2020秋高数上期末(回忆版)

一

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

1-1

The number of the real roots for the equation $x^3-3x+3=0$ is

(A) 0

(B) 1

(C) 2

(D) 3

1-2

If $f(x)$ is continuous on $(-\mathrm{\infty },+\mathrm{\infty })$, which of the following statements is wrong?

(A) ${\int }_0^{1}f(x)dx={\int }_0^{1}f(t)dt$

(B) ${\int }_0^{1}f(x)dx={\int }_0^{1}f(\sin x)d(\sin x)$

(C) $d({\int }_0^{x}f(t)dt)=f(x)dx$

(D) $d({\int }_0^{x^2}f(t)dt)=f(x^2)d(x^2)$

1-3

Let $f(x)=\{\begin{array}{ll}{x}^4\sin \frac{1}{x}& x\ne 0\\ 0,& x=0\end{array}$

Then the largest positive integer $n$, for which $f^{(n)}(0)$ exists, is

(A) 1

(B) 2

(C) 3

(D) 4

1-4

If $f(x)$ is twice-differentiable on $(-\mathrm{\infty },+\mathrm{\infty })$, and $g(x)=(1-x)f(0)+xf(1)$, then which of the following statements is correct on $(0,1)$ ?

(A) $f(x)>g(x)$ if $f^{\prime }(x)>0$

(B) $f(x)>g(x)$ if $f^{\prime }(x)<0$

(C) $f(x)>g(x)$ if $f^{\prime \prime }(x)>0$

(D) $f(x)>g(x)$ if $f^{\prime \prime }(x)<0$

1-5

If the improper integal ${\int }_0^{+\mathrm{\infty }}\frac{{\tan }^{-1}(x^2)}{x^k}dx$ converges, then the constant $k$ must satisfy

(A) $k<1$

(B) $k>3$

(C) $1<k<2$

(D) $1<k<3$

二

Fill in the blanks

2-1

function $f(x)=x^2$ has a tangent line $y=kx-1$ if $k=$ $\underset{―}{}$, or $\underset{―}{}$

2-2

Assume that $f^{\prime }(0)=3,f^{\prime \prime }(0)=5,f^{\prime }(1)=-4$, and $f^{\prime \prime }(1)=-7$ .Let $g(x)=f(\ln x)$ .Then $g^{\prime \prime }(1)=$ $\underset{―}{}$

2-3

The average value for $f(x)={\sin }^{3}x$ on $[0,\pi ]$ is $\underset{―}{}$

2-4

Let $y=(\cos x{)}^x$ for $0<x<\frac{\pi }{2}$, then $y^{\prime }(x)=$ $\underset{―}{}$

2-5

If $f^{\prime \prime }(a)$ exists, and $f^{\prime }(a)\ne 0$, then $\underset{x\to a}{lim}(\frac{1}{f^{\prime }(a)(x-a)}-\frac{1}{f(x)-f(a)})=$ $\underset{―}{}$

三

The region $D$ is enclosed by the curve $y=\ln \sqrt{x-1}$, the straight line $x=5$, and the $x$-axis

3-1

Find the area of the region $D$

3-2

Find the volumes generated by revolving the region $D$ about the line $x=5$

四

Find the particular solution of

$$x{y}^{\prime }+(x-2)y=3x^3{e}^{-x},x>0$$

satisfying $y(1)=0$

五

Evaluate the following limits

5-1

$\underset{n\to +\mathrm{\infty }}{lim}(\frac{n}{2n^2+3n+1^2}+\frac{n}{2n^2+6n+2^2}+\cdots \frac{n}{2n^2+3nk+k^2}+\cdots +\frac{n}{2n^2+3n^2+n^2})$

5-2

$\underset{x\to 0}{lim}(\frac{\ln (1+x)}{x}{)}^{\frac{1}{e^x-1}}$

六

6-1

For $y=\frac{x^2+1}{x+1}$, identify the coordinates of any local and absolute extreme points and inflection points that may exist

6-2

Sketch the graph of the above function. (Please identify all the asymptotes and some specific points, such as local maximum and minimum points, inflection points, and intercepts.)

七

Find $\frac{dy}{dx}$ if $y={\int }_{x^2+1}^{2x^2+3}t\tan \sqrt{x+t}dt$

八

Evaluate the integrals

8-1

${\int }_{\frac{\pi }{3}}^{\frac{\pi }{6}}{\csc }^{3}xxdx$

8-2

$\int \sqrt{\frac{x}{x-2}}dx$, where $x>2$

8-3

${\int }_1^e{\ln }^{3}xdx$

8-4

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

九

Let $f(n)=\sum _{m=1}^n{\int }_0^m\cos \frac{2\pi n\lfloor x+1\rfloor }{m}dx$, here $\lfloor x+1\rfloor$ is the largest integer which is less than or equal to $x+1$. Evaluate $f(2021)$