AMC12 2003 B

AMC12 2003 B · Q23

AMC12 2003 B · Q23. It mainly tests Functions basics, Inequalities with floors/ceilings (basic).

The number of $x$-intercepts on the graph of $y=\sin(1/x)$ in the interval $(0.0001,0.001)$ is closest to

函数 $y=\sin(1/x)$ 的图像在区间 $(0.0001,0.001)$ 中的 $x$ 截距个数最接近于

(A) 2900 2900

(B) 3000 3000

(D) 3200 3200

(E) 3300 3300

Answer

Correct choice: (A)

正确答案：（A）

Solution

The function $f(x) = \sin x$ has roots in the form of $\pi n$ for all integers $n$. Therefore, we want $\frac{1}{x} = \pi n$ on $\frac{1}{10000} \le x \le \frac{1}{1000}$, so $1000 \le \frac 1x = \pi n \le 10000$. There are $\frac{10000-1000}{\pi} \approx \boxed{2900} \Rightarrow \mathrm{(A)}$ solutions for $n$ on this interval.

函数 $f(x)=\sin x$ 的零点形如 $\pi n$，其中 $n$ 为任意整数。因此在 $\frac{1}{10000} \le x \le \frac{1}{1000}$ 上，我们需要 $\frac{1}{x}=\pi n$，即 $1000 \le \frac 1x = \pi n \le 10000$。在该区间内满足条件的 $n$ 的个数为 $\frac{10000-1000}{\pi} \approx \boxed{2900} \Rightarrow \mathrm{(A)}$。

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