2019春高数下期末试题(回忆版)

1

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

1-(1)

If $a_n>0,{\mathrm{\forall }}_n$, and $\underset{n\to \mathrm{\infty }}{lim}n{a}_n=0$, then $\sum _{n=1}^{\mathrm{\infty }}{a}_n$ converges

1-(2)

The plane $x+y-2z=1$ is perpendicular to the plane $x+y+z=1$

1-(3)

If $f(x,y)$ has two local maxima, then $f$ must have a local minimum

2

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

2-(1)

Which one of the following series diverges?

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

(B) $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^{1+\frac{1}{n}}}$

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

(D) $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n{(3+(-1{)}^n\cdot 2)}^n}{6^n}$

2-(2)

The iterated integal ${\int }_0^{\frac{\pi }{2}}{\int }_0^{\cos \theta }f(r\cos \theta ,r\sin \theta )rdrd\theta$ can be written as

(A) ${\int }_0^1{\int }_0^{\sqrt{y-y^2}}f(x,y)dxdy$

(B) ${\int }_0^1{\int }_0^{\sqrt{1-y^2}}f(x,y)dxdy$

(C) ${\int }_0^1{\int }_0^{1}f(x,y)dydx$

(D) ${\int }_0^1{\int }_0^{\sqrt{x-x^2}}f(x,y)dydx$

2-(3)

For the function, $f(x,y)=\{\begin{array}{ll}\frac{2xy}{\sqrt{x^2+y^2}},& (x,y)\ne (0,0)\text{ ；which of the following statements is correct }\\ 0,& (x,y)=(0,0)\end{array}$

(A) $f$ is not continuous at $(0,0)$

(B) $f$ is continuous at $(0,0)$, but its partial derivative $f_x$ and $f_y$ do not exist at $(0,0)$

(C) Both partial derivatives $f_x$ and $f_y$ exist everywhere and are also continuous at $(0,0)$

(D) $f$ is not differentiable at $(0,0)$

2-(4)

For the critical points of the function $f(x,y)=2x^4+y^4-2x^2-2y^2$, which one of the following statements is correct?

(A) $(0,0)$ is a local minima

(B) $(0,1)$ is a local maxima

(C) $(0,-1)$ is a saddle point

(D) There are no local maxima among all the critical points

2-(5)

If the function $f(x,y)$ has the continuous first partial derivatives $\frac{\partial f}{\partial x}>0$ and $\frac{\partial f}{\partial y}<0,\mathrm{\forall }(x,y)\in R^2$, which one of the following statements is correct?

(A) $f(0,0)>f(1,1)$

(B) $f(0,0)<f(1,1)$

(C) $f(0,1)>f(1,0)$

(D) $f(0,1)<f(1,0)$

3

Please fill in the blank for the questions below

3-(1)

Compute the limit: $\underset{(x,y)\to (0,0)}{lim}\frac{\sqrt{x^2+y^2+1}-1}{x^2+y^2}=$ $\underset{―}{}$

3-(2)

The direction(unit vector) in which the function $f(x,y)=x^2+xy+y^2-y$ increases most rapidly at the point $(-1,+2)$ is $\underset{―}{}$

3-(3)

${\int }_0^1{\int }_y^1\frac{\tan x}{x}dxdy=$ $\underset{―}{}$

4

4-(1)

Find the interval of convergence of the series $\sum _{n=2}^{\mathrm{\infty }}\frac{(-1{)}^n(x-1{)}^{2n+1}}{\sqrt{n+9012}\ln n}$

4-(2)

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

5

The region $D$ is bounded by $z=\sqrt{x^2+y^2}$ and $z=\sqrt{1-x^2-y^2}$ .Consider the following integral ${\iiint }_D(x+z)dxdydz$,

5-(1)

Convert the above integral to an equivalent iterated integral in cylindrical coordinates;

5-(2)

Convert the above integral to an equivalent iterated integral in spherical coordinates

6

Assume we can put a cuboid into the ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$. Use the mothed of Lagrange multipliers to find the length, width and height of the cuboid such that it achieve the maximum volume

7

Find the equation of the osculating circle for the parabola $y=x^2$ at $x=1$

8

A solid in the first octant is bounded by the planes $y=0$ and $z=0$ and by the surfaces $z=4-x^2$ and $x=y^2$ (see the figure below). Its density function is $\delta (x,y,z)=xy$. Find the center of the mass for the solid

not graph, draw it by yourself

9

Use the substitution in double integal (please find the transformation by yourself) to evaluate the integral ${\iint }_D{e}^{\frac{y-x}{y+x}}dxdy$,

here $D$ is the triangular region bounded by the lines $x=0,y=0$, and $x+y=2$

10

Consider the line integral ${\int }_{(1,1,1)}^{(1,3,\pi )}{e}^x\ln ydx+(\frac{e^x}{y}+\sin z)dy+y\cos zdz$

10-(1)

Show that the differential form in the integral is exact

10-(2)

Evaluate the integral

11

Evaluate ${\iint }_s\mathrm{\nabla }x(4xj)\cdot nd\sigma$

where $S$ is the hemisphere $x^2+y^2+z^2=16,z⩾0$. Use the normal vectors pointed away from the origin

12

Find the outward flux of $F=(6x+y)i-(x+z)j+4yzk$ across the boundary of $D$, where $D$ is the region in the first octant bounded by the cone $z=\sqrt{x^2+y^2}$, the cylinder $x^2+y^2=1$, and the coordinate planes

13

The sequences $\left\{a_n\right\}$ and $\left\{b_n\right\}$ satisfy $0<a_n<\frac{\pi }{2},0<b_n<\frac{\pi }{2}$, and $\cos a_n-a_n=\cos b_n$, $n=1,2,3,\cdots$. The series $\sum _{n=1}^{\mathrm{\infty }}{b}_n$ converges. Show that $\underset{n\to \mathrm{\infty }}{lim}{a}_n=0$