2023春高数下期中试题(回忆版)

1.Multiple Choice Questions: (only one correct answer for each of the following questions.) (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) $\frac{\pi }{6}$

(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 $\frac{1}{4}⩽t⩽\frac{1}{2}$ is

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

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

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

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

(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 $(0,1)$

(4) If $0⩽a_n⩽\frac{1}{n}(n=1,2,\cdots )$, 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) $\sum _{n=1}^{\mathrm{\infty }}(-1{)}^n\frac{a_n}{1+\ln n}$

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

converges and has the sum

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

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

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

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

2.Fill in the blanks

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

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

(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 $\underset{―}{}$

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

(5) The series

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

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

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

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

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

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

5. 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

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

(A) Determine whether the series

where

$$\frac{a}{\sqrt{(1+a)(1+b)}}+\frac{b^2}{\sqrt{(1+a^2)(1+b^2)}}+\frac{a^3}{\sqrt{(1+a^3)(1+b^3)}}+\frac{b^4}{\sqrt{(1+a^4)(1+b^4)}}+\cdots +\frac{c_n}{\sqrt{(1+a^n)(1+b^n)}}+\cdots \text{, }$$$$C_n=\{\begin{array}{ll}{a}^n,& \text{ if }n\text{ is odd. }\\ b^n,& \text{ if }n\text{ is even. }\end{array}$$

converges or diverges. Justify your answer !

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

7. Let $C$ be the curve given by

$$r(t)=\sqrt{2}\cos ti+\sin tj+\sin tk$$

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

(B) Find the curvature $k(t)$

8. Using known Taylor series expansions, write the Taylor series ofor the function

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

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