MA117 2022秋 期中

2022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

Which of the following functions is differentiable at $x=0$ ?

(A) $|x|\sqrt{\sin x+2}$

(B) $|x|+\sqrt{\sin x+2}$

(C) $|x|\sin x$

(D) $|x|+\sin x$

1-2

Which of the following functions has an oblique asymptote?

(A) $\frac{\sqrt{2x^3+x+1}}{x+1}$

(B) $\frac{x^4+1}{x^3+\sin x}$

(C) $x+\sin x$

(D) $x+\frac{1}{2+\sin x}$

1-3

$${\int }_{\frac{\pi }{6}}^{\frac{\pi }{2}}(\frac{\cos x}{{\sin }^{2}x}+\cos x)dx=$$

(A) $\frac{3}{2}$

(B) $\frac{2}{3}$

(C) $-\frac{3}{2}$

(D) $-\frac{2}{3}$

1-4

Let $f(x)=\{\begin{array}{ll}\frac{1-\cos x}{x},& x>0\\ x\sin \frac{1}{x-1},& x\le 0\end{array}$

Which of the following must be true?

(A) $\underset{x\to 0}{lim}f(x)$ does not exist

(B) $\underset{x\to 0}{lim}f(x)$ exists and $f$ is not continuous at $x=0$

(C) $f$ is continuous at $x=0$ and $f$ is not differentiable at $x=0$

(D) $f$ is differentiable at $x=0$

1-5

If there is a jump discontinuity for the function $y=f(x)$ at $x=0$, then which of the following limits must exist?

(A) $\underset{x\to 0}{lim}f(x^2)$

(B) $\underset{x\to 0}{lim}(f(x){)}^2$

(C) $\underset{x\to 0}{lim}f(x^3)$

(D) $\underset{x\to 0}{lim}(f(x)-f(-x))$

二

Fill in the blanks

2-1

$$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}(\sin \frac{\pi }{n}+\sin \frac{2\pi }{n}+\cdots +\sin \frac{n\pi }{n})=\underset{―}{}$$

2-2

If the line $y=9x+b$ is a tangent line of the curve $y=x^3-3x$, then $b=$ $\underset{―}{}$

2-3

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

2-4

$$\underset{x\to 0}{lim}\frac{x\tan x}{1-\cos x}=\underset{―}{}$$

2-5

Let $f(x)=\tan x$ .Then $f^{(4)}(0)=$ $\underset{―}{}$

三

Prove that there is only one real root for the equation $x^5+2x-100=0$

四

Compute

$${\int }_0^1(1+x{)}^2(1-x{)}^{5}dx$$

五

Find the linear approximation of $f(x)=\frac{2}{1-x}+\sqrt{1+x}$ at $x=0$

六

Find the constants $a$ and $b$ such that the function $f(x)=\{\begin{array}{ll}\frac{2x^2-x+b}{x-1},& x>1\\ a,& x\le 1\end{array}$ is continuous at $x=1$

七

Let $f(x)=\frac{x^3}{2(x-1{)}^2}$

7-1

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

7-2

Identify the horizontal, vertical and oblique asymptotes that may exist

7-3

Sketch the graph

八

Let $y=f(x)$ be an implicit function defined by the equation $2y^3-y^2+3xy-2x^2-2=0$. Find the equation of the tangent line to the curve $y=f(x)$ at $x=1$

九

Let $f$ be continuous on $(-\mathrm{\infty },\mathrm{\infty })$ and define $F(x)={\int }_0^{x}xtf(x^2-t^2)dt$. Find $F^{\prime }(x)$