大學高數(shù)b1期中考試試題及答案

一、單項選擇題（每題2分，共10題）1.函數(shù)\(y=\frac{1}{\ln(x-1)}\)的定義域是（）A.\((1,2)\cup(2,+\infty)\)B.\((1,+\infty)\)C.\([1,+\infty)\)D.\((2,+\infty)\)答案：A2.\(\lim_{x\rightarrow0}\frac{\sin2x}{x}=()\)A.0B.1C.2D.不存在答案：C3.函數(shù)\(y=x\cosx\)在\(x=0\)處的導(dǎo)數(shù)是（）A.0B.1C.-1D.不存在答案：A4.設(shè)\(y=e^{2x}\)，則\(y''=()\)A.\(2e^{2x}\)B.\(4e^{2x}\)C.\(e^{2x}\)D.\(e^{x}\)答案：B5.曲線\(y=x^{3}-3x^{2}+1\)在點\((1,-1)\)處的切線方程是（）A.\(y=-3x+2\)B.\(y=3x-4\)C.\(y=-x\)D.\(y=x-2\)答案：C6.\(\intx\sinxdx=()\)A.\(-x\cosx+\sinx+C\)B.\(x\cosx+\sinx+C\)C.\(-x\cosx-\sinx+C\)D.\(x\cosx-\sinx+C\)答案：A7.\(\int_{0}^{1}e^{x}dx=()\)A.\(e-1\)B.\(1-e\)C.\(e\)D.\(1\)答案：A8.函數(shù)\(y=x^{2}\)在區(qū)間\([0,1]\)上的平均值為（）A.\(\frac{1}{3}\)B.\(\frac{1}{2}\)C.\(\frac{2}{3}\)D.1答案：A9.若\(f(x)\)的一個原函數(shù)是\(\sinx\)，則\(f'(x)=()\)A.\(\cosx\)B.\(-\sinx\)C.\(-\cosx\)D.\(\sinx\)答案：B10.下列函數(shù)中，在區(qū)間\((-1,1)\)上滿足羅爾定理條件的是（）A.\(y=\frac{1}{x}\)B.\(y=|x|\)C.\(y=x^{2}-1\)D.\(y=x^{3}\)答案：C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.\(y=x^{3}\sinx\)B.\(y=\frac{\sinx}{x}\)C.\(y=x\cosx\)D.\(y=\frac{x^{2}\sinx}{1+x^{2}}\)答案：AC2.當\(x\rightarrow0\)時，與\(x\)等價的無窮小量有（）A.\(\sinx\)B.\(\tanx\)C.\(\ln(1+x)\)D.\(e^{x}-1\)答案：ABCD3.下列求導(dǎo)正確的有（）A.\((\sin^{2}x)'=\sin2x\)B.\((\ln\cosx)'=-\tanx\)C.\((e^{-x^{2}})'=-2xe^{-x^{2}}\)D.\((\arctan\frac{1}{x})'=\frac{-1}{1+x^{2}}\)答案：ABC4.設(shè)\(y=f(u)\)，\(u=g(x)\)，且\(f(u)\)，\(g(x)\)均可導(dǎo)，則\(\frac{dy}{dx}=()\)A.\(f'(u)g'(x)\)B.\(f(g(x))'\)C.\(y'_{u}\cdotu'_{x}\)D.\(f'(g(x))g'(x)\)答案：ACD5.下列函數(shù)在給定區(qū)間上單調(diào)遞增的有（）A.\(y=x^{2}\)在\((0,+\infty)\)B.\(y=\sinx\)在\((-\frac{\pi}{2},\frac{\pi}{2})\)C.\(y=e^{x}\)在\((-\infty,+\infty)\)D.\(y=\lnx\)在\((0,+\infty)\)答案：ABCD6.以下定積分的值為0的有（）A.\(\int_{-1}^{1}x^{3}dx\)B.\(\int_{-\pi}^{\pi}\sinxdx\)C.\(\int_{-a}^{a}f(x)dx\)（\(f(x)\)為奇函數(shù)）D.\(\int_{-2}^{2}(x^{2}+1)dx\)答案：ABC7.函數(shù)\(y=f(x)\)的極值可能出現(xiàn)在（）A.\(f'(x)=0\)的點B.\(f'(x)\)不存在的點C.區(qū)間端點D.任意點答案：AB8.設(shè)\(F(x)\)是\(f(x)\)的一個原函數(shù)，則（）A.\(\intf(x)dx=F(x)+C\)B.\((\intf(x)dx)'=f(x)\)C.\(\intF(x)dx=\frac{1}{2}F^{2}(x)+C\)D.\((F(x))'=f(x)\)答案：ABD9.對于曲線\(y=x^{3}-3x^{2}+1\)，以下說法正確的有（）A.在\(x=0\)處取得極大值B.在\(x=2\)處取得極小值C.有兩個拐點D.在\((-\infty,0)\)上單調(diào)遞增答案：ABC10.若\(\intf(x)dx=F(x)+C\)，則\(\intf(ax+b)dx=()\)（\(a\neq0\)）A.\(\frac{1}{a}F(ax+b)+C\)B.\(F(ax+b)+C\)C.\(aF(ax+b)+C\)D.\(\frac{1}{a}F(x)+C\)答案：A三、判斷題（每題2分，共10題）1.若\(\lim_{x\rightarrowx_{0}}f(x)\)存在，則\(f(x)\)在\(x=x_{0}\)處一定有定義。（×）2.函數(shù)\(y=\frac{1}{x}\)在區(qū)間\((0,1)\)上無界。（√）3.若\(y=f(x)\)在\(x=a\)處可導(dǎo)，則\(y=f(x)\)在\(x=a\)處連續(xù)。（√）4.\((uv)'=u'v+uv'\)對任意函數(shù)\(u(x)\)，\(v(x)\)都成立。（√）5.函數(shù)\(y=x^{3}\)的二階導(dǎo)數(shù)\(y''=6x\)。（√）6.\(\int\frac{1}{x^{2}}dx=\lnx^{2}+C\)。（×）7.若\(f(x)\)在\([a,b]\)上連續(xù)，則\(\int_{a}^{b}f(x)dx=\int_{a}^{b}f(t)dt\)。（√）8.函數(shù)\(y=\sinx\)在\([0,2\pi]\)上只有一個極值點。（×）9.若\(F(x)\)是\(f(x)\)的原函數(shù)，則\(F(x)+1\)也是\(f(x)\)的原函數(shù)。（√）10.曲線\(y=x^{2}\)在點\((1,1)\)處的曲率為\(2\)。（×）四、簡答題（每題5分，共4題）1.求極限\(\lim_{x\rightarrow1}\frac{x^{2}-1}{x-1}\)。答案：\(\lim_{x\rightarrow1}\frac{x^{2}-1}{x-1}=\lim_{x\rightarrow1}\frac{(x+1)(x-1)}{x-1}=\lim_{x\rightarrow1}(x+1)=2\)。2.求函數(shù)\(y=x^{3}-3x^{2}+1\)的單調(diào)區(qū)間。答案：\(y'=3x^{2}-6x=3x(x-2)\)。令\(y'=0\)，得\(x=0\)或\(x=2\)。當\(x<0\)或\(x>2\)時，\(y'>0\)，函數(shù)單調(diào)遞增；當\(0<x<2\)時，\(y'<0\)，函數(shù)單調(diào)遞減。3.計算\(\int_{0}^{\pi}\sin^{2}xdx\)。答案：\(\sin^{2}x=\frac{1-\cos2x}{2}\)，\(\int_{0}^{\pi}\sin^{2}xdx=\int_{0}^{\pi}\frac{1-\cos2x}{2}dx=\frac{1}{2}\int_{0}^{\pi}(1-\cos2x)dx=\frac{1}{2}(x-\frac{\sin2x}{2})\big|_{0}^{\pi}=\frac{\pi}{2}\)。4.求函數(shù)\(y=\ln(x+1)\)的二階導(dǎo)數(shù)。答案：\(y'=\frac{1}{x+1}\)，\(y''=-\frac{1}{(x+1)^{2}}\)。五、討論題（每題5分，共4題）1.討論函數(shù)\(y=\frac{x^{3}}{3}-x^{2}+2\)的凹凸性。答案：\(y'=x^{2}-2x\)，\(y''=2x-2\)。令\(y''=0\)，得\(x=1\)。當\(x<1\)時，\(y''<0\)，函數(shù)為凸函數(shù)；當\(x>1\)時，\(y''>0\)，函數(shù)為凹函數(shù)。2.設(shè)\(y=f(x)\)在\(x=a\)處可導(dǎo)，討論\(\lim_{h\rightarrow0}\frac{f(a+h)-f(a-h)}{2h}\)與\(f'(a)\)的關(guān)系。答案：\(\lim_{h\rightarrow0}\frac{f(a+h)-f(a-h)}{2h}=f'(a)\)，根據(jù)導(dǎo)數(shù)定義可推出。3.討論定積分\(\int_{0}^{1}\frac{1}{x^{p}}dx\)（\(p\neq1\)）的斂散性。答案：當\(p<1\)時，\(\int_{0}^{1}\frac{1}{x^{p}}dx=\frac{1}{1-p}x^{1-p}\big