MA127-2019春 期中

1

单项选择题说明...

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

2

填空题说明...

2-1

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

(A) $|a+b|=|a|+|b|$ ；

(B) $|a-b|=|a|-|b|$ ；

(C) $|a+b|=|a-b|$ ；

(D) $a+b=a-b$

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) parallel；

(B) orthogonal；

(C) intersect with each other；(D)skew

2-3

Suppose $0⩽a_n⩽\frac{1}{n},(n=1,2,\cdots )$, 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) $\sum _{n=1}^{\mathrm{\infty }}(-1{)}^n{a}_n^2$

3

解答题说明...

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

$\sum _{n=2020}^{\mathrm{\infty }}(-1{)}^n\frac{1}{\sqrt{n^2-2019n+1}}$

4

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?

5

Let $x={\cos }^{3}t,y={\sin }^{3}t$, where $0⩽t⩽\frac{\pi }{2}$, 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

6

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

7

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 $i+2j+k$. Find its position vector $r(t)$ at time $t$

8

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

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

9

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

10

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