高數(shù)下冊(cè)試卷及答案

一、單項(xiàng)選擇題1.空間曲線(xiàn)\(x=t\)，\(y=t^2\)，\(z=t^3\)在點(diǎn)\((1,1,1)\)處的切線(xiàn)方程為（）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.函數(shù)\(z=f(x,y)\)在點(diǎn)\((x_0,y_0)\)處可微的充分條件是（）A.\(f(x,y)\)在點(diǎn)\((x_0,y_0)\)處連續(xù)B.\(f_x(x_0,y_0)\)與\(f_y(x_0,y_0)\)都存在C.\(\lim\limits_{\Deltax\to0,\Deltay\to0}[\Deltaz-f_x(x_0,y_0)\Deltax-f_y(x_0,y_0)\Deltay]=0\)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\(zhòng)(\rho=\sqrt{(\Deltax)^2+(\Deltay)^2}\)答案：D3.設(shè)\(D\)是由\(x=0\)，\(y=0\)，\(x+y=1\)所圍成的區(qū)域，則\(\iint_Dxyd\sigma\)的值為（）A.\(\frac{1}{24}\)B.\(\frac{1}{12}\)C.\(\frac{1}{6}\)D.\(\frac{1}{4}\)答案：A4.設(shè)\(\varOmega\)是由\(z=x^2+y^2\)與\(z=1\)所圍成的閉區(qū)域，則\(\iiint_{\varOmega}dxdydz\)的值為（）A.\(\frac{\pi}{2}\)B.\(\frac{\pi}{3}\)C.\(\frac{\pi}{4}\)D.\(\frac{\pi}{6}\)答案：D5.冪級(jí)數(shù)\(\sum_{n=1}^{\infty}\frac{x^n}{n}\)的收斂半徑為（）A.\(0\)B.\(1\)C.\(2\)D.\(+\infty\)答案：B6.下列級(jí)數(shù)中，絕對(duì)收斂的是（）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.設(shè)向量\(\vec{a}=(1,-2,3)\)，\(\vec=(2,1,0)\)，則\(\vec{a}\cdot\vec\)等于（）A.\(0\)B.\(-1\)C.\(1\)D.\(2\)答案：B9.設(shè)\(z=\ln(x+y)\)，則\(\frac{\partialz}{\partialx}\)在點(diǎn)\((1,1)\)處的值為（）A.\(\frac{1}{2}\)B.\(1\)C.\(2\)D.\(0\)答案：A10.曲線(xiàn)積分\(\int_{L}xdy-ydx\)，其中\(zhòng)(L\)是單位圓\(x^2+y^2=1\)正向一周，則其值為（）A.\(0\)B.\(2\pi\)C.\(\pi\)D.\(-\pi\)答案：B二、多項(xiàng)選擇題1.下列關(guān)于多元函數(shù)極限的說(shuō)法正確的是（）A.若\(\lim\limits_{(x,y)\to(x_0,y_0)}f(x,y)\)存在，則\(f(x,y)\)在\((x_0,y_0)\)處一定有定義B.若\(f(x,y)\)在\((x_0,y_0)\)處的兩個(gè)累次極限都存在且相等，則\(\lim\limits_{(x,y)\to(x_0,y_0)}f(x,y)\)一定存在C.若\(\lim\limits_{(x,y)\to(x_0,y_0)}f(x,y)=A\)，則\(\lim\limits_{x\tox_0}[\lim\limits_{y\toy_0}f(x,y)]=A\)（假設(shè)累次極限存在）D.多元函數(shù)極限存在的充要條件是沿任意路徑趨近于某點(diǎn)時(shí)極限都存在且相等答案：CD2.設(shè)函數(shù)\(z=f(x,y)\)具有二階連續(xù)偏導(dǎo)數(shù)，下列說(shuō)法正確的是（）A.\(f_{xy}(x,y)=f_{yx}(x,y)\)B.\((f_x(x,y))_y=f_{xy}(x,y)\)C.函數(shù)\(z\)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\(AC-B^2\gt0\)，\(A\gt0\)，則\(f(x_0,y_0)\)為極小值答案：ABCD3.下列二重積分計(jì)算正確的是（）A.若\(D\)是由\(x=0\)，\(y=0\)，\(x+y=1\)圍成，則\(\iint_Dx^2yd\sigma=\int_{0}^{1}dx\int_{0}^{1-x}x^2ydy\)B.若\(D\)是由\(x^2+y^2\leq1\)圍成，則\(\iint_Df(x,y)d\sigma=\int_{0}^{2\pi}d\theta\int_{0}^{1}f(r\cos\theta,r\sin\theta)rdr\)C.若\(D\)是由\(y=x\)，\(y=x^2\)圍成，則\(\iint_D(x+y)d\sigma=\int_{0}^{1}dx\int_{x^2}^{x}(x+y)dy\)D.若\(D\)是由\(x=1\)，\(x=2\)，\(y=0\)，\(y=x\)圍成，則\(\iint_Dxyd\sigma=\int_{1}^{2}dx\int_{0}^{x}xydy\)答案：ABCD4.關(guān)于三重積分，下列說(shuō)法正確的是（）A.直角坐標(biāo)系下\(\iiint_{\varOmega}f(x,y,z)dxdydz=\int_{a}^dx\int_{c(x)}^{d(x)}dy\int_{e(x,y)}^{f(x,y)}f(x,y,z)dz\)B.柱面坐標(biāo)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.球面坐標(biāo)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\(\varOmega\)關(guān)于\(x=0\)對(duì)稱(chēng)，\(f(x,y,z)\)關(guān)于\(x\)為奇函數(shù)，則\(\iiint_{\varOmega}f(x,y,z)dxdydz=0\)答案：ABCD5.下列冪級(jí)數(shù)的收斂區(qū)間正確的是（）A.冪級(jí)數(shù)\(\sum_{n=0}^{\infty}x^n\)的收斂區(qū)間是\((-1,1)\)B.冪級(jí)數(shù)\(\sum_{n=1}^{\infty}\frac{x^n}{n!}\)的收斂區(qū)間是\((-\infty,+\infty)\)C.冪級(jí)數(shù)\(\sum_{n=1}^{\infty}n^2x^n\)的收斂區(qū)間是\((-1,1)\)D.冪級(jí)數(shù)\(\sum_{n=1}^{\infty}\frac{(x-1)^n}{n}\)的收斂區(qū)間是\((0,2)\)答案：ABCD6.下列關(guān)于常數(shù)項(xiàng)級(jí)數(shù)斂散性的說(shuō)法正確的是（）A.若\(\lim\limits_{n\to\infty}u_n\neq0\)，則\(\sum_{n=1}^{\infty}u_n\)發(fā)散B.若\(\sum_{n=1}^{\infty}u_n\)與\(\sum_{n=1}^{\infty}v_n\)都收斂，則\(\sum_{n=1}^{\infty}(u_n+v_n)\)收斂C.若\(\sum_{n=1}^{\infty}u_n\)收斂，\(\sum_{n=1}^{\infty}v_n\)發(fā)散，則\(\sum_{n=1}^{\infty}(u_n+v_n)\)發(fā)散D.正項(xiàng)級(jí)數(shù)\(\sum_{n=1}^{\infty}u_n\)收斂的充要條件是其部分和數(shù)列\(zhòng)(\{S_n\}\)有界答案：ABCD7.下列微分方程是線(xiàn)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.設(shè)向量\(\vec{a}=(x_1,y_1,z_1)\)，\(\vec=(x_2,y_2,z_2)\)，下列運(yùn)算正確的是（）A.\(\vec{a}+\vec=(x_1+x_2,y_1+y_2,z_1+z_2)\)B.\(\vec{a}\cdot\vec=x_1x_2+y_1y_2+z_1z_2\)C.\(\vec{a}\times\vec=(y_1z_2-y_2z_1,z_1x_2-z_2x_1,x_1y_2-x_2y_1)\)D.\(k\vec{a}=(kx_1,ky_1,kz_1)\)（\(k\)為常數(shù)）答案：ABCD9.下列關(guān)于曲線(xiàn)積分的說(shuō)法正確的是（）A.第一類(lèi)曲線(xiàn)積分\(\int_{L}f(x,y)ds\)與曲線(xiàn)\(L\)的方向無(wú)關(guān)B.第二類(lèi)曲線(xiàn)積分\(\int_{L}P(x,y)dx+Q(x,y)dy\)與曲線(xiàn)\(L\)的方向有關(guān)C.若\(L\)是封閉曲線(xiàn)，\(P\)，\(Q\)在\(L\)所圍成區(qū)域\(D\)上具有一階連續(xù)偏導(dǎo)數(shù)，則\(\oint_{L}Pdx+Qdy=\iint_{D}(\frac{\partialQ}{\partialx}-\frac{\partialP}{\partialy})d\sigma\)（格林公式）D.曲線(xiàn)積分\(\int_{L}f(x,y)ds\)可以通過(guò)將曲線(xiàn)\(L\)參數(shù)化后轉(zhuǎn)化為定積分計(jì)算答案：ABCD10.關(guān)于曲面積分，下列說(shuō)法正確的是（）A.第一類(lèi)曲面積分\(\iint_{\varSigma}f(x,y,z)dS\)與曲面\(\varSigma\)的側(cè)無(wú)關(guān)B.第二類(lèi)曲面積分\(\iint_{\varSigma}P(x,y,z)dydz+Q(x,y,z)dzdx+R(x,y,z)dxdy\)與曲面\(\varSigma\)的側(cè)有關(guān)C.若\(\varSigma\)是封閉曲面，\(P\)，\(Q\)，\(R\)在\(\varSigma\)所圍成區(qū)域\(\varOmega\)上具有一階連續(xù)偏導(dǎo)數(shù)，則\(\oiint_{\varSigma}Pdydz+Qdzdx+Rdxdy=\iiint_{\varOmega}(\frac{\partialP}{\partialx}+\frac{\partialQ}{\partialy}+\frac{\partialR}{\partialz})dxdydz\)（高斯公式）D.計(jì)算第一類(lèi)曲面積分可以通過(guò)將曲面方程代入并轉(zhuǎn)化為二重積分計(jì)算答案：ABCD三、判斷題1.若函數(shù)\(z=f(x,y)\)在點(diǎn)\((x_0,y_0)\)處的偏導(dǎo)數(shù)\(f_x(x_0,y_0)\)和\(f_y(x_0,y_0)\)都存在，則\(z=f(x,y)\)在點(diǎn)\((x_0,y_0)\)處一定連續(xù)。（×）2.函數(shù)\(z=f(x,y)\)在點(diǎn)\((x_0,y_0)\)處的全微分\(dz=f_x(x_0,y_0)\Deltax+f_y(x_0,y_0)\Deltay\)。（×）3.二