大學數(shù)學考試試題及答案

一、單項選擇題（每題2分，共10題）1.若函數(shù)\(y=f(x)\)在點\(x_0\)處可導，則\(\lim\limits_{\Deltax\to0}\frac{f(x_0+\Deltax)-f(x_0-\Deltax)}{\Deltax}=(\)\()\)A.\(f'(x_0)\)B.\(2f'(x_0)\)C.\(0\)D.不存在答案：B2.設\(f(x)=\lnx\)，則\(\intf'(x^2)dx=(\)\()\)A.\(\frac{1}{2x}+C\)B.\(\lnx^2+C\)C.\(\frac{1}{x}+C\)D.\(\frac{1}{2}\lnx^2+C\)答案：A3.函數(shù)\(y=x^3-3x\)的單調(diào)遞減區(qū)間是（）A.\((-\infty,-1)\)B.\((-1,1)\)C.\((1,+\infty)\)D.\((-\infty,+\infty)\)答案：B4.設\(z=e^{xy}\)，則\(\frac{\partialz}{\partialy}=(\)\()\)A.\(xe^{xy}\)B.\(ye^{xy}\)C.\(xye^{xy}\)D.\(e^{xy}\)答案：A5.定積分\(\int_{0}^{1}e^xdx=(\)\()\)A.\(e-1\)B.\(1-e\)C.\(e\)D.\(-e\)答案：A6.下列函數(shù)中，是奇函數(shù)的是（）A.\(y=x^2\cosx\)B.\(y=\sinx+\cosx\)C.\(y=x^3\sinx\)D.\(y=\frac{\sinx}{x}\)答案：D7.若向量\(\vec{a}=(1,2,3)\)，\(\vec=(3,-1,2)\)，則\(\vec{a}\cdot\vec=(\)\()\)A.\(11\)B.\(7\)C.\(5\)D.\(1\)答案：B8.級數(shù)\(\sum_{n=1}^{\infty}\frac{1}{n(n+1)}\)的和為（）A.\(1\)B.\(\frac{1}{2}\)C.\(2\)D.不存在答案：A9.微分方程\(y''+y=0\)的通解是（）A.\(y=C_1\cosx+C_2\sinx\)B.\(y=C_1e^x+C_2e^{-x}\)C.\(y=(C_1+C_2x)e^x\)D.\(y=C_1x+C_2\)答案：A10.設\(A=\begin{pmatrix}1&2\\3&4\end{pmatrix}\)，則\(\vertA\vert=(\)\()\)A.\(-2\)B.\(2\)C.\(-10\)D.\(10\)答案：A二、多項選擇題（每題2分，共10題）1.下列函數(shù)在\(x=0\)處連續(xù)的有（）A.\(y=\frac{\sinx}{x}\)B.\(y=\left\{\begin{array}{ll}x\sin\frac{1}{x}&(x\neq0)\\0&(x=0)\end{array}\right.\)C.\(y=\frac{1}{x}\)D.\(y=\left\{\begin{array}{ll}x^2&(x\geq0)\\-x^2&(x<0)\end{array}\right.\)答案：ABD2.設\(f(x)\)在\([a,b]\)上可積，則（）A.\(\int_{a}^kf(x)dx=k\int_{a}^f(x)dx\)（\(k\)為常數(shù)）B.\(\int_{a}^[f(x)+g(x)]dx=\int_{a}^f(x)dx+\int_{a}^g(x)dx\)C.\(\int_{a}^f(x)dx=\int_{a}^{c}f(x)dx+\int_{c}^f(x)dx\)（\(a<c<b\)）D.若\(f(x)\geq0\)，\(x\in[a,b]\)，則\(\int_{a}^f(x)dx\geq0\)答案：ABCD3.曲線\(y=x^3-3x^2+2x\)的拐點可能為（）A.\((1,0)\)B.\((0,0)\)C.\((2,-2)\)D.\((-1,-6)\)答案：AC4.設\(z=f(x,y)\)，則全微分\(dz\)等于（）A.\(\frac{\partialz}{\partialx}dx+\frac{\partialz}{\partialy}dy\)B.\(\frac{\partialz}{\partialy}dx+\frac{\partialz}{\partialx}dy\)C.\(z_x'dx+z_y'dy\)D.\(z_y'dx+z_x'dy\)答案：AC5.對于向量\(\vec{a},\vec\)，以下等式成立的有（）A.\(\vert\vec{a}+\vec\vert=\vert\vec{a}\vert+\vert\vec\vert\)B.\(\vert\vec{a}-\vec\vert=\vert\vec{a}\vert-\vert\vec\vert\)C.\((\vec{a}+\vec)\cdot(\vec{a}-\vec)=\vert\vec{a}\vert^2-\vert\vec\vert^2\)D.\(\vec{a}\cdot\vec=\vert\vec{a}\vert\vert\vec\vert\cos\theta\)（\(\theta\)為\(\vec{a}\)與\(\vec\)的夾角）答案：CD6.下列級數(shù)收斂的有（）A.\(\sum_{n=1}^{\infty}\frac{1}{n}\)B.\(\sum_{n=1}^{\infty}\frac{1}{n^2}\)C.\(\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n}\)D.\(\sum_{n=1}^{\infty}\frac{1}{\sqrt{n}}\)答案：BC7.函數(shù)\(y=\sinx\)的麥克勞林展開式為（）A.\(\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)!}x^{2n+1}\)B.\(\sum_{n=0}^{\infty}\frac{1}{(2n)!}x^{2n}\)C.\(\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}x^{n}\)D.\(\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n!}x^{2n}\)答案：A8.設\(A,B\)為\(n\)階方陣，則下列等式正確的有（）A.\((AB)^T=B^TA^T\)B.\((A+B)^T=A^T+B^T\)C.\(\vertAB\vert=\vertA\vert\vertB\vert\)D.\(\vertA+B\vert=\vertA\vert+\vertB\vert\)答案：ABC9.下列函數(shù)中，可作為某二階線性齊次微分方程的通解的是（）A.\(y=C_1e^x+C_2e^{-x}\)B.\(y=C_1\cosx+C_2\sinx\)C.\(y=(C_1+C_2x)e^x\)D.\(y=C_1x+C_2\)答案：ABC10.設\(y=f(x)\)在\(x=x_0\)處取得極值，則（）A.\(f'(x_0)=0\)B.\(f''(x_0)\neq0\)C.\(f'(x_0)\)可能不存在D.\(f(x)\)在\(x=x_0\)處連續(xù)答案：ACD三、判斷題（每題2分，共10題）1.若函數(shù)\(f(x)\)在\(x=a\)處不可導，則\(f(x)\)在\(x=a\)處不連續(xù)。（×）2.\(\int_{-a}^{a}f(x)dx=0\)，則\(f(x)\)為奇函數(shù)。（×）3.若\(y=f(x)\)在\((a,b)\)內(nèi)單調(diào)遞增，則\(f'(x)>0\)，\(x\in(a,b)\)。（×）4.對于函數(shù)\(z=f(x,y)\)，\(\frac{\partial^2z}{\partialx\partialy}=\frac{\partial^2z}{\partialy\partialx}\)恒成立。（×）5.兩個向量\(\vec{a},\vec\)垂直的充要條件是\(\vec{a}\cdot\vec=0\)。（√）6.若級數(shù)\(\sum_{n=1}^{\infty}u_n\)收斂，則\(\lim\limits_{n\rightarrow\infty}u_n=0\)。（√）7.二階常系數(shù)非齊次線性微分方程\(y''+py'+qy=P_n(x)e^{\lambdax}\)（\(p,q\)為常數(shù)，\(P_n(x)\)為\(n\)次多項式）的特解\(y^\)的形式為\(Q_n(x)e^{\lambdax}\)（\(Q_n(x)\)為\(n\)次多項式）。（×）8.若\(A,B\)為\(n\)階方陣，且\(AB=I\)（\(I\)為單位矩陣），則\(BA=I\)。（√）9.函數(shù)\(y=x^3\)的圖像在\((-\infty,+\infty)\)上是凹的。（√）10.設\(y=f(x)\)在\([a,b]\)上可導，則\(f(x)\)在\([a,b]\)上一定可積。（√）四、簡答題（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\(y'=3x^2-6x-9=3(x^2-2x-3)=3(x+1)(x-3)\)。令\(y'=0\)，得\(x=-1\)或\(x=3\)。當\(x<-1\)時，\(y'>0\)；當\(-1<x<3\)時，\(y'<0\)；當\(x>3\)時，\(y'>0\)。所以\(x=-1\)時取得極大值\(y(-1)=10\)，\(x=3\)時取得極小值\(y(3)=-22\)。2.計算\(\int\frac{1}{x^2-1}dx\)。答案：\(\int\frac{1}{x^2-1}dx=\frac{1}{2}\int(\frac{1}{x-1}-\frac{1}{x+1})dx=\frac{1}{2}\ln\vert\frac{x-1}{x+1}\vert+C\)。3.已知向量\(\vec{a}=(1,2,-1)\)，\(\vec=(2,-1,3)\)，求\(\vec{a}\times\vec\)。答案：\(\vec{a}\times\vec=\begin{vmatrix}\vec{i}&\vec{j}&\vec{k}\\1&2&-1\\2&-1&3\end{vmatrix}=\vec{i}(6-1)-\vec{j}(3+2)+\vec{k}(-1-4)=5\vec{i}-5\vec{j}-5\vec{k}=(5,-5,-5)\)。4.求冪級數(shù)\(\sum_{n=1}^{\infty}\frac{x^n}{n}\)的收斂半徑。答案：設\(a_n=\frac{1}{n}\)，根據(jù)\(R=\lim\limits_{n\rightarrow\infty}\vert\frac{a_n}{a_{n+1}}\vert=\lim\limits_{n\rightarrow\infty}\vert\frac{\frac{1}{n}}{\frac{1}{n+1}}\vert=1\)，所以收斂半徑\(R=1\)。五、討論題（每題5分，共4題）1.討論函數(shù)\(y=\frac{x^2-1}{x-1}\)在\(x=1\)處的極限、連續(xù)性和可導性。答案：\(\lim\limits_{x\rightarrow1}\frac{x^2-1}{x-1}=\lim\limits_{x\rightarrow1}(x+1)=2\)。函數(shù)在\(x=1\)處不連續(xù)，因為\(f(1)\)無定義。不可導，因為不連續(xù)必然不可導。2.討論級數(shù)\(\sum_{n=1}^{\infty}\