高等数学2试题及答案
一、单项选择题(每题2分,共20分)
1.函数\(z=\ln(x+y)\)的定义域是()
A.\(x+y\geq0\)B.\(x+y0\)C.\(x+y\neq0\)D.\(x+y\leq0\)
2.设\(z=x^2y\),则\(\frac{\partialz}{\partialx}\)=()
A.\(2xy\)B.\(x^2\)C.\(y\)D.\(2x\)
3.二重积分\(\iint_{D}1dxdy\)(\(D\)是由\(x=0\),\(x=1\),\(y=0\),\(y=1\)围成的区域)的值为()
A.0B.1C.2D.\(\frac{1}{2}\)
4.级数\(\sum_{n=1}^{\infty}\frac{1}{n}\)是()
A.收敛B.发散C.条件收敛D.绝对收敛
5.设\(y=e^{2x}\),则\(y^\prime\)=()
A.\(e^{2x}\)B.\(2e^{2x}\)C.\(\frac{1}{2}e^{2x}\)D.\(e^{x}\)
6.向量\(\vec{a}=(1,2,3)\)与向量\(\vec{b}=(2,4,6)\)的关系是()
A.垂直B.平行C.相交D.异面
7.曲线\(y=x^3\)在点\((1,1)\)处的切线斜率为()
A.1B.2C.3D.4
8.设\(z=f(x^2+y^2)\),则\(\frac{\partialz}{\partialy}\)=()
A.\(f^\prime(x^2+y^2)\)B.\(2yf^\prime(x^2+y^2)\)C.\(2xf^\prime(x^2+y^2)\)D.\(f^\prime(2y)\)
9.已知\(\intf(x)dx=F(x)+C\),则\(\intf(2x)dx\)=()
A.\(F(2x)+C\)B.\(\frac{1}{2}F(2x)+C\)C.\(2F(2x)+C\)D.\(F(x)+C\)
10.微分方程\(y^\prime-2y=0\)的通解是()
A.\(y=Ce^{2x}\)B.\(y=Ce^{-2x}\)C.\(y=Cx\)D.\(y=C\)
二、多项选择题(每题2分,共20分)
1.下列函数中,是多元函数的有()
A.\(z=x+y\)B.\(u=xyz\)C.\(y=x^2\)D.\(w=\sin(x+y+z)\)
2.下列级数中,收敛的有()
A.\(\sum_{n=1}^{\infty}\frac{1}{n^2}\)B.\(\sum_{n=1}^{\infty}\frac{1}{n!}\)C.\(\sum_{n=1}^{\infty}\frac{1}{n^3}\)D.\(\sum_{n=1}^{\infty}\frac{1}{2^n}\)
3.设\(z=x^3y^2\),则()
A.\(\frac{\partialz}{\partialx}=3x^2y^2\)B.\(\frac{\partialz}{\partialy}=2x^3y\)C.\(\frac{\partial^2z}{\partialx\partialy}=6x^2y\)D.\(\frac{\partial^2z}{\partialy\partialx}=6x^2y\)
4.下列向量中,与向量\(\vec{a}=(1,-1,0)\)垂直的有()
A.\(\vec{b}=(1,1,0)\)B.\(\vec{c}=(0,0,1)\)C.\(\vec{d}=(1,0,1)\)D.\(\vec{e}=(-1,-1,0)\)
5.曲线\(y=\sinx\)的性质有()
A.是奇函数B.周期是\(2\pi\)C.值域是\([-1,1]\)D.在\([0,\frac{\pi}{2}]\)上单调递增
6.下列积分中,能用牛顿-莱布尼茨公式计算的有()
A.\(\int_{0}^{1}x^2dx\)B.\(\int_{-1}^{