19數(shù)學(xué)二試題及答案

單項(xiàng)選擇題(每題2分,共10題)1.函數(shù)\(y=x^2\)在\(x=1\)處的導(dǎo)數(shù)是(）A.1B.2C.3D.42.\(\lim\limits_{x\to0}\frac{\sinx}{x}\)的值為()A.0B.1C.-1D.不存在3.曲線\(y=e^x\)的單調(diào)遞增區(qū)間是()A.\((-\infty,+\infty)\)B.\((-\infty,0)\)C.\((0,+\infty)\)D.\((1,+\infty)\)4.函數(shù)\(f(x)=x^3-3x\)的極大值點(diǎn)是()A.-1B.0C.1D.25.\(\intx^2dx\)等于()A.\(\frac{1}{3}x^3+C\)B.\(x^3+C\)C.\(\frac{1}{2}x^2+C\)D.\(2x+C\)6.若\(y=\lnx\),則\(y^\prime\)是()A.\(\frac{1}{x}\)B.\(x\)C.\(-\frac{1}{x}\)D.\(x^2\)7.函數(shù)\(y=\cosx\)的周期是()A.\(\frac{\pi}{2}\)B.\(\pi\)C.\(2\pi\)D.\(4\pi\)8.直線\(y=2x+1\)的斜率是()A.1B.2C.\(\frac{1}{2}\)D.-29.拋物線\(y=x^2\)的焦點(diǎn)坐標(biāo)是()A.\((0,\frac{1}{4})\)B.\((\frac{1}{4},0)\)C.\((0,1)\)D.\((1,0)\)10.已知向量\(\vec{a}=(1,2)\),\(\vec=(2,-1)\),則\(\vec{a}\cdot\vec\)等于()A.0B.1C.2D.3多項(xiàng)選擇題(每題2分,共10題)1.以下哪些是基本初等函數(shù)(）A.冪函數(shù)B.指數(shù)函數(shù)C.對(duì)數(shù)函數(shù)D.三角函數(shù)2.函數(shù)\(y=f(x)\)在點(diǎn)\(x_0\)處可導(dǎo)的充分條件有()A.左導(dǎo)數(shù)存在B.右導(dǎo)數(shù)存在C.左導(dǎo)數(shù)等于右導(dǎo)數(shù)D.函數(shù)在該點(diǎn)連續(xù)3.下列積分計(jì)算正確的是()A.\(\int1dx=x+C\)B.\(\int\cosxdx=\sinx+C\)C.\(\inte^xdx=e^x+C\)D.\(\intxdx=\frac{1}{2}x^2+C\)4.曲線\(y=x^3-3x^2+2\)的駐點(diǎn)有()A.\(x=0\)B.\(x=1\)C.\(x=2\)D.\(x=3\)5.以下哪些函數(shù)是奇函數(shù)()A.\(y=x\)B.\(y=x^3\)C.\(y=\sinx\)D.\(y=\cosx\)6.直線\(Ax+By+C=0\)(\(B\neq0\))的斜截式方程可以寫成()A.\(y=-\frac{A}{B}x-\frac{C}{B}\)B.\(y=kx+b\)（\(k=-\frac{A}{B}\),\(b=-\frac{C}{B}\)）C.\(y=\frac{A}{B}x+\frac{C}{B}\)D.\(y=-\frac{A}{B}x+\frac{C}{B}\)7.橢圓\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)(\(a\gtb\gt0\))的性質(zhì)有()A.長軸長為\(2a\)B.短軸長為\(2b\)C.焦距為\(2c\)(\(c^2=a^2-b^2\))D.離心率\(e=\frac{c}{a}\)8.下列極限存在的是(）A.\(\lim\limits_{x\to0}\frac{1}{x}\)B.\(\lim\limits_{x\to\infty}\frac{1}{x}\)C.\(\lim\limits_{x\to0}x\sin\frac{1}{x}\)D.\(\lim\limits_{x\to\infty}x\sin\frac{1}{x}\)9.函數(shù)\(f(x)\)在區(qū)間\([a,b]\)上滿足羅爾定理的條件有()A.在\([a,b]\)上連續(xù)B.在\((a,b)\)內(nèi)可導(dǎo)C.\(f(a)=f(b)\)D.\(f(a)\neqf(b)\)10.向量\(\vec{a}=(x_1,y_1)\)與向量\(\vec=(x_2,y_2)\)平行的充要條件是()A.\(x_1y_2-x_2y_1=0\)B.\(\vec{a}=\lambda\vec\)（\(\lambda\)為實(shí)數(shù)）C.\(x_1x_2+y_1y_2=0\)D.\(\frac{x_1}{x_2}=\frac{y_1}{y_2}\)（\(x_2\neq0\),\(y_2\neq0\)）判斷題(每題2分,共10題)1.函數(shù)\(y=\frac{1}{x}\)在定義域內(nèi)是單調(diào)遞減函數(shù)。()2.若\(f^\prime(x_0)=0\),則\(x_0\)一定是函數(shù)\(f(x)\)的極值點(diǎn)。（)3.\(\int_{-a}^{a}f(x)dx=0\)(\(f(x)\)為奇函數(shù))。()4.直線\(y=kx+b\)與\(y\)軸的交點(diǎn)坐標(biāo)是\((0,b)\)。()5.函數(shù)\(y=x^2+1\)的圖像是一條拋物線。()6.兩個(gè)向量的數(shù)量積為零,則這兩個(gè)向量一定垂直。()7.函數(shù)\(y=\tanx\)的定義域是\(x\neqk\pi+\frac{\pi}{2}\),\(k\inZ\)。(）8.若\(\lim\limits_{x\tox_0}f(x)\)存在,則\(f(x)\)在\(x_0\)處一定連續(xù)。（)9.橢圓\(\frac{x^2}{4}+\frac{y^2}{9}=1\)的長軸長為4。()10.函數(shù)\(y=e^{-x}\)的導(dǎo)數(shù)是\(y^\prime=e^{-x}\)。(）簡(jiǎn)答題(每題5分,共4題)1.求函數(shù)\(y=x^3-2x+1\)的導(dǎo)數(shù)。答案:根據(jù)求導(dǎo)公式\((x^n)^\prime=nx^{n-1}\),\(y^\prime=3x^2-2\)。2.計(jì)算\(\int(2x+3)dx\)。答案:由積分公式\(\intx^ndx=\frac{1}{n+1}x^{n+1}+C\),\(\int(2x+3)dx=\int2xdx+\int3dx=x^2+3x+C\)。3.求曲線\(y=x^2\)在點(diǎn)\((1,1)\)處的切線方程。答案:先求導(dǎo)\(y^\prime=2x\),在點(diǎn)\((1,1)\)處切線斜率\(k=2\times1=2\),由點(diǎn)斜式得切線方程\(y-1=2(x-1)\),即\(y=2x-1\)。4.已知向量\(\vec{a}=(3,4)\),\(\vec=(-1,2)\),求\(\vec{a}+2\vec\)。答案：\(2\vec=2(-1,2)=(-2,4)\),則\(\vec{a}+2\vec=(3-2,4+4)=(1,8)\)。討論題(每題5分,共4題)1.討論函數(shù)\(y=x^2-4x+3\)的單調(diào)性。答案:先求導(dǎo)\(y^\prime=2x-4\),令\(y^\prime=0\),得\(x=2\)。當(dāng)\(x\lt2\)時(shí),\(y^\prime\lt0\),函數(shù)遞減;當(dāng)\(x\gt2\)時(shí),\(y^\prime\gt0\),函數(shù)遞增。2.闡述定積分的幾何意義。答案:定積分\(\int_{a}^f(x)dx\),當(dāng)\(f(x)\geq0\)時(shí),表示由曲線\(y=f(x)\),直線\(x=a\),\(x=b\)及\(x\)軸圍成的曲邊梯形面積;當(dāng)\(f(x)\)有正有負(fù)時(shí),是\(x\)軸上方圖形面積減去\(x\)軸下方圖形面積。3.如何判斷直線與圓的位置關(guān)系?答案：設(shè)圓的方程為\((x-a)^2+(y-b)^2=r^2\),直線方程為\(Ax+By+C=0\)。計(jì)算圓心\((a,b)\)到直線的距離\(d=\frac{\vertAa+Bb+C\vert}{\sqrt{A^2+B^2}}\),\(d\gtr\)時(shí),直線與圓相離；\(d=r\)時(shí),相切；\(d\ltr\)時(shí),相交。4.舉例說明函數(shù)的極值與最值的區(qū)別。答案:例如\(y=x^3-3x\),在\(x=-1\)處取得極大值2,在\(x=1\)處取得極小值-2。但在整個(gè)定義域\((-\infty,+\infty)\)內(nèi)無最大值和最小值。極值是局部概念,最值是整體概念。