外國微積分考試題及答案

單項選擇題（每題2分，共10題）1.函數(shù)\(y=x^2\)在點\((1,1)\)處的切線斜率是？A.1B.2C.3D.42.定積分\(\int_{0}^{1}xdx\)的值是？A.\(\frac{1}{2}\)B.1C.\(\frac{3}{2}\)D.23.函數(shù)\(f(x)=\sinx\)的導數(shù)是？A.\(\cosx\)B.\(-\cosx\)C.\(\sinx\)D.\(-\sinx\)4.極限\(\lim_{x\to0}\frac{\sinx}{x}\)等于？A.0B.1C.2D.35.曲線\(y=e^x\)在點\((0,1)\)處的切線方程是？A.\(y=x+1\)B.\(y=-x+1\)C.\(y=2x+1\)D.\(y=-2x+1\)6.定積分\(\int_{-1}^{1}x^3dx\)的值是？A.0B.1C.2D.37.函數(shù)\(y=\lnx\)的導數(shù)是？A.\(\frac{1}{x}\)B.\(-\frac{1}{x}\)C.\(x\)D.\(-x\)8.極限\(\lim_{x\to\infty}(1+\frac{1}{x})^x\)等于？A.\(e\)B.\(e^2\)C.\(1\)D.\(0\)9.曲線\(y=x^3\)在點\((1,1)\)處的切線斜率是？A.1B.2C.3D.410.定積分\(\int_{0}^{2}(x^2+1)dx\)的值是？A.\(\frac{10}{3}\)B.\(\frac{11}{3}\)C.\(\frac{13}{3}\)D.\(\frac{14}{3}\)答案：1.B2.A3.A4.B5.A6.A7.A8.A9.C10.C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.\(y=x^2\)D.\(y=\cosx\)2.下列哪些是基本初等函數(shù)？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)3.定積分的性質(zhì)包括？A.\(\int_{a}^kf(x)dx=k\int_{a}^f(x)dx\)B.\(\int_{a}^[f(x)+g(x)]dx=\int_{a}^f(x)dx+\int_{a}^g(x)dx\)C.\(\int_{a}^f(x)dx=-\int_^{a}f(x)dx\)D.\(\int_{a}^{a}f(x)dx=0\)4.函數(shù)\(y=\frac{1}{x}\)的定義域是？A.\(x\neq0\)B.\((-\infty,0)\)C.\((0,+\infty)\)D.\(R\)5.下列極限存在的有？A.\(\lim_{x\to0}\frac{1}{x}\)B.\(\lim_{x\to\infty}e^x\)C.\(\lim_{x\to0}x^2\)D.\(\lim_{x\to1}\frac{x-1}{x^2-1}\)6.函數(shù)\(y=\sqrt{x}\)的導數(shù)是？A.\(\frac{1}{2\sqrt{x}}\)B.\(2\sqrt{x}\)C.\(\frac{1}{2x}\)D.\(-\frac{1}{2\sqrt{x}}\)7.下列屬于積分公式的是？A.\(\intx^ndx=\frac{x^{n+1}}{n+1}+C(n\neq-1)\)B.\(\int\sinxdx=-\cosx+C\)C.\(\int\cosxdx=\sinx+C\)D.\(\inte^xdx=e^x+C\)8.曲線\(y=x^2\)與直線\(y=1\)圍成圖形的面積可表示為？A.\(\int_{-1}^{1}(1-x^2)dx\)B.\(2\int_{0}^{1}(1-x^2)dx\)C.\(\int_{-1}^{0}(1-x^2)dx+\int_{0}^{1}(1-x^2)dx\)D.\(\int_{-1}^{1}x^2dx-2\)9.函數(shù)\(y=\tanx\)的定義域是？A.\(x\neqk\pi+\frac{\pi}{2},k\inZ\)B.\(R\)C.\((-\frac{\pi}{k},\frac{\pi}{k})\)D.\((k\pi-\frac{\pi}{2},k\pi+\frac{\pi}{2}),k\inZ\)10.下列函數(shù)在\(x=0\)處連續(xù)的有？A.\(y=x\)B.\(y=\frac{\sinx}{x}\)C.\(y=e^x\)D.\(y=\ln(1+x)\)答案：1.AB2.ABCD3.ABCD4.ABC5.CD6.A7.ABCD8.BC9.AD10.ACD判斷題（每題2分，共10題）1.函數(shù)\(y=x^3\)是偶函數(shù)。（）2.定積分\(\int_{a}^f(x)dx\)一定大于0。（）3.函數(shù)\(y=\lnx\)在定義域內(nèi)單調(diào)遞增。（）4.極限\(\lim_{x\to\infty}\frac{1}{x^2}=0\)。（）5.曲線\(y=x^2\)與\(y=x\)在\((0,0)\)處相切。（）6.函數(shù)\(y=\cos^2x\)的導數(shù)是\(-2\cosx\sinx\)。（）7.\(\int_{0}^{1}xdx=\int_{0}^{1}(x+1)dx\)。（）8.函數(shù)\(y=\frac{1}{x^2}\)在\(x=0\)處有定義。（）9.極限\(\lim_{x\to0}\frac{\ln(1+x)}{x}=1\)。（）10.曲線\(y=e^{-x}\)在\(x=0\)處的切線斜率為-1。（）答案：1.×2.×3.√4.√5.×6.√7.×8.×9.√10.√簡答題（總4題，每題5分）1.簡述導數(shù)的幾何意義。答案：函數(shù)在某點處導數(shù)的幾何意義是該點處切線的斜率。2.寫出基本積分公式\(\intx^ndx\)（\(n\neq-1\)）。答案：\(\intx^ndx=\frac{x^{n+1}}{n+1}+C\)。3.什么是函數(shù)的連續(xù)性？答案：函數(shù)在某點處極限值等于函數(shù)值，則函數(shù)在該點連續(xù)。4.簡述定積分\(\int_{a}^f(x)dx\)與曲邊梯形面積的關(guān)系。答案：當\(f(x)\geq0\)時，定積分等于曲邊梯形面積；當\(f(x)\)有正有負時，定積分是各部分面積代數(shù)和。討論題（總4題，每題5分）1.