MA127 2025 春 期末考试

1. Multiple Choice Questions (only one correct answer for each of the following questions)

(1) The divergence of the vector field $F=⟨x{y}^2,y{e}^z,xln(1+z^2)⟩$ at the point $(1,0,1)$ is:

• (A) $-1$
• (B) $0$
• (C) $1$
• (D) $2$

(2) Let $f(x,y)$ be a function defined on the entire plane such that $f_x(x,y)<0$ and $f_y(x,y)>0$. Which of the following conditions guarantees that $f(x_1,y_1)<f(x_2,y_2)$?

• (A) $x_1<x_2$, $y_1<y_2$
• (B) $x_1<x_2$, $y_1>y_2$
• (C) $x_1>x_2$, $y_1<y_2$
• (D) $x_1>x_2$, $y_1>y_2$

(3) ${\int }_{-1}^0{\int }_{-x}^{2-x^2}(1-xy)dydx+{\int }_0^1{\int }_x^{2-x^2}(1-xy)dydx=?$

• (A) $\frac{7}{3}$
• (B) $\frac{7}{6}$
• (C) $\frac{5}{3}$
• (D) $\frac{5}{6}$

(4) $\underset{(x,y)\to (0,0)}{lim}\frac{\sin (x^4{y}^2)}{(x^4+y^2{)}^2}=?$

• (A) $0$
• (B) $1$
• (C) $\frac{1}{2}$
• (D) The limit does not exist

(5) Consider the following four properties of a function with two variables:

1. $f(x,y)$ is continuous at point $P(x_0,y_0)$
2. The partial derivatives of $f(x,y)$ at point $(x_0,y_0)$ are continuous
3. $f(x,y)$ is differentiable at point $(x_0,y_0)$
4. The partial derivatives of $f(x,y)$ exist at point $(x_0,y_0)$

If "$P\Rightarrow Q$" denotes that property $P$ implies property $Q$, which of the following relationships is correct?

• (A) 2 $\Rightarrow$ 1 $\Rightarrow$ 3
• (B) 2 $\Rightarrow$ 2 $\Rightarrow$ 3
• (C) 3 $\Rightarrow$ 4 $\Rightarrow$ 1
• (D) 3 $\Rightarrow$ 1 $\Rightarrow$ 4

2 Fill in the blanks

(1) The equation for the tangent plane at the point $(1,1,e)$ on the surface $z=x^2(x-\sin y)+y^2(1-\sin x)$ is $\underset{―}{}$

(2) Let $D$ be the region $\{(x,y)\mid x^2+y^2\le 1\}$, then ${\iint }_D{x}^2+y^{2}dxdy$ = $\underset{―}{}$

(3) The maximum directional derivative of $f(x,y,z)=x^2{e}^y+\cos (y^2+z^2)$ at the point $(1,0,0)$ is $\underset{―}{}$

(4) $\underset{n\to 0}{lim}\frac{e^{-x-\frac{x^2}{2}}-1+2x-\sin x}{x^3}$ = $\underset{―}{}$

(5) If $\frac{(x+ay)dx+ydy}{(x+y{)}^2}$ is the total differential for some function , then $a$ = $\underset{―}{}$

3

曲线 C 由参数方程定义：$x=t-\sin t,y=1-\cos t$

(1) 求曲线 C 在点 ? 处的切线方程（对应 $t=\frac{\pi }{3}$）

(2) 求 $\frac{d^{2}y}{d{x}^2}$

4

判断下列级数是绝对收敛、条件收敛还是发散，并说明理由：

(1) ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n{n}^3}{(2{)}^n}}$

(2) ${\displaystyle \sum _{n=2}^{\mathrm{\infty }}(-1{)}^{n+1}\frac{1}{n\ln n}}$

5

求函数 $f(x,y)=x{e}^{-\frac{x^2+y^2}{2}}$ 的所有局部极值。

6

求区域 $D=\{(x,y,z)\mid x^2+y^2\le z\le 1\}$ 的形心。

7

求向量场 $\mathbf{F}=(3x{y}^2+e^{y^2}\cos (y^2+z^2))\mathbf{i}+(6x^{2}y-y{z}^2)\mathbf{j}+(z^3/3+\ln (x^2+y^2))\mathbf{k}$ 穿过区域 $\mathrm{\Omega }=\{(x,y,z)\mid 0\le z\le 1-2x^2-y^2\}$ 边界的向外通量。

8

计算线积分

$${\int }_C(\frac{\sin x}{1+x^2}-y)dx+(x-z+e^{y^2})dy+(x-y+\ln (a+2z^2))dz$$

其中曲线 C 由以下分段定义：

$$\{\begin{array}{l}{x}^2+y^2-2y=0\\ x-y+z=2\end{array}$$

从上方观察时，逆时针方向为正方向 (couterclockswise)