D.\(\frac{\partial^2z}{\partialx^2}=-\sin(x+y)\)3.函数\(z=x^3y^2\)的偏导数有()A.\(\frac{\partialz}{\partialx}=3x^2y^2\)B.\(\frac{\partialz}{\partialy}=2x^3y\)C.\(\frac{\partial^2z}{\partialx\partialy}=6x^2y\)D.\(\frac{\partial^2z}{\partialy^2}=2x^3\)4.已知\(z=e^{xy}\),下列正确的是()A.\(\frac{\partialz}{\partialx}=ye^{xy}\)B.\(\frac{\partialz}{\partialy}=xe^{xy}\)C.\(\frac{\partial^2z}{\partialx^2}=y^2e^{xy}\)D.\(\frac{\partial^2z}{\partialx\partialy}=(1+xy)e^{xy}\)5.对于函数\(z=\ln(x^2-y^2)\),()A.定义域为\(x^2>y^2\)B.\(\frac{\partialz}{\partialx}=\frac{2x}{x^2-y^2}\)C.\(\frac{\partialz}{\partialy}=\frac{-2y}{x^2-y^2}\)D.偏导数在定义域内处处连续6.设\(z=\sqrt{xy}\),则()A.\(\frac{\partialz}{\partialx}=\frac{\sqrt{y}}{2\sqrt{x}}\)B.\(\frac{\partialz}{\partialy}=\frac{\sqrt{x}}{2\sqrt{y}}\)C.\(\frac{\partial^2z}{\partialx\partialy}=\frac{1}{4\sqrt{xy}}\)D.\(\frac{\partial^2z}{\partialx^2}=-\frac{\sqrt{y}}{4x^{\frac{3}{2}}}\)7.函数\(z=x^y\)的偏导数()A.\(\frac{\partialz}{\partialx}=yx^{y-1}\)B.\(\frac{\partialz}{\partialy}=x^y\lnx\)C.\(\frac{\partial^2z}{\partialx\partialy}=x^{y-1}(1+y\lnx)\)D.\(\frac{\partial^2z}{\partialy^2}=x^y(\lnx)^2\)8.已知\(z=\arctan(\frac{x}{y})\),则()A.\(\frac{\partialz}{\partialx}=\frac{y}{x^2+y^2}\)B.\(\frac{\partialz}{\partialy}=\frac{-x}{x^2+y^2}\)C.\(\frac{\partial^2z}{\partialx^2}=\frac{-2xy}{(x^2+y^2)^2}\)D.\(\frac{\partial^2z}{\partialx\partialy}=\frac{x^2-y^2}{(x^2+y^2)^2}\)9.对于函数\(z=x\siny+y\cosx\),()A.\(\frac{\partialz}{\partialx}=\siny-y\sinx\)B.\(\frac{\partialz}{\partialy}=x\cosy+\cosx\)C.\(\frac{\partial^2z}{\partialx^2}=-y\cosx\)D.\(\frac{\partial^2z}{\partialy^2}=-x\siny\)10.设\(z=e^{x^2+y^2}\),则()A.\(\frac{\partialz}{\partialx}=2xe^{x^2+y^2}\)B.\(\frac{\partialz}{\partialy}=2ye^{x^2+y^2}\)C.\(\frac{\partial^2z}{\partialx^2}=(2+4x^2)e^{x^2+y^2}\)D.\(\frac{\partial^2z}{\partialx\partialy}=4xye^{x^2+y^2}\)三、判断题(每题2分,共10题)1.函数\(z=x+y\)关于\(x\)的偏导数就是\(1\)。()2.若\(z=f(x,y)\),则\(\frac{\partialz}{\partialx}\)与\(\frac{\partialz}{\partialy}\)一定不相等。（)3.函数\(z=x^2y\)的二阶混合偏导数\(\frac{\partial^2z}{\partialx\partialy}\)和\(\frac{\partial^2z}{\partialy\partialx}\)相等。()4.对于\(z=\ln(xy)\),\(\frac{\partialz}{\partialx}=\frac{1}{x}\)。（)5.函数\(z=\sin(x-y)\)关于\(y\)的偏导数是\(-\cos(x-y)\)。()6.若\(z=e^{x^2y}\),则\(\frac{\partialz}{\partialy}=x^2e^{x^2y}\)。（)7.函数\(z=\sqrt{x^2-y}\)在其定义域内偏导数都存在。()8.对于\(z=x^y\),\(\frac{\partialz}{\partialx}\)在\((0,0)\)处无定义。(）9.已知\(z=\arctan(\frac{x+y}{x-y})\),则\(\frac{\partialz}{\partialx}+\frac{\partialz}{\partialy}=0\)。（)10.函数\(z=x^3+y^3-3xy\)的驻点是\((0,0)\)和\((1,1)\)。(）四、简答题(每题5分,共4题)1.简述求函数\(z=f(x,y)\)偏导数\(\frac{\partialz}{\partialx}\)的方法。将\(y\)视为常数,对\(x\)求导。2.已知\(z=x^2\siny\),求\(\frac{\partialz}{\partialy}\)。\(\frac{\partialz}{\partialy}=x^2\cosy\)。3.说明函数\(z=\ln(x^2+y^2)\)的定义域。\(x^2+y^2>0\),即除原点外的整个平面。4.若\(z=e^{2x+3y}\),求\(\frac{\partial^2z}{\partialx\partialy}\)。先求\(\frac{\partialz}{\partialx}=2e^{2x+3y}\),再对\(y\)求导得\(\frac{\partial^2z}{\partialx\partialy}=6e^{2x+3y}\)。五、讨论题(每题5分,共4题)1.讨论函数\(z=x^3-3x^2-9x+y^2\)的极值情况。求偏导并令其为0,得驻点。再求二阶偏导判断,\(A=6x-6\),\(B=0\),\(C=2\)。根据判别式判断极值。2.对于函数\(z=\frac{x^2}{y}\),讨论\(x\)、\(y\)变化时偏导数的变化。\(\frac{\partialz}{\partialx}=\frac{2x}{y}\),\(\frac{\partialz}{\partialy}=-\frac{x^2}{y^2}\)。\(x\)增大\(\frac{\partialz}{\partialx}\)增大,\(y\)增大\(\frac{\partialz}{\partialx}\)不变,\(\frac{\partialz}{\partialy}\)绝对值减小。3.讨论函数\(z=\sin(x+y)\)的偏导数的周期性。\(\frac{\partialz}{\partialx}=\cos(x+y)\),\(\frac{\partialz}{\partialy}=\cos(x+y)\),都以\(2\pi\)为周期。4.已知\(z=x^y\),讨论其偏导数在不同象限的正负情况。\(\frac{\partialz}{\partialx}=yx^{y-1}\),\(\frac{\partialz}{\partialy}=x^y\lnx\)。在第一象限\(x>0,y>0\)时,\(\frac{\partialz}{\p