微積分期末考試題及答案

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.\(\intx^2dx\)等于（）A.\(\frac{1}{3}x^3+C\)B.\(x^3+C\)C.\(\frac{1}{2}x^2+C\)D.\(2x+C\)答案：A3.極限\(\lim\limits_{x\rightarrow0}\frac{\sinx}{x}\)的值為（）A.0B.1C.不存在D.\(\infty\)答案：B4.函數(shù)\(y=e^x\)在點(diǎn)\((0,1)\)處的切線方程為（）A.\(y=x+1\)B.\(y=-x+1\)C.\(y=0\)D.\(y=2x+1\)答案：A5.定積分\(\int_{0}^{1}xdx\)的值為（）A.\(\frac{1}{2}\)B.1C.0D.\(\frac{1}{3}\)答案：A6.函數(shù)\(y=\lnx\)的定義域是（）A.\((0,+\infty)\)B.\([0,+\infty)\)C.\((-\infty,0)\)D.\((-\infty,+\infty)\)答案：A7.若\(y=f(x)\)在\(x=a\)處可導(dǎo)，則\(\lim\limits_{h\rightarrow0}\frac{f(a+h)-f(a-h)}{2h}\)等于（）A.\(f'(a)\)B.\(2f'(a)\)C.\(\frac{1}{2}f'(a)\)D.0答案：A8.函數(shù)\(y=x^3-3x\)的單調(diào)遞增區(qū)間是（）A.\((-\infty,-1)\cup(1,+\infty)\)B.\((-1,1)\)C.\((-\infty,+\infty)\)D.\((0,+\infty)\)答案：A9.下列函數(shù)中，是奇函數(shù)的是（）A.\(y=x^2\)B.\(y=\sinx\)C.\(y=e^x\)D.\(y=\lnx\)答案：B10.\(\fracewagkqc{dx}(x\cosx)\)等于（）A.\(\cosx-x\sinx\)B.\(\cosx+x\sinx\)C.\(-\sinx\)D.\(\sinx\)答案：A二、多項(xiàng)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.\(y=\sinx\)B.\(y=\frac{\sinx}{x}\)C.\(y=e^x\)D.\(y=\left\{\begin{array}{ll}x+1,x\geqslant0\\x-1,x<0\end{array}\right.\)答案：AC2.以下關(guān)于定積分性質(zhì)的說(shuō)法正確的是（）A.\(\int_{a}^[f(x)+g(x)]dx=\int_{a}^f(x)dx+\int_{a}^g(x)dx\)B.\(\int_{a}^kf(x)dx=k\int_{a}^f(x)dx\)（\(k\)為常數(shù)）C.\(\int_{a}^f(x)dx=-\int_^{a}f(x)dx\)D.\(\int_{a}^{a}f(x)dx=f(a)\)答案：ABC3.函數(shù)\(y=f(x)\)的極值點(diǎn)可能出現(xiàn)在（）A.\(f'(x)=0\)的點(diǎn)B.\(f'(x)\)不存在的點(diǎn)C.區(qū)間端點(diǎn)D.函數(shù)的駐點(diǎn)答案：AB4.下列求導(dǎo)正確的有（）A.\((\sin2x)'=2\cos2x\)B.\((\lnx^2)'=\frac{2}{x}\)C.\((e^{-x})'=-e^{-x}\)D.\((x^3)'=3x^2\)答案：ABCD5.以下關(guān)于函數(shù)單調(diào)性的說(shuō)法正確的是（）A.若\(f'(x)>0\)在區(qū)間\((a,b)\)上恒成立，則\(y=f(x)\)在\((a,b)\)上單調(diào)遞增B.若\(f'(x)<0\)在區(qū)間\((a,b)\)上恒成立，則\(y=f(x)\)在\((a,b)\)上單調(diào)遞減C.函數(shù)的單調(diào)性與導(dǎo)數(shù)無(wú)關(guān)D.函數(shù)的單調(diào)性只與函數(shù)的表達(dá)式有關(guān)答案：AB6.定積分\(\int_{-1}^{1}x^3dx\)的值為（）A.0B.\(\frac{1}{4}\)C.\(-\frac{1}{4}\)D.1答案：A7.下列函數(shù)中，是偶函數(shù)的有（）A.\(y=x^2\)B.\(y=\cosx\)C.\(y=e^{x^2}\)D.\(y=\frac{1}{x}\)答案：ABC8.函數(shù)\(y=\sqrt{x}\)在\(x=4\)處的導(dǎo)數(shù)為（）A.\(\frac{1}{4}\)B.\(\frac{1}{2\sqrt{4}}\)C.\(\frac{1}{2}\)D.\(\frac{1}{2x}\)答案：A9.對(duì)于函數(shù)\(y=\frac{1}{x}\)，以下說(shuō)法正確的是（）A.在\((0,+\infty)\)上單調(diào)遞減B.在\((-\infty,0)\)上單調(diào)遞減C.在\((-\infty,0)\cup(0,+\infty)\)上單調(diào)遞減D.在\((-\infty,0)\)和\((0,+\infty)\)上分別單調(diào)遞減答案：D10.若\(y=f(x)\)在\(x=a\)處二階可導(dǎo)，且\(f'(a)=0,f''(a)>0\)，則（）A.\(x=a\)是函數(shù)\(y=f(x)\)的極小值點(diǎn)B.\(x=a\)是函數(shù)\(y=f(x)\)的極大值點(diǎn)C.函數(shù)\(y=f(x)\)在\(x=a\)處的切線平行于\(x\)軸D.函數(shù)\(y=f(x)\)在\(x=a\)處單調(diào)遞增答案：AC三、判斷題（每題2分，共10題）1.函數(shù)\(y=\tanx\)的定義域是\((-\frac{\pi}{2},\frac{\pi}{2})\)。（）答案：錯(cuò)誤2.若\(f(x)\)在\(x=a\)處可導(dǎo)，則\(f(x)\)在\(x=a\)處連續(xù)。（）答案：正確3.\(\int_{0}^{\pi}\sinxdx=0\)。（）答案：錯(cuò)誤4.函數(shù)\(y=x^2\)在\((-\infty,+\infty)\)上是凹函數(shù)。（）答案：正確5.若\(y=f(x)\)是奇函數(shù)，則\(\int_{-a}^{a}f(x)dx=0\)。（）答案：正確6.\(y=\lnx\)的導(dǎo)數(shù)是\(\frac{1}{x}\)，則\(y=\ln(x+1)\)的導(dǎo)數(shù)是\(\frac{1}{x+1}\)。（）答案：正確7.函數(shù)\(y=e^x\)是單調(diào)遞增函數(shù)。（）答案：正確8.函數(shù)\(y=\frac{1}{x}\)在\(x=0\)處的極限不存在。（）答案：正確9.若\(f(x)\)和\(g(x)\)在\(x=a\)處都不可導(dǎo)，則\(f(x)+g(x)\)在\(x=a\)處也不可導(dǎo)。（）答案：錯(cuò)誤10.\(\fracmwcqwku{dx}(\cosx)=-\sinx\)。（）答案：正確四、簡(jiǎn)答題（每題5分，共4題）1.求函數(shù)\(y=x^3-3x^2+2\)的單調(diào)區(qū)間。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)\(x<0\)或\(x>2\)時(shí)，\(y'>0\)，函數(shù)單調(diào)遞增；當(dāng)\(0<x<2\)時(shí)，\(y'<0\)，函數(shù)單調(diào)遞減。2.計(jì)算\(\int_{0}^{1}(x^2+1)dx\)。答案：\(\int_{0}^{1}(x^2+1)dx=\int_{0}^{1}x^2dx+\int_{0}^{1}1dx=\left[\frac{1}{3}x^3\right]_{0}^{1}+[x]_{0}^{1}=\frac{1}{3}+1=\frac{4}{3}\)。3.求函數(shù)\(y=\sinx\)在\(x=\frac{\pi}{4}\)處的切線方程。答案：\(y'=\cosx\)，當(dāng)\(x=\frac{\pi}{4}\)時(shí)，\(y'=\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}\)，\(y=\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}\)。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(jiǎn)述函數(shù)極值與最值的區(qū)別。答案：極值是局部概念，是函數(shù)在某點(diǎn)鄰域內(nèi)的最值；最值是函數(shù)在整個(gè)定義域或指定區(qū)間上的最大或最小值。極值點(diǎn)可能有多個(gè)，最值若存在則唯一（在區(qū)間上），最值點(diǎn)可能是極值點(diǎn)也可能是區(qū)間端點(diǎn)。五、討論題（每題5分，共4題）1.討論函數(shù)\(y=\frac{x}{1+x^2}\)的單調(diào)性。答案：對(duì)\(y=\frac{x}{1+x^2}\)求導(dǎo)得\(y'=\frac{1-x^2}{(1+x^2)^2}\)。令\(y'=0\)，得\(x=\pm1\)。當(dāng)\(x\in(-\infty,-1)\cup(1,+\infty)\)時(shí)，\(y'<0\)，函數(shù)單調(diào)遞減；當(dāng)\(x\in(-1,1)\)時(shí)，\(y'>0\)，函數(shù)單調(diào)遞增。2.討論函數(shù)\(y=e^x-x-1\)的零點(diǎn)個(gè)數(shù)。答案：求導(dǎo)得\(y'=e^x-1\)。令\(y'=0\)，得\(x=0\)。當(dāng)\(x<0\)時(shí)，\(y'<0\)，函數(shù)遞減；當(dāng)\(x>0\)時(shí)，\(y'>0\)，函數(shù)遞增。\(y(0)=0\)，所以函數(shù)有且僅有一個(gè)零點(diǎn)。3.討論定積分\(\int_{a}^f(x)dx\)的幾何意義。答案：當(dāng)\(f(x)\geqslant0\)時(shí)，\(\int_{a}^f(x)dx\)表示由曲線\(y=f(x)\)，\(x=a\)，\(x=b\)以及\(x\)軸所圍成的曲邊梯形的面積；當(dāng)\(f(x)\leqslant0