2025 秋高数上期中试题（回忆版）

1. 单选

1. Equation $x^4-x-1=0$ has one root on the interval

• (A) $(0,\frac{1}{2})$
• (B) $(\frac{1}{2},1)$
• (C) $(1,2)$
• (D) $(2,3)$

2. Let $f(0)=0$. Which of the following conditions ensures that $f^{\prime }(0)$ exists?

• (A) ${\displaystyle \underset{h\to 0}{lim}\frac{f(1-\sqrt{1-h^2})}{h^2}}$ exists.
• (B) ${\displaystyle \underset{h\to 0}{lim}\frac{f(2h)-f(h)}{h}}$ exists.
• (C) ${\displaystyle \underset{h\to 0}{lim}\frac{f(h)-f(-h)}{2h}}$ exists.
• (D) ${\displaystyle \underset{h\to 0}{lim}\frac{f(h^2-h)}{h}}$ exists.

3. Assume that $y=f(x)$ and $y=\sin x$ have the same tangent line at the origin. Then

$$\underset{x\to +\mathrm{\infty }}{lim}\left(2xf\left(\frac{1}{3x}\right)\right)=$$

• (A) ${\displaystyle \frac{1}{36}}$
• (B) ${\displaystyle \frac{4}{9}}$
• (C) ${\displaystyle \frac{9}{4}}$
• (D) $36$

4. Suppose $f(x)>0$, $f^{\prime }(x)<0$, and $f^{″}(x)>0$ on $[a,b]$. Let

$$A_1={\int }_a^{b}f(x)dx,A_2=f(b)(b-a),A_3=\frac{1}{2}[f(a)+f(b)](b-a)$$

Then

• (A) $A_1<A_2<A_3$
• (B) $A_2<A_1<A_3$
• (C) $A_3<A_1<A_2$
• (D) $A_2<A_3<A_1$

5. Suppose $f(x)$ is differentiable at $x=a$, and $|f(x)|$ is not differentiable at $x=a$. Then

• (A) $f(a)=0,f^{\prime }(a)=0$
• (B) $f(a)=0,f^{\prime }(a)\ne 0$
• (C) $f(a)\ne 0,f^{\prime }(a)=0$
• (D) $f(a)\ne 0,f^{\prime }(a)\ne 0$

2. 填空

(1) $f(x)={\displaystyle {\int }_1^x(2-\frac{1}{t^{1/2}})dt}$ $(x>0)$ is decreasing on the open interval $\underset{―}{}$

(2) The linearization of $f(x)=(1-x{)}^{-5}(x+1{)}^{1/2}$ at $x=0$ is $L(x)=$ $\underset{―}{}$

(3) The asymptotes of the graph of the function $f(x)={\displaystyle \frac{x^2-1}{x}+x\sin \frac{1}{x}}$ are $\underset{―}{}$

(4) Given that $f(1)=8$ and $f^{\prime }(1)=3$. If

$$h(x)=\sqrt[3]{xf(x^2)},$$

then $h^{\prime }(1)=$ $\underset{―}{}$

(5) $f(x)$ is continuous on $(0,\mathrm{\infty })$, and

$${\int }_0^{x^2(1+x)}f(t)dt=x.$$

Then $f(2)=$ $\underset{―}{}$

3. Compute the first derivative of the following functions

(1) $y=\sin \sqrt{1+\sqrt{x}}$

(2) $y={\displaystyle \frac{x^3+1}{1-x^2}}$

4. Find all non-differentiable points of the function

$$f(x)=x^2|x^3-x|$$

on $\mathbb{R}$.

5. If $\sin x+xy+y^2=1$, find the value of ${\displaystyle \frac{d^{2}y}{d{x}^2}}$ at the point $(0,1)$

6. Let $f(x)={\displaystyle \frac{x}{1+x^2}}$

(1) Identify where the local extrema of $f$ occur. Find the function's local extreme values.

(2) Find the open intervals where the graph of $f$ is concave up and where it is concave down.

(3) Sketch the graph.

7. Find the constants $a$ and $b$ such that the function

$$f(x)=\{\begin{array}{ll}{\displaystyle \frac{{\sin }^2(3x)}{x^2+4x^3},}& x>0,\\ a,& x=0,\\ {\displaystyle \frac{\sqrt{|1+2x|}-1-x}{b{x}^2},}& x<0\end{array}$$

is continuous at $x=0$.

8. The equation

$$x^5-5x+k=0$$

has three distinct real roots. Find the range of $k$.