189 lines
7.3 KiB
Plaintext
189 lines
7.3 KiB
Plaintext
ans
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024739
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$\{x|x\ne \dfrac{\pi}{6}+k\pi, \ k\in \mathbf{Z}\}$
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024740
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$[-1,\dfrac{5}{4}]$
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024742
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充分非必要
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024741
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$\dfrac{5\pi}{6}$
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024743
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$0$
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024744
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$[\dfrac{13\pi}{6},+\infty)$
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011448
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(1) $[0,\dfrac{\pi}{3}]$; (2) $\dfrac{24}{25}$
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004308
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(1) $AN=\dfrac{4\sqrt{3}}{3}\sin\theta$, $AM=\dfrac{4\sqrt{3}}{3}\sin(\theta+\dfrac{\pi}{3})$, $\theta\in (0,\dfrac{2\pi}{3})$; (2) 使$AM=AN=2\text{km}$, 才能使工厂产生的噪声对居民的影响最小
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018453
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(1) 最大值是$6$, 当且仅当$x=\pi+2k\pi$, $k\in \mathbf{Z}$时取到; 最小值是$-2$, 当且仅当$x=2k\pi$, $k\in \mathbf{Z}$时取到;\\
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(2) 最大值是$1$, 当且仅大哥$x=0$时取到; 最小值是$-\dfrac{1}{2}$, 当且仅当$x=-\dfrac{4\pi}{3}$时取到
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018454
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最小正周期为$\pi$, 单调增区间是$[k\pi-\dfrac{\pi}{3}k\pi+\dfrac{\pi}{6}]$, $k\in \mathbf{Z}$
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018456
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(1) $\{x|\dfrac{\pi}{2}+2k\pi<x<\dfrac{3\pi}{2}+2k\pi, \ k\in \mathbf{Z}\}$; (2) $\{x|-\dfrac{3\pi}{4}+2k\pi<x<\dfrac{3\pi}{4}+2k\pi, \ k\in \mathbf{Z}\}$
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009604
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$\dfrac{1}{2}$
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009605
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(1) 奇函数, 理由略; (2) 奇函数, 理由略; (3) 既不是奇函数, 又不是偶函数, 理由略
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009606
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最小正周期是$4\pi$, 单调减区间为$[\dfrac{\pi}{3}+4k\pi,\dfrac{7\pi}{3}+4k\pi]$, $k\in \mathbf{Z}$, 单调增区间为$[-\dfrac{5\pi}{3}+4k\pi,\dfrac{\pi}{3}+4k\pi]$, $k\in \mathbf{Z}$
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010289
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(1) \begin{tikzpicture}[>=latex, scale = 0.6]
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\draw [->] (-0.5,0) -- (7,0) node [below] {$x$};
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\draw [->] (0,-3.5) -- (0,1.5) node [left] {$y$};
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\draw (0,0) node [below left] {$O$};
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\draw [domain = 0:2*pi, samples = 100] plot (\x,{cos(\x/pi*180)*2-1});
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\draw [dashed] (2*pi,1) -- (0,1) node [left] {$1$};
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\draw [dashed] (2*pi,1) -- (2*pi,0) node [below] {$2\pi$};
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\draw [dashed] (pi,0) node [above] {$\pi$} -- (pi,-3) -- (0,-3) node [left] {$-3$};
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\end{tikzpicture}\\
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(2) \begin{tikzpicture}[>=latex]
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\draw [->] (-7,0) -- (7,0) node [below] {$x$};
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\draw [->] (0,-0.5) -- (0,1.5) node [left] {$y$};
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\draw (0,0) node [below left] {$O$};
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\draw [domain = -pi/2:pi/2, samples = 100] plot (\x,{cos(\x/pi*180)});
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\draw [domain = pi/2:3*pi/2, samples = 100] plot (\x,{-cos(\x/pi*180)});
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\draw [domain = 3*pi/2:7, samples = 100] plot (\x,{cos(\x/pi*180)});
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\draw [domain = -3*pi/2:-pi/2, samples = 100] plot (\x,{-cos(\x/pi*180)});
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\draw [domain = -7:-3*pi/2, samples = 100] plot (\x,{cos(\x/pi*180)});
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\draw (pi/2,0) node [below] {$\frac{\pi}{2}$};
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\draw (0,1) node [above left] {$1$};
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\end{tikzpicture}
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023606
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$\dfrac{\pi}{6}$, $\pi$
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010291
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(1) 最大值为$3$, 取得最大值时$x$的集合为$\{x|x=k\pi, \ k\in \mathbf{Z}\}$; 最小值为$\dfrac{1}{3}$, 取得最小值时$x$的集合为$\{x|x=\dfrac{\pi}{2}+k\pi, \ k\in \mathbf{Z}\}$;\\
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(2) 最大值为$1$, 取得最大值时$x$的集合为$\{x|x=2k\pi, \ k\in \mathbf{Z}\}$; 最小值为$-\dfrac{5}{4}$, 取最小值时$x$的集合为$\{x|x=\dfrac{2\pi}{3}+2k\pi\text{ 或 }\dfrac{4\pi}{3}+2k\pi, \ k\in \mathbf{Z}\}$
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023607
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在$[-\dfrac{\pi}{6},\dfrac{\pi}{3}]$上严格减, 在$[\dfrac{\pi}{3},\dfrac{2\pi}{3}]$上严格增, 值域为$[-1,1]$
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018457
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\begin{tikzpicture}[>=latex]
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\draw [->] (-7,0) -- (7,0) node [below] {$x$};
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\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};
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\draw (0,0) node [below right] {$O$};
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\draw [domain = -7:7, samples = 200, ultra thick] plot (\x,{2*sin(\x/pi*180)});
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\draw [domain = -7:7, samples = 200, thick] plot (\x,{1*sin(2*\x/pi*180)});
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\draw [domain = -7:7, samples = 200, dashed, ultra thick] plot (\x,{sin(\x/pi*180+90)});
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\draw [domain = -7:7, samples = 200, dashed] plot (\x,{sin(\x/pi*180)});
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\draw [dashed] (-6,-3) -- (-5.5,-3) node [right] {$y=\sin x$};
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\draw [ultra thick] (-3,-3) -- (-2.5,-3) node [right] {$y=2\sin x$};
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\draw [thick] (0,-3) -- (0.5,-3) node [right] {$y=\sin 2x$};
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\draw [ultra thick, dashed] (3,-3) -- (3.5,-3) node [right] {$y=\sin (x+\dfrac{\pi}{2})$};
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\end{tikzpicture}
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018458
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大致图像: \begin{tikzpicture}[>=latex, scale = 0.5]
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\draw [->] (-5,0) -- (5,0) node [below] {$x$};
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\draw [->] (0,-3.5) -- (0,3.5) node [left] {$y$};
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\draw (0,0) node [below right] {$O$};
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\draw [domain = -5:5, samples = 100] plot (\x,{3*sin(2*\x/pi*180+45)});
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\foreach \i in {-3,-2,-1,1,2,3}
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{\draw (0.1,\i) -- (0,\i) node [left] {$\i$};};
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\foreach \i in {{-pi/8},{pi/8},{3*pi/8},{5*pi/8},{7*pi/8}}
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{\draw (\i,0.2) -- (\i,0);};
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\draw (-pi/8,0) node [above left] {$-\frac{\pi}{8}$};
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\draw (pi/8,0) node [above] {$\frac{\pi}{8}$};
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\draw (3*pi/8,0) node [above right] {$\frac{3\pi}{8}$};
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\draw (5*pi/8,0) node [below] {$\frac{5\pi}{8}$};
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\draw (7*pi/8,0) node [below right] {$\frac{7\pi}{8}$};
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\end{tikzpicture}\\
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振幅为$3$, 频率为$\dfrac{1}{\pi}$, 初始相位为$\dfrac{\pi}{4}$
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018459
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周期$T=0.02\text{s}$, 频率$f=50\text{Hz}$, 电流的最大值为$10\text{A}$; $I_0=10$, $\omega = 100\pi$, $\varphi = 0$
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009607
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(1) \begin{tikzpicture}[>=latex]
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\draw [->] (-5,0) -- (5,0) node [below] {$x$};
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\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};
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\draw (0,0) node [below left] {$O$};
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\draw [domain = -5:5, samples = 100] plot (\x,{sin(\x/pi*180+30)});
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\draw [dashed] (pi/3,0) node [below] {$\frac{\pi}{3}$}-- (pi/3,1) -- (0,1) node [left] {$1$};
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\draw [dashed] (-2*pi/3,0) node [below] {$-\frac{2\pi}{3}$}-- (-2*pi/3,-1) -- (0,-1) node [right] {$-1$};
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\end{tikzpicture}\\
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(2) \begin{tikzpicture}[>=latex, scale = 0.5]
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\draw [->] (-5,0) -- (5,0) node [below] {$x$};
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\draw [->] (0,-4) -- (0,4) node [left] {$y$};
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\draw (0,0) node [below left] {$O$};
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\draw [domain = -5:5, samples = 100] plot (\x,{3*sin(2*\x/pi*180-60)});
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\draw [dashed] (5*pi/12,0) node [below] {$\frac{5\pi}{12}$} --++ (0,3) -- (0,3) node [left] {$3$};
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\draw [dashed] (-pi/12,0) node [above] {$\frac{\pi}{12}$} --++ (0,-3) -- (0,-3) node [right] {$-3$};
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\end{tikzpicture}
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009608
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D
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009609
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$y=\dfrac{3}{2}\sin (2x+\dfrac{\pi}{6})$
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018460
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$\dfrac{4\pi}{3}$
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018461
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$\varphi=\dfrac{\pi}{2}$, $\omega = \dfrac{3}{4}$
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010298
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振幅为$\sqrt{2}$, 频率为$30\pi$, 初始相位为$-\dfrac{\pi}{12}$
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010301
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\begin{tikzpicture}[>=latex]
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\draw [->] (-0.5,0) -- (3.5,0) node [below] {$x$};
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\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};
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\draw (0,0) node [below left] {$O$};
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\draw [domain = 0:3.5, samples = 100] plot (\x,{2*sin(\x*180+45)});
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\draw [dashed] (0.25,0) node [below] {$\frac{1}{4}$} --++ (0,2) -- (0,2) node [left] {$2$};
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\draw [dashed] (1.25,0) node [above] {$\frac{5}{4}$} --++ (0,-2) -- (0,-2) node [left] {$-2$};
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\end{tikzpicture}\\
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(1) 开始振动的位置在平衡位置上方$\sqrt{2}\text{cm}$处; (2) 最高点和最低点与平衡位置的距离都是$2$; (3) 经过$2\text{s}$小球往复振动一次; (4) 每秒小球往复振动$0.5$次
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010303
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$y=2\cos(\dfrac{1}{2} x+\dfrac{5\pi}{3})$
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018462
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定义域为$\{x|x\in \mathbf{R}, \ x\ne 6k+1, \ k\in \mathbf{Z}\}$, 单调增区间为$(6k-5,6k+1)$, $k\in \mathbf{Z}$
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018463
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$\dfrac{\pi}{2}$
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018464
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$2\sqrt{2}$
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009610
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$\{\alpha|\alpha = \dfrac{\pi}{3}+k\pi, \ k\in \mathbf{Z}\}$
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009611
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(1) $\tan (-\dfrac 27\pi )>\tan (-\dfrac 25\pi )$, 理由略; (2) $\cot 231^\circ>\cot 237^\circ$, 理由略; (3) $\tan (k\pi -\dfrac\pi 3)<\tan (k\pi +\dfrac\pi 3)$, 理由略
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018465
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$n=3$, 在$(-\dfrac{\pi}{18}+\dfrac{k\pi}{3}, \dfrac{5\pi}{18}+\dfrac{k\pi}{3})$, $k\in \mathbf{Z}$上是严格增函数
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010308
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(1) 奇函数, 理由略; (2) 偶函数, 理由略; (3) 奇函数, 理由略; (4) 偶函数, 理由略
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024747
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$\pi$
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024748
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面积$S=2\sqrt{2}\sin\theta\sin(\dfrac{\pi}{4}-\theta)(=\sqrt{2}\sin(2\theta+\dfrac{\pi}{4})-1)$, $\theta\in (0,\dfrac{\pi}{4})$, 面积的最大值为$\sqrt{2}-1$
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