题目

Problem

Figure 1 shows a sketch of part of the curve $C_1$ with equation $y=4\cos x^\circ$ .

Figure 1

The point $P$ and the point $Q$ lie on $C_1$ and are shown in Figure 1.

(a) State

(i) the coordinates of $P$ ,

(ii) the coordinates of $Q$ .

(3)

The curve $C_2$ has equation $y=4\cos x^\circ+k$ , where $k$ is a constant.

Curve $C_2$ has a minimum $y$ value of $-1$ .

The point $R$ is the maximum point on $C_2$ with the smallest positive $x$ coordinate.

(b) State the coordinates of $R$ .

(2)

解答

展开

$y=4\cos x^\circ$ 的振幅是 $4$ 。最小值 $-4$ 在 $x=180^\circ+360^\circ n$ ，图中左侧最小点是 $x=-180^\circ$ 。右侧的 $x$ 轴交点是 $x=450^\circ$ 。

展开

The point $P$ is

\begin{align*} P=(-180,-4). \end{align*}

The point $Q$ is

\begin{align*} Q=(450,0). \end{align*}

展开

原函数 $4\cos x^\circ$ 的最小值是 $-4$ 。加上 $k$ 后最小值为 $-1$ ，所以 $k=3$ 。最大值因此是 $4+3=7$ 。最小正 $x$ 的最大点在 $x=360^\circ$ 。

展开

The minimum value of $4\cos x^\circ$ is $-4$ . Since the minimum value of $C_2$ is $-1$ ,

\begin{align*} -4+k=&\,-1\\ k=&\,3. \end{align*}

So the maximum value of $C_2$ is

\begin{align*} 4+3=7. \end{align*}

The maximum point with the smallest positive $x$ coordinate is at $x=360$ . Hence

\begin{align*} R=(360,7). \end{align*}