世界最難高數(shù)題目及答案

一、單項(xiàng)選擇題（每題2分，共10題）1.函數(shù)\(y=\frac{1}{x-1}\)的間斷點(diǎn)是（）A.\(x=0\)B.\(x=1\)C.\(x=-1\)D.無(wú)間斷點(diǎn)2.極限\(\lim_{x\to0}\frac{\sinx}{x}\)的值是（）A.0B.1C.不存在D.\(\infty\)3.函數(shù)\(y=x^2\)在點(diǎn)\(x=1\)處的導(dǎo)數(shù)是（）A.1B.2C.3D.44.不定積分\(\intxdx\)等于（）A.\(\frac{1}{2}x^2+C\)B.\(x^2+C\)C.\(\frac{1}{3}x^3+C\)D.\(x+C\)5.定積分\(\int_{0}^{1}x^2dx\)的值是（）A.\(\frac{1}{2}\)B.\(\frac{1}{3}\)C.\(\frac{1}{4}\)D.\(\frac{1}{5}\)6.向量\(\vec{a}=(1,2)\)與向量\(\vec=(2,-1)\)的關(guān)系是（）A.平行B.垂直C.夾角為\(45^{\circ}\)D.夾角為\(60^{\circ}\)7.函數(shù)\(z=x^2+y^2\)在點(diǎn)\((1,1)\)處的全微分\(dz\)是（）A.\(2dx+2dy\)B.\(dx+dy\)C.\(4dx+4dy\)D.\(0\)8.級(jí)數(shù)\(\sum_{n=1}^{\infty}\frac{1}{n}\)是（）A.收斂B.發(fā)散C.條件收斂D.絕對(duì)收斂9.曲線\(y=e^x\)在點(diǎn)\((0,1)\)處的切線方程是（）A.\(y=x+1\)B.\(y=x-1\)C.\(y=-x+1\)D.\(y=-x-1\)10.設(shè)\(f(x)\)的一個(gè)原函數(shù)是\(x^2\)，則\(f(x)\)等于（）A.\(2x\)B.\(x^2\)C.\(\frac{1}{2}x^3\)D.\(x\)二、多項(xiàng)選擇題（每題2分，共10題）1.下列函數(shù)中，在\(x=0\)處連續(xù)的有（）A.\(y=|x|\)B.\(y=\frac{1}{x}\)C.\(y=x^2\)D.\(y=\sinx\)2.下列求導(dǎo)公式正確的有（）A.\((x^n)^\prime=nx^{n-1}\)B.\((\sinx)^\prime=\cosx\)C.\((\lnx)^\prime=\frac{1}{x}\)D.\((e^x)^\prime=e^x\)3.下列積分計(jì)算正確的有（）A.\(\int_{0}^{2\pi}\sinxdx=0\)B.\(\int_{0}^{1}e^xdx=e-1\)C.\(\intx^3dx=\frac{1}{4}x^4+C\)D.\(\int\frac{1}{x^2}dx=-\frac{1}{x}+C\)4.向量\(\vec{a}=(1,-1,2)\)與下列向量垂直的有（）A.\(\vec=(1,1,0)\)B.\(\vec{c}=(2,2,-1)\)C.\(\vecwmaekug=(0,0,0)\)D.\(\vec{e}=(-2,2,2)\)5.多元函數(shù)\(z=f(x,y)\)在點(diǎn)\((x_0,y_0)\)處可微的充分條件有（）A.\(z=f(x,y)\)在點(diǎn)\((x_0,y_0)\)處偏導(dǎo)數(shù)連續(xù)B.\(z=f(x,y)\)在點(diǎn)\((x_0,y_0)\)處兩個(gè)偏導(dǎo)數(shù)都存在C.\(\lim_{\Deltax\to0,\Deltay\to0}\frac{\Deltaz-f_x(x_0,y_0)\Deltax-f_y(x_0,y_0)\Deltay}{\sqrt{(\Deltax)^2+(\Deltay)^2}}=0\)D.\(z=f(x,y)\)在點(diǎn)\((x_0,y_0)\)處連續(xù)6.下列級(jí)數(shù)收斂的有（）A.\(\sum_{n=1}^{\infty}\frac{1}{n^2}\)B.\(\sum_{n=1}^{\infty}\frac{(-1)^n}{n}\)C.\(\sum_{n=1}^{\infty}\frac{1}{n!}\)D.\(\sum_{n=1}^{\infty}n\)7.曲線\(y=x^3-3x\)的極值點(diǎn)有（）A.\(x=-1\)B.\(x=0\)C.\(x=1\)D.\(x=2\)8.設(shè)\(f(x)\)在區(qū)間\([a,b]\)上可積，則下列說(shuō)法正確的有（）A.\(\int_{a}^f(x)dx=-\int_^{a}f(x)dx\)B.\(\int_{a}^[f(x)+g(x)]dx=\int_{a}^f(x)dx+\int_{a}^g(x)dx\)C.\(\int_{a}^kf(x)dx=k\int_{a}^f(x)dx\)（\(k\)為常數(shù)）D.\(\int_{a}^f(x)dx=\int_{a}^{c}f(x)dx+\int_{c}^f(x)dx\)（\(a<c<b\)）9.二元函數(shù)\(z=x^2+y^2-2x+4y\)的駐點(diǎn)有（）A.\((1,-2)\)B.\((-1,2)\)C.\((1,2)\)D.\((-1,-2)\)10.下列關(guān)于函數(shù)\(y=\cosx\)的性質(zhì)正確的有（）A.周期是\(2\pi\)B.是偶函數(shù)C.值域是\([-1,1]\)D.在\([0,\pi]\)上單調(diào)遞減三、判斷題（每題2分，共10題）1.函數(shù)\(y=\sqrt{x}\)的定義域是\(x\geq0\)。（）2.若\(f^\prime(x_0)=0\)，則\(x_0\)一定是\(f(x)\)的極值點(diǎn)。（）3.定積分\(\int_{a}^f(x)dx\)的值與積分變量用什么字母表示無(wú)關(guān)。（）4.向量\(\vec{a}=(1,2)\)與向量\(\vec=(2,4)\)平行。（）5.函數(shù)\(z=\ln(x+y)\)的定義域是\(x+y>0\)。（）6.級(jí)數(shù)\(\sum_{n=1}^{\infty}a_n\)收斂，則\(\lim_{n\to\infty}a_n=0\)。（）7.函數(shù)\(y=x^3\)在\(R\)上是單調(diào)遞增的。（）8.若\(f(x)\)在\([a,b]\)上連續(xù)，則\(f(x)\)在\([a,b]\)上一定可積。（）9.函數(shù)\(z=x^2+y^2\)的圖形是一個(gè)拋物面。（）10.兩個(gè)非零向量的點(diǎn)積為\(0\)，則這兩個(gè)向量垂直。（）四、簡(jiǎn)答題（每題5分，共4題）1.求函數(shù)\(y=\frac{1}{x^2-4}\)的定義域。答案：要使函數(shù)有意義，則\(x^2-4\neq0\)，即\((x+2)(x-2)\neq0\)，解得\(x\neq\pm2\)，定義域?yàn)閈(x\in(-\infty,-2)\cup(-2,2)\cup(2,+\infty)\)。2.求函數(shù)\(y=x^3-3x^2+2\)的導(dǎo)數(shù)。答案：根據(jù)求導(dǎo)公式\((x^n)^\prime=nx^{n-1}\)，\(y^\prime=3x^2-6x\)。3.計(jì)算定積分\(\int_{0}^{1}(x^2+1)dx\)。答案：\(\int_{0}^{1}(x^2+1)dx=(\frac{1}{3}x^3+x)\big|_{0}^{1}=(\frac{1}{3}×1^3+1)-(\frac{1}{3}×0^3+0)=\frac{4}{3}\)。4.求函數(shù)\(z=xy\)在點(diǎn)\((1,2)\)處的偏導(dǎo)數(shù)\(z_x\)和\(z_y\)。答案：\(z_x=y\)，將\((1,2)\)代入得\(z_x(1,2)=2\)；\(z_y=x\)，將\((1,2)\)代入得\(z_y(1,2)=1\)。五、討論題（每題5分，共4題）1.討論函數(shù)\(y=\frac{1}{x}\)在\((0,+\infty)\)上的單調(diào)性。答案：對(duì)\(y=\frac{1}{x}=x^{-1}\)求導(dǎo)，\(y^\prime=-x^{-2}=-\frac{1}{x^2}\)。在\((0,+\infty)\)上，\(y^\prime=-\frac{1}{x^2}<0\)，所以函數(shù)\(y=\frac{1}{x}\)在\((0,+\infty)\)上單調(diào)遞減。2.討論級(jí)數(shù)\(\sum_{n=1}^{\infty}\frac{1}{n^p}\)（\(p\)為常數(shù)）的斂散性。答案：當(dāng)\(p>1\)時(shí)，級(jí)數(shù)\(\sum_{n=1}^{\infty}\frac{1}{n^p}\)收斂；當(dāng)\(p\leq1\)時(shí)，級(jí)數(shù)\(\sum_{n=1}^{\infty}\frac{1}{n^p}\)發(fā)散?？赏ㄟ^(guò)積分判別法等證明。3.討論二元函數(shù)\(z=x^2+y^2\)的幾何意義及最值情況。答案：幾何意義是開(kāi)口向上的旋轉(zhuǎn)拋物面。因?yàn)閈(x^2\geq0\)，\(y^2\geq0\)，所以\(z=x^2+y^2\geq0\)，當(dāng)\(x=y=0\)時(shí)，\(z\)取得最小值\(0\)，無(wú)最大值。4.討論函數(shù)\(y=\sinx\)與\(y=\cosx\)在\([0,2\pi]\)上的交點(diǎn)情況。答案：令\(\sinx=\cosx\)，即\(\tanx=1\)，在\([0,2\pi]\)上，\(x=\frac{\pi}{4}\)或\(x=\frac{5\pi}{4}\)，所以兩函數(shù)在\([0,2\pi]\)上的交點(diǎn)為\((\frac{\pi}{4},\frac{\sqrt{2}}{2})\)和\((\frac{5\pi}{4},-\frac{\