已知 $f(x)=2\sin\frac{x}{2}$ 且 $g(x)=3\cos x+1$ ，其中 $0 \le x \le 2\pi$ 。

(a)
写出 $f(x)$ 的周期。
(b)
写出 $f(x)$ 的最小值。
(c)
写出 $g(x)$ 的最大值。
(d)
在同一坐标系中，画出 $0 \le x \le 2\pi$ 范围内 $y=f(x)$ 和 $y=g(x)$ 的函数图像。请清晰地标注你的图像。
(e)
方程 $f(x)=g(x)$ 在 $0 \\le x \\le 2\\pi$ 范围内的解为 $a$ 和 $b$ ，其中 $a<b$ 。用 $a$ 和 $b$ 表示满足 $f(x)>g(x)$ 的 $x$ 的取值范围。

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完整解法

10(a)

1. Identify the coefficient of $x$ and calculate the period

For $f(x) = 2\sin\left(\frac{1}{2}x\right)$ , the coefficient of $x$ is $\frac{1}{2}$ .
$\text{Period} = \frac{2\pi}{1/2} = 4\pi$

答案

$4\pi$

10(b)

1. Determine the smallest value of the function in the given domain

For $0 \le x \le 2\pi$ , the angle $\frac{x}{2}$ ranges from $0$ to $\pi$ .
In this interval, $0 \le \sin\frac{x}{2} \le 1$ .
Therefore, the smallest value of $2\sin\frac{x}{2}$ is $2(0) = 0$ .

答案

$0$

10(c)

1. Determine the maximum value of the cosine function

The maximum value of $\cos x$ is $1$ .

2. Calculate the largest value of $g(x)$

Largest value of $g(x) = 3(1) + 1 = 4$

答案

$4$

10(d)

1. Determine the key features of $y=f(x)=2\sin\frac{x}{2}$ for $0 \le x \le 2\pi$

Amplitude $= 2$ . Period $= \frac{2\pi}{1/2} = 4\pi$ .
For $0 \le x \le 2\pi$ , the graph is half of a sine wave.
Key points: $(0,0)$ , maximum at $(\pi, 2)$ , and $(2\pi, 0)$ .

2. Determine the key features of $y=g(x)=3\cos x+1$ for $0 \le x \le 2\pi$

Amplitude $= 3$ . Period $= 2\pi$ . Vertical shift $= +1$ .
Maximum value $= 3+1=4$ , minimum value $= -3+1=-2$ .
Key points: $(0,4)$ , $(\frac{\pi}{2}, 1)$ , $(\pi, -2)$ , $(\frac{3\pi}{2}, 1)$ , $(2\pi, 4)$ .

3. Sketch both graphs on the same axes and label them

Draw the axes. Plot the key points for $f(x)$ and draw a smooth curve. Plot the key points for $g(x)$ and draw a smooth curve. Label the curves $y=f(x)$ and $y=g(x)$ .

答案

A sketch showing $y=2\sin\frac{x}{2}$ as a half-sine wave from $(0,0)$ to $(2\pi,0)$ with a peak at $(\pi,2)$ , and $y=3\cos x+1$ as a full cosine wave shifted up by 1, starting at $(0,4)$ , passing through $(\frac{\pi}{2},1)$ , reaching a minimum at $(\pi,-2)$ , passing through $(\frac{3\pi}{2},1)$ , and ending at $(2\pi,4)$ . Both curves are labelled.

10(e)

1. Interpret the inequality $f(x) > g(x)$ graphically

The inequality $f(x) > g(x)$ corresponds to the region where the graph of $y = f(x)$ lies above the graph of $y = g(x)$ .

2. Determine the range of $x$ using the intersection points

From the sketch in part (d), the curve $y = 2\sin\frac{x}{2}$ starts at $(0,0)$ , reaches a maximum of $2$ at $x=\pi$ , and ends at $(2\pi, 0)$ . The curve $y = 3\cos x + 1$ starts at $(0,4)$ , goes down to a minimum of $-2$ at $x=\pi$ , and goes back up to $(2\pi, 4)$ .
The graphs intersect at $x=a$ and $x=b$ (where $a < b$ ). Between these two points, the sine curve $f(x)$ is above the cosine curve $g(x)$ .
Therefore, $f(x) > g(x)$ for $a < x < b$ .

答案

$a < x < b$

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已知 f(x)=2sin⁡x2f(x)=2\sin\frac{x}{2}f(x)=2sin2x​ 且 g(x)=3cos⁡x+1g(x)=3\cos x+1g(x)=3cosx+1，其中 0≤x≤2π0 \le x \le 2\pi0≤x≤2π。