大一期末高數(shù)試卷及答案一、選擇題（每題5分，共20分）1.函數(shù)f(x)=x^2+1的導(dǎo)數(shù)為：A.2x+1B.2xC.x^2D.12.定積分∫(0到1)x^2dx的值為：A.1/3B.1/2C.2/3D.13.極限lim(x→0)(sin(x)/x)的值為：A.0B.1C.πD.∞4.以下哪個(gè)函數(shù)是偶函數(shù)？A.f(x)=x^3B.f(x)=x^2C.f(x)=x^2+xD.f(x)=x^2-x二、填空題（每題5分，共20分）5.函數(shù)f(x)=3x-2的反函數(shù)為_(kāi)_______。6.函數(shù)f(x)=x^3-3x+2的極值點(diǎn)為_(kāi)_______。7.曲線y=x^2+1在點(diǎn)(1,2)處的切線斜率為_(kāi)_______。8.函數(shù)f(x)=e^x的不定積分為_(kāi)_______。三、計(jì)算題（每題15分，共30分）9.計(jì)算極限lim(x→∞)((x^2+3x+2)/(x^2-x+1))。10.計(jì)算定積分∫(0到π)sin(x)dx。四、證明題（每題15分，共30分）11.證明函數(shù)f(x)=x^3在(-∞,+∞)上是增函數(shù)。12.證明函數(shù)f(x)=x^2+2x+1的圖像關(guān)于直線x=-1對(duì)稱。五、解答題（每題15分，共30分）13.已知函數(shù)f(x)=x^2-4x+3，求：(1)函數(shù)的極值點(diǎn)；(2)函數(shù)的單調(diào)區(qū)間；(3)函數(shù)的凹凸區(qū)間。14.已知函數(shù)f(x)=ln(x)，求：(1)函數(shù)的導(dǎo)數(shù)；(2)函數(shù)的極值點(diǎn)；(3)函數(shù)的拐點(diǎn)。答案一、選擇題1.B2.A3.B4.B二、填空題5.f^(-1)(x)=(x+2)/36.x=0,x=±√37.28.e^x+C三、計(jì)算題9.極限lim(x→∞)((x^2+3x+2)/(x^2-x+1))=110.定積分∫(0到π)sin(x)dx=2四、證明題11.證明：函數(shù)f(x)=x^3的導(dǎo)數(shù)為f'(x)=3x^2，因?yàn)閷?duì)于所有x∈(-∞,+∞)，f'(x)>0，所以函數(shù)f(x)在(-∞,+∞)上是增函數(shù)。12.證明：函數(shù)f(x)=x^2+2x+1可以重寫為f(x)=(x+1)^2，因?yàn)閷?duì)于所有x，f(-2-x)=f(-2+x)，所以函數(shù)f(x)的圖像關(guān)于直線x=-1對(duì)稱。五、解答題13.(1)函數(shù)f(x)=x^2-4x+3的導(dǎo)數(shù)為f'(x)=2x-4，令f'(x)=0，解得x=2，所以極值點(diǎn)為x=2。(2)當(dāng)x<2時(shí)，f'(x)<0，函數(shù)單調(diào)遞減；當(dāng)x>2時(shí)，f'(x)>0，函數(shù)單調(diào)遞增。(3)函數(shù)f(x)的二階導(dǎo)數(shù)為f''(x)=2，因?yàn)閒''(x)>0，所以函數(shù)在整個(gè)定義域內(nèi)都是凸函數(shù)。14.(1)函數(shù)f(x)=ln(x)的導(dǎo)數(shù)為f'(x)=1/x。(2)函數(shù)f(x)的極值點(diǎn)為x=1，因?yàn)閒'(x)在x=1處