MA127-2020春 期末考试 2（回忆版）

一

Determine whether the following statements are true or false? No justification is necessary

1-1

Parametric curves $x(t)=\cos t,y(t)=\sin t$ and $x(t)=\sin t,y(t)=\cos t$ have the same graph

1-2

If $x(t)=f(t)$ and $y(t)=g(t)$ are twice differentiable, then

$$\frac{d^{2}x}{d{x}^2}=\frac{d^{2}y(t)/d{t}^2}{d^{2}g(t)/d{t}^2}$$

二

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

2-1

If $|u|=2,|v|=\sqrt{2}$, and $u\cdot v=2$, then $|u\times v|$ is

(A) 2

(B) $2\sqrt{2}$

(C) $\frac{\sqrt{2}}{2}$

(D) 1

2-2

How many points of interesection do the curves $r=\frac{1}{2}$ and $r=\cos 2\theta$ have?

(A) 2

(B) 4

(C) 6

(D) 8

2-3

If $f(x+y,x-y)=x^2-y^2$, then $\frac{\partial f(x,y)}{\partial x}+\frac{\partial f(x,y)}{\partial y}=$

(A) $2x-2y$

(B) $2x+2y$

(C) $x-y$

(D) $x+y$

三

Please fill in the blank for the questions below

3-1

If the plane $3x+\lambda y-3z+16=0$ is tangent to the surface $3x^2+y^2+z^2=16$, then $\lambda =$ $\underset{―}{}$

3-2

Let $z=\ln \sqrt{x^2+y^2}+{\tan }^{-1}\frac{x+y}{x-y}$, then $dz=$ $\underset{―}{}$

3-3

The distance from the point $P(1,4,0)$ to the plane through $A(0,0,0),B(2,0,1)$ and $C(2,-1,0)$ is $\underset{―}{}$

3-4

A closed path $C$ consists of three curves:

$$\begin{array}{ll}{C}_1=r(t)=(\cos t)i+(\sin t)j+tk,& 0\le t\le \frac{\pi }{2}\\ C_2=r(t)=i+\left(\frac{\pi }{2}\right)(1-t)k,& 0\le t\le 1\\ C_3=r(t)=ti+(1-t)j,& 0\le t\le 1\end{array}$$

Then the circulation of $F=2xi+2zj+2yk$ around path $C$ traversed in the direction of increasing is $\underset{―}{}$

请自行画图理解

四

Determine the length of polar curve $r={\sin }^3\left(\frac{\theta }{3}\right),0\le \theta \le \frac{\pi }{4}$

五

Given a curve $r(t)=({\cos }^{3}t,{\sin }^{3}t,0),0<t<\frac{\pi }{2}$ in $R^3$, find its curvature and principal unit normal

六

The sequence $\left\{a_n\right\}$ is defined by $a_{2k-1}=\frac{1}{k},a_{2k}=-\frac{1}{k+2}$( $k$ can be any positive integer).Is the series $\sum _{n=1}^{\mathrm{\infty }}{a}_n$ convergent or divergent? Prove your conclusion

七

Find the Maclaurin series for $f(x)=\frac{1}{(1+x{)}^3}$

八

Find the absolute maximum and minimum values of $f(x,y)=4x{y}^2-x^2{y}^2-x{y}^3$ on the close triangular region in the $xy$-plane with verties $(0,0),(0,6)$ and $(6,0)$

九

Find the centroid of the solid bounded above by the surface $z=\sqrt{r}$, on the sides by the cylinder $r=4$, and below by the $xy$-plane

十

Use the Stokes' Theorem to compute the surface integral ${\iint }_s\mathrm{\nabla }\times F\cdot nd\sigma$, here $F=xz\mathbf{i}+yz\mathbf{j}+xy\mathbf{k}$, and $S$ is the part of the surface sphere $x^2+y^2+z^2=4$ that lies inside the cylinder $x^2+y^2=1$ and above the $xy$-plane (the boundary is counterclockwise when viewed from above)

十一

Use the Divergence Theorem to find the outward flux of $F$ across the boundary of the region $D$, here $F=xy\mathbf{i}+(y^2+e^{x{z}^2})\mathbf{j}+\sin (xy)\mathbf{k}$; and $D$ is the region bounded by the parabolic cylinder $z=1-x^2$, and the planes $z=0,y=0$, and $y+z=2$

自行绘图理解