MA117 2019秋 期中

2019秋高数上期中试题(回忆版)

一

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

1-1

If $f^{\prime }(x)$ is bound on $(0,1)$, so is $f(x)$

1-2

Let $f(x)$ be defined on $(-\mathrm{\infty },+\mathrm{\infty })$. There must be a local maximum point of $f(x)$ between two local minimum points of $f(x)$

1-3

If $f(x)$ is differentiable on $(-1,1)$, and $f(-1)=f(1)$, then $f^{\prime }(c)=0$ for some number $|c|<1$

1-4

If $f(x)$ is a continuous, even function on $[-1,1]$, then $g(x)={\int }_0^{x}f(t)dt$ is odd and differentiable on $[-1,1]$

1-5

If $f(x)$ is a continuous, periodic function on $\mathbb{R}$ (T is the period), then $g(x)={\int }_0^{x}f(t)dt$ is also a periodic function with the period $T$

二

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

2-1

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

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

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

(C) $\cos |x|$

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

2-2

Suppose that $f(x)$ is differentiable at $x=0$ and $f(0)=0$. Then $\underset{x\to 0}{lim}\frac{x^{2}f(x)-2f(x^3)}{x^3}=$

(A) $-2f^{\prime }(0)$

(B) $-f^{\prime }(0)$

(C) $f^{\prime }(0)$

(D) 0

2-3

Suppose that $f(x)$ has a second derivative and $f^{\prime }(0)=0$, $\underset{x\to 0}{lim}\frac{f^{\prime }\prime (x)}{x}=1$. Then

(A) $f(0)$ is a local minimum value

(B) $f(0)$ is a local maximum value

(C) $(0,f(0))$ is a point of inflection of the curve

(D) $(0,f(0))$ is neither a local extrema nor a point of inflection of the curve

2-4

Suppose that $f(x)$ is defined on $(-\mathrm{\infty },+\mathrm{\infty })$. Which of the following statements is equivalent to the statement that "$f(x)$ is differentiable at $x=a$"?

(A) $\underset{h\to 0}{lim}(f(a+h)+f(a-h)-2f(a))=0$

(B) $\underset{h\to 0}{lim}\frac{f(a+h)-f(a-h)}{2h}$ exists

(C) $\underset{h\to 0}{lim}\frac{f(a+h^2)-f(a)}{h^2}$ exists

(D) $\underset{h\to 0}{lim}\frac{f(a+h^3)-f(a)}{h^3}$ exists

2-5

Suppose that $f(x)>0$, $f^{\prime }(x)>0$, and $f^{\prime }\prime (x)>0$ for all $x\in [a,b]$. Let $M={\int }_a^{b}f(x)dx$, $N=f(a)(b-a)$, and $P=\frac{f(a)+f(b)}{2}(b-a)$. Then

(A) $N<P<M$

(B) $N<M<P$

(C) $M<N<P$

(D) $M<P<N$

三

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

3-1

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

3-2

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

3-3

Graph the function

四

Find the limits

4-1

$$\underset{x\to 1}{lim}(\frac{\sin 5x}{x}+\frac{x^3+x^2-2}{x^2+2x-3})$$

4-2

$$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}(\sqrt{1-(\frac{1}{n}{)}^2}+\sqrt{1-(\frac{2}{n}{)}^2}+\cdots +\sqrt{1-(\frac{n}{n}{)}^2})$$

五

Evaluate the definite integral

5-1

$${\int }_{-1}^1|a-t|t^{2}dt,\text{where }a\in (-1,1)$$

5-2

$${\int }_0^{\pi }\frac{\sin 2x}{\sqrt{1-\cos x}}dx$$

六

Find the volume of the solid generated by revolving the region bounded by $x=12(y^2-y^3)$ $(0\le y\le 1)$ and $y$-axis about the line $y=2$

七

Use the linear approximation of $f(x)=\tan x$ at $a=\frac{\pi }{6}$ to estimate the value of $\tan \frac{11\pi }{60}$. Comparing the estimation with the true value, which one is larger?

八

Find the area of the region in the first quadrant bounded on the left by the $y$-axis, below by the curve $x=2\sqrt{y}$, above left by the curve $x=(y-1{)}^2$, and above right by the line $x=3-y$

九

Let $F(x)={\int }_{2019}^{x^2}\cos (2t^2)dt$. Find all the critical points for $F(x)$ on $[-1,1]$

十

(Use Rolle's theorem to prove the mean value theorem.) If the function $f(x)$ is continuous on $[a,b]$, and differentiable on $(a,b)$, prove that there exists a number $c$ in $(a,b)$, such that

$$f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}$$