2019秋高数上期末(回忆版)

一 Determine whether the following statements are true or false? No justification is necessary

(1) If $k>0$, then ${\ln }^{100}x<x^{0.0001}<2^{kx}$ for sufficiently large $x$

(2) If $f$ is continuous on $R$, then ${\int }_0^{a}f(a-x)dx={\int }_0^{a}f(x)dx$

(3) If the graph of a differentiable function $f(x)$ is concave up on an open interval $(a,b)$, then $f(x)$ has a local minimum value at a point $c\in (a,b)$ if and only if $f^{\prime }(c)=0$

(4) If $|f(x)|$ is continuous at $x=a$, then so is $(f(x){)}^2$

(5) Suppose that $f(a)=g(a)=0$, that $f$ and $g$ are differentiable on an open interval I containing $a$, and that $g^{\prime }(x)\ne 0$ on I if $x\ne a$ .If $\underset{x\to a}{lim}\frac{f^{\prime }(x)}{g^{\prime }(x)}$ does not exist, then neither does $\underset{x\to a}{lim}\frac{f(x)}{g(x)}$

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

2-1

If $g(x)$ is one-to-one, and $g(1)=3,g(3)=1,g^{\prime }(1)=4,g^{\prime }(3)=28$, then $(g^{-1}{)}^{\prime }(3)=$

(A) $\frac{1}{4}$

(B) $\frac{1}{28}$

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

(D) 4

2-2

Let $c>0$ .How many real roots are there for the equation $x^3-6x^2+9x+c=0$ ?

(A) 0

(B) 1

(C) 2

(D) 3

2-3

Suppose $\underset{x\to 0^{+}}{lim}f(x)=a,\underset{x\to 0^{-}}{lim}f(x)=b$, then $\underset{x\to 0^{-}}{lim}(f(x-\sin x)+2f(x^2+x))=$

(A) $a+2b$

(B) $b+2a$

(C) $3a$

(D) $3b$

2-4

If $f(x)=\frac{\ln |x|}{|x-1|}\sin x$, then the function $f(x)$ has

(A) 1 removable discontinuity and 1 jump discontinuity

(B) 2 removable discontinuities

(C) 1 removable discontinuity and 1 infinite discontinuity

(D) 2 jump discontinuities

2-5

Let $f(x)$ be a continuous function, and $a$ is a nonzero constant.Which of the following function is an odd function?

(A) ${\int }_a^x({\int }_0^{u}tf(t^2)dt)du$

(B) ${\int }_0^x({\int }_a^{u}f(t^3)dt)du$

(C) ${\int }_0^x({\int }_a^{u}tf(t^2)dt)du$

(D) ${\int }_a^x({\int }_0^u(f(t){)}^{2}dt)du$

三

If the function $f(x)=\{\begin{array}{l}a\cdot \sin x,x\le \frac{\pi }{4}\\ 1+b\cdot \tan x,\frac{\pi }{4}<x<\frac{\pi }{2}\end{array}$

is differentiable at $x=\frac{\pi }{4}$, find the values of $a$ and $b$

四 Evaluate the following limits

(1) $\underset{x\to 0}{lim}\frac{{\tan }^{-1}x-x}{x{\tan }^{2}x}$

(2) $\underset{x\to \mathrm{\infty }}{lim}\frac{(x+100{)}^{100x}}{x^{100x}}$

五

Find the area of the region enclosed by the curve $y=|x^2-4|$ and $y=\frac{x^2}{2}+4$

六

The graph of the equation $x^{\frac{2}{3}}+y^{\frac{2}{3}}=1$ is an astroid. Find the area of the surface generated by revolving the curve about the $x$-axis

七

The point $P(a,b)$ lies on the curve $l:(y-x{)}^3=y+x$, and the slope of the tangent line of $l$ at $P(a,b)$ is 3 . Find the values of $a$ and $b$

八

Find $f^{\prime }(2)$ if $f(x)=e^{g(x)}$ and $g(x)={\int }_2^{\frac{x^2}{2}}\frac{t}{1+t^4}dt$

九 Evaluate the integrals

(1) $\int \frac{dx}{\sqrt{1+e^x}}$

(2) $\int \frac{3x+6}{(x-1{)}^2(x^2+x+1)}dx$

(3) ${\int }_{\frac{\pi }{6}}^{\frac{\pi }{3}}{\tan }^{2}x\sec xdx$

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

十

An $1600-L$ tank is half full of fresh water; i.e., contains $800-L$ of fresh water. At the time $t=0$, a solution containing $0.0625\mathrm{kg}/\mathrm{L}$ of salt runs into the tank at the rate of $16\mathrm{L}/\min$, and the mixture is pumped out of the tank at the rate of $8\mathrm{L}/\min$. At the time the tank is full, how many kilograms of salt will it contain?

十一

$f(x)$ is differentiable, and $f^{\prime }(x)>0$ on $(0,+\mathrm{\infty })$. Let $F(x)={\int }_{\frac{1}{x}}^{1}xf(u)du+{\int }_1^{\frac{1}{x}}\frac{f(u)}{u^2}du$

11-1

Identify the open intervals on which $F(x)$ is decreasing and the open intervals on which $F(x)$ is increasing

11-2

Find the open intervals on which the graph of $y=F(x)$ is concave up and the open intervals on which it is concave down

十二

Let $g$ be a function that is differentiable throughout an open interval containing the origin. Suppose $g$ has the following properties:

(i) $g(x+y)=\frac{g(x)+g(y)}{1-g(x)g(y)}$ for all real numbers $x,y$, and $x+y$ in the domain of $g$

(ii) $\underset{h\to 0}{lim}g(h)=0$

(iii) $\underset{h\to 0}{lim}\frac{g(h)}{h}=1$

Find $g(x)$