2022秋高数上期末试题(回忆版)

1

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

1-(1)

If $f(x)=\frac{ax+b}{x^2-1}$ has a local extreme of 1 at $x=3$, then

(A) $a=3,b=-1$

(B) $a=4,b=-4$

(C) $a=5,b=-7$

(D) $a=b,b=-10$

1-(2)

Let $f(x)=\frac{\tan x}{|x|{(x-\frac{\pi }{2})}^4}$ .Which of the following statements must be correct?

(A) $f$ is continuous at $x=0$ and $f$ has a jump discontinuity at $x=\frac{\pi }{2}$

(B) $f$ has a jump discontinuity at $x=0$ and $f$ is continuous at at $x=\frac{\pi }{2}$

(C) $f$ has an infinite discontinuity at $x=0$ and $f$ has an oscillating discontinuity at $x=\frac{\pi }{2}$

(D) f has a jump discontinuity at $x=0$ and $f$ has an infinite discontinuity at $x=\frac{\pi }{2}$

1-(3)

The number of asymptotes of $y=e^{\frac{1}{x}}\arctan \frac{x^2+x+1}{(x-1)(x+2)}$ is

(A) 1

(B) 2

(C) 3

(D) 4

1-(4)

Which of the following improper integrals is divergent?

(A) ${\int }_{-1}^1\ln (x)dx$

(B) ${\int }_0^1\frac{e^x}{\sqrt{1-x}}dx$

(C) ${\int }_0^{+\mathrm{\infty }}\frac{1}{x^2}dx$

(D) ${\int }_e^{+\mathrm{\infty }}\frac{1}{x{\ln }^{2}x}dx$

1-(5)

Suppose that $a<0<b$, and $f(x)$ is continuous on $(a,b)$ .Let $F(x)=\{\begin{array}{ll}\frac{{\int }_0^{x}f(t)dt}{x},& x\ne 0\\ 0,& x=0.\end{array}$ Which of the following statements must be correct?

(A) $F$ is differentiable on $(a,b)$ and $F^{\prime }$ is not continuous at $x=0$

(B) $F$ is differentiable on $(a,b)$ and $F^{\prime }$ is continuous at $x=0$

(C) $F$ is not differentiable on $(a,b)$ and $F$ is continuous at $x=0$

(D) None of the above statements is correct

2.Fill in the blanks

2-(1)

The number of the real roots for the equation $x^3-4x^2+x+1=0$ is $\underset{―}{}$

2-(2)

If $f(x)=(1+x)(1+2x)\cdots (1+10x)$, then $f^{\prime }(0)=$ $\underset{―}{}$

2-(3)

Use Euler＇s method to find the approximation for the solution of

$$y^{\prime }=1+xy,y(0)=1$$

Take $dx=0.5$, and start at $x_0=0,y_0=1$ .Then $y_2=$ $\underset{―}{}$

2-(4)

If $f(x)=\arctan \frac{1+x}{1-x}$, then $f^{\prime }(0)=$ $\underset{―}{}$

2-(5)

If ${\int }_0^1\frac{e^x}{x+1}\cdot dx=a$, then ${\int }_0^1\frac{e^x}{(x+1{)}^2}dx=$ $\underset{―}{}$

3

Solve the following first-order linear differential equation

$$x{y}^{\prime }+2y=x^2+1,x>0$$

4

4-(1)

Find the area of the region enclosed by the curves $y=x^2-2x,y=0,x=1$, and $x=3$

4-(2)

Find the volume of the solid generated by revolving the region in(1)about the $y$-axis

5

Assume that $f$ is differentiable at $x=1$, and $f(1)=1,f^{\prime }(1)=2$ .Find the value of

$\underset{n\to \mathrm{\infty }}{lim}{\left(f(1+\frac{1}{n})\right)}^n$

6

Evaluate the following limits

6-(1)

$\underset{x\to 0^{+}}{lim}\frac{\ln \tan 7x}{\ln \tan 2x}$

6-(2)

$\underset{n\to \mathrm{\infty }}{lim}\sum _{k=1}^n\frac{k}{n^2}{\sin }^2(1+\frac{k}{n})$

7

Evaluate the integrals

7-(1)

$\int x{\tan }^{2}xdx$

7-(2)

${\int }_0^{+\mathrm{\infty }}\frac{{\tan }^{-1}{x}^x}{e^{2x}}dx$

7-(3)

${\int }_1^{\sqrt{3}}\frac{1}{x{(1+x^2)}^2}dx$

7-(4)

$\int \frac{\ln (1-x^2)}{x^2\sqrt{1-x^2}}dx$

8

Assume $f(x)$ is continuous on $[0,1]$ and differentiable on $(0,1)$. If $f(0)=f(1)=0$, $f\left(\frac{1}{2}\right)=1$, prove that:

8-(1)

there exists $c\in (\frac{1}{2},1)$, such that $f(c)=c$

8-(2)

For any real number $k$, there always exists $\xi \in (0,c)$, such that $f^{\prime }(\xi )-k|f(\xi )-\xi |=1$