高等數(shù)學(xué)考試題及答案

一、單項(xiàng)選擇題（每題2分，共10題）1.函數(shù)y=sinx的導(dǎo)數(shù)是（）A.cosxB.-cosxC.sinxD.-sinx答案：A2.定積分\(\int_{0}^{1}xdx\)的值為（）A.\(\frac{1}{2}\)B.1C.2D.\(\frac{1}{3}\)答案：A3.極限\(\lim_{x\rightarrow0}\frac{\sinx}{x}\)的值為（）A.0B.1C.不存在D.\(\infty\)答案：B4.函數(shù)y=\(e^{x}\)的圖象在點(diǎn)(0,1)處的切線(xiàn)方程為（）A.y=x+1B.y=-x+1C.y=xD.y=-x答案：A5.下列函數(shù)中為奇函數(shù)的是（）A.y=\(x^{2}\)B.y=\(\cosx\)C.y=\(\sinx\)D.y=\(e^{x}\)答案：C6.函數(shù)y=\(\lnx\)的定義域是（）A.\((-\infty,+\infty)\)B.\((0,+\infty)\)C.\([0,+\infty)\)D.\((-\infty,0)\)答案：B7.二次函數(shù)y=\(ax^{2}+bx+c\)（\(a\neq0\)）的對(duì)稱(chēng)軸為（）A.\(x=-\frac{2a}\)B.\(x=\frac{2a}\)C.\(x=-\frac{a}\)D.\(x=\frac{a}\)答案：A8.若\(f(x)=3x+2\)，則\(f(2)\)的值為（）A.8B.6C.4D.10答案：A9.導(dǎo)數(shù)\(y'=(x^{3})'\)等于（）A.\(3x^{2}\)B.\(x^{2}\)C.\(3x\)D.\(x^{3}\)答案：A10.設(shè)\(y=\sqrt{x}\)，則\(y'\)等于（）A.\(\frac{1}{2\sqrt{x}}\)B.\(\frac{1}{\sqrt{x}}\)C.\(2\sqrt{x}\)D.\(\sqrt{x}\)答案：A二、多項(xiàng)選擇題（每題2分，共10題）1.下列函數(shù)在\((-\infty,+\infty)\)上是單調(diào)遞增函數(shù)的有（）A.y=\(x^{3}\)B.y=\(2x+1\)C.y=\(\sinx\)D.y=\(\lnx\)答案：AB2.以下關(guān)于函數(shù)極值的說(shuō)法正確的是（）A.可導(dǎo)函數(shù)的極值點(diǎn)處導(dǎo)數(shù)為0B.導(dǎo)數(shù)為0的點(diǎn)一定是極值點(diǎn)C.函數(shù)在極值點(diǎn)兩側(cè)單調(diào)性不同D.函數(shù)可能有多個(gè)極值點(diǎn)答案：ACD3.下列積分結(jié)果正確的是（）A.\(\intx^{2}dx=\frac{1}{3}x^{3}+C\)B.\(\int\sinxdx=\cosx+C\)C.\(\inte^{x}dx=e^{x}+C\)D.\(\int\frac{1}{x}dx=\ln|x|+C\)答案：ACD4.關(guān)于函數(shù)y=\(\cosx\)的性質(zhì)，正確的有（）A.是偶函數(shù)B.周期為\(2\pi\)C.值域?yàn)閈([-1,1]\)D.在\((0,\pi)\)上單調(diào)遞減答案：ABCD5.下列函數(shù)是復(fù)合函數(shù)的有（）A.\(y=(2x+1)^{2}\)B.\(y=\sin(x^{2})\)C.\(y=e^{2x}\)D.\(y=\ln(3x-1)\)答案：ABCD6.若函數(shù)\(y=f(x)\)在點(diǎn)\(x=a\)處連續(xù)，則（）A.\(\lim_{x\rightarrowa^{-}}f(x)=\lim_{x\rightarrowa^{+}}f(x)\)B.\(\lim_{x\rightarrowa}f(x)=f(a)\)C.函數(shù)在\(x=a\)處有定義D.函數(shù)在\(x=a\)處可導(dǎo)答案：ABC7.下列關(guān)于極限的計(jì)算正確的是（）A.\(\lim_{n\rightarrow\infty}\frac{1}{n}=0\)B.\(\lim_{x\rightarrow1}(x^{2}-1)=0\)C.\(\lim_{x\rightarrow\infty}\frac{\sinx}{x}=0\)D.\(\lim_{x\rightarrow0}\frac{e^{x}-1}{x}=1\)答案：ABCD8.設(shè)函數(shù)\(y=f(x)\)的圖象關(guān)于原點(diǎn)對(duì)稱(chēng)，則（）A.\(f(x)\)是奇函數(shù)B.\(f(-x)=-f(x)\)C.對(duì)于任意\(x\)，\(f(x)+f(-x)=0\)D.\(y=f(x)\)的圖象過(guò)原點(diǎn)答案：ABC9.當(dāng)\(x\rightarrow0\)時(shí)，與\(x\)等價(jià)的無(wú)窮小量有（）A.\(\sinx\)B.\(\tanx\)C.\(e^{x}-1\)D.\(\ln(1+x)\)答案：ABCD10.下列關(guān)于定積分的幾何意義說(shuō)法正確的是（）A.\(\int_{a}^f(x)dx\)表示由曲線(xiàn)\(y=f(x)\)，直線(xiàn)\(x=a\)，\(x=b\)以及\(x\)軸所圍成的曲邊梯形的面積B.當(dāng)\(f(x)\leqslant0\)在\([a,b]\)上時(shí)，\(\int_{a}^f(x)dx\)表示曲邊梯形面積的相反數(shù)C.當(dāng)\(f(x)\)在\([a,b]\)上有正有負(fù)時(shí)，\(\int_{a}^f(x)dx\)表示位于\(x\)軸上方的曲邊梯形面積減去位于\(x\)軸下方的曲邊梯形面積D.\(\int_{a}^f(x)dx\)與\(\int_^{a}f(x)dx\)的值互為相反數(shù)答案：ABCD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.函數(shù)\(y=x^{3}\)的二階導(dǎo)數(shù)\(y''=6x\)。（）答案：正確3.若\(\lim_{x\rightarrowa}f(x)=\lim_{x\rightarrowa}g(x)\)，則\(f(x)=g(x)\)。（）答案：錯(cuò)誤4.定積分\(\int_{a}^{a}f(x)dx=0\)。（）答案：正確5.函數(shù)\(y=\lnx\)在\((0,1)\)上是單調(diào)遞增函數(shù)。（）答案：錯(cuò)誤6.所有的連續(xù)函數(shù)都是可導(dǎo)函數(shù)。（）答案：錯(cuò)誤7.若函數(shù)\(y=f(x)\)在\(x=a\)處可導(dǎo)，則函數(shù)在\(x=a\)處一定連續(xù)。（）答案：正確8.函數(shù)\(y=\cos2x\)的周期是\(\pi\)。（）答案：正確9.對(duì)于函數(shù)\(y=f(x)\)，若\(f'(x_{0})=0\)，則\(x_{0}\)一定是函數(shù)的極值點(diǎn)。（）答案：錯(cuò)誤10.\(\inte^{x}\sinxdx=\frac{1}{2}e^{x}(\sinx-\cosx)+C\)。（）答案：正確四、簡(jiǎn)答題（每題5分，共4題）1.求函數(shù)\(y=x^{2}+2x-3\)的頂點(diǎn)坐標(biāo)。答案：對(duì)于二次函數(shù)\(y=ax^{2}+bx+c\)（\(a\neq0\)），其頂點(diǎn)橫坐標(biāo)\(x=-\frac{2a}\)，這里\(a=1\)，\(b=2\)，則\(x=-1\)，將\(x=-1\)代入函數(shù)得\(y=(-1)^{2}+2\times(-1)-3=-4\)，頂點(diǎn)坐標(biāo)為\((-1,-4)\)。2.求函數(shù)\(y=\sinx\)在\([0,2\pi]\)上的最大值和最小值。答案：\(y=\sinx\)在\([0,2\pi]\)上，當(dāng)\(x=\frac{\pi}{2}\)時(shí)，\(y=1\)為最大值；當(dāng)\(x=\frac{3\pi}{2}\)時(shí)，\(y=-1\)為最小值。3.簡(jiǎn)述導(dǎo)數(shù)的定義。答案：函數(shù)\(y=f(x)\)在點(diǎn)\(x_{0}\)處的導(dǎo)數(shù)\(f'(x_{0})\)定義為\(f'(x_{0})=\lim_{\Deltax\rightarrow0}\frac{f(x_{0}+\Deltax)-f(x_{0})}{\Deltax}\)，它表示函數(shù)在該點(diǎn)處的瞬時(shí)變化率。4.計(jì)算定積分\(\int_{0}^{1}(2x+1)dx\)。答案：\(\int_{0}^{1}(2x+1)dx=\left[x^{2}+x\right]_{0}^{1}=(1^{2}+1)-(0^{2}+0)=2\)。五、討論題（每題5分，共4題）1.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(shí)，\(y'>0\)，函數(shù)單調(diào)遞增；當(dāng)\(x<0\)時(shí)，\(y'<0\)，函數(shù)單調(diào)遞減。2.討論函數(shù)\(y=\frac{x^{2}-1}{x-1}\)在\(x=1\)處的極限情況。答案：化簡(jiǎn)\(y=\frac{x^{2}-1}{x-1}=\frac{(x+1)(x-1)}{x-1}=x+1(x\neq1)\)，\(\lim_{x\rightarrow1}(x+1)=2\)，所以\(x\rightarrow1\)時(shí)極限為2。3.討論定積分\(\int_{-1}^{1}x^{3}dx\)的計(jì)算結(jié)果及幾何意義。答案：計(jì)算\(\int_{-1}^{1}x^{3}dx=\left[\frac{1}{4}x^{4}\right]_{-1}^{1}=\frac{1}{4}\times1^{4}-\frac{1}{4}\times(-1)^{4}=0\)。幾何意義是\(y=x^{3}\)在