大學(xué)高數(shù)上冊(cè)考試題及答案

大學(xué)高數(shù)上冊(cè)考試題及答案

一、單項(xiàng)選擇題（每題2分，共10題）1.函數(shù)\(y=\sinx\)的導(dǎo)數(shù)是（）A.\(\cosx\)B.\(-\cosx\)C.\(\sinx\)D.\(-\sinx\)答案：A2.\(\lim\limits_{x\rightarrow0}\frac{\sinx}{x}=\)（）A.0B.1C.\(\infty\)D.不存在答案：B3.函數(shù)\(y=x^{2}\)在點(diǎn)\((1,1)\)處的切線斜率是（）A.1B.2C.\(-1\)D.\(-2\)答案：B4.\(\intxdx=\)（）A.\(\frac{1}{2}x^{2}+C\)B.\(x^{2}+C\)C.\(\frac{1}{3}x^{3}+C\)D.\(2x+C\)答案：A5.函數(shù)\(y=\lnx\)的定義域是（）A.\((-\infty,+\infty)\)B.\((0,+\infty)\)C.\([0,+\infty)\)D.\((-\infty,0)\)答案：B6.下列函數(shù)中是奇函數(shù)的是（）A.\(y=x^{2}\)B.\(y=\sinx\)C.\(y=\cosx\)D.\(y=e^{x}\)答案：B7.\(\lim\limits_{x\rightarrow\infty}(1+\frac{1}{x})^{x}=\)（）A.\(e\)B.1C.\(\infty\)D.0答案：A8.函數(shù)\(y=\sqrt{x}\)在\(x=4\)處的導(dǎo)數(shù)是（）A.\(\frac{1}{4}\)B.\(\frac{1}{2}\)C.\(\frac{1}{8}\)D.\(\frac{1}{16}\)答案：A9.\(\int\cosxdx=\)（）A.\(\sinx+C\)B.\(-\sinx+C\)C.\(\cosx+C\)D.\(-\cosx+C\)答案：A10.函數(shù)\(y=e^{x}\)的二階導(dǎo)數(shù)是（）A.\(e^{x}\)B.\(-e^{x}\)C.\(e^{-x}\)D.\(\frac{1}{e^{x}}\)答案：A二、多項(xiàng)選擇題（每題2分，共10題）1.下列函數(shù)在\((-\infty,+\infty)\)上連續(xù)的是（）A.\(y=x\)B.\(y=\sinx\)C.\(y=\frac{1}{x}\)D.\(y=e^{x}\)答案：ABD2.下列求導(dǎo)正確的是（）A.\((x^{3})'=3x^{2}\)B.\((\sinx)'=\cosx\)C.\((\lnx)'=\frac{1}{x}\)D.\((e^{x})'=e^{x}\)答案：ABCD3.下列積分正確的是（）A.\(\int2xdx=x^{2}+C\)B.\(\int\sinxdx=-\cosx+C\)C.\(\int\frac{1}{x}dx=\ln|x|+C\)D.\(\inte^{x}dx=e^{x}+C\)答案：ABCD4.下列函數(shù)是偶函數(shù)的是（）A.\(y=x^{2}\)B.\(y=\cosx\)C.\(y=|x|\)D.\(y=e^{-x}\)答案：ABC5.下列極限存在的是（）A.\(\lim\limits_{x\rightarrow0}\frac{1}{x}\)B.\(\lim\limits_{x\rightarrow1}(x+1)\)C.\(\lim\limits_{x\rightarrow\infty}\frac{x^{2}}{x+1}\)D.\(\lim\limits_{x\rightarrow0}\sinx\)答案：BD6.函數(shù)\(y=f(x)\)在點(diǎn)\(x=a\)處可導(dǎo)，則（）A.函數(shù)在點(diǎn)\(x=a\)處連續(xù)B.函數(shù)在點(diǎn)\(x=a\)處有定義C.\(\lim\limits_{x\rightarrowa}f(x)\)存在D.\(f(x)\)在\(x=a\)的鄰域內(nèi)有定義答案：ABCD7.下列關(guān)于定積分\(\int_{a}^f(x)dx\)說法正確的是（）A.它是一個(gè)常數(shù)B.它的值與被積函數(shù)\(f(x)\)有關(guān)C.它的值與積分區(qū)間\([a,b]\)有關(guān)D.它的值與積分變量的符號(hào)無關(guān)答案：ABCD8.函數(shù)\(y=\tanx\)的定義域是（）A.\(\{x|x\neqk\pi+\frac{\pi}{2},k\inZ\}\)B.\((-\infty,+\infty)\)除去\(x=k\pi+\frac{\pi}{2},k\inZ\)C.\((-\frac{\pi}{2},\frac{\pi}{2})\)D.\((-\infty,+\infty)\)答案：AB9.若\(y=f(x)\)的導(dǎo)數(shù)\(y'=0\)，則（）A.函數(shù)\(y=f(x)\)在該點(diǎn)有極值B.函數(shù)\(y=f(x)\)在該點(diǎn)可能有極值C.函數(shù)\(y=f(x)\)在該點(diǎn)是常數(shù)函數(shù)D.函數(shù)\(y=f(x)\)在該點(diǎn)的切線平行于\(x\)軸答案：BD10.下列函數(shù)中，單調(diào)遞增的函數(shù)是（）A.\(y=x^{3}\)B.\(y=\sinx(x\in[-\frac{\pi}{2},\frac{\pi}{2}])\)C.\(y=e^{x}\)D.\(y=\lnx(x\in(0,+\infty))\)答案：ABCD三、判斷題（每題2分，共10題）1.函數(shù)\(y=\frac{1}{x}\)在\(x=0\)處連續(xù)。（）答案：錯(cuò)誤2.\((\cosx)'=\sinx\)。（）答案：錯(cuò)誤3.\(\int_{a}^f(x)dx=-\int_^{a}f(x)dx\)。（）答案：正確4.函數(shù)\(y=x^{3}\)是奇函數(shù)。（）答案：正確5.\(\lim\limits_{x\rightarrow\infty}\frac{1}{x^{2}}=0\)。（）答案：正確6.函數(shù)\(y=\lnx\)在\((0,+\infty)\)上單調(diào)遞增。（）答案：正確7.若\(y=f(x)\)在\(x=a\)處不可導(dǎo)，則\(y=f(x)\)在\(x=a\)處不連續(xù)。（）答案：錯(cuò)誤8.\(\inte^{x}\sinxdx\)不能用分部積分法計(jì)算。（）答案：錯(cuò)誤9.函數(shù)\(y=\sqrt{x}\)的定義域是\([0,+\infty)\)。（）答案：正確10.\(\lim\limits_{x\rightarrow0}(1-x)^{\frac{1}{x}}=e^{-1}\)。（）答案：正確四、簡(jiǎn)答題（每題5分，共4題）1.求函數(shù)\(y=x^{3}-3x^{2}+2\)的極值。答案：首先求導(dǎo)\(y'=3x^{2}-6x=3x(x-2)\)。令\(y'=0\)，得\(x=0\)或\(x=2\)。當(dāng)\(x<0\)時(shí)，\(y'>0\)；當(dāng)\(0<x<2\)時(shí)，\(y'<0\)；當(dāng)\(x>2\)時(shí)，\(y'>0\)。所以\(x=0\)時(shí)，\(y\)有極大值\(y(0)=2\)；\(x=2\)時(shí)，\(y\)有極小值\(y(2)=-2\)。2.計(jì)算\(\int_{0}^{1}(x^{2}+1)dx\)。答案：\(\int_{0}^{1}(x^{2}+1)dx=\left[\frac{1}{3}x^{3}+x\right]_{0}^{1}=\frac{1}{3}+1=\frac{4}{3}\)。3.簡(jiǎn)述函數(shù)連續(xù)與可導(dǎo)的關(guān)系。答案：若函數(shù)在某點(diǎn)可導(dǎo)，則函數(shù)在該點(diǎn)一定連續(xù)；但函數(shù)在某點(diǎn)連續(xù)，不一定在該點(diǎn)可導(dǎo)。例如\(y=|x|\)在\(x=0\)處連續(xù)但不可導(dǎo)。4.求曲線\(y=\sinx\)在\(x=\frac{\pi}{2}\)處的切線方程。答案：首先求導(dǎo)\(y'=\cosx\)，當(dāng)\(x=\frac{\pi}{2}\)時(shí)，\(y'=0\)，\(y=1\)。切線方程為\(y-1=0\times(x-\frac{\pi}{2})\)，即\(y=1\)。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(dǎo)性。答案：\(y=\frac{x^{2}-1}{x-1}=\frac{(x+1)(x-1)}{x-1}=x+1(x\neq1)\)。\(\lim\limits_{x\rightarrow1}y=\lim\limits_{x\rightarrow1}(x+1)=2\)。函數(shù)在\(x=1\)處不連續(xù)，因?yàn)楹瘮?shù)在\(x=1\)無定義。不可導(dǎo)，因?yàn)椴贿B續(xù)就不可導(dǎo)。2.討論函數(shù)\(y=e^{x}-x-1\)的單調(diào)性。答案：求導(dǎo)得\(y'=e^{x}-1\)。當(dāng)\(x>0\)時(shí)，\(y'>0\)，函數(shù)單調(diào)遞增；當(dāng)\(x<0\)時(shí)，\(y'<0\)，函數(shù)單調(diào)遞減。3.如何利用定積分求平面圖形的面積？答案：若平面圖形由\(y=f(x)\)，\(y=g(x)\)以及\(x=a\)，\(x=b\)所圍成