25-26-1-高等数学A（上）-期中

Question 1（35 marks）

1. $\lim_{x\to0}\frac{x}{\mathrm{e}^x-1}=$ 【暂无答案】

2. $\lim_{n\to+\infty}\frac{4n^5+5n^4}{5n^5+3n^4+1}=$ 【暂无答案】

3.

$$f(x)= \begin{cases} x+a,&\text{if }x>0\\ \frac{\sin x}x,&x<0 \end{cases}$$

if $f(x)$ is continuous at $x=0$, find $a=$ 【暂无答案】

4. Find the tangent line of the equation $\dfrac{x^2}{16}+\dfrac{y^2}{9}=1$ at the point $(0,3)$.

5. $y=x^2\mathrm{e}^x$, find $f^{(5)}(0)=$ 【暂无答案】

6.

$$f(x)= \begin{cases} \mathrm e^{-\frac1{x^2}},&x\ne0\\ 1,&x=0 \end{cases}$$

$x=0$ is a 【暂无答案】 discontinuous point.

Question 2（15 marks）

Suppose that $\alpha,\beta$ are the roots of the equation $x^2 - x - 1 = 0$, and $a_n = \alpha^n + \beta^n$.

a) Prove that $a_{n+2}=a_{n+1}+a_n$. [5 marks]

b) Determine whether the sequence $\{a_n\}$ is convergent. If so, find its limit; otherwise, prove that it is not convergent. [5 marks]

c) Determine whether the sequence $\left\{\dfrac{a_{n+1}}{a_n}\right\}$ is convergent. If so, find its limit; otherwise, prove that it is not convergent. [5 marks]

Question 3（15 marks）

a)

$y = x^{\tan(x)} + \frac{\sqrt{x-1}}{(x+2)^2\sqrt[3]{2x-1}}$, find $y'$.

b)

$$f(x)= \begin{cases} x=\ln(1+t^2),\\ y=t-\arctan t, \end{cases}$$

if $f(x)$ is continuous at $x=0$, find $a=$【暂无答案】. Find the first derivative $\left.\frac{\mathrm{d}y}{\mathrm{d}x}\right|_{t=1}=$【暂无答案】, then find the second derivative $\left.\frac{\mathrm{d}^2y}{\mathrm{d}x^2}\right|_{t=1}=$【暂无答案】.

Question 4（15 marks）

Suppose that $f(x)$ is continuous on the closed interval $[a,b]$, differentiable in the open interval $(a,b)$, and $ab>0$. Prove that there exists at least one point $\xi\in(a,b)$ such that

$$\frac{ab}{b-a}\left|\begin{matrix}b&a\\f(a)&f(b)\end{matrix}\right|=\xi^{2}[f(\xi)+\xi f'(\xi)].$$

Question 5（20 marks）

Suppose that $f(x)$ has a continuous derivative, $f''(0)$ exists, and

$$\lim_{x\to0}\left(1+x+\frac{f(x)}{x}\right)^{1/x}=\mathrm{e}^3.$$

Find $f(0)$, $f'(0)$, $f''(0)$.