高中微積分考試題及答案

一、單項(xiàng)選擇題1.函數(shù)\(y=x^{2}\)在區(qū)間\([1,2]\)上的平均變化率為（）A.\(4\)B.\(3\)C.\(2\)D.\(1\)答案：B2.曲線\(y=x^{3}\)在點(diǎn)\((1,1)\)處的切線斜率為（）A.\(1\)B.\(2\)C.\(3\)D.\(4\)答案：C3.若\(f(x)=\sinx\)，則\(f^\prime(x)\)等于（）A.\(\cosx\)B.\(-\cosx\)C.\(\sinx\)D.\(-\sinx\)答案：A4.已知\(f(x)=x^{n}\)，且\(f^\prime(1)=2\)，則\(n\)的值為（）A.\(1\)B.\(2\)C.\(3\)D.\(4\)答案：B5.函數(shù)\(f(x)=x^{3}-3x^{2}+2\)在區(qū)間\([-1,1]\)上的最大值為（）A.\(0\)B.\(2\)C.\(-2\)D.\(4\)答案：B6.若\(\int_{0}^{a}x^{2}dx=9\)，則\(a\)的值為（）A.\(1\)B.\(2\)C.\(3\)D.\(4\)答案：C7.定積分\(\int_{-1}^{1}(x^{3}+\sinx)dx\)的值為（）A.\(0\)B.\(2\)C.\(-2\)D.\(1\)答案：A8.曲線\(y=x^{2}\)與直線\(y=x\)所圍成的圖形面積為（）A.\(\frac{1}{6}\)B.\(\frac{1}{3}\)C.\(\frac{1}{2}\)D.\(1\)答案：A9.已知函數(shù)\(f(x)\)的導(dǎo)函數(shù)\(f^\prime(x)=2x-9\)，且\(f(0)\)的值為整數(shù)，當(dāng)\(x\in[n,n+1](n\inN^{})\)時(shí)，\(f(x)\)是單調(diào)遞減函數(shù)，則\(n\)的值為（）A.\(2\)B.\(3\)C.\(4\)D.\(5\)答案：B10.若函數(shù)\(f(x)=x^{3}-ax^{2}+3x+1\)在區(qū)間\((\frac{1}{2},1)\)上單調(diào)遞減，則實(shí)數(shù)\(a\)的取值范圍為（）A.\([\frac{15}{4},+\infty)\)B.\((3,+\infty)\)C.\((\frac{9}{2},+\infty)\)D.\((0,3)\)答案：A二、多項(xiàng)選擇題1.下列求導(dǎo)正確的是（）A.\((x^{3})^\prime=3x^{2}\)B.\((\lnx)^\prime=\frac{1}{x}\)C.\((\cosx)^\prime=\sinx\)D.\((e^{x})^\prime=e^{x}\)答案：ABD2.對(duì)于函數(shù)\(y=f(x)\)，下列說(shuō)法正確的是（）A.函數(shù)的極大值一定大于極小值B.函數(shù)的極值點(diǎn)處導(dǎo)數(shù)為\(0\)C.函數(shù)的導(dǎo)數(shù)為\(0\)的點(diǎn)不一定是極值點(diǎn)D.函數(shù)在某區(qū)間上單調(diào)遞增，則其導(dǎo)數(shù)大于\(0\)答案：BC3.下列積分值為\(0\)的是（）A.\(\int_{-1}^{1}x^{5}dx\)B.\(\int_{-1}^{1}x^{4}dx\)C.\(\int_{-1}^{1}\sinxdx\)D.\(\int_{-1}^{1}(x^{3}+\cosx)dx\)答案：AC4.已知函數(shù)\(f(x)\)的導(dǎo)數(shù)\(f^\prime(x)\)的圖象如圖所示（）A.函數(shù)\(f(x)\)在\(x=1\)處取得極大值B.函數(shù)\(f(x)\)在\(x=3\)處取得極小值C.函數(shù)\(f(x)\)在區(qū)間\((1,3)\)上單調(diào)遞減D.函數(shù)\(f(x)\)在區(qū)間\((3,5)\)上單調(diào)遞增答案：ABCD5.設(shè)函數(shù)\(f(x)=x^{3}+ax^{2}+bx+c\)，以下說(shuō)法正確的是（）A.\(f(x)\)可能沒(méi)有極值B.若\(f(x)\)有極值，則\(a^{2}-3b\gt0\)C.\(f(x)\)的圖象可能與\(x\)軸有三個(gè)交點(diǎn)D.\(f(x)\)的圖象一定與\(x\)軸有交點(diǎn)答案：ABCD6.已知函數(shù)\(f(x)\)滿足\(f(x)=f^\prime(1)e^{x-1}-f(0)x+\frac{1}{2}x^{2}\)，則（）A.\(f(0)=1\)B.\(f^\prime(1)=e\)C.\(f(x)\)在\((1,+\infty)\)上單調(diào)遞增D.\(f(x)\)的最小值為\(1\)答案：ABCD7.下列關(guān)于定積分的幾何意義說(shuō)法正確的是（）A.\(\int_{a}^f(x)dx\)表示由直線\(x=a\)，\(x=b\)，\(y=0\)和曲線\(y=f(x)\)所圍成的曲邊梯形的面積B.若\(f(x)\)在\([a,b]\)上有正有負(fù)，則\(\int_{a}^f(x)dx\)表示\(x\)軸上方的曲邊梯形面積減去\(x\)軸下方的曲邊梯形面積C.\(\int_{a}^|f(x)|dx\)表示由直線\(x=a\)，\(x=b\)，\(y=0\)和曲線\(y=|f(x)|\)所圍成的曲邊梯形的面積D.\(\int_{a}^f(x)dx\)可能為負(fù)答案：BCD8.已知函數(shù)\(f(x)=x^{3}-3x\)，則（）A.\(f(x)\)的極大值為\(2\)B.\(f(x)\)的極小值為\(-2\)C.\(f(x)\)在\((-\infty,-1)\)上單調(diào)遞增D.\(f(x)\)在\((1,+\infty)\)上單調(diào)遞增答案：ABCD9.若函數(shù)\(f(x)\)在\(R\)上可導(dǎo)，且\(f(x)=x^{2}+2f^\prime(2)x+m\)（\(m\inR\)），則（）A.\(f^\prime(0)=-4\)B.\(f(0)=m\)C.\(f(x)\)的對(duì)稱軸為\(x=2\)D.\(f(x)\)在\((-\infty,2)\)上單調(diào)遞減答案：ABC10.已知函數(shù)\(f(x)=\frac{1}{3}x^{3}-ax^{2}+(a^{2}-1)x+b\)（\(a,b\inR\)），則（）A.函數(shù)\(f(x)\)的圖象一定經(jīng)過(guò)原點(diǎn)B.當(dāng)\(a=0\)時(shí)，函數(shù)\(f(x)\)在\(R\)上單調(diào)遞增C.函數(shù)\(f(x)\)可能存在兩個(gè)極值點(diǎn)D.若\(f(x)\)在區(qū)間\((1,2)\)上單調(diào)遞增，則\(a\leqslant1\)或\(a\geqslant3\)答案：BCD三、判斷題1.函數(shù)\(y=f(x)\)在\(x=x_{0}\)處的導(dǎo)數(shù)\(f^\prime(x_{0})\)的幾何意義是曲線\(y=f(x)\)在點(diǎn)\((x_{0},f(x_{0}))\)處的切線斜率。（）答案：對(duì)2.若函數(shù)\(y=f(x)\)在區(qū)間\((a,b)\)內(nèi)有\(zhòng)(f^\prime(x)\gt0\)，則\(y=f(x)\)在區(qū)間\((a,b)\)上單調(diào)遞增。（）答案：對(duì)3.函數(shù)的極大值一定比極小值大。（）答案：錯(cuò)4.若\(f^\prime(x_{0})=0\)，則\(x_{0}\)是函數(shù)\(y=f(x)\)的極值點(diǎn)。（）答案：錯(cuò)5.定積分\(\int_{a}^f(x)dx\)的值一定是一個(gè)正數(shù)。（）答案：錯(cuò)6.若\(f(x)\)是偶函數(shù)，則\(\int_{-a}^{a}f(x)dx=2\int_{0}^{a}f(x)dx\)。（）答案：對(duì)7.曲線\(y=f(x)\)與直線\(x=a\)，\(x=b\)，\(y=0\)所圍成的圖形面積\(S=\int_{a}^|f(x)|dx\)。（）答案：對(duì)8.函數(shù)\(f(x)\)在某區(qū)間上的最大值一定是該區(qū)間上的極大值。（）答案：錯(cuò)9.若函數(shù)\(f(x)\)在區(qū)間\([a,b]\)上單調(diào)遞增，則\(\int_{a}^f(x)dx\gt0\)。（）答案：錯(cuò)10.若\(f^\prime(x)\)在區(qū)間\((a,b)\)上恒大于\(0\)，則\(f(x)\)在\((a,b)\)上的圖象是下凸的。（）答案：對(duì)四、簡(jiǎn)答題1.求函數(shù)\(y=x^{3}-3x^{2}-9x+5\)的單調(diào)區(qū)間和極值。對(duì)函數(shù)\(y=x^{3}-3x^{2}-9x+5\)求導(dǎo)得\(y^\prime=3x^{2}-6x-9=3(x^{2}-2x-3)=3(x-3)(x+1)\)。令\(y^\prime\gt0\)，即\(3(x-3)(x+1)\gt0\)，解得\(x\lt-1\)或\(x\gt3\)，此時(shí)函數(shù)單調(diào)遞增；令\(y^\prime\lt0\)，即\(3(x-3)(x+1)\lt0\)，解得\(-1\ltx\lt3\)，此時(shí)函數(shù)單調(diào)遞減。當(dāng)\(x=-1\)時(shí)，\(y=(-1)^{3}-3\times(-1)^{2}-9\times(-1)+5=10\)，為極大值；當(dāng)\(x=3\)時(shí)，\(y=3^{3}-3\times3^{2}-9\times3+5=-22\)，為極小值。2.計(jì)算定積分\(\int_{0}^{2}(x^{2}+1)dx\)。根據(jù)定積分的運(yùn)算法則\(\int_{0}^{2}(x^{2}+1)dx=\int_{0}^{2}x^{2}dx+\int_{0}^{2}1dx\)。由定積分公式\(\intx^{n}dx=\frac{1}{n+1}x^{n+1}+C(n\neq-1)\)可得\(\int_{0}^{2}x^{2}dx=\left[\frac{1}{3}x^{3}\right]_{0}^{2}=\frac{8}{3}\)，\(\int_{0}^{2}1dx=\left[x\right]_{0}^{2}=2\)。所以\(\int_{0}^{2}(x^{2}+1)dx=\frac{8}{3}+2=\frac{14}{3}\)。3.已知函數(shù)\(f(x)=x^{3}+ax^{2}+bx+c\)在\(x=1\)處取得極值，且其圖象在\(x=0\)處的切線斜率為\(-3\)。求\(a\)，\(b\)的值。首先對(duì)\(f(x)\)求導(dǎo)得\(f^\prime(x)=3x^{2}+2ax+b\)。因?yàn)楹瘮?shù)在\(x=1\)處取得極值，所以\(f^\prime(1)=3+2a+b=0\)。又因?yàn)楹瘮?shù)圖象在\(x=0\)處的切線斜率為\(-3\)，而切線斜率就是函數(shù)在該點(diǎn)的導(dǎo)數(shù)值，所以\(f^\prime(0)=b=-3\)。將\(b=-3\)代入\(3+2a+b=0\)，可得\(3+2a-3=0\)，解得\(a=0\)。綜上，\(a=0\)，\(b=-3\)。4.簡(jiǎn)述定積分與不定積分的聯(lián)系與區(qū)別。聯(lián)系：若\(F^\prime(x)=f(x)\)，則\(\int_{a}^f(x)dx=F(b)-F(a)\)，不定積分\(\intf(x)dx=F(x)+C\)，定積分的值可通過(guò)不定積分求出原函數(shù)后計(jì)算得到。區(qū)別：不定積分是一個(gè)函數(shù)族，它表示的是所有原函數(shù)；而定積分是一個(gè)數(shù)值，它表示的是由曲線、直線圍成的曲邊梯形面積的代數(shù)和。不定積分側(cè)重于求原函數(shù)，定積分側(cè)重于計(jì)算特定區(qū)間上的數(shù)值。五、討論題1.討論函數(shù)\(f(x)=x^{3}-ax\)的單調(diào)性與極值情況，其中\(zhòng)(a\)為實(shí)數(shù)。對(duì)\(f(x)=x^{3}-ax\)求導(dǎo)得\(f^\prime(x)=3x^{2}-a\)。當(dāng)\(a\leqslant0\)時(shí)，\(f^\prime(x)=3x^{2}-a\geqslant0\)恒成立，\(f(x)\)在\(R\)上單調(diào)遞增，無(wú)極值。當(dāng)\(a\gt0\)時(shí)，令\(f^\prime(x)=0\)，即\(3x^{2}-a=0\)，解得\(x=\pm\sqrt{\frac{a}{3}}\)。當(dāng)\(x\lt-\sqrt{\frac{a}{3}}\)或\(x\gt\sqrt{\frac{a}{3}}\)時(shí)，\(f^\prime(x)\gt0\)，\(f(x)\)單調(diào)遞增；當(dāng)\(-\sqrt{\frac{a}{3}}\ltx\lt\sqrt{\frac{a}{3}}\)時(shí)，\(f^\prime(x)\lt0\)，\(f(x)\)單調(diào)遞減。所以\(f(x)\)在\(x=-\sqrt{\frac{a}{3}}\)處取得極大值\(f(-\sqrt{\frac{a}{3}})=\frac{2a}{3}\sqrt{\frac{a}{3}}\)；在\(x=\sqrt{\frac{a}{3}}\)處取得極小值\(f(\sqrt{