Sichuan University 2008 years of Mathematical analysis of postgraduate examination questions

One, limit (7 points per small question, total 28 points)

1.$\displaystyle \lim\limits_{x\to +\infty} e^{-x}\left (1+\frac{1}{x}\right) ^{x^{2}}$

2.$\displaystyle \lim\limits_{n\to \infty} ne^{\frac{1}{n}}-n^{2}\ln (1+\frac{1}{n}) $

3.$\displaystyle \lim\limits_{n\to \infty}\left (n!\right) ^{\frac{1}{n^{2}}}$

4.$\displaystyle \lim\limits_{x \to 0}\frac{\cos x-e^{-\frac{x^{2}}{2}}}{x^{2}[x+\ln (1-x)]}$

Second, calculate or prove the following questions (10 points per small question, total 60 points).

1. When $x\le 0$, $f (x) =1+x^{2}$; when $x>0$, $f (x) =xe^{-x}$. $\displaystyle \int_{1}^{3}f (x-2) dx$.

2. Set $\displaystyle f ' (2^{x}) =x2^{-x},f (1) =0$, seeking $f (x) $.

3. Calculate curvature area $\displaystyle i= \iint\limits_{s} (x+y+z) ds$, where surface $s=\{(x, Y, z) \in R^{3}\mid
X^{2}+y^{2}+z^{2}=a^{2},z\ge 0\}$

4. Calculate the curve integral $\displaystyle i=\int\limits_{amb}\left (\varphi (y) e^{x}-my\right) dx+\left (\varphi ' (y) e^{x}-m\right) dy$. where $\varphi (y) $, $\varphi ' (y) $ is a continuous function on $r$, $AmB $ for the connection point $ A ($), B (3,4) for any path (direction from A to B), but it is set to $ab$ ($p) $ for the area of the line p>0.


5. Calculate the surface area of $\displaystyle i=\iint\limits_{s}\left (x^{2} \cos \alpha +y^{2}\cos \beta +z^{2}\cos \gamma \right) dS$, where $s$ For conical faces $x^{2}+y^{2}=z^{2},0\le z \le h,\cos \alpha, \cos\beta, \cos \gamma$ is the direction cosine of the outer normal vector $\overrightarrow{n}$ of the surface.

6. Function $z=z (x, y) $ with second-order continuous biasing and satisfying equation

$ $q (1+q) \frac{\partial ^{2} z}{\partial x^{2}}-(1+P+Q+2PQ) \frac{\partial ^{2} z}{\partial x \partial y}+p (1+p) \frac{\ Partial ^{2} z}{\partial y^{2}}=0$$

where $\displaystyle p=\frac{\partial z}{\partial x},q=\frac{\partial z}{\partial y}$. Under the assumption $u=x+y,v=y+z,w=x+y+z$, prove that:

$$\displaystyle \frac{\partial ^{2} w}{\partial u\partial v}=0$$

Third, (10 points) set $f (x) $ on $[0,1]$ with a continuous derivative, proving that:

$$\lim\limits_{n\to \infty}n\int_{0}^{1}x^{n}f (x) dx=f (1) $$



Iv. (10 points) set $f (x) $ in $ (a, a) $ Nei micro, Proof: Existence $c\in (A, a) $ makes $ $f (a) -2f\left (\frac{a+b}{2}\right)-F (b) =\frac{(b-a) ^{2}}{4}f ' ' (c) $$

V. (10 points) set $f (x) $ within $ (b) $ with continuous derivative and $f (a) =f (b) =0$, proving: $$\max\limits_{a\le x\le b}\left|f ' (x) \right| \ge \frac{4}{(b-a) ^{2}}\int_{a}^{b}\left| F (x) \right| dx.$$

Six, (12 points) (error) Set $x>0,y>0,z>0$, Proof: $$3\left (x+y+z+\frac{1}{x+y+z}\right) ^{2}\le \left (x+\frac{1}{x}\right) ^{2}+\left (y+\frac{1}{y}\right) ^{2}+\left (z+\frac{1}{z}\right) ^{2}.$$

Seven (20 points) set $f (x) $ on $-\infty<x<+\infty$, with a second-order continuous derivative in a field of $x=0$, and $\displaystyle \lim\limits_{x\to 0}\frac{f (x) }{x}=a\in R $. Proof:

1. If $a>0$, then the series $ \sum\limits_{n=1}^{\infty} ( -1) ^{n}f\left (\frac{1}{n}\right) $ convergence, series $ \sum\limits_{n=1}^{\infty}f\ Left (\frac{1}{n}\right) $ divergence.

2. If $a=0$, then the series $ \sum\limits_{n=1}^{\infty}f\left (\frac{1}{n}\right) $ absolute convergence.

Sichuan University 2008 years of Mathematical analysis of postgraduate examination questions