c級高數(shù)期末考試試題及答案

c級高數(shù)期末考試試題及答案

一、單項選擇題（每題2分，共10題）1.函數(shù)\(y=\sinx\)的導(dǎo)數(shù)是（）A.\(\cosx\)B.\(-\cosx\)C.\(\sinx\)D.\(-\sinx\)答案：A2.\(\intxdx=\)（）A.\(\frac{1}{2}x^{2}+C\)B.\(x^{2}+C\)C.\(\frac{1}{3}x^{3}+C\)D.\(2x+C\)答案：A3.極限\(\lim\limits_{x\rightarrow0}\frac{\sinx}{x}=\)（）A.0B.1C.\(\infty\)D.不存在答案：B4.函數(shù)\(y=x^{2}\)在\(x=1\)處的切線斜率是（）A.1B.2C.\(-1\)D.-2答案：B5.下列函數(shù)中是奇函數(shù)的是（）A.\(y=x^{2}\)B.\(y=\sinx\)C.\(y=e^{x}\)D.\(y=\lnx\)答案：B6.\(\lim\limits_{x\rightarrow\infty}(1+\frac{1}{x})^{x}=\)（）A.\(e\)B.1C.\(\infty\)D.0答案：A7.函數(shù)\(y=\ln(x+1)\)的定義域是（）A.\((-1,+\infty)\)B.\((-\infty,-1)\)C.\((0,+\infty)\)D.\((-\infty,0)\)答案：A8.若\(y=f(x)\)在\(x=a\)處可導(dǎo)，則\(\lim\limits_{h\rightarrow0}\frac{f(a+h)-f(a-h)}{2h}=\)（）A.\(f'(a)\)B.\(2f'(a)\)C.\(\frac{1}{2}f'(a)\)D.0答案：A9.\(\int\frac{1}{x}dx=\)（）A.\(\ln|x|+C\)B.\(\frac{1}{x^{2}}+C\)C.\(e^{x}+C\)D.\(\sinx+C\)答案：A10.函數(shù)\(y=3x^{3}-2x^{2}+x-1\)的二階導(dǎo)數(shù)\(y''=\)（）A.\(18x-4\)B.\(9x^{2}-4x+1\)C.\(6x-4\)D.\(3x^{2}-2x+1\)答案：A二、多項選擇題（每題2分，共10題）1.下列函數(shù)在\((0,+\infty)\)上單調(diào)遞增的是（）A.\(y=x^{2}\)B.\(y=\lnx\)C.\(y=e^{x}\)D.\(y=\sinx\)答案：ABC2.下列積分正確的是（）A.\(\inte^{x}dx=e^{x}+C\)B.\(\int\cosxdx=\sinx+C\)C.\(\int\frac{1}{\sqrt{1-x^{2}}}dx=\arcsinx+C\)D.\(\intx^{3}dx=\frac{1}{4}x^{4}+C\)答案：ABCD3.下列關(guān)于函數(shù)極限的說法正確的是（）A.\(\lim\limits_{x\rightarrowa}f(x)\)存在當(dāng)且僅當(dāng)\(\lim\limits_{x\rightarrowa^{+}}f(x)=\lim\limits_{x\rightarrowa^{-}}f(x)\)B.\(\lim\limits_{x\rightarrow\infty}\frac{1}{x}=0\)C.\(\lim\limits_{x\rightarrow0}\frac{1}{x}=\infty\)D.若\(\lim\limits_{x\rightarrowa}f(x)=A\)，\(\lim\limits_{x\rightarrowa}g(x)=B\)，則\(\lim\limits_{x\rightarrowa}(f(x)+g(x))=A+B\)答案：ABD4.函數(shù)\(y=f(x)\)的極值點可能是（）A.駐點B.不可導(dǎo)點C.端點D.任意點答案：AB5.下列函數(shù)是偶函數(shù)的有（）A.\(y=x^{2}\cosx\)B.\(y=e^{x}+e^{-x}\)C.\(y=\ln|x|\)D.\(y=\frac{1}{x}\)答案：AB6.對于定積分\(\int_{a}^f(x)dx\)，以下說法正確的是（）A.它表示的是由\(y=f(x)\)，\(x=a\)，\(x=b\)以及\(x\)軸所圍成的曲邊梯形的面積B.當(dāng)\(f(x)\geqslant0\)在\([a,b]\)上時，它表示的是由\(y=f(x)\)，\(x=a\)，\(x=b\)以及\(x\)軸所圍成的曲邊梯形的面積C.\(\int_{a}^f(x)dx=-\int_^{a}f(x)dx\)D.\(\int_{a}^{a}f(x)dx=0\)答案：BCD7.若\(y=\sin(2x)\)，則\(y'\)為（）A.\(2\cos(2x)\)B.\(\cos(2x)\)C.\(-2\cos(2x)\)D.\(-\cos(2x)\)答案：A8.下列等式成立的是（）A.\(\ln(a\cdotb)=\lna+\lnb\)B.\(\ln\frac{a}=\lna-\lnb\)C.\(\lna^{n}=n\lna\)D.\(\lne=1\)答案：ABCD9.設(shè)\(y=f(u)\)，\(u=g(x)\)，則復(fù)合函數(shù)\(y=f(g(x))\)的導(dǎo)數(shù)\(y'=\)（）A.\(f'(u)g'(x)\)B.\(f'(g(x))g'(x)\)C.\(f(g'(x))\)D.\(f'(x)g(x)\)答案：AB10.以下函數(shù)在\(x=0\)處連續(xù)的是（）A.\(y=\frac{\sinx}{x}\)B.\(y=x^{2}\)C.\(y=\left\{\begin{array}{ll}x+1,&x\geqslant0\\x-1,&x<0\end{array}\right.\)D.\(y=e^{x}\)答案：BD三、判斷題（每題2分，共10題）1.函數(shù)\(y=\frac{1}{x}\)在\((0,+\infty)\)上單調(diào)遞減。（）答案：正確2.若\(f(x)\)在\(x=a\)處可導(dǎo)，則\(f(x)\)在\(x=a\)處連續(xù)。（）答案：正確3.\(\int_{0}^{1}x^{2}dx=\frac{1}{3}\)。（）答案：正確4.函數(shù)\(y=x^{3}\)是奇函數(shù)。（）答案：正確5.極限\(\lim\limits_{x\rightarrow0}\frac{\cosx-1}{x}=0\)。（）答案：正確6.函數(shù)\(y=\lnx\)的導(dǎo)數(shù)是\(\frac{1}{x}\)，其定義域為\((0,+\infty)\)。（）答案：正確7.若\(f(x)\)和\(g(x)\)在\(x=a\)處都不可導(dǎo)，則\(f(x)+g(x)\)在\(x=a\)處也不可導(dǎo)。（）答案：錯誤8.\(\inte^{-x}dx=-e^{-x}+C\)。（）答案：正確9.函數(shù)\(y=\sqrt{x}\)在\([0,+\infty)\)上單調(diào)遞增。（）答案：正確10.\(\lim\limits_{x\rightarrow1}\frac{x^{2}-1}{x-1}=2\)。（）答案：正確四、簡答題（每題5分，共4題）1.求函數(shù)\(y=x^{3}-3x^{2}+2\)的單調(diào)區(qū)間。答案：首先求導(dǎo)\(y'=3x^{2}-6x=3x(x-2)\)。令\(y'=0\)，得\(x=0\)或\(x=2\)。當(dāng)\(x<0\)或\(x>2\)時，\(y'>0\)，函數(shù)單調(diào)遞增；當(dāng)\(0<x<2\)時，\(y'<0\)，函數(shù)單調(diào)遞減。2.計算\(\int_{0}^{\pi}\sinxdx\)。答案：根據(jù)定積分公式\(\int\sinxdx=-\cosx+C\)，\(\int_{0}^{\pi}\sinxdx=-\cosx|_{0}^{\pi}=-(\cos\pi-\cos0)=-(-1-1)=2\)。3.求函數(shù)\(y=\ln(x^{2}+1)\)的導(dǎo)數(shù)。答案：設(shè)\(u=x^{2}+1\)，則\(y=\lnu\)。根據(jù)復(fù)合函數(shù)求導(dǎo)法則，\(y'=\frac{1}{u}\cdotu'=\frac{2x}{x^{2}+1}\)。4.簡述函數(shù)極限的定義。答案：設(shè)函數(shù)\(y=f(x)\)，如果對于任意給定的正數(shù)\(\varepsilon\)（不論它多么?。偞嬖谡龜?shù)\(\delta\)，使得當(dāng)\(x\)滿足不等式\(0<|x-a|<\delta\)時，對應(yīng)的函數(shù)值\(f(x)\)都滿足不等式\(|f(x)-A|<\varepsilon\)，那么常數(shù)\(A\)就叫做函數(shù)\(y=f(x)\)當(dāng)\(x\rightarrowa\)時的極限，記作\(\lim\limits_{x\rightarrowa}f(x)=A\)。五、討論題（每題5分，共4題）1.討論函數(shù)\(y=\frac{x^{2}-1}{x-1}\)在\(x=1\)處的極限。答案：將函數(shù)化簡為\(y=\frac{(x+1)(x-1)}{x-1}=x+1(x\neq1)\)，當(dāng)\(x\rightarrow1\)時，\(\lim\limits_{x\rightarrow1}y=\lim\limits_{x\rightarrow1}(x+1)=2\)。2.討論函數(shù)\(y=e^{x}-x-1\)的單調(diào)性。答案：求導(dǎo)得\(y'=e^{x}-1\)。當(dāng)\(x>0\)時，\(y'>0\)，函數(shù)單調(diào)遞增；當(dāng)\(x<0\)時，\(y'<0\)，函數(shù)單調(diào)遞減。3.討