高數下考試試題及答案

一、單項選擇題（每題2分，共10題）1.設向量\(\vec{a}=(1,2,3)\)，\(\vec{b}=(2,-1,1)\)，則\(\vec{a}\cdot\vec{b}=(\)\)A.7B.8C.9D.10答案：A2.函數\(z=\ln(x+y)\)的定義域是(\)A.\(x+y>0\)B.\(x+y\neq0\)C.\(x>0,y>0\)D.\(x\geqslant0,y\geqslant0\)答案：A3.二重積分\(\iint_Ddxdy\)（\(D\)為\(x^{2}+y^{2}\leqslant1\)）的值為(\)A.\(\pi\)B.\(2\pi\)C.\(3\pi\)D.\(4\pi\)答案：A4.級數\(\sum_{n=1}^{\infty}\frac{1}{n(n+1)}\)的和為(\)A.0B.1C.2D.發散答案：B5.設\(y=e^{x}\sinx\)，則\(y''=(\)\)A.\(2e^{x}\cosx\)B.\(2e^{x}\sinx\)C.\(-2e^{x}\cosx\)D.\(-2e^{x}\sinx\)答案：A6.曲線\(y=\frac{1}{x}\)在點\((1,1)\)處的切線方程為(\)A.\(y=-x+2\)B.\(y=x\)C.\(y=-x\)D.\(y=x+2\)答案：A7.若\(\intf(x)dx=F(x)+C\)，則\(\intf(2x)dx=(\)\)A.\(F(2x)+C\)B.\(\frac{1}{2}F(2x)+C\)C.\(2F(2x)+C\)D.\(\frac{1}{2}F(x)+C\)答案：B8.已知\(z=f(x,y)\)，\(\frac{\partialz}{\partialx}=x+y\)，\(\frac{\partialz}{\partialy}=x-y\)，則\(dz=(\)\)A.\((x+y)dx+(x-y)dy\)B.\((x-y)dx+(x+y)dy\)C.\((x+y)dx-(x-y)dy\)D.\((x-y)dx-(x+y)dy\)答案：A9.冪級數\(\sum_{n=0}^{\infty}nx^{n}\)的收斂半徑\(R=(\)\)A.0B.1C.2D.\(\infty\)答案：B10.設\(L\)為從\((0,0)\)到\((1,1)\)的直線段，則\(\int_{L}xyds=(\)\)A.\(\frac{1}{3}\)B.\(\frac{1}{4}\)C.\(\frac{1}{5}\)D.\(\frac{1}{6}\)答案：A二、多項選擇題（每題2分，共10題）1.下列函數中是二元函數的有(\)A.\(z=x^{2}+y^{2}\)B.\(z=\sqrt{x}+\sqrt{y}\)C.\(z=\frac{1}{x-y}\)D.\(z=\sin(x-y)\)E.\(z=e^{x}+e^{y}\)答案：ABCDE2.下列級數收斂的有(\)A.\(\sum_{n=1}^{\infty}\frac{1}{n^{2}}\)B.\(\sum_{n=1}^{\infty}\frac{1}{n}\)C.\(\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n}\)D.\(\sum_{n=1}^{\infty}\frac{1}{n!}\)E.\(\sum_{n=1}^{\infty}\frac{n}{2^{n}}\)答案：ACDE3.設\(z=f(x,y)\)，則全微分\(dz\)存在的充分條件是(\)A.\(f_{x}(x,y)\)，\(f_{y}(x,y)\)在點\((x,y)\)連續B.\(z=f(x,y)\)在點\((x,y)\)可微C.\(\lim\limits_{\Deltax\rightarrow0}\frac{\Deltaz-f_{x}(x,y)\Deltax-f_{y}(x,y)\Deltay}{\sqrt{(\Deltax)^{2}+(\Deltay)^{2}}}=0\)D.\(z=f(x,y)\)的一階偏導數存在E.\(z=f(x,y)\)在點\((x,y)\)連續答案：ABC4.下列向量場是保守場的有(\)A.\(\vec{F}(x,y)=(y,x)\)B.\(\vec{F}(x,y)=(x+y,x-y)\)C.\(\vec{F}(x,y)=(x^{2},y^{2})\)D.\(\vec{F}(x,y)=(y\cosx,y\sinx)\)E.\(\vec{F}(x,y)=(e^{x}\cosy,e^{x}\siny)\)答案：AE5.下列曲線積分與路徑無關的有(\)A.\(\int_{L}(x+y)dx+(x-y)dy\)在整個\(xOy\)平面內B.\(\int_{L}(x^{2}+y^{2})dx+(2xy)dy\)在整個\(xOy\)平面內C.\(\int_{L}\frac{y}{x^{2}+y^{2}}dx-\frac{x}{x^{2}+y^{2}}dy\)在\(x^{2}+y^{2}>0\)內D.\(\int_{L}(2x\siny)dx+(x^{2}\cosy)dy\)在整個\(xOy\)平面內E.\(\int_{L}(e^{x}\siny)dx+(e^{x}\cosy)dy\)在整個\(xOy\)平面內答案：ADE6.對于二重積分\(\iint_Df(x,y)dxdy\)，若\(D\)是由\(y=x^{2}\)，\(y=x\)所圍成的區域，則化為累次積分正確的有(\)A.\(\int_{0}^{1}dx\int_{x^{2}}^{x}f(x,y)dy\)B.\(\int_{0}^{1}dy\int_{y}^{\sqrt{y}}f(x,y)dx\)C.\(\int_{0}^{1}dx\int_{x}^{\sqrt{x}}f(x,y)dy\)D.\(\int_{0}^{1}dy\int_{y^{2}}^{y}f(x,y)dx\)E.\(\int_{-1}^{1}dx\int_{x^{2}}^{x}f(x,y)dy\)答案：AB7.設\(y=e^{x}\)，則下列說法正確的是(\)A.\(y'=e^{x}\)B.\(y''=e^{x}\)C.\(\inte^{x}dx=e^{x}+C\)D.\(\int_{0}^{1}e^{x}dx=e-1\)E.\(y=e^{x}\)是單調遞增函數答案：ABCDE8.下列關于冪級數的說法正確的有(\)A.冪級數\(\sum_{n=0}^{\infty}a_{n}x^{n}\)在\(x=0\)處一定收斂B.若冪級數\(\sum_{n=0}^{\infty}a_{n}x^{n}\)在\(x=x_{0}\)處收斂，則在\(\vertx\vert<\vertx_{0}\vert\)處絕對收斂C.若冪級數\(\sum_{n=0}^{\infty}a_{n}x^{n}\)在\(x=x_{0}\)處發散，則在\(\vertx\vert>\vertx_{0}\vert\)處發散D.冪級數\(\sum_{n=0}^{\infty}a_{n}x^{n}\)的收斂區間是關于原點對稱的區間E.冪級數\(\sum_{n=0}^{\infty}a_{n}x^{n}\)的和函數在收斂區間內連續答案：ABCE9.設\(u=x^{2}+y^{2}+z^{2}\)，則(\)A.\(\frac{\partialu}{\partialx}=2x\)B.\(\frac{\partialu}{\partialy}=2y\)C.\(\frac{\partialu}{\partialz}=2z\)D.\(du=2xdx+2ydy+2zdz\)E.\(\nablau=(2x,2y,2z)\)答案：ABCDE10.設\(L\)為\(x^{2}+y^{2}=1\)的正向圓周，則(\)A.\(\oint_{L}xdy-ydx=2\pi\)B.\(\oint_{L}x^{2}dx+y^{2}dy=0\)C.\(\oint_{L}(x+y)ds=\sqrt{2}\cdot2\pi\)D.\(\oint_{L}\frac{1}{x^{2}+y^{2}}ds=2\pi\)E.\(\oint_{L}(x^{2}-y^{2})dx+(2xy)dy=0\)答案：ABCDE三、判斷題（每題2分，共10題）1.若\(\sum_{n=1}^{\infty}a_{n}\)收斂，則\(\sum_{n=1}^{\infty}\verta_{n}\vert\)一定收斂。（×）2.函數\(z=\frac{1}{x-y}\)在\(x=y\)處連續。（×）3.對于向量\(\vec{a}=(1,2,3)\)，\(\vec{b}=(2,-1,1)\)，\(\vec{a}\times\vec{b}=(-1,5,-5)\)。（√）4.若\(f(x)\)在\([a,b]\)上可積，則\(f(x)\)在\([a,b]\)上連續。（×）5.冪級數\(\sum_{n=0}^{\infty}n!x^{n}\)的收斂半徑\(R=0\)。（√）6.設\(z=f(x,y)\)，若\(\frac{\partialz}{\partialx}\)，\(\frac{\partialz}{\partialy}\)在點\((x,y)\)存在，則\(z=f(x,y)\)在點\((x,y)\)可微。（×）7.曲線\(y=\lnx\)在\((1,0)\)處的切線方程為\(y=x-1\)。（√）8.若\(\int_{a}^{b}f(x)dx=\int_{a}^{b}g(x)dx\)，則\(f(x)=g(x)\)。（×）9.對于二重積分\(\iint_Df(x,y)dxdy\)，當\(D\)關于\(y=x\)對稱時，\(\iint_Df(x,y)dxdy=\iint_Df(y,x)dxdy\)。（√）10.向量場\(\vec{F}(x,y)=(x^{2},y^{2})\)是有勢場。（×）四、簡答題（每題5分，共4題）1.求函數\(z=x^{3}-3x^{2}-3y^{2}\)的駐點。答案：令\(\frac{\partialz}{\partialx}=3x^{2}-6x=0\)，解得\(x=0\)或\(x=2\)；\(\frac{\partialz}{\partialy}=-6y=0\)，解得\(y=0\)。所以駐點為\((0,0)\)和\((2,0)\)。2.簡述冪級數收斂半徑的求法。答案：對于冪級數\(\sum_{n=0}^{\infty}a_{n}x^{n}\)，若\(\lim\limits_{n\rightarrow\infty}\vert\frac{a_{n+1}}{a_{n}}\vert=\rho\)（\(\rho\neq0\)），則收斂半徑\(R=\frac{1}{\rho}\)；當\(\rho=0\)時，\(R=\infty\)；當\(\rho=\infty\)時，\(R=0\)。3.計算\(\int_{0}^{1}x^{2}e^{x}dx\)。答案：利用分部積分法，設\(u=x^{2}\)，\(dv=e^{x}dx\)，則\(du=2xdx\)，\(v=e^{x}\)。\(\int_{0}^{1}x^{2}e^{x}dx=\left[x^{2}e^{x}\right]_{0}^{1}-2\int_{0}^{1}xe^{x}dx\)，再對\(\int_{0}^{1}xe^{x}dx\)用分部積分法可得結果為\(e-2\)。4.求曲線\(y=\frac{1}{x}\)在點\((1,1)\)處的曲率。答案：首先\(y'=-\frac{1}{x^{2}}\)，\(y''=\frac{2}{x^{3}}\)，在點\((1,1)\)處\(y'=-1\)，\(y''=2\)。根據曲率公式\(k=\frac{\verty''\vert}{(1+y'^{2})^{\frac{3}{2}}}\)，可得\(k=\frac{\sqrt{2}}{2}\)。五、討論題（每題5分，共4題）1.討論函數\(z=\frac{1}{(x-1)(y-1)}\)的間斷點。答案：函數\(z=\frac{1}{(x-1)(y-1)}\)在\