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高数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.\(x+y\gt0\)C.\(x+y\neq0\)D.\(x+y\leq0\)

2.设\(z=x^2y\),则\(\frac{\partialz}{\partialx}\)等于()

A.\(2xy\)B.\(x^2\)C.\(2x\)D.\(y\)

3.二重积分\(\iint_{D}d\sigma\),其中\(D\)是由\(x=0\),\(y=0\),\(x+y=1\)围成的区域,其值为()

A.\(\frac{1}{2}\)B.\(1\)C.\(\frac{1}{3}\)D.\(\frac{1}{4}\)

4.级数\(\sum_{n=1}^{\infty}\frac{1}{n^p}\)收敛的条件是()

A.\(p\leq1\)B.\(p\gt1\)C.\(p\geq1\)D.\(p\lt1\)

5.设向量\(\vec{a}=(1,-2,3)\),\(\vec{b}=(2,k,6)\),若\(\vec{a}\)与\(\vec{b}\)平行,则\(k\)的值为()

A.\(-4\)B.\(4\)C.\(-1\)D.\(1\)

6.函数\(z=f(x,y)\)在点\((x_0,y_0)\)处可微的必要条件是()

A.\(f(x,y)\)在点\((x_0,y_0)\)处连续

B.\(f_x(x_0,y_0)\)和\(f_y(x_0,y_0)\)都存在

C.\(f(x,y)\)在点\((x_0,y_0)\)处有连续的偏导数

D.\(A\)和\(B\)同时成立

7.曲线\(x=t\),\(y=t^2\),\(z=t^3\)在点\((1,1,1)\)处的切线方程为()

A.\(\frac{x-1}{1}=\frac{y-1}{2}=\frac{z-1}{3}\)B.\(\frac{x-1}{1}=\frac{y-1}{1}=\frac{z-1}{1}\)

C.\(\frac{x-1}{3}=\frac{y-1}{2}=\frac{z-1}{1}\)D.\(\frac{x-1}{1}=\frac{y-1}{3}=\frac{z-1}{2}\)

8.设\(D\)是由\(x^2+y^2\leq4\)所围成的区域,则\(\iint_{D}(x^2+y^2)d\sigma\)的值为()

A.\(4\pi\)B.\(8\pi\)C.\(16\pi\)D.\(32\pi\)

9.幂级数\(\sum_{n=0}^{\infty}a_n(x-2)^n\)在\(x=0\)处收敛,则该幂级数在\(x=3\)处()

A.绝对收敛B.条件收敛C.发散D.敛散性不确定

10.设\(z=e^{xy}\),则\(dz\)等于()

A.\(e^{xy}(xdx+ydy)\)B.\(e^{xy}(ydx+xdy)\)

C.\(e^{xy}dx+e^{xy}dy\)D.\(e^{xy}(dx+dy)\)

二、多项选择题(每题2分,共20分)

1.下列关于多元函数极限的说法正确的是()

A.若\(\lim_{(x,y)\to(x_0,y_0)}f(x,y)\)存在,则\(f(x,y)\)在\((x_0,y_0)\)处有定义

B.若\(f(x,y)\)在\((x_0,y_0)\)处的两个累次极限都存在且相等,则\(\lim_{(x,y)\to(x_0,y_0)}f(x,y)\)存在

C.若\(\lim_{(x,y)\to(x_0,y_0)}f(x,y)\)存在,则\(f(x,y)\)在\((x_0,y_0)\)处的两个累次极限可能不存在

D.若\(f(x,y)\)在\((x_0,y_0)\)处的两个累次极限都不存在,则\(\lim_{(x,y)\to(x_0,y_0)}f(x,y)\)可能存在

2.设\(z=f(x,y)\),则下列说法正确的是()

A.\(\frac{\partial^2z}{\partialx\partialy}=\frac{\partial^2z}{\partialy\partialx}\)在