高数下册试卷及答案

高数下册试卷及答案

一、单项选择题1.空间曲线\(x=t\)，\(y=t^2\)，\(z=t^3\)在点\((1,1,1)\)处的切线方程为（）A.\(\frac{x-1}{1}=\frac{y-1}{2}=\frac{z-1}{3}\)B.\(\frac{x-1}{1}=\frac{y-1}{1}=\frac{z-1}{1}\)C.\(\frac{x-1}{3}=\frac{y-1}{2}=\frac{z-1}{1}\)D.\(\frac{x-1}{1}=\frac{y-1}{3}=\frac{z-1}{2}\)答案：A2.函数\(z=f(x,y)\)在点\((x_0,y_0)\)处可微的充分条件是（）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.\(\lim\limits_{\Deltax\to0,\Deltay\to0}[\Deltaz-f_x(x_0,y_0)\Deltax-f_y(x_0,y_0)\Deltay]=0\)D.\(\lim\limits_{\rho\to0}\frac{\Deltaz-f_x(x_0,y_0)\Deltax-f_y(x_0,y_0)\Deltay}{\rho}=0\)，其中\(\rho=\sqrt{(\Deltax)^2+(\Deltay)^2}\)答案：D3.设\(D\)是由\(x=0\)，\(y=0\)，\(x+y=1\)所围成的区域，则\(\iint_Dxyd\sigma\)的值为（）A.\(\frac{1}{24}\)B.\(\frac{1}{12}\)C.\(\frac{1}{6}\)D.\(\frac{1}{4}\)答案：A4.设\(\varOmega\)是由\(z=x^2+y^2\)与\(z=1\)所围成的闭区域，则\(\iiint_{\varOmega}dxdydz\)的值为（）A.\(\frac{\pi}{2}\)B.\(\frac{\pi}{3}\)C.\(\frac{\pi}{4}\)D.\(\frac{\pi}{6}\)答案：D5.幂级数\(\sum_{n=1}^{\infty}\frac{x^n}{n}\)的收敛半径为（）A.\(0\)B.\(1\)C.\(2\)D.\(+\infty\)答案：B6.下列级数中，绝对收敛的是（）A.\(\sum_{n=1}^{\infty}(-1)^{n-1}\frac{1}{n}\)B.\(\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{\sqrt{n}}\)C.\(\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^2}\)D.\(\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{0.5}}\)答案：C7.微分方程\(y''-2y'+y=0\)的通解是（）A.\(y=C_1e^x+C_2xe^x\)B.\(y=C_1e^{-x}+C_2xe^{-x}\)C.\(y=C_1e^x+C_2e^{-x}\)D.\(y=C_1+C_2e^x\)答案：A8.设向量\(\vec{a}=(1,-2,3)\)，\(\vec{b}=(2,1,0)\)，则\(\vec{a}\cdot\vec{b}\)等于（）A.\(0\)B.\(-1\)C.\(1\)D.\(2\)答案：B9.设\(z=\ln(x+y)\)，则\(\frac{\partialz}{\partialx}\)在点\((1,1)\)处的值为（）A.\(\frac{1}{2}\)B.\(1\)C.\(2\)D.\(0\)答案：A10.曲线积分\(\int_{L}xdy-ydx\)，其中\(L\)是单位圆\(x^2+y^2=1\)正向一周，则其值为（）A.\(0\)B.\(2\pi\)C.\(\pi\)D.\(-\pi\)答案：B二、多项选择题1.下列关于多元函数极限的说法正确的是（）A.若\(\lim\limits_{(x,y)\to(x_0,y_0)}f(x,y)\)存在，则\(f(x,y)\)在\((x_0,y_0)\)处一定有定义B.若\(f(x,y)\)在\((x_0,y_0)\)处的两个累次极限都存在且相等，则\(\lim\limits_{(x,y)\to(x_0,y_0)}f(x,y)\)一定存在C.若\(\lim\limits_{(x,y)\to(x_0,y_0)}f(x,y)=A\)，则\(\lim\limits_{x\tox_0}[\lim\limits_{y\toy_0}f(x,y)]=A\)（假设累次极限存在）D.多元函数极限存在的充要条件是沿任意路径趋近于某点时极限都存在且相等答案：CD2.设函数\(z=f(x,y)\)具有二阶连续偏导数，下列说法正确的是（）A.\(f_{xy}(x,y)=f_{yx}(x,y)\)B.\((f_x(x,y))_y=f_{xy}(x,y)\)C.函数\(z\)的全微分\(dz=f_x(x,y)dx+f_y(x,y)dy\)D.若\(f_{xx}(x_0,y_0)A\)，\(f_{xy}(x_0,y_0)=B\)，\(f_{yy}(x_0,y_0)=C\)，且\(AC-B^2\gt0\)，\(A\gt0\)，则\(f(x_0,y_0)\)为极小值答案：ABCD3.下列二重积分计算正确的是（）A.若\(D\)是由\(x=0\)，\(y=0\)，\(x+y=1\)围成，则\(\iint_Dx^2yd\sigma=\int_{0}^{1}dx\int_{0}^{1-x}x^2ydy\)B.若\(D\)是由\(x^2+y^2\leq1\)围成，则\(\iint_Df(x,y)d\sigma=\int_{0}^{2\pi}d\theta\int_{0}^{1}f(r\cos\theta,r\sin\theta)rdr\)C.若\(D\)是由\(y=x\)，\(y=x^2\)围成，则\(\iint_D(x+y)d\sigma=\int_{0}^{1}dx\int_{x^2}^{x}(x+y)dy\)D.若\(D\)是由\(x=1\)，\(x=2\)，\(y=0\)，\(y=x\)围成，则\(\iint_Dxyd\sigma=\int_{1}^{2}dx\int_{0}^{x}xydy\)答案：ABCD4.关于三重积分，下列说法正确的是（）A.直角坐标系下\(\iiint_{\varOmega}f(x,y,z)dxdydz=\int_{a}^{b}dx\int_{c(x)}^{d(x)}dy\int_{e(x,y)}^{f(x,y)}f(x,y,z)dz\)B.柱面坐标系下\(\iiint_{\varOmega}f(x,y,z)dxdydz=\int_{\alpha}^{\beta}d\theta\int_{r_1(\theta)}^{r_2(\theta)}rdr\int_{z_1(r\theta)}^{z_2(r\theta)}f(r\cos\theta,r\sin\theta,z)dz\)C.球面坐标系下\(\iiint_{\varOmega}f(x,y,z)dxdydz=\int_{0}^{2\pi}d\theta\int_{0}^{\pi}d\varphi\int_{r_1(\theta,\varphi)}^{r_2(\theta,\varphi)}f(r\sin\varphi\cos\theta,r\sin\varphi\sin\theta,r\cos\varphi)r^2\sin\varphidr\)D.若\(\varOmega\)关于\(x=0\)对称，\(f(x,y,z)\)关于\(x\)为奇函数，则\(\iiint_{\varOmega}f(x,y,z)dxdydz=0\)答案：ABCD5.下列幂级数的收敛区间正确的是（）A.幂级数\(\sum_{n=0}^{\infty}x^n\)的收敛区间是\((-1,1)\)B.幂级数\(\sum_{n=1}^{\infty}\frac{x^n}{n!}\)的收敛区间是\((-\infty,+\infty)\)C.幂级数\(\sum_{n=1}^{\infty}n^2x^n\)的收敛区间是\((-1,1)\)D.幂级数\(\sum_{n=1}^{\infty}\frac{(x-1)^n}{n}\)的收敛区间是\((0,2)\)答案：ABCD6.下列关于常数项级数敛散性的说法正确的是（）A.若\(\lim\limits_{n\to\infty}u_n\neq0\)，则\(\sum_{n=1}^{\infty}u_n\)发散B.若\(\sum_{n=1}^{\infty}u_n\)与\(\sum_{n=1}^{\infty}v_n\)都收敛，则\(\sum_{n=1}^{\infty}(u_n+v_n)\)收敛C.若\(\sum_{n=1}^{\infty}u_n\)收敛，\(\sum_{n=1}^{\infty}v_n\)发散，则\(\sum_{n=1}^{\infty}(u_n+v_n)\)发散D.正项级数\(\sum_{n=1}^{\infty}u_n\)收敛的充要条件是其部分和数列\(\{S_n\}\)有界答案：ABCD7.下列微分方程是线性微分方程的是（）A.\(y'+xy=e^x\)B.\(y''+y^2=0\)C.\(y''+2y'+y=\sinx\)D.\(xy'-y=x^2\)答案：ACD8.设向量\(\vec{a}=(x_1,y_1,z_1)\)，\(\vec{b}=(x_2,y_2,z_2)\)，下列运算正确的是（）A.\(\vec{a}+\vec{b}=(x_1+x_2,y_1+y_2,z_1+z_2)\)B.\(\vec{a}\cdot\vec{b}=x_1x_2+y_1y_2+z_1z_2\)C.\(\vec{a}\times\vec{b}=(y_1z_2-y_2z_1,z_1x_2-z_2x_1,x_1y_2-x_2y_1)\)D.\(k\vec{a}=(kx_1,ky_1,kz_1)\)（\(k\)为常数）答案：ABCD9.下列关于曲线积分的说法正确的是（）A.第一类曲线积分\(\int_{L}f(x,y)ds\)与曲线\(L\)的方向无关B.第二类曲线积分\(\int_{L}P(x,y)dx+Q(x,y)dy\)与曲线\(L\)的方向有关C.若\(L\)是封闭曲线，\(P\)，\(Q\)在\(L\)所围成区域\(D\)上具有一阶连续偏导数，则\(\oint_{L}Pdx+Qdy=\iint_{D}(\frac{\partialQ}{\partialx}-\frac{\partialP}{\partialy})d\sigma\)（格林公式）D.曲线积分\(\int_{L}f(x,y)ds\)可以通过将曲线\(L\)参数化后转化为定积分计算答案：ABCD10.关于曲面积分，下列说法正确的是（）A.第一类曲面积分\(\iint_{\varSigma}f(x,y,z)dS\)与曲面\(\varSigma\)的侧无关B.第二类曲面积分\(\iint_{\varSigma}P(x,y,z)dydz+Q(x,y,z)dzdx+R(x,y,z)dxdy\)与曲面\(\varSigma\)的侧有关C.若\(\varSigma\)是封闭曲面，\(P\)，\(Q\)，\(R\)在\(\varSigma\)所围成区域\(\varOmega\)上具有一阶连续偏导数，则\(\oiint_{\varSigma}Pdydz+Qdzdx+Rdxdy=\iiint_{\varOmega}(\frac{\partialP}{\partialx}+\frac{\partialQ}{\partialy}+\frac{\partialR}{\partialz})dxdydz\)（高斯公式）D.计算第一类曲面积分可以通过将曲面方程代入并转化为二重积分计算答案：ABCD三、判断题1.若函数\(z=f(x,y)\)在点\((x_0,y_0)\)处的偏导数\(f_x(x_0,y_0)\)和\(f_y(x_0,y_0)\)都存在，则\(z=f(x,y)\)在点\((x_0,y_0)\)处一定连续。（×）2.函数\(z=f(x,y)\)在点\((x_0,y_0)\)处的全微分\(dz=f_x(x_0,y_0)\Deltax+f_y(x_0,y_0)\Deltay\)。（×）3.二