MA127-2019春 期中考试

一

1-1

If $a_n<0$ for $n>1000$, and $\sum _{n=1}^{\mathrm{\infty }}{a}_n$ converges, then $\sum _{n=1}^{\mathrm{\infty }}{a}_n^2$ may diverge

1-2

Suppose that the power series $\sum _{n=1}^{\mathrm{\infty }}{a}_n(x-1{)}^n$ converges at $x=0$, and diverges at $x=2$, then the interval of convergence of this series is $[0,2)$

1-3

If $f(x,y)$ has partial derivatives $f_x(x,y)$ and $f_y(x,y)$ at $(x_0,y_0)$, then $\underset{x\to x_0}{lim}f(x,y_0)=\underset{y\to y_0}{lim}f(x_0,y)=f(x_0,y_0)$

1-4

If a vector function $r(t)$ is always perpendicular to its derivative $\frac{dr}{dt}$, then $|r(t)|$ must be constant

1-5

The curvature of a unit circle is greater than the curvature of the parabola $y=x^2$ at the origin

二 Multiply Choice Questions

Only one Correct answer for each of the following questions

2-1

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two nonzero orthogonal vectors, which of the following must be true?

(A) $|\overrightarrow{a}+\overrightarrow{b}|=|\overrightarrow{a}|+|\overrightarrow{b}|$

(B) $|\overrightarrow{a}-\overrightarrow{b}|=|\overrightarrow{a}|-|\overrightarrow{b}|$

(C) $|\overrightarrow{a}+\overrightarrow{b}|=|\overrightarrow{a}-\overrightarrow{b}|$

(D) $\overrightarrow{a}+\overrightarrow{b}=\overrightarrow{a}-\overrightarrow{b}$

2-2

The equations of two lines are $l_1:x=t,y=2t,z=-t$, and $l_2:x=1-2t,y=t,z=-1+t$. Then $l_1$ and $l_2$ are

(A) paralle

(B) orthogonal

(C) intersect with each other

(D) skew

2-3

Suppose $0\le a_n\le \frac{1}{n},(n=1,2,\cdots )$, then which of the following series must converge?

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

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

(C) $\sum _{n=1}^{\mathrm{\infty }}\sqrt{a_n}$

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

三

Does the following series absolutely converge, conditionally converge or diverge? Give reasons for your answers

3-1

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

3-2

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

3-3

$\sum _{n=2019}^{\mathrm{\infty }}(-1{)}^n\frac{1}{\sqrt{n^2-2019n+1}}$

四

4-1

Find the radius and interval of convergence of the series

$$\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^{n+1}{2}^n{x}^n}{\sqrt{n^2+n+1}}$$

4-2

For what values of $x$ does the series converge absolutely, or conditionally?

五

Let $x={\cos }^{3}t,y={\sin }^{3}t$, where $0\le t\le \frac{\pi }{2}$, be a parametrization of a curve

5-1

Find the length of the curve

5-2

Find the area of the surface generated by revolving the curve about the $x$-axis

六

Find the equation of the plane through the points $(2,-1,-1)$ and $(1,0,-1)$ perpendicular to the plane $2x+3y-5z+6=0$

七

A particle is located at the point $(1,0,-2)$. Its initial speed is $|v(0)|=3$ at time $t=0$, and the direction of its initial velocity is toward the point $(2,-1,3)$. The particle moves with constant acceleration $\overrightarrow{i}+2\overrightarrow{j}+\overrightarrow{k}$. Find its position vector $\overrightarrow{r}(t)$ at time $t$

八

Is the following function, $f(x,y)$ continuous at $(0,0)$? Give reasons for your answer.

$$f(x,y)=\{\begin{array}{ll}\frac{\sin (x^3+y^2)}{x^2+y^2},& (x,y)\ne (0,0)\\ 0,& (x,y)=(0,0)\end{array}$$

九

Let $f(u)$ be differentiable, $z=f(e^{x}y)$ $(y\ne 0)$, and $\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}=1$. If $f(1)=0$, find $f(u)$.

十

Find the Taylor series for $f(x)=\ln (x+\sqrt{x^2+1})$ at $x=0$