MA127-2021春 期中考试

一、单项选择题

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

1-1

If $f$ is differentiable, and $z=z(x,y)$ is determined by $f(x-az,y-bz)=0$, then $a\frac{\partial z}{\partial x}+b\frac{\partial z}{\partial y}=$

(A) 1

(B) -1

(C) $a$

(D) $b$

1-2

Let $a_n>0$ for all $n$. Which of the following statements must be true?

(A) If $\underset{n\to \mathrm{\infty }}{lim}n{a}_n=0$, then the series $\sum _{n=1}^{\mathrm{\infty }}{a}_n$ converges

(B) If $\underset{n\to \mathrm{\infty }}{lim}n{a}_n=l$ and $l\ne 0$, then the series $\sum _{n=1}^{\mathrm{\infty }}{a}_n$ converges

(C) If $\underset{n\to \mathrm{\infty }}{lim}n{a}_n=l$ and $l\ne 0$, then the series $\sum _{n=1}^{\mathrm{\infty }}{a}_n$ diverges

(D) None of the above statements is correct

1-3

Identify the surface of $2x^2+y^2=z^2$

(A) Hyperboloid of two sheets

(B) Elliptical Cone

(C) Hyperboloid of one sheet

(D) Elliptical paraboloid

1-4

If $f(x,y)=\phi (x+y)+\phi (x-y)+{\int }_{x-y}^{x+y}\psi (t)dt$, where $\phi$ and $\psi$ are twice differentiable functions, then

(A) $\frac{{\partial }^{2}f}{\partial x\partial y}=\frac{{\partial }^{2}f}{\partial x^2}$

(B) $\frac{{\partial }^{2}f}{\partial x\partial y}=\frac{{\partial }^{2}f}{\partial y^2}$

(C) $\frac{{\partial }^{2}f}{\partial x^2}=-\frac{{\partial }^{2}f}{\partial y^2}$

(D) $\frac{{\partial }^{2}f}{\partial x^2}=\frac{{\partial }^{2}f}{\partial y^2}$

1-5

$\underset{(x,y)\to (0,0)}{lim}(1+xy{)}^{\frac{1}{x^2+y^2}}$

(A) 0

(B) 1

(C) e

(D) does not exist

二、填空题

Fill in the blanks

2-1

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are unit vectors and $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}$, then $\overrightarrow{a}\cdot \overrightarrow{b}+\overrightarrow{b}\cdot \overrightarrow{c}+\overrightarrow{c}\cdot \overrightarrow{a}=$ $\underset{―}{}$

2-2

If the vector $\overrightarrow{c}$ is perpendicular to $\overrightarrow{a}=⟨1,2,1⟩$ and $\overrightarrow{b}=⟨-1,1,1⟩$ and $\overrightarrow{c}\cdot ⟨\overrightarrow{i}+2\overrightarrow{j}+\overrightarrow{k}⟩=16$, then $\overrightarrow{c}=$ $\underset{―}{}$

2-3

If $\sum _{n=2}^{\mathrm{\infty }}(\tan \frac{1}{n}-k\ln (1-\frac{1}{n}))$ converges, then $k=$ $\underset{―}{}$

2-4

The maximum curvature $\kappa$ of function $y(x)=\sin x$ is $\underset{―}{}$

2-5

If $(z+y{)}^x=xy$, then $\frac{\partial z}{\partial x}(1,2)=$ $\underset{―}{}$

三

Given a cardioid $r=a(1+\cos \theta )$, $a>0$ and a circle $r=a$

3-1

Find the area of the region that lies inside the cardioid and outside the circle

3-2

Find the area of the region that lies inside the cardioid and inside the circle

四

Assume $\overrightarrow{r}(t)=f(t)\overrightarrow{i}+g(t)\overrightarrow{j}+h(t)\overrightarrow{k}$, where

$$f(t)={\int }_0^t\cos (x^2)dx,g(t)=-t\cos t,h(t)=\sum _{n=1}^{\mathrm{\infty }}\frac{t^n}{n}$$

Calculate ${\overrightarrow{r}}^{\prime }(0)$

五

Let $f(x,y)=\{\begin{array}{ll}y\sin \frac{1}{x^2+y^2},& (x,y)\ne (0,0)\\ 0,& (x,y)=(0,0)\end{array}$

5-1

Is $f(x,y)$ continuous at $(0,0)$ ?

5-2

Find $f_x(0,0)$ and $f_y(0,0)$, if they exist

六

Find the limit, if it exists, or show that the limit does not exist

6-1

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

6-2

$\underset{(x,y)\to (0,0)}{lim}\frac{x{y}^3+2x^2{y}^4}{x^2+y^6}$

七

For the power series $f(x)=\sum _{n=1}^{\mathrm{\infty }}\frac{n+2}{n(n+1)}{x}^n$,

7-1

For what values of $x$ does the power series converge?

7-2

Find the sum of the series within the interval of convergence

八

Determine if the series, $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^{n+1}}{n^p(\ln n{)}^2}$ $(p>0)$, converges absolutely, or converges conditionally or diverges. Give reasons for your answer

九

Find $\underset{n\to \mathrm{\infty }}{lim}((n^2-n)e^{\frac{1}{n}}-\sqrt{n^4+1})$