MA117 2020秋 期中

2020秋高数上期中试题(回忆版)

一

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

1-1

Let $f(x)=|x|\sin x$ .The greatest value of $n$, for which $f^{(n)}(0)$ exist, is $\underset{―}{}$

(A) 0

(B) 1

(C) 2

(D) 3

1-2

If $\underset{x\to \mathrm{\infty }}{lim}(\frac{x^2+1}{x+1}-ax-b)=\frac{1}{2}$, then the values of $a,b$ are $\underset{―}{}$

(A) $a=1,b=-\frac{3}{2}$

(B) $a=-1,b=\frac{3}{2}$

(C) $a=-1,b=1$

(D) $a=1,b=-1$

1-3

The average value of function $g(x)=x^2+6$, for $0\le x\le 6$ is $\underset{―}{}$

(A) 12

(B) 18

(C) 16

(D) 10

1-4

Which one of the following functions is not differentiable at $x=0$ ?

(A) $f(x)=|x|\sin |x|$

(B) $f(x)=|x|\sin \sqrt{|x|}$

(C) $f(x)=\cos |x|$

(D) $f(x)=\cos \sqrt{|x|}$

1-5

What is the derivative of $f(x)=\frac{1-\sin x}{1+\sin x}$ at $x=\frac{\pi }{6}$ ?

(A) $\frac{4\sqrt{3}}{9}$

(B) $-\frac{\sqrt{3}}{3}$

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

(D) $\frac{1}{3}$

二

Please fill in the blank for the questions below

2-1

The integration ${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}{\sin }^{5}x{\cos }^{3}xdx$ equals $\underset{―}{}$

2-2

If $f$ is continuous and ${\int }_0^{x^3-1}f(t)dt=x$, then $f(7)=$ $\underset{―}{}$

2-3

If $f^{\prime }(x)=(x+\frac{1}{x}{)}^2$ and $f(1)=1$, then $f(x)=$ $\underset{―}{}$

2-4

A particle is moving on the sphere $x^2+y^2+z^2={13}^2$ .While $t=t_0,x(t_0)=3,y(t_0)=4,z(t_0)=12$, $y^{\prime }(t_0)=3$, then $z^{\prime }(t_0)=$ $\underset{―}{}$

2-5

$\underset{s\to a}{lim}\frac{\sqrt{s^2+1}-\sqrt{a^2+1}}{s-a}=$ $\underset{―}{}$

三

Find the limits(DO NOT apply l'Hôpital's Rule.)

3-1

$\underset{x\to \mathrm{\infty }}{lim}\frac{\sqrt{3+x}-\sqrt{x+1}}{x^2+x-2}$

3-2

$\underset{x\to 0}{lim}\frac{\cos x-{\sec }^{2}x}{x\sin x}$

四

Evaluate the integral

4-1

${\int }_0^{2\pi }|{\sin }^{2}x-{\cos }^{2}x|dx$

4-2

${\int }_0^1(x+2)\sqrt{1-x^2}dx$

五

Let $f(x)=\frac{x^3+x-2}{x-x^2}$

5-1

Indentify the inflection points and local maxima and minima of the function that may exist

5-2

Identify the horizontal, vertical, and oblique asymptotes that may exist

5-3

Graph the function

六

6-1

Find $\frac{dy}{dx}$ if

$$y(x)={\int }_1^{1+2x}\sqrt{t^2-1}dt,x>0$$

6-2

Find the equation of the line that is tangent to the curve $x^2-y^2=9$ at point $(5,-4)$

七

Find the area of the region bounded by curves $y=x^2$ and $y=2x-x^2$

八

Find the volume of the solid generated by revolving the region bounded by $y=2x-1$, $y=\sqrt{x}$ and $y$-axis about the line $x=-1$

九

Assume that $f$ is continuous on $[0,1]$. Show that there exists a number $c\in (0,1)$ such that $f(c)={\int }_0^{1}f(x)dx$