數(shù)學(xué)分析二考試題及答案

一、單項(xiàng)選擇題（每題2分，共10題）1.設(shè)函數(shù)\(f(x)\)在\(x=a\)處可導(dǎo)，則\(\lim\limits_{h\rightarrow0}\frac{f(a+h)-f(a-h)}{h}=()\)A.\(f'(a)\)B.\(2f'(a)\)C.\(0\)D.不存在答案：B2.函數(shù)\(y=x^{3}-3x^{2}+1\)的單調(diào)遞減區(qū)間是()A.\((2,+\infty)\)B.\((-\infty,0)\)C.\((0,2)\)D.\((-\infty,+\infty)\)答案：C3.設(shè)\(f(x)=\frac{\sinx}{x}\)，則\(x=0\)是\(f(x)\)的()A.可去間斷點(diǎn)B.跳躍間斷點(diǎn)C.無窮間斷點(diǎn)D.連續(xù)點(diǎn)答案：A4.\(\int\frac{1}{x^{2}+4}dx=()\)A.\(\frac{1}{2}\arctan\frac{x}{2}+C\)B.\(\arctan\frac{x}{2}+C\)C.\(\frac{1}{2}\arctanx+C\)D.\(\arctanx+C\)答案：A5.設(shè)\(y=e^{x}\cosx\)，則\(y''=()\)A.\(2e^{x}\sinx\)B.\(-2e^{x}\sinx\)C.\(2e^{x}\cosx\)D.\(-2e^{x}\cosx\)答案：B6.曲線\(y=\frac{1}{x}\)在點(diǎn)\((1,1)\)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.函數(shù)\(y=x^{2}\lnx\)的極小值點(diǎn)為()A.\(x=e^{-\frac{1}{2}}\)B.\(x=1\)C.\(x=0\)D.\(x=e\)答案：A9.\(\lim\limits_{x\rightarrow0}\frac{\ln(1+x)}{x}=()\)A.\(0\)B.\(1\)C.\(-1\)D.不存在答案：B10.設(shè)\(y=\sqrt{x}\)，則\(y'\)在\(x=4\)處的值為()A.\(\frac{1}{4}\)B.\(\frac{1}{2}\)C.\(\frac{1}{8}\)D.\(\frac{1}{16}\)答案：A二、多項(xiàng)選擇題（每題2分，共10題）1.下列函數(shù)中，在區(qū)間\((0,+\infty)\)上是增函數(shù)的是()A.\(y=x^{2}\)B.\(y=\lnx\)C.\(y=e^{x}\)D.\(y=\cosx\)答案：ABC2.下列函數(shù)的導(dǎo)數(shù)正確的是()A.若\(y=x^{n}\)，則\(y'=nx^{n-1}\)B.若\(y=\sinx\)，則\(y'=\cosx\)C.若\(y=\lnx\)，則\(y'=\frac{1}{x}\)D.若\(y=e^{x}\)，則\(y'=e^{x}\)答案：ABCD3.下列定積分的值為\(0\)的有()A.\(\int_{-1}^{1}x^{3}dx\)B.\(\int_{-a}^{a}\sinxdx\)C.\(\int_{-2}^{2}(x^{2}+1)dx\)D.\(\int_{-\pi}^{\pi}\cosxdx\)答案：ABD4.函數(shù)\(y=f(x)\)在\(x=x_{0}\)處取得極值的必要條件是()A.\(f'(x_{0})=0\)B.\(f(x)\)在\(x=x_{0}\)處連續(xù)C.\(f''(x_{0})=0\)D.\(f(x)\)在\(x=x_{0}\)處可導(dǎo)答案：AD5.下列函數(shù)在\(x=0\)處連續(xù)的有()A.\(y=\frac{\sinx}{x}\)B.\(y=x\sin\frac{1}{x}\)C.\(y=\left\{\begin{array}{ll}x+1,x\geqslant0\\x-1,x<0\end{array}\right.\)D.\(y=\frac{1}{x}\)答案：AB6.設(shè)\(y=f(u)\)，\(u=g(x)\)，則復(fù)合函數(shù)\(y=f(g(x))\)的求導(dǎo)法則正確的是()A.\(y'=f'(u)g'(x)\)B.\(y'=\frac{dy}{du}\cdot\frac{du}{dx}\)C.\(y'=f(g(x))g'(x)\)D.\(y'=f'(g(x))\)答案：AB7.下列廣義積分收斂的有()A.\(\int_{1}^{+\infty}\frac{1}{x^{2}}dx\)B.\(\int_{0}^{1}\frac{1}{\sqrt{x}}dx\)C.\(\int_{1}^{+\infty}\frac{1}{x}dx\)D.\(\int_{0}^{+\infty}e^{-x}dx\)答案：AD8.設(shè)\(f(x)\)在\([a,b]\)上可積，則()A.\(\int_{a}^f(x)dx=\int_{a}^f(t)dt\)B.\(\int_{a}^f(x)dx=-\int_^{a}f(x)dx\)C.若\(f(x)\geqslant0\)，\(x\in[a,b]\)，則\(\int_{a}^f(x)dx\geqslant0\)D.若\(m\leqslantf(x)\leqslantM\)，\(x\in[a,b]\)，則\(m(b-a)\leqslant\int_{a}^f(x)dx\leqslantM(b-a)\)答案：ABCD9.曲線\(y=f(x)\)的凹凸性與下列哪些因素有關(guān)()A.\(f''(x)\)的符號(hào)B.\(f'(x)\)的單調(diào)性C.\(f(x)\)的單調(diào)性D.\(f(x)\)的極值答案：AB10.若\(\lim\limits_{x\rightarrowa}f(x)=\lim\limits_{x\rightarrowa}g(x)=0\)，則\(\lim\limits_{x\rightarrowa}\frac{f(x)}{g(x)}\)可能為()A.\(0\)B.\(1\)C.\(\infty\)D.不存在答案：ABCD三、判斷題（每題2分，共10題）1.若函數(shù)\(y=f(x)\)在\(x=a\)處可導(dǎo)，則\(y=f(x)\)在\(x=a\)處連續(xù)。（對(duì)）2.函數(shù)\(y=\frac{1}{x}\)在區(qū)間\((0,1)\)上是有界函數(shù)。（錯(cuò)）3.若\(f(x)\)在\([a,b]\)上連續(xù)，則\(\int_{a}^f(x)dx=\int_{a}^{c}f(x)dx+\int_{c}^f(x)dx\)（對(duì)）4.若\(f'(x)>0\)在\((a,b)\)上恒成立，則\(y=f(x)\)在\((a,b)\)上單調(diào)遞增。（對(duì)）5.函數(shù)\(y=x^{3}\)沒有極值點(diǎn)。（對(duì)）6.\(\lim\limits_{x\rightarrow+\infty}\frac{\sinx}{x}=1\)。（錯(cuò)）7.若\(y=f(x)\)是偶函數(shù)，則\(y=f'(x)\)也是偶函數(shù)。（錯(cuò)）8.\(\inte^{x}dx=e^{x}+C\)。（對(duì)）9.函數(shù)\(y=\lnx\)在\((0,+\infty)\)上的圖像是下凸的。（錯(cuò)）10.若\(\int_{a}^f(x)dx=0\)，則\(f(x)=0\)在\([a,b]\)上恒成立。（錯(cuò)）四、簡(jiǎn)答題（每題5分，共4題）1.求函數(shù)\(y=x^{3}-3x^{2}-9x+5\)的極值。答案：首先求導(dǎo)\(y'=3x^{2}-6x-9=3(x^{2}-2x-3)=3(x+1)(x-3)\)。令\(y'=0\)，得\(x=-1\)或\(x=3\)。當(dāng)\(x<-1\)時(shí)，\(y'>0\)；當(dāng)\(-1<x<3\)時(shí)，\(y'<0\)；當(dāng)\(x>3\)時(shí)，\(y'>0\)。所以極大值為\(y(-1)=10\)，極小值為\(y(3)=-22\)。2.計(jì)算\(\int\frac{x^{2}}{x^{2}+1}dx\)。答案：\(\int\frac{x^{2}}{x^{2}+1}dx=\int\frac{x^{2}+1-1}{x^{2}+1}dx=\int(1-\frac{1}{x^{2}+1})dx=x-\arctanx+C\)。3.簡(jiǎn)述函數(shù)\(y=f(x)\)在\(x=x_{0}\)處可導(dǎo)的定義。答案：函數(shù)\(y=f(x)\)在\(x=x_{0}\)處可導(dǎo)，即\(\lim\limits_{\Deltax\rightarrow0}\frac{f(x_{0}+\Deltax)-f(x_{0})}{\Deltax}\)存在，這個(gè)極限值稱為函數(shù)\(y=f(x)\)在\(x=x_{0}\)處的導(dǎo)數(shù)，記作\(f'(x_{0})\)。4.求曲線\(y=\lnx\)在點(diǎn)\((e,1)\)處的切線方程。答案：首先求導(dǎo)\(y'=\frac{1}{x}\)，在\(x=e\)處的導(dǎo)數(shù)\(y'(e)=\frac{1}{e}\)。切線方程為\(y-1=\frac{1}{e}(x-e)\)，即\(y=\frac{1}{e}x\)。五、討論題（每題5分，共4題）1.討論函數(shù)\(y=\frac{1}{x^{2}}\)的單調(diào)性和凹凸性。答案：\(y=\frac{1}{x^{2}}=x^{-2}\)，\(y'=-2x^{-3}=-\frac{2}{x^{3}}\)，\(y''=6x^{-4}=\frac{6}{x^{4}}\)。當(dāng)\(x>0\)時(shí)，\(y'<0\)，函數(shù)單調(diào)遞減；當(dāng)\(x<0\)時(shí)，\(y'>0\)，函數(shù)單調(diào)遞增。\(y''>0\)對(duì)于\(x\neq0\)恒成立，函數(shù)圖像是下凸的。2.討論廣義積分\(\int_{1}^{+\infty}\frac{1}{x^{p}}dx\)（\(p>0\)）的斂散性。答案：當(dāng)\(p=1\)時(shí)，\(\int_{1}^{+\infty}\frac{1}{x}dx=\lim\limits_{t\rightarrow+\infty}\lnt-\ln1=+\infty\)，發(fā)散；當(dāng)\(p>1\)時(shí)，\(\int_{1}^{+\infty}\frac{1}{x^{p}}dx=\lim\limits_{t\rightarrow+\infty}\frac{1}{1-p}t^{1-p}-\frac{1}{1-p}=\frac{1}{p-1}\)，收斂；當(dāng)\(0<p<1\)時(shí)，\(\int_{1}^{+\infty}\frac{1}{x^{p}}dx=\lim\limits_{t\rightarrow+\infty}\frac{1}{1-p}t^{1-p}-\frac{1}{1-p}=+\infty\)，發(fā)散。3.討論函數(shù)\(y=\sinx\)在\([0,2\pi]\)上的極值情況。答案：\(y'=\cosx\)，令\(y'=0\)，得\(x=\frac{\pi}{2}\)或\(x=\frac{3\pi}{2}\)。\(y''=-\sinx\)，當(dāng)\(x=\frac{\pi}{2}\)時(shí)，\(y''>0\)，極小值為\(y(\frac{\pi}{2})=1\)；當(dāng)\(x=\frac{3\p