MA127-2023春 期中考试（回忆版）

一、单项选择题

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

1-1

If $P_1$ and $P_2$ are the following planes:

$$\begin{array}{rl}& P_1:=\{(x,y,z):x+2y-z=\frac{\sqrt{2}}{2}\},\\ & P_2:=\{(x,y,z):x+y=-\frac{\sqrt{2}}{2}\}\end{array}$$

then the angle $\theta$ between $P_1$ and $P_2$ is:

(A) $\frac{\pi }{2}$

(B) $\frac{\pi }{3}$

(C) $\frac{\pi }{4}$

(D) $\frac{\pi }{6}$

1-2

The arc length of the curve $r(t)=\frac{2}{3}{t}^{\frac{3}{2}}i+\frac{2}{3}(2-t{)}^{\frac{3}{2}}j+(t-1)k$ for $\frac{1}{4}\le t\le \frac{1}{2}$ is

(A) $\frac{1}{4}$

(B) $\frac{\sqrt{2}}{4}$

(C) $\frac{\sqrt{3}}{4}$

(D) $\frac{1}{2}$

1-3

Let $f(x,y)=\{\begin{array}{ll}\ln (1+xy),& x\ne 0\\ a,& x=0\end{array}$, then

(A) If $a=1$, then $f(x,y)$ is continuous at $(0,2)$

(B) If $a=2$, then $f(x,y)$ is discontinuous at $(0,2)$

(C) If $a=1$, then $f(x,y)$ is continuous at $(0,1)$

(D) If $a=2$, then $f(x,y)$ is continuous at $(0,1)$

1-4

If $0\le a_n\le \frac{1}{n}(n=1,2,\cdots )$, which of the following series must be convergent?

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

(B) $\sum _{n=1}^{\mathrm{\infty }}(-1{)}^n{a}_n$

(C) $\sum _{n=1}^{\mathrm{\infty }}{a}_n$

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

1-5

If $\sum _{n=1}^{\mathrm{\infty }}{a}_n{x}^n$ converges at $x=-2$, then

(A) $\sum _{n=1}^{\mathrm{\infty }}{a}_n{2}^n$ converges

(B) $\sum _{n=1}^{\mathrm{\infty }}{a}_n{2}^n$ diverges

(C) $\sum _{n=1}^{\mathrm{\infty }}n{a}_n$ converges

(D) $\sum _{n=1}^{\mathrm{\infty }}n{a}_n$ diverges

二、填空题

Fill in the blanks

2-1

Let the polar equation be $r=\csc \theta e^{r\cos \theta }$. Its equivalent Cartesian equation is $\underset{―}{}$

2-2

The distance from the point $S(1,1,1)$ to the plane $x+2y+3z+4=0$ is $\underset{―}{}$

2-3

A particle is traveling with acceleration $a(t)=⟨e^t,t,\sin 2t⟩$. At $t=0$, it was at the origin and its initial velocity is $v(0)=⟨1,0,0⟩$. The position function of the particle is $\underset{―}{}$

2-4

$\underset{x\to 0}{lim}\frac{3\sin (2x)-2\sin (3x)}{6x^3+x^4}=$ $\underset{―}{}$

2-5

The series

$$S=-\ln 2+\frac{{\ln }^{2}2}{2!}+\cdots +\frac{(-1{)}^n{\ln }^{n}2}{n!}+\cdots =\underset{―}{}$$

三

3-1

Find the interval of convergence for the series $\sum _{n=1}^{\mathrm{\infty }}\frac{\ln n}{n}(x-1{)}^n$

3-2

For what value of $x$ the above series converges absolutely, or conditionally?

四

Find the limit if it exists. If it does not exist, explain why

4-1

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

4-2

$\underset{(x,y)\to (0,0)}{lim}\frac{2xy}{\sqrt{x^2+2y^2}}$

五

Let $L_1:x=t,y=1,z=1+t$ and $L_2:x=1+2s,y=0,z=2s$. Is there one plane in which both lines lie? If so, find the equation of the plane. If not, give your reason

六

Assume that $a$ and $b$ are real numbers, and $0<b<a<1$

6-1

Determine whether the series

$$\begin{array}{l}\frac{a}{\sqrt{(1+a)(1+b)}}\\ +\frac{b^2}{\sqrt{(1+a^2)(1+b^2)}}\\ +\frac{a^3}{\sqrt{(1+a^3)(1+b^3)}}\\ +\frac{b^4}{\sqrt{(1+a^4)(1+b^4)}}\\ +\cdots +\frac{c_n}{\sqrt{(1+a^n)(1+b^n)}}\\ +\cdots \end{array}$$

where

$$C_n=\{\begin{array}{ll}{a}^n,& \text{ if }n\text{ is odd}\\ b^n,& \text{ if }n\text{ is even}\end{array}$$

converges or diverges. Justify your answer!

6-2

Find the limit: $\underset{n\to \mathrm{\infty }}{lim}(\frac{a^n}{n^2}+\frac{b^n}{n}{)}^{\frac{1}{n}}$

七

Let $C$ be the curve given by

$$r(t)=\sqrt{2}\cos t\mathbf{i}+\sin t\mathbf{j}+\sin t\mathbf{k}$$

7-1

Find the unit tangent vector $T(t)$ and principal unit normal vector $N(t)$ for $C$

7-2

Find the curvature $\kappa (t)$

八

Using known Taylor series expansions, write the Taylor series for the function

$$f(x)=\frac{4}{x^2-2x+5}+\ln x$$

at $x=1$ in the interval $(0,2)$