高等數(shù)學(xué)普通題庫及答案

單項(xiàng)選擇題(每題2分,共10題)1.函數(shù)\(y=\frac{1}{\ln(x-1)}\)的定義域是(）A.\((1,+\infty)\)B.\((1,2)\cup(2,+\infty)\)C.\([1,+\infty)\)D.\([1,2)\cup(2,+\infty)\)2.當(dāng)\(x\to0\)時(shí),與\(x\)等價(jià)的無窮小是()A.\(\sinx-x\)B.\(x^2+x\)C.\(e^x-1\)D.\(1-\cosx\)3.設(shè)\(f(x)\)在\(x=a\)處可導(dǎo),則\(\lim\limits_{h\to0}\frac{f(a+h)-f(a-h)}{h}=\)()A.\(f'(a)\)B.\(2f'(a)\)C.\(0\)D.\(f'(2a)\)4.曲線\(y=x^3-3x^2+1\)的拐點(diǎn)是()A.\((0,1)\)B.\((1,-1)\)C.\((2,-3)\)D.\((3,1)\)5.函數(shù)\(f(x)=x^3-3x\)在區(qū)間\([-2,2]\)上的最大值是()A.\(2\)B.\(-2\)C.\(1\)D.\(-1\)6.若\(\intf(x)dx=F(x)+C\),則\(\intf(2x+1)dx=\)()A.\(F(2x+1)+C\)B.\(\frac{1}{2}F(2x+1)+C\)C.\(2F(2x+1)+C\)D.\(F(x)+C\)7.定積分\(\int_{-1}^{1}x^3dx=\)()A.\(0\)B.\(1\)C.\(2\)D.\(4\)8.級數(shù)\(\sum\limits_{n=1}^{\infty}\frac{1}{n^2}\)是()A.發(fā)散的B.條件收斂的C.絕對收斂的D.斂散性不確定9.向量\(\vec{a}=(1,2,3)\),\(\vec=(3,2,1)\),則\(\vec{a}\cdot\vec=\)()A.\(10\)B.\(6\)C.\(8\)D.\(12\)10.方程\(x^2+y^2+z^2=4\)表示的曲面是()A.圓柱面B.圓錐面C.球面D.拋物面多項(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.\(y=x^2+1\)D.\(y=\frac{1}{x}\)E.\(y=\cosx\)2.下列極限存在的有()A.\(\lim\limits_{x\to0}\frac{\sinx}{x}\)B.\(\lim\limits_{x\to\infty}(1+\frac{1}{x})^x\)C.\(\lim\limits_{x\to0}(1-x)^{\frac{1}{x}}\)D.\(\lim\limits_{x\to0}\frac{\ln(1+x)}{x}\)E.\(\lim\limits_{x\to\infty}\frac{x^2+1}{x}\)3.函數(shù)\(y=f(x)\)在點(diǎn)\(x_0\)處可導(dǎo)的充分必要條件是()A.函數(shù)在點(diǎn)\(x_0\)處連續(xù)B.函數(shù)在點(diǎn)\(x_0\)處左導(dǎo)數(shù)存在C.函數(shù)在點(diǎn)\(x_0\)處右導(dǎo)數(shù)存在D.函數(shù)在點(diǎn)\(x_0\)處左導(dǎo)數(shù)和右導(dǎo)數(shù)都存在且相等E.函數(shù)在點(diǎn)\(x_0\)處有定義4.下列函數(shù)中,在給定區(qū)間上滿足羅爾定理?xiàng)l件的有()A.\(f(x)=x^2-1\),\([-1,1]\)B.\(f(x)=x^3\),\([0,1]\)C.\(f(x)=\sinx\),\([0,\pi]\)D.\(f(x)=\frac{1}{x^2}\),\([-1,1]\)E.\(f(x)=e^x\),\([0,1]\)5.下列函數(shù)中,單調(diào)遞增的有()A.\(y=x^3\)B.\(y=\sinx\),\((0,\frac{\pi}{2})\)C.\(y=e^x\)D.\(y=\lnx\),\((0,+\infty)\)E.\(y=-x^2\)6.下列積分中,計(jì)算正確的有（）A.\(\intx^2dx=\frac{1}{3}x^3+C\)B.\(\int\sinxdx=-\cosx+C\)C.\(\inte^xdx=e^x+C\)D.\(\int\frac{1}{x}dx=\ln|x|+C\)E.\(\int\cosxdx=\sinx+C\)7.下列級數(shù)中,收斂的有()A.\(\sum\limits_{n=1}^{\infty}\frac{1}{n}\)B.\(\sum\limits_{n=1}^{\infty}\frac{(-1)^{n-1}}{n}\)C.\(\sum\limits_{n=1}^{\infty}\frac{1}{n^3}\)D.\(\sum\limits_{n=1}^{\infty}\frac{1}{2^n}\)E.\(\sum\limits_{n=1}^{\infty}n\)8.向量\(\vec{a}=(1,2,3)\),\(\vec=(2,3,4)\),則()A.\(\vec{a}+\vec=(3,5,7)\)B.\(\vec{a}-\vec=(-1,-1,-1)\)C.\(2\vec{a}=(2,4,6)\)D.\(\vec{a}\cdot\vec=20\)E.\(|\vec{a}|=\sqrt{14}\)9.下列方程中,是一階線性微分方程的有()A.\(y'+2y=x\)B.\(y''+y=0\)C.\(y'=\sinx\)D.\(y'-xy=x\)E.\(y^2y'+x=0\)10.下列曲面中,是二次曲面的有()A.球面B.圓柱面C.圓錐面D.拋物面E.橢球面判斷題(每題2分,共10題)1.函數(shù)\(y=\sqrt{x-1}\)的定義域是\([1,+\infty)\)。()2.若\(\lim\limits_{x\tox_0}f(x)=A\),則\(f(x_0)=A\)。（)3.函數(shù)\(y=x^2\)在\(x=0\)處的導(dǎo)數(shù)為\(0\)。()4.若函數(shù)\(f(x)\)在區(qū)間\((a,b)\)內(nèi)單調(diào)遞增,則\(f'(x)>0\)在\((a,b)\)內(nèi)恒成立。（)5.曲線\(y=\sinx\)在\(x=0\)處的切線方程為\(y=x\)。()6.定積分\(\int_{a}^f(x)dx\)的值與積分變量的選取無關(guān)。()7.級數(shù)\(\sum\limits_{n=1}^{\infty}a_n\)收斂,則\(\lim\limits_{n\to\infty}a_n=0\)。(）8.向量\(\vec{a}=(1,0,0)\),\(\vec=(0,1,0)\),則\(\vec{a}\times\vec=(0,0,1)\)。（)9.方程\(x^2+y^2-z^2=1\)表示的曲面是雙葉雙曲面。()10.若\(y=f(x)\)是微分方程\(y'=2x\)的解,則\(f(x)=x^2+C\)（\(C\)為任意常數(shù)）。（）簡答題(每題5分,共4題)1.簡述函數(shù)極限的定義。當(dāng)\(x\)無限趨近于\(x_0\)(或\(x\)趨于無窮)時(shí),函數(shù)\(f(x)\)無限趨近于一個確定的常數(shù)\(A\),則稱\(A\)為\(f(x)\)當(dāng)\(x\)趨近于\(x_0\)(或\(x\)趨于無窮)時(shí)的極限。2.求函數(shù)\(y=x^3-3x^2+5\)的單調(diào)區(qū)間。\(y'=3x^2-6x=3x(x-2)\),令\(y'>0\)得\(x<0\)或\(x>2\),令\(y'<0\)得\(0<x<2\)。所以單調(diào)遞增區(qū)間為\((-\infty,0)\)和\((2,+\infty)\),單調(diào)遞減區(qū)間為\((0,2)\)。3.計(jì)算定積分\(\int_{0}^{1}x^2dx\)。\(\int_{0}^{1}x^2dx=\frac{1}{3}x^3|_{0}^{1}=\frac{1}{3}\)。4.簡述級數(shù)收斂的必要條件。若級數(shù)\(\sum\limits_{n=1}^{\infty}a_n\)收斂,則\(\lim\limits_{n\to\infty}a_n=0\)。討論題(每題5分,共4題)1.討論函數(shù)\(y=\frac{1}{x-1}\)的間斷點(diǎn)類型。\(x=1\)是間斷點(diǎn),\(\lim\limits_{x\to1^+}\frac{1}{x-1}=+\infty\),\(\lim\limits_{x\to1^-}\frac{1}{x-1}=-\infty\),所以\(x=1\)是無窮間斷點(diǎn)。2.討論函數(shù)\(y=x^3\)的凹凸性。\(y'=3x^2\),\(y''=6x\),令\(y''>0\)得\(x>0\),令\(y''<0\)得\(x<0\)。所以在\((-\infty,0)\)上是凸的,在\((0,+\infty)\)上是凹的。3.討論級數(shù)\(\sum\limits_{n=1}^{\infty}\frac{(-1)^{n-1}}{n}\)的斂散性。這是交錯級數(shù),\(a_n=\frac{1}{n}\)單調(diào)遞減且\(\lim\limits_{n\to\infty}a_n=0\),根據(jù)萊布尼茨判別法,