数分2期末考试题及答案

数分2期末考试题及答案

一、单项选择题（每题2分，共10题）1.设函数\(f(x)\)在\(x=a\)处可导，则\(\lim\limits_{h\rightarrow0}\frac{f(a+h)-f(a-h)}{h}=(\)\)A.\(f'(a)\)B.\(2f'(a)\)C.\(0\)D.不存在答案：B2.函数\(y=\ln(x+\sqrt{x^{2}+1})\)是（）A.偶函数B.奇函数C.非奇非偶函数D.既是奇函数又是偶函数答案：B3.若\(f(x)\)在\([a,b]\)上连续，在\((a,b)\)内可导，且\(f(a)=f(b)\)，则在\((a,b)\)内（）A.至少存在一点\(\xi\)，使得\(f'(\xi)=0\)B.一定不存在点\(\xi\)，使得\(f'(\xi)=0\)C.恰存在一点\(\xi\)，使得\(f'(\xi)=0\)D.对任意的\(\xi\in(a,b)\)，\(f'(\xi)\neq0\)答案：A4.设\(y=e^{x}\cosx\)，则\(y^{(4)}=(\)\)A.\(-4e^{x}\cosx\)B.\(4e^{x}\cosx\)C.\(e^{x}\cosx\)D.\(-e^{x}\cosx\)答案：A5.\(\int\frac{dx}{x^{2}+4x+5}=(\)\)A.\(\arctan(x+2)+C\)B.\(\ln(x^{2}+4x+5)+C\)C.\(\frac{1}{2}\arctan\frac{x+2}{2}+C\)D.\(\frac{1}{2}\ln(x^{2}+4x+5)+C\)答案：A6.设\(I=\int_{0}^{1}\frac{x^{4}}{\sqrt{1-x^{2}}}dx\)，令\(x=\sint\)，则\(I=(\)\)A.\(\int_{0}^{\frac{\pi}{2}}\sin^{4}tdt\)B.\(\int_{0}^{\frac{\pi}{2}}\cos^{4}tdt\)C.\(\int_{0}^{\frac{\pi}{2}}\sin^{4}t\costdt\)D.\(\int_{0}^{\frac{\pi}{2}}\cos^{4}t\sintdt\)答案：A7.下列广义积分收敛的是（）A.\(\int_{1}^{+\infty}\frac{dx}{x}\)B.\(\int_{1}^{+\infty}\frac{dx}{x^{2}}\)C.\(\int_{1}^{+\infty}\sqrt{x}dx\)D.\(\int_{1}^{+\infty}\frac{dx}{\sqrt{x}}\)答案：B8.函数\(z=f(x,y)\)在点\((x_{0},y_{0})\)处可微的充分条件是（）A.\(f_{x}(x_{0},y_{0})\)与\(f_{y}(x_{0},y_{0})\)都存在B.\(f_{x}(x,y)\)与\(f_{y}(x,y)\)在点\((x_{0},y_{0})\)的某邻域内连续C.\(\Deltaz=f(x_{0}+\Deltax,y_{0}+\Deltay)-f(x_{0},y_{0})\)D.\(\lim\limits_{\rho\rightarrow0}\frac{\Deltaz-f_{x}(x_{0},y_{0})\Deltax-f_{y}(x_{0},y_{0})\Deltay}{\rho}=0\)（其中\(\rho=\sqrt{\Deltax^{2}+\Deltay^{2}}\)）答案：B9.设\(D=\{(x,y)|x^{2}+y^{2}\leqslant1\}\)，则\(\iint_{D}dxdy=(\)\)A.\(\pi\)B.\(2\pi\)C.\(1\)D.\(4\pi\)答案：A10.设幂级数\(\sum_{n=0}^{\infty}a_{n}x^{n}\)的收敛半径为\(R\)，则幂级数\(\sum_{n=0}^{\infty}a_{n}(x-1)^{n}\)的收敛区间为（）A.\((-R+1,R+1)\)B.\((1-R,1+R)\)C.\((-R,R)\)D.\((0,2R)\)答案：B二、多项选择题（每题2分，共10题）1.下列函数中，在\(x=0\)处可导的是（）A.\(y=\vertx\vert\)B.\(y=x\vertx\vert\)C.\(y=\sqrt[3]{x}\)D.\(y=\sin\vertx\vert\)答案：B2.设\(f(x)\)在\((a,b)\)内可导，\(x_{1},x_{2}\in(a,b)\)且\(x_{1}<x_{2}\)，则（）A.若\(f(x)\)在\((a,b)\)内单调增加，则\(f'(x)\geqslant0\)B.若\(f'(x_{1})>0\)，则\(f(x)\)在\(x_{1}\)的某个邻域内单调增加C.若\(f(x)\)在\((a,b)\)内为凸函数，则\(f''(x)\geqslant0\)D.若\(f''(x_{1})>0\)，则\(f(x)\)在\(x_{1}\)的某个邻域内为凸函数答案：AB3.下列等式正确的是（）A.\(\int\sin^{2}xdx=\frac{1}{2}(x-\sinx\cosx)+C\)B.\(\int\cos^{2}xdx=\frac{1}{2}(x+\sinx\cosx)+C\)C.\(\int\sec^{2}xdx=\tanx+C\)D.\(\int\csc^{2}xdx=\cotx+C\)答案：ABCD4.设\(I=\int_{a}^{b}f(x)dx\)，\(J=\int_{a}^{b}\vertf(x)\vertdx\)，\(K=\vert\int_{a}^{b}f(x)dx\vert\)，则（）A.\(J\geqslantK\)B.\(J=K\)C.当\(f(x)\geqslant0\)时，\(J=K\)D.当\(f(x)\)在\([a,b]\)上恒为常数时，\(J=K\)答案：ACD5.下列广义积分中，收敛的是（）A.\(\int_{0}^{+\infty}e^{-x}dx\)B.\(\int_{1}^{+\infty}\frac{1}{x^{3}}dx\)C.\(\int_{0}^{1}\frac{1}{\sqrt{1-x^{2}}}dx\)D.\(\int_{0}^{1}\frac{1}{x(1-x)}dx\)答案：ABC6.设\(z=f(x,y)\)，则\(\frac{\partialz}{\partialx}\vert_{(x_{0},y_{0})}\)的几何意义是（）A.曲线\(\left\{\begin{array}{l}z=f(x,y)\\y=y_{0}\end{array}\right.\)在点\((x_{0},y_{0},f(x_{0},y_{0}))\)处的切线对\(x\)轴的斜率B.曲线\(\left\{\begin{array}{l}z=f(x,y)\\x=x_{0}\end{array}\right.\)在点\((x_{0},y_{0},f(x_{0},y_{0}))\)处的切线对\(y\)轴的斜率C.曲面\(z=f(x,y)\)在点\((x_{0},y_{0},f(x_{0},y_{0}))\)处的切平面对\(x\)轴的斜率D.曲面\(z=f(x,y)\)在点\((x_{0},y_{0},f(x_{0},y_{0}))\)处的切平面对\(y\)轴的斜率答案：AC7.设\(D=\{(x,y)\vertx^{2}+y^{2}\leqslant4\}\)，则\(\iint_{D}(x^{2}+y^{2})dxdy\)可通过（）计算。A.直角坐标B.极坐标C.先对\(x\)积分再对\(y\)积分D.先对\(y\)积分再对\(x\)积分答案：AB8.对于幂级数\(\sum_{n=0}^{\infty}a_{n}x^{n}\)，若\(\lim\limits_{n\rightarrow\infty}\vert\frac{a_{n+1}}{a_{n}}\vert=\rho\)，则（）A.当\(\rho=0\)时，收敛半径\(R=+\infty\)B.当\(\rho=+\infty\)时，收敛半径\(R=0\)C.当\(0<\rho<+\infty\)时，收敛半径\(R=\frac{1}{\rho}\)D.收敛半径\(R\)与\(\rho\)无关答案：ABC9.设\(y=f(x)\)是二阶常系数线性微分方程\(y''+py'+qy=0\)的解，\(y=g(x)\)是\(y''+py'+qy=r(x)\)（\(r(x)\neq0\)）的解，则（）A.\(y=f(x)+g(x)\)是\(y''+py'+qy=r(x)\)的解B.\(y=f(x)-g(x)\)是\(y''+py'+qy=-r(x)\)的解C.\(ay=f(x)+bg(x)\)（\(a,b\)为常数）是\(y''+py'+qy=ar(x)\)的解D.\(y=cf(x)\)（\(c\)为常数）是\(y''+py'+qy=0\)的解答案：AD10.下列函数在\((0,+\infty)\)上单调递增的是（）A.\(y=x^{2}\)B.\(y=\lnx\)C.\(y=e^{x}\)D.\(y=\frac{1}{x}\)答案：ABC三、判断题（每题2分，共10题）1.若函数\(f(x)\)在\(x=a\)处不可导，则\(f(x)\)在\(x=a\)处不连续。（×）2.若\(f(x)\)在\([a,b]\)上连续，则\(\int_{a}^{b}f(x)dx=\int_{a}^{b}f(t)dt\)。（√）3.若\(\sum_{n=1}^{\infty}a_{n}\)收敛，则\(\sum_{n=1}^{\infty}\verta_{n}\vert\)也收敛。（×）4.设\(z=f(x,y)\)，则\(\frac{\partial^{2}z}{\partialx\partialy}=\frac{\partial^{2}z}{\partialy\partialx}\)恒成立。（×）5.若\(f(x)\)在\((a,b)\)内的导数\(f'(x)>0\)，则\(f(x)\)在\((a,b)\)内单调递增。（√）6.\(\int_{-\infty}^{+\infty}\frac{dx}{1+x^{2}}=\pi\)。（√）7.幂级数\(\sum_{n=0}^{\infty}n!x^{n}\)的收敛半径\(R=0\)。（√）8.若\(y=f(x)\)在\(x=x_{0}\)处可微，则\(y=f(x)\)在\(x=x_{0}\)处连续。（√）9.设\(D=\{(x,y)\vert0\leqslantx\leqslant1,0\leqslanty\leqslant1\}\)，则\(\iint_{D}xydxdy=\frac{1}{4}\)。（√）10.若\(y=f(x)\)是\(y''-y=0\)的解，则\(y=cf(x)\)（\(c\)为常数）也是\(y''-y=0\)的解。（√）四、简答题（每题5分，共4题）1.简述函数\(y=f(x)\)在点\(x_{0}\)处可导与可微的关系。答案：函数\(y=f(x)\)在点\(x_{0}\)处可导等价于函数\(y=f(x)\)在点\(x_{0}\)处可微，且\(dy=f'(x_{0})\Deltax\)。2.求\(\intx\cosxdx\)。答案：利用分部积分法，设\(u=x\)，\(dv=\cosxdx\)，则\(du=dx\)，\(v=\sinx\)。\(\intx\cosxdx=x\sinx-\int\sinxdx=x\sinx+\cosx+C\)。3.设\(z=f(u,v)\)，\(u=x^{2}+y^{2}\)，\(v=xy\)，求\(\frac{\partialz}{\partialx}\)。答案：根据复合函数求导法则，\(\frac{\partialz}{\partialx}=\frac{\partialz}{\partialu}\cdot\frac{\partialu}{\partialx}+\frac{\partialz}{\partialv}\cdot\frac{\partialv}{\partialx}=2x\frac{\partialz}{\partialu}+y\frac{\partialz}{\partialv}\)。4.求幂级数\(\sum_{n=1}^{\infty}\frac{x^{n}}{n}\)的收敛区间。答