高數(shù)上冊期末考試題及答案

一、單項選擇題（每題2分，共10題）1.函數(shù)\(y=\ln(x+1)\)的定義域是（）A.\((-1,+\infty)\)B.\([-1,+\infty)\)C.\((0,+\infty)\)D.\((-\infty,-1)\)答案：A2.若\(f(x)=x^2\)，則\(f'(1)\)等于（）A.1B.2C.3D.4答案：B3.\(\intx\mathrm66o6maox=\)（）A.\(\frac{1}{2}x^{2}+C\)B.\(x^{2}+C\)C.\(\frac{1}{3}x^{3}+C\)D.\(2x+C\)答案：A4.函數(shù)\(y=\sinx\)在\(x=\frac{\pi}{2}\)處的切線斜率是（）A.0B.1C.-1D.\(\frac{1}{2}\)答案：A5.\(\lim\limits_{x\rightarrow0}\frac{\sinx}{x}=\)（）A.0B.1C.不存在D.-1答案：B6.設(shè)\(y=e^{x}\cosx\)，則\(y'\)是（）A.\(e^{x}\cosx-e^{x}\sinx\)B.\(e^{x}\cosx+e^{x}\sinx\)C.\(-e^{x}\sinx\)D.\(e^{x}\sinx\)答案：A7.\(\int_{0}^{1}x^{2}\mathrmooc6s6ix=\)（）A.\(\frac{1}{3}\)B.\(\frac{1}{2}\)C.1D.3答案：A8.函數(shù)\(y=x^{3}-3x^{2}+1\)的單調(diào)遞增區(qū)間是（）A.\((-\infty,0)\cup(2,+\infty)\)B.\((0,2)\)C.\((-\infty,1)\cup(3,+\infty)\)D.\((1,3)\)答案：A9.下列函數(shù)中是奇函數(shù)的是（）A.\(y=x^{2}+1\)B.\(y=\cosx\)C.\(y=\sinx\)D.\(y=e^{x}\)答案：C10.\(\lim\limits_{x\rightarrow+\infty}\frac{\lnx}{x}=\)（）A.0B.1C.\(+\infty\)D.\(-\infty\)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)的是（）A.\(y=\frac{\sinx}{x}\)B.\(y=x^{2}+1\)C.\(y=\begin{cases}x+1,x\geq0\\x-1,x<0\end{cases}\)D.\(y=\frac{1}{x}\)答案：ABC2.下列求導(dǎo)正確的是（）A.\((x^{3})'=3x^{2}\)B.\((\sin2x)'=2\cos2x\)C.\((\lnx)'=\frac{1}{x}\)D.\((e^{-x})'=e^{-x}\)答案：ABC3.下列積分正確的是（）A.\(\int\cosx\mathrmwc6y66yx=\sinx+C\)B.\(\inte^{x}\mathrmua6a6s4x=e^{x}+C\)C.\(\int\frac{1}{x}\mathrm46eiw6wx=\ln|x|+C\)D.\(\intx^{n}\mathrmsqmqea6x=\frac{1}{n+1}x^{n+1}+C(n\neq-1)\)答案：ABCD4.函數(shù)\(y=f(x)\)在點\(x_{0}\)處可導(dǎo)的必要條件是（）A.函數(shù)在\(x_{0}\)處連續(xù)B.函數(shù)在\(x_{0}\)處有極限C.函數(shù)在\(x_{0}\)的某鄰域內(nèi)有定義D.函數(shù)在\(x_{0}\)處的左右導(dǎo)數(shù)相等答案：ABC5.下列函數(shù)中，在區(qū)間\((0,+\infty)\)上單調(diào)遞增的是（）A.\(y=x^{2}\)B.\(y=\lnx\)C.\(y=e^{x}\)D.\(y=\sinx\)答案：ABC6.設(shè)\(y=f(u)\)，\(u=g(x)\)，則復(fù)合函數(shù)\(y=f(g(x))\)的求導(dǎo)公式為（）A.\(y'=f'(u)g'(x)\)B.\(y'=\frac{\mathrmsok6yyiy}{\mathrmiwwcicsu}\cdot\frac{\mathrmoa6g666u}{\mathrmiw6qwaox}\)C.\(y'=f(g(x))g'(x)\)D.\(y'=f'(g(x))\)答案：AB7.下列關(guān)于定積分性質(zhì)正確的是（）A.\(\int_{a}^[f(x)+g(x)]\mathrm6sg666wx=\int_{a}^f(x)\mathrmqomqwaox+\int_{a}^g(x)\mathrmeiei666x\)B.\(\int_{a}^kf(x)\mathrm6c6ay6wx=k\int_{a}^f(x)\mathrmyqgs6kyx(k為常數(shù))\)C.\(\int_{a}^f(x)\mathrme6ee6ugx=\int_{a}^{c}f(x)\mathrm666skysx+\int_{c}^f(x)\mathrmmekwc6ox(a<c<b)\)D.\(\int_{a}^{a}f(x)\mathrme6qcy6ex=f(a)\)答案：ABC8.函數(shù)\(y=\frac{1}{x}\)的漸近線有（）A.\(x=0\)B.\(y=0\)C.\(x=1\)D.\(y=1\)答案：AB9.若\(y=f(x)\)是偶函數(shù)，則（）A.\(f(-x)=f(x)\)B.\(y=f(x)\)的圖象關(guān)于\(y\)軸對稱C.\(f'(x)\)是奇函數(shù)D.\(\int_{-a}^{a}f(x)\mathrmgq666wux=2\int_{0}^{a}f(x)\mathrmeqes6qex\)答案：ABCD10.設(shè)\(y=\arctanx\)，則（）A.\(y'=\frac{1}{1+x^{2}}\)B.函數(shù)在\((-\infty,+\infty)\)上單調(diào)遞增C.函數(shù)是奇函數(shù)D.\(\lim\limits_{x\rightarrow+\infty}y=\frac{\pi}{2}\)答案：ABCD三、判斷題（每題2分，共10題）1.若函數(shù)\(f(x)\)在\(x=a\)處可導(dǎo)，則\(f(x)\)在\(x=a\)處連續(xù)。（）答案：對2.\(\inte^{x}\sinx\mathrmq6ciek6x\)不能用分部積分法求解。（）答案：錯3.函數(shù)\(y=\sqrt{x}\)在\(x=0\)處的導(dǎo)數(shù)不存在。（）答案：對4.若\(f(x)\)是周期為\(T\)的函數(shù)，則\(\int_{a}^{a+T}f(x)\mathrmy66q4c6x=\int_{0}^{T}f(x)\mathrmyiu6666x\)。（）答案：對5.函數(shù)\(y=x^{3}\)是奇函數(shù)。（）答案：對6.若\(y=f(x)\)在\((a,b)\)內(nèi)單調(diào)遞增，則\(f'(x)>0\)在\((a,b)\)內(nèi)恒成立。（）答案：錯7.\(\lim\limits_{x\rightarrow0}\frac{\tanx}{x}=1\)。（）答案：對8.\(\int_{0}^{\pi}\sinx\mathrmamsaesgx=0\)。（）答案：錯9.函數(shù)\(y=\frac{1}{x^{2}}\)的二階導(dǎo)數(shù)\(y''=\frac{6}{x^{4}}\)。（）答案：對10.若\(f(x)\)在\(x_{0}\)處的左右極限都存在，則\(f(x)\)在\(x_{0}\)處極限存在。（）答案：錯四、簡答題（每題5分，共4題）1.求函數(shù)\(y=x^{3}-3x^{2}+2\)的極值。答案：先求導(dǎo)\(y'=3x^{2}-6x=3x(x-2)\)，令\(y'=0\)，得\(x=0\)或\(x=2\)。當(dāng)\(x<0\)時，\(y'>0\)；當(dāng)\(0<x<2\)時，\(y'<0\)；當(dāng)\(x>2\)時，\(y'>0\)。所以極大值為\(y(0)=2\)，極小值為\(y(2)=-2\)。2.計算\(\int\frac{x^{2}}{1+x^{2}}\mathrm6eswokqx\)。答案：\(\int\frac{x^{2}}{1+x^{2}}\mathrmo6kymymx=\int\frac{x^{2}+1-1}{1+x^{2}}\mathrmo666666x=\int(1-\frac{1}{1+x^{2}})\mathrmigu6w4gx=x-\arctanx+C\)。3.求曲線\(y=\sinx\)在\(x=\frac{\pi}{4}\)處的切線方程。答案：\(y'=\cosx\)，當(dāng)\(x=\frac{\pi}{4}\)時，\(y'=\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}\)，\(y=\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}\)。切線方程為\(y-\frac{\sqrt{2}}{2}=\frac{\sqrt{2}}{2}(x-\frac{\pi}{4})\)，即\(y=\frac{\sqrt{2}}{2}x+\frac{\sqrt{2}}{2}-\frac{\sqrt{2}\pi}{8}\)。4.簡述函數(shù)連續(xù)與可導(dǎo)的關(guān)系。答案：可導(dǎo)一定連續(xù)，連續(xù)不一定可導(dǎo)。例如\(y=|x|\)在\(x=0\)處連續(xù)但不可導(dǎo)。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\((0,1)\)和\((1,+\infty)\)上的單調(diào)性。答案：\(y'=-\frac{1}{x^{2}}\)，在\((0,1)\)和\((1,+\infty)\)上\(y'<0\)，所以函數(shù)\(y=\frac{1}{x}\)在\((0,1)\)和\((1,+\infty)\)上單調(diào)遞減。2.討論定積分\(\int_{-1}^{1}x^{3}\mathrmeaqg6ggx\)的值。答案：因為\(y=x^{3}\)是奇函數(shù)，根據(jù)定積分性質(zhì)\(\int_{-a}^{a}f(x)\mathrm4msqu6mx=0\)（\(f(x)\)為奇函數(shù)），所以\(\int_{-1}^{1}x^{3}\mathrmcqguqe6x=0\)。3.討論函數(shù)\(y=e^{x}-x-1\)的零點個數(shù)。答案：求導(dǎo)得\(y'=e^{x}-1\)，令\(y'=0\