經(jīng)濟(jì)數(shù)學(xué)2考試題及答案

一、單項(xiàng)選擇題（每題2分，共10題）1.若函數(shù)\(y=f(x)\)在點(diǎn)\(x_0\)處可導(dǎo)，則\(\lim\limits_{\Deltax\rightarrow0}\frac{f(x_0+\Deltax)-f(x_0)}{\Deltax}\)等于（）A.\(f'(x_0)\)B.\(f(x_0)\)C.\(0\)D.不存在答案：A2.設(shè)\(y=\sinx\)，則\(y^{(5)}=\)（）A.\(\cosx\)B.\(-\cosx\)C.\(\sinx\)D.\(-\sinx\)答案：B3.函數(shù)\(y=x^3-3x\)的單調(diào)遞減區(qū)間是（）A.\((-1,1)\)B.\((-\infty,-1)\)C.\((1,+\infty)\)D.\((-\infty,+\infty)\)答案：A4.設(shè)\(f(x)=\int_{0}^{x}(t^2-1)dt\)，則\(f'(x)=\)（）A.\(x^2-1\)B.\(2x\)C.\(x^2\)D.\(-1\)答案：A5.\(\inte^{2x}dx=\)（）A.\(\frac{1}{2}e^{2x}+C\)B.\(e^{2x}+C\)C.\(2e^{2x}+C\)D.\(\frac{1}{2}e^{x}+C\)答案：A6.已知\(z=x^2y\)，則\(\frac{\partialz}{\partialx}=\)（）A.\(2xy\)B.\(x^2\)C.\(2y\)D.\(y\)答案：A7.若\(A=\begin{pmatrix}1&2\\3&4\end{pmatrix}\)，則\(\vertA\vert=\)（）A.-2B.2C.-10D.10答案：A8.線性方程組\(\begin{cases}x+y=1\\2x+2y=2\end{cases}\)的解的情況是（）A.無(wú)解B.有唯一解C.有無(wú)窮多解D.不能確定答案：C9.設(shè)\(L\)是從點(diǎn)\((0,0)\)到點(diǎn)\((1,1)\)的直線段，則\(\int_{L}(x+y)ds=\)（）A.\(\sqrt{2}\)B.\(1\)C.\(2\)D.\(\frac{1}{2}\)答案：A10.設(shè)\(D=\{(x,y)\vertx^2+y^2\leqslant1\}\)，則\(\iint_{D}dxdy=\)（）A.\(\pi\)B.\(2\pi\)C.\(\frac{\pi}{2}\)D.\(1\)答案：A二、多項(xiàng)選擇題（每題2分，共10題）1.下列函數(shù)中，在區(qū)間\((0,+\infty)\)上是增函數(shù)的是（）A.\(y=x^2\)B.\(y=\lnx\)C.\(y=e^x\)D.\(y=\sinx\)答案：ABC2.設(shè)\(y=f(u)\)，\(u=g(x)\)，則\(y=f(g(x))\)的導(dǎo)數(shù)\(y'=\)（）A.\(f'(u)g'(x)\)B.\(f'(g(x))g'(x)\)C.\(f(g'(x))\)D.\(f'(g(x))\)答案：AB3.下列積分中，能用牛頓-萊布尼茨公式計(jì)算的是（）A.\(\int_{0}^{1}\frac{1}{x}dx\)B.\(\int_{0}^{1}x^2dx\)C.\(\int_{-1}^{1}\frac{1}{1+x^2}dx\)D.\(\int_{0}^{1}\sqrt{x}dx\)答案：BD4.設(shè)\(z=f(x,y)\)，則全微分\(dz=\)（）A.\(\frac{\partialz}{\partialx}dx+\frac{\partialz}{\partialy}dy\)B.\(z_xdx+z_ydy\)C.\(\frac{\partialz}{\partialy}dx+\frac{\partialz}{\partialx}dy\)D.\(z_ydx+z_xdy\)答案：AB5.關(guān)于矩陣\(A=\begin{pmatrix}1&0\\0&0\end{pmatrix}\)，下列說(shuō)法正確的是（）A.\(A\)是對(duì)稱矩陣B.\(A\)是可逆矩陣C.\(A\)的秩為1D.\(A\)的行列式為0答案：ACD6.對(duì)于線性方程組\(Ax=b\)（\(A\)為系數(shù)矩陣，\(x\)為未知數(shù)向量，\(b\)為常數(shù)項(xiàng)向量），當(dāng)（）時(shí)，方程組有無(wú)窮多解。A.\(r(A)=r(A,b)<n\)（\(n\)為未知數(shù)個(gè)數(shù)）B.\(r(A)=r(A,b)=n\)C.\(r(A)<r(A,b)\)D.\(A\)為零矩陣答案：A7.下列曲線積分與路徑無(wú)關(guān)的是（）A.\(\int_{L}(x+y)dx+(x-y)dy\)，其中\(zhòng)(L\)在\(xOy\)平面內(nèi)的單連通區(qū)域\(D\)內(nèi)，且\(D\)是開(kāi)區(qū)域B.\(\int_{L}\frac{y}{x^2+y^2}dx-\frac{x}{x^2+y^2}dy\)，其中\(zhòng)(L\)在\(xOy\)平面內(nèi)除原點(diǎn)外的單連通區(qū)域\(D\)內(nèi)C.\(\int_{L}(x^2+y^2)dx+(2xy)dy\)，其中\(zhòng)(L\)在\(xOy\)平面內(nèi)的單連通區(qū)域\(D\)內(nèi)，且\(D\)是開(kāi)區(qū)域D.\(\int_{L}e^x\sinydx+e^x\cosydy\)，其中\(zhòng)(L\)在\(xOy\)平面內(nèi)的單連通區(qū)域\(D\)內(nèi)，且\(D\)是開(kāi)區(qū)域答案：ACD8.設(shè)\(D=\{(x,y)\vert0\leqslantx\leqslant1,0\leqslanty\leqslant1\}\)，則\(\iint_{D}(x+y)dxdy=\)（）A.\(\int_{0}^{1}dx\int_{0}^{1}(x+y)dy\)B.\(\int_{0}^{1}dy\int_{0}^{1}(x+y)dx\)C.\(\frac{3}{2}\)D.\(1\)答案：AB9.函數(shù)\(y=f(x)\)的駐點(diǎn)可能是（）A.極大值點(diǎn)B.極小值點(diǎn)C.拐點(diǎn)D.單調(diào)區(qū)間的分界點(diǎn)答案：ABD10.設(shè)\(f(x)\)是可導(dǎo)函數(shù)，則\((\intf(x)dx)'=\)（）A.\(f(x)\)B.\(f'(x)\)C.\(0\)D.\(\intf'(x)dx\)答案：A三、判斷題（每題2分，共10題）1.若函數(shù)\(y=f(x)\)在點(diǎn)\(x_0\)處不可導(dǎo)，則函數(shù)\(y=f(x)\)在點(diǎn)\(x_0\)處不連續(xù)。（）答案：錯(cuò)誤2.若\(f(x)\)和\(g(x)\)在區(qū)間\(I\)上可導(dǎo)，且\(f'(x)=g'(x)\)，則\(f(x)=g(x)\)。（）答案：錯(cuò)誤3.\(\int_{a}^f(x)dx=-\int_^{a}f(x)dx\)。（）答案：正確4.若\(z=f(x,y)\)在點(diǎn)\((x_0,y_0)\)處可微，則\(z=f(x,y)\)在點(diǎn)\((x_0,y_0)\)處連續(xù)。（）答案：正確5.矩陣\(A\)的秩\(r(A)\)等于\(A\)的行向量組的秩。（）答案：正確6.對(duì)于齊次線性方程組\(Ax=0\)（\(A\)為系數(shù)矩陣），當(dāng)\(r(A)=n\)（\(n\)為未知數(shù)個(gè)數(shù)）時(shí)，方程組只有零解。（）答案：正確7.曲線積分\(\int_{L}F(x,y)\cdotds\)的值與曲線\(L\)的方向有關(guān)。（）答案：錯(cuò)誤8.若\(D=\{(x,y)\vertx^2+y^2\leqslantR^2\}\)，則\(\iint_{D}dxdy=\piR^2\)。（）答案：正確9.函數(shù)\(y=f(x)\)的極值點(diǎn)一定是駐點(diǎn)。（）答案：錯(cuò)誤10.若\(y=f(x)\)是\([a,b]\)上的連續(xù)函數(shù)，則\(\int_{a}^f(x)dx\)一定存在。（）答案：正確四、簡(jiǎn)答題（每題5分，共4題）1.求函數(shù)\(y=x^3-3x^2-9x+5\)的極值。答案：首先求導(dǎo)\(y'=3x^2-6x-9=3(x^2-2x-3)=3(x+1)(x-3)\)。令\(y'=0\)，得\(x=-1\)或\(x=3\)。當(dāng)\(x<-1\)時(shí)，\(y'>0\)；當(dāng)\(-1<x<3\)時(shí)，\(y'<0\)；當(dāng)\(x>3\)時(shí)，\(y'>0\)。所以\(x=-1\)時(shí)，\(y\)有極大值\(y(-1)=10\)；\(x=3\)時(shí)，\(y\)有極小值\(y(3)=-22\)。2.計(jì)算\(\int_{0}^{1}x\sqrt{1-x^2}dx\)。答案：令\(u=1-x^2\)，則\(du=-2xdx\)。當(dāng)\(x=0\)時(shí)，\(u=1\)；當(dāng)\(x=1\)時(shí)，\(u=0\)。原式\(=-\frac{1}{2}\int_{1}^{0}\sqrt{u}du=\frac{1}{2}\int_{0}^{1}\sqrt{u}du=\frac{1}{3}\)。3.設(shè)\(z=e^{xy}\sin(x+y)\)，求\(\frac{\partialz}{\partialx}\)。答案：根據(jù)復(fù)合函數(shù)求導(dǎo)法則，\(\frac{\partialz}{\partialx}=ye^{xy}\sin(x+y)+e^{xy}\cos(x+y)\)。4.求線性方程組\(\begin{cases}x+2y-z=1\\2x-y+z=2\\x+y+z=3\end{cases}\)的解。答案：對(duì)增廣矩陣\(\begin{pmatrix}1&2&-1&1\\2&-1&1&2\\1&1&1&3\end{pmatrix}\)進(jìn)行初等行變換化為\(\begin{pmatrix}1&0&0&1\\0&1&0&1\\0&0&1&1\end{pmatrix}\)，所以方程組的解為\(x=1,y=1,z=1\)。五、討論題（每題5分，共4題）1.討論函數(shù)\(y=\frac{1}{x}\)在區(qū)間\((0,+\infty)\)和\((-\infty,0)\)上的單調(diào)性。答案：在\((0,+\infty)\)上，\(y'=-\frac{1}{x^2}<0\)，函數(shù)單調(diào)遞減；在\((-\infty,0)\)上，\(y'=-\frac{1}{x^2}<0\)，函數(shù)單調(diào)遞減。2.討論\(\int_{0}^{+\infty}e^{-x}dx\)的斂散性。答案：\(\int_{0}^{+\infty}e^{-x}dx=\lim\limits_{b\rightarrow+\infty}\int_{0}^e^{-x}dx=\lim\limits_{b\rightarrow+\infty}(1-e^{-b})=1\)，所以該積分收斂。3.討論矩陣\(A=\begin{pmatrix}1&1&1\\1&2&3\\1&3&6\end{pmatrix}\)的可逆性。答案：計(jì)算\(\vertA\vert=\begin{vmatrix}1&1&1\\1&2&3\\1&3&6\end{vmatrix}=1\times\begin{vmatrix}2&3\\3&6\end{vmatrix}-1\times\begin{vmatrix}1&3\\1&6\end{vmatrix}+1\times\begin{vmatrix}1&2\\1&3\end{vmatrix}=1\times(12-9)-1\times(6-3)+1\ti