高數(shù)一考試試題及答案

高數(shù)一考試試題及答案

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.\(-\sinx\)2.定積分\(\int_{0}^{1}xdx=\)（）A.\(\frac{1}{2}\)B.1C.2D.\(\frac{1}{3}\)3.極限\(\lim_{x\rightarrow0}\frac{\sinx}{x}=\)（）A.0B.1C.\(\infty\)D.不存在4.函數(shù)\(y=x^{2}\)在\(x=1\)處的切線斜率是（）A.1B.2C.3D.45.設(shè)\(y=e^{x}\)，則\(y^{(n)}=\)（）A.\(e^{x}\)B.\(ne^{x}\)C.\(x^{n}e^{x}\)D.\(e^{nx}\)6.函數(shù)\(y=\lnx\)的定義域是（）A.\((0,+\infty)\)B.\((-\infty,0)\)C.\((-\infty,+\infty)\)D.\([0,+\infty)\)7.若\(f(x)=\frac{1}{x}\)，則\(f'(x)=\)（）A.\(\frac{1}{x^{2}}\)B.\(-\frac{1}{x^{2}}\)C.\(x^{2}\)D.\(-x^{2}\)8.無窮積分\(\int_{1}^{+\infty}\frac{1}{x^{2}}dx=\)（）A.1B.\(\infty\)C.0D.\(-1\)9.設(shè)\(z=x+y\)，則\(\frac{\partialz}{\partialx}=\)（）A.0B.1C.2D.\(x\)10.函數(shù)\(y=3x^{3}-2x+1\)的二階導(dǎo)數(shù)\(y''=\)（）A.\(18x\)B.\(18x-2\)C.\(9x^{2}-2\)D.\(18\)答案1.A2.A3.B4.B5.A6.A7.B8.A9.B10.A二、多項(xiàng)選擇題（每題2分，共10題）1.下列函數(shù)中，是奇函數(shù)的有（）A.\(y=x^{3}\)B.\(y=\sinx\)C.\(y=e^{x}\)D.\(y=\frac{1}{x}\)2.下列等式正確的有（）A.\((\sinx)'=\cosx\)B.\((\cosx)'=-\sinx\)C.\((\lnx)'=\frac{1}{x}\)D.\((e^{x})'=e^{x}\)3.以下哪些函數(shù)在\(x=0\)處連續(xù)（）A.\(y=x\)B.\(y=\frac{\sinx}{x}\)C.\(y=\left\{\begin{array}{ll}x+1,x\geqslant0\\x-1,x<0\end{array}\right.\)D.\(y=\ln(x+1)\)4.下列哪些是定積分的性質(zhì)（）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.\(\int_{a}^{a}f(x)dx=0\)D.\(\int_{a}^f(x)dx=-\int_^{a}f(x)dx\)5.對(duì)于函數(shù)\(y=f(x)\)，如果\(f'(x_{0})=0\)，則（）A.\(x_{0}\)一定是函數(shù)的極值點(diǎn)B.\(x_{0}\)可能是函數(shù)的極值點(diǎn)C.函數(shù)在\(x_{0}\)處的切線平行于\(x\)軸D.函數(shù)在\(x_{0}\)處一定不可導(dǎo)6.下列函數(shù)中，在\((0,+\infty)\)上單調(diào)遞增的有（）A.\(y=x^{2}\)B.\(y=\lnx\)C.\(y=e^{x}\)D.\(y=\sinx\)7.以下關(guān)于極限的說法正確的是（）A.\(\lim_{x\rightarrow+\infty}\frac{1}{x}=0\)B.\(\lim_{x\rightarrow0}\frac{1-\cosx}{x}=0\)C.\(\lim_{x\rightarrow1}\frac{x^{2}-1}{x-1}=2\)D.\(\lim_{x\rightarrow\infty}(1+\frac{1}{x})^{x}=e\)8.下列關(guān)于導(dǎo)數(shù)的幾何意義說法正確的是（）A.函數(shù)\(y=f(x)\)在點(diǎn)\(x_{0}\)處的導(dǎo)數(shù)\(f'(x_{0})\)就是曲線\(y=f(x)\)在點(diǎn)\((x_{0},f(x_{0}))\)處的切線斜率B.若函數(shù)\(y=f(x)\)在點(diǎn)\(x_{0}\)處的導(dǎo)數(shù)不存在，則曲線\(y=f(x)\)在點(diǎn)\((x_{0},f(x_{0}))\)處沒有切線C.若曲線\(y=f(x)\)在點(diǎn)\((x_{0},f(x_{0}))\)處的切線垂直于\(x\)軸，則\(f'(x_{0})=\infty\)D.若曲線\(y=f(x)\)在點(diǎn)\((x_{0},f(x_{0}))\)處的切線平行于\(x\)軸，則\(f'(x_{0})=0\)9.下列關(guān)于不定積分的說法正確的是（）A.\(\intkdx=kx+C\)（\(k\)為常數(shù)）B.\(\intx^{n}dx=\frac{x^{n+1}}{n+1}+C\)（\(n\neq-1\)）C.\(\int\sinxdx=-\cosx+C\)D.\(\int\cosxdx=\sinx+C\)10.設(shè)\(z=f(x,y)\)，則全微分\(dz=\)（）A.\(\frac{\partialz}{\partialx}dx+\frac{\partialz}{\partialy}dy\)B.\(\frac{\partialz}{\partialy}dx+\frac{\partialz}{\partialx}dy\)C.\(z_{x}dx+z_{y}dy\)D.\(z_{y}dx+z_{x}dy\)答案1.ABD2.ABCD3.AD4.ABCD5.BC6.ABC7.ACD8.ACD9.ABCD10.AC三、判斷題（每題2分，共10題）1.函數(shù)\(y=\sqrt{x}\)在\([0,+\infty)\)上可導(dǎo)。（）2.若\(f(x)\)在\(x=a\)處連續(xù)，則\(f(x)\)在\(x=a\)處一定可導(dǎo)。（）3.\(\int_{-1}^{1}x^{3}dx=0\)。（）4.函數(shù)\(y=e^{-x}\)是單調(diào)遞減函數(shù)。（）5.若\(f(x)\)是偶函數(shù)，則\(f'(x)\)是奇函數(shù)。（）6.\(\lim_{x\rightarrow0}\frac{e^{x}-1}{x}=1\)。（）7.函數(shù)\(y=\frac{1}{x}\)在\((0,+\infty)\)上沒有最大值。（）8.對(duì)于函數(shù)\(z=x^{2}+y^{2}\)，\(\frac{\partialz}{\partialx}=2x\)，\(\frac{\partialz}{\partialy}=2y\)。（）9.無窮積分\(\int_{0}^{+\infty}\sinxdx\)收斂。（）10.若\(y=f(x)\)在\(x_{0}\)處的二階導(dǎo)數(shù)\(f''(x_{0})>0\)，則\(y=f(x)\)在\(x_{0}\)處取得極小值。（）答案1.False2.False3.True4.True5.True6.True7.True8.True9.False10.False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：首先求導(dǎo)\(y'=3x^{2}-6x=3x(x-2)\)，令\(y'=0\)，解得\(x=0\)或\(x=2\)。當(dāng)\(x<0\)時(shí)，\(y'>0\)，函數(shù)遞增；當(dāng)\(0<x<2\)時(shí)，\(y'<0\)，函數(shù)遞減；當(dāng)\(x>2\)時(shí)，\(y'>0\)，函數(shù)遞增。所以\(x=0\)時(shí)取得極大值\(y(0)=2\)，\(x=2\)時(shí)取得極小值\(y(2)=-2\)。2.計(jì)算定積分\(\int_{0}^{\pi}\sinxdx\)。-答案：根據(jù)定積分公式\(\int\sinxdx=-\cosx+C\)，\(\int_{0}^{\pi}\sinxdx=-\cosx|_{0}^{\pi}=-(\cos\pi-\cos0)=-(-1-1)=2\)。3.求函數(shù)\(y=\ln(x^{2}+1)\)的導(dǎo)數(shù)。-答案：設(shè)\(u=x^{2}+1\)，則\(y=\lnu\)。根據(jù)復(fù)合函數(shù)求導(dǎo)法則\(y'=\frac{1}{u}\cdotu'=\frac{2x}{x^{2}+1}\)。4.簡述導(dǎo)數(shù)的定義。-答案：設(shè)函數(shù)\(y=f(x)\)在點(diǎn)\(x_{0}\)的某個(gè)鄰域內(nèi)有定義，當(dāng)自變量\(x\)在\(x_{0}\)處取得增量\(\Deltax\)（點(diǎn)\(x_{0}+\Deltax\)仍在該鄰域內(nèi)）時(shí)，相應(yīng)地函數(shù)取得增量\(\Deltay=f(x_{0}+\Deltax)-f(x_{0})\)；如果\(\Deltay\)與\(\Deltax\)之比當(dāng)\(\Deltax\rightarrow0\)時(shí)的極限存在，則稱函數(shù)\(y=f(x)\)在點(diǎn)\(x_{0}\)處可導(dǎo)，這個(gè)極限值稱為函數(shù)\(y=f(x)\)在點(diǎn)\(x_{0}\)處的導(dǎo)數(shù)，記作\(f'(x_{0})\)。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：\(\lim_{x\rightarrow1}\frac{x^{2}-1}{x-1}=\lim_{x\rightarrow1}(x+1)=2\)。函數(shù)在\(x=1\)處不連續(xù)，因?yàn)楹瘮?shù)在\(x=1\)無定義。由于不連續(xù)，所以不可導(dǎo)。2.討論函數(shù)\(y=x^{3}-3x\)的單調(diào)性。-答案：求導(dǎo)得\(y'=3x^{2}-3=3(x+1)(x-1)\)。當(dāng)\(x<-1\)或\(x>1\)時(shí)，\(y'>0\)，函數(shù)單調(diào)遞增；當(dāng)\(-1<x<1\)時(shí)，\(y'<0\)，函數(shù)單調(diào)遞減。3.討論函數(shù)\(y=e^{x}-x-1\)的零點(diǎn)個(gè)數(shù)。-答案：求導(dǎo)得\(y'=e^{x}-1\)。當(dāng)\(x=0\)時(shí)