川大高數(shù)期末試題和答案

一、單項選擇題(每題2分,共10題)1.函數(shù)\(y=\sqrt{x-1}\)的定義域是(）A.\(x\geq0\)B.\(x\geq1\)C.\(x\gt1\)D.\(x\gt0\)2.\(\lim_{x\to0}\frac{\sinx}{x}\)的值為()A.0B.1C.\(\infty\)D.不存在3.函數(shù)\(y=x^3\)的導(dǎo)數(shù)\(y'\)是()A.\(x^2\)B.\(3x^2\)C.\(3x\)D.\(x^3\)4.\(\intx^2dx\)等于()A.\(\frac{1}{3}x^3+C\)B.\(3x^3+C\)C.\(\frac{1}{2}x^3+C\)D.\(2x^3+C\)5.曲線\(y=x^2\)在點(diǎn)\((1,1)\)處的切線斜率是()A.1B.2C.3D.46.已知\(f(x)\)的一個原函數(shù)是\(x^2\),則\(f(x)\)=()A.\(2x\)B.\(x^2\)C.\(\frac{1}{2}x^3\)D.\(x\)7.\(\lim_{x\to\infty}(1+\frac{1}{x})^x\)的值是()A.\(e\)B.\(e^2\)C.1D.08.函數(shù)\(y=\lnx\)的導(dǎo)數(shù)是()A.\(\frac{1}{x}\)B.\(x\)C.\(-\frac{1}{x}\)D.\(x^2\)9.若\(f(x)\)在\([a,b]\)上連續(xù),則\(\int_{a}^f(x)dx\)與\(\int_{a}^f(t)dt\)()A.相等B.不相等C.互為相反數(shù)D.不確定10.函數(shù)\(y=\cosx\)的一個周期是()A.\(\frac{\pi}{2}\)B.\(\pi\)C.\(2\pi\)D.\(4\pi\)二、多項選擇題(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.\(y=\cosx\)D.\(y=x+1\)2.極限存在的情況有()A.\(\lim_{x\to0}\frac{1}{x}\)B.\(\lim_{x\to0}\frac{\sinx}{x}\)C.\(\lim_{x\to\infty}(1+\frac{1}{x})^x\)D.\(\lim_{x\to\infty}\frac{x^2+1}{x}\)3.以下求導(dǎo)正確的是()A.\((x^n)^\prime=nx^{n-1}\)B.\((\sinx)^\prime=\cosx\)C.\((\lnx)^\prime=\frac{1}{x}\)D.\((e^x)^\prime=e^x\)4.下列積分運(yùn)算正確的是()A.\(\int1dx=x+C\)B.\(\intx^ndx=\frac{1}{n+1}x^{n+1}+C(n\neq-1)\)C.\(\int\sinxdx=-\cosx+C\)D.\(\inte^xdx=e^x+C\)5.函數(shù)\(y=f(x)\)在\(x_0\)處可導(dǎo)的等價條件有()A.函數(shù)在\(x_0\)處連續(xù)B.左右導(dǎo)數(shù)存在且相等C.函數(shù)在\(x_0\)處有切線D.極限\(\lim_{\Deltax\to0}\frac{f(x_0+\Deltax)-f(x_0)}{\Deltax}\)存在6.以下屬于基本初等函數(shù)的有()A.冪函數(shù)B.指數(shù)函數(shù)C.對數(shù)函數(shù)D.三角函數(shù)7.對于定積分\(\int_{a}^f(x)dx\),下列說法正確的是()A.與積分變量的符號無關(guān)B.當(dāng)\(a=b\)時,\(\int_{a}^f(x)dx=0\)C.\(\int_{a}^f(x)dx=-\int_^{a}f(x)dx\)D.若\(f(x)\geq0\)在\([a,b]\)上,則\(\int_{a}^f(x)dx\geq0\)8.函數(shù)\(y=x^2-2x+3\)的性質(zhì)有(）A.開口向上B.對稱軸為\(x=1\)C.有最小值D.最小值為29.若\(f(x)\)的一個原函數(shù)是\(F(x)\),則()A.\(F^\prime(x)=f(x)\)B.\(\intf(x)dx=F(x)+C\)C.\((\intf(x)dx)^\prime=f(x)\)D.\(\intF^\prime(x)dx=F(x)+C\)10.下列函數(shù)中,在其定義域內(nèi)單調(diào)遞增的有()A.\(y=x\)B.\(y=e^x\)C.\(y=\lnx\)(\(x\gt0\)）D.\(y=x^2\)(\(x\gt0\)）三、判斷題(每題2分,共10題)1.函數(shù)\(y=\frac{1}{x}\)在\(x=0\)處連續(xù)。()2.\(\lim_{x\to\infty}x\sin\frac{1}{x}=1\)。()3.若\(f(x)\)在\(x_0\)處可導(dǎo),則\(f(x)\)在\(x_0\)處一定連續(xù)。（)4.函數(shù)\(y=x^3\)的二階導(dǎo)數(shù)\(y^{\prime\prime}=6x\)。()5.\(\int_{0}^{1}x^2dx=\frac{1}{3}\)。()6.函數(shù)\(y=\cos^2x\)的導(dǎo)數(shù)是\(-\sin2x\)。()7.常數(shù)的導(dǎo)數(shù)為0。()8.若\(f(x)\)為偶函數(shù),則\(\int_{-a}^{a}f(x)dx=0\)。（)9.函數(shù)\(y=\ln(x^2+1)\)的定義域為\(R\)。()10.極限\(\lim_{x\to0}\frac{1-\cosx}{x^2}=\frac{1}{2}\)。(）四、簡答題(每題5分,共4題)1.求函數(shù)\(y=x^3-3x^2+5\)的導(dǎo)數(shù)。答案:根據(jù)求導(dǎo)公式\((x^n)^\prime=nx^{n-1}\),\(y^\prime=3x^2-6x\)。2.計算定積分\(\int_{0}^{2}x^2dx\)。答案:由積分公式\(\intx^ndx=\frac{1}{n+1}x^{n+1}+C(n\neq-1)\),\(\int_{0}^{2}x^2dx=[\frac{1}{3}x^3]_{0}^{2}=\frac{8}{3}\)。3.求函數(shù)\(y=\sin(2x+1)\)的導(dǎo)數(shù)。答案:令\(u=2x+1\),則\(y=\sinu\),根據(jù)復(fù)合函數(shù)求導(dǎo)法則,\(y^\prime=(\sinu)^\prime\cdotu^\prime=2\cos(2x+1)\)。4.已知函數(shù)\(f(x)\)的一個原函數(shù)是\(x^2e^x\),求\(f(x)\)。答案：因為原函數(shù)的導(dǎo)數(shù)就是函數(shù)本身,對\(x^2e^x\)求導(dǎo),\(f(x)=(x^2e^x)^\prime=2xe^x+x^2e^x=e^x(x^2+2x)\)。五、討論題(每題5分,共4題)1.討論函數(shù)\(y=\frac{1}{x-1}\)的單調(diào)性與間斷點(diǎn)。答案:對\(y=\frac{1}{x-1}\)求導(dǎo)得\(y^\prime=-\frac{1}{(x-1)^2}\lt0\),在\((-\infty,1)\)和\((1,+\infty)\)單調(diào)遞減。\(x=1\)時分母為0,是間斷點(diǎn)。2.探討導(dǎo)數(shù)與函數(shù)單調(diào)性之間的關(guān)系。答案:若函數(shù)\(f(x)\)在區(qū)間\(I\)內(nèi)導(dǎo)數(shù)\(f^\prime(x)\gt0\),則\(f(x)\)在\(I\)上單調(diào)遞增;若\(f^\prime(x)\lt0\),則\(f(x)\)在\(I\)上單調(diào)遞減。導(dǎo)數(shù)為0的點(diǎn)可能是函數(shù)單調(diào)性的轉(zhuǎn)折點(diǎn)。3.說明不定積分與定積分的聯(lián)系與區(qū)別。答案:聯(lián)系:定積分可通過不定積分計算。區(qū)別:不定積分是原函數(shù)的集合,結(jié)果含常數(shù)\(C\);定積分是一個數(shù)值,有積分上下限,與積分區(qū)間有關(guān)。4.談?wù)勅绾卫脤?dǎo)數(shù)求函數(shù)的極值。答案:先求函數(shù)\(f(x)\)的導(dǎo)數(shù)\(f^\prime(x)\),令\(f^\prime(x)=0\),求出駐點(diǎn)。再判斷駐點(diǎn)兩側(cè)導(dǎo)數(shù)的符號,若左正右負(fù),駐點(diǎn)為極大值點(diǎn);若左負(fù)右正,駐點(diǎn)為極小值點(diǎn)。答案一、單項選擇題1.B