2021 秋高数上期末试题(回忆版)

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

1-1

Let $f(x)$ be a continuous function on $[-a,a],a>0$, then ${\int }_{-a}^{a}f(x)dx=$

(A) ${\int }_0^a(f(x)+f(-x))dx$

(B) ${\int }_0^a(f(x)-f(-x))dx$

(C) 0

(D) $2{\int }_0^{a}f(x)dx$

1-2

If $f(x)=\{\begin{array}{ll}\frac{2+e^{\frac{1}{x}}}{1+e^{\frac{1}{x}}},& x\ne 0,\text{ then at }x=0\text{, it is a }\\ 0,& x=0\end{array}$

(A) jump discontinuity

(B) removable discontinuity

(C) infinite discontinuity

(D) continuous point

1-3

If the function $f(x)$ has the third derivative at $x=x_0$, and $f^{\prime }(x_0)=f^{\prime \prime }(x_0)=0,f^{(3)}(x_0)>0$, then

(A) $f(x)$ has a local minimum at $x_0$

(B) $f(x)$ has a local maximum at $x_0$

(C) $f(x)$ has no local extremum at $x_0$

(D) None of $(A),(B)$ and $(C)$ is correct

1-4

If ${\int }_0^{x}f(t)dt=\frac{x^4}{2}$, then ${\int }_0^4\frac{1}{\sqrt{x}}f(\sqrt{x})dx=$

(A) 8

(B) 16

(C) 128

(D) 256

1-5

The number of real roots in $(0,1)$ for $5x-2-{\int }_0^x\frac{dt}{1+t^8}=0$ is

(A) 0

(B) 1

(C) 2

(D) greater than 2

二 Fill in the blanks

(1) If $f(x)=(x^2+1)(x^2+2)(x^2+3)(x^2+4)$, then $f^{(6)}(0)=$ $\underset{―}{}$

(2) The average value for $f(x)={\cos }^{4}x$ on $[0,\pi ]$ is $\underset{―}{}$

(3) Using Simpson's Rule with $n=4$ to estimate ${\int }_2^4\frac{1}{x-1}dx$, the approximation is $\underset{―}{}$

(4) If $\underset{x\to \mathrm{\infty }}{lim}(\frac{x+a}{x-a}{)}^x=8$, then $a=$ $\underset{―}{}$

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

三

Find the area of the surface generated by revering the curve $4y=x^2(1\le y\le 3)$ about the $y$-axis

四

Solve the following first-order linear differential equation

$$x{y}^{\prime }-y=2x\ln x,x>0$$

五

If the line $y=x$ is tangent to the curve $y={\log }_{a}x$, find the value of $a$

六

A isosceles triangle is to be inscribed in a circle of radius $R$ .What is the largest perimeter possible for the isosceles triangle? Please provide the reason

七

Find all values for $p$ such that the improper integral ${\int }_0^{\mathrm{\infty }}\frac{e^{-x}}{x^P}dx$ converges

八 Evaluate the following limits

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

(2) $\underset{x\to 0}{lim}\frac{3\sin x+x^2\cos \frac{1}{x}}{(1+\cos x)\ln (1+x)}$

九 Evaluate the intergrals

(1) ${\int }_{\frac{1}{e}}^e\frac{{\ln }^{2}x}{x}dx$

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

(3) ${\int }_1^{\mathrm{\infty }}\frac{1}{x^6(x^5+4)}dx$

(4) $\int \frac{1}{(1+x+x^2{)}^2}dx$