第一章  预备知识
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.. contents:: :local:

.. _ex-chap1:

课后习题解答
================================

.. _ex-chap1-sec1:

§1.1 函数的概念与性质
--------------------------------

这节题目大部分比较简单, 容易错的主要是以下几题:

.. _ex-chap1-sec1-ex3:

3. 判断函数奇偶性:

(3). :math:`y = \ln(\sqrt{1+x^2}-x)`

.. proof:solution::

   计算 :math:`f(-x) = \ln(\sqrt{1+(-x)^2}-(-x)) = \ln(\sqrt{1+x^2}+x) = \ln\frac{1}{\sqrt{1+x^2}-x} = -\ln(\sqrt{1+x^2}-x) = -f(x)`, 故为奇函数.

.. _ex-chap1-sec2:

§1.2 反函数与复合函数
--------------------------------

这节题目大部分比较简单, 容易错的主要是以下几题:

.. _ex-chap1-sec2-ex3:

3. 设 :math:`f(x-2) = e^{x^2}`, 求 :math:`f(x)`.

.. proof:solution::

   由 :math:`f(x-2) = e^{x^2} = e^{((x-2)+2)^2} = e^{(x-2)^2 + 4(x-2) + 4}`, 知 :math:`f(x) = e^{x^2 + 4x + 4}`.

4. 设 :math:`f(x) = \begin{cases} 2x, & 0 \leqslant x \leqslant 1 \\ x^2, & 1 < x \leqslant 2, \end{cases}`
   :math:`g(x) = \ln x`, 求 :math:`f[g(x)]`.

.. proof:solution::

   引入中间变量 :math:`u`, 将题目重写为 :math:`f(u) = \begin{cases} 2u, & 0 \leqslant u \leqslant 1 \\ u^2, & 1 < u \leqslant 2, \end{cases}`,
   :math:`u = g(x) = \ln x`, 那么

   .. math::
      f[g(x)] = \begin{cases} 2\ln x, & 1 \leqslant x \leqslant e \\ (\ln x)^2, & e < u \leqslant e^2. \end{cases}

   这里关键是定义域要计算清楚.

.. _ex-chap1-sec3:

§1.3 基本初等函数、初等函数
--------------------------------

这节题目大部分比较简单, 容易错的主要是以下几题:

.. _ex-chap1-sec3-ex2:

2. 设 :math:`f\left( x - \dfrac{1}{x} \right) = x^2 + \dfrac{1}{x^2}`, 求 :math:`f(x)`.

.. proof:solution::

   将原式变形:

   .. math::
      f\left( x - \dfrac{1}{x} \right) = x^2 + \dfrac{1}{x^2}
      = x^2 + \dfrac{1}{x^2} - 2 + 2 = \left( x - \dfrac{1}{x} \right)^2 + 2,

   所以 :math:`f(x) = x^2 + 2`.

.. _ex-chap1-sec4:

§1.4 函数的极坐标方程与参数方程
--------------------------------

.. _extra-chap1:

补充内容
================================

.. _extra-chap1-topic1:

1. 三角函数

   课程中涉及的三角函数图示如下:

   .. tikz:: 三角函数图示
      :align: center
      :xscale: 85
      :libs: angles, quotes, arrows.meta, positioning, fit, calc, decorations.pathreplacing, shapes.misc
      :packages: amsfonts, amsmath, amssymb

      \tikzset{>=Stealth, scale=3.6}

      \definecolor{sinColor}{RGB}{255,0,0}
      \definecolor{cosColor}{RGB}{0,0,255}
      \definecolor{versinColor}{RGB}{0,96,0}
      \definecolor{exsecColor}{RGB}{255,0,255}
      \definecolor{secColor}{RGB}{0,168,192}
      \definecolor{tanColor}{RGB}{165,42,42}
      \definecolor{cotColor}{RGB}{255,165,0}
      \definecolor{excscColor}{RGB}{0,255,0}
      \definecolor{cvsColor}{RGB}{0,255,255}
      \definecolor{cscColor}{RGB}{255,192,203}
      \definecolor{crdColor}{RGB}{128,128,128}
      \definecolor{vercosColor}{RGB}{0,112,192}
      \definecolor{covercosColor}{RGB}{153,50,204}

      \def\myangle{60}
      \pgfmathsetmacro{\costheta}{cos(\myangle)}
      \pgfmathsetmacro{\sintheta}{sin(\myangle)}
      \pgfmathsetmacro{\sectheta}{1/cos(\myangle)}
      \pgfmathsetmacro{\csctheta}{1/sin(\myangle)}
      \pgfmathsetmacro{\cottheta}{cos(\myangle)/sin(\myangle)}
      \pgfmathsetmacro{\versin}{1 - \costheta}
      \pgfmathsetmacro{\exsec}{\sectheta - 1}
      \pgfmathsetmacro{\excsc}{\csctheta - 1}

      \draw[thick] (0,0) circle (1);

      \coordinate (O) at (0,0) node[below left] {$O$};
      \coordinate (A) at (\costheta, \sintheta) node[above right=-0.5ex and 0.1em of A] {$A$};
      \coordinate (B) at (\costheta, -\sintheta) node[below right=0.1ex and 0.1em of B] {$B$};
      \coordinate (C) at (\costheta, 0) node[above right=0.1ex and 0.1em of C] {$C$};
      \coordinate (D) at (1, 0) node[below right=0.1ex and 0.1em of D] {$D$};
      \coordinate (E) at (\sectheta, 0) node[right=0.1em of E] {$E$};
      \coordinate (F) at (0, \csctheta) node[above=0.1ex of F] {$F$};
      \coordinate (G) at (0, \sintheta) node[below left=-0.6ex and -0.2em of G] {$G$};
      \coordinate (H) at (0, 1) node[below right=-0.6ex and -0.3em of H] {$H$};
      \coordinate (K) at (-1, 0) node[left =-0.3ex of K] {$K$};
      \coordinate (L) at (0, -1) node[below =-0.3ex of L] {$L$};
      \coordinate (Z1) at ({1.4*cos(\myangle)},{1.4*sin(\myangle)});
      \coordinate (Z2) at (-0.4, 0);

      \draw[line width=3.2pt] (D) arc[start angle=0, end angle=\myangle, radius=1] node[near start, right] {$\mathrm{arc}$};

      \draw[thick] (O) -- (0.1,0) arc[start angle=0, end angle=\myangle, radius=0.1];
      \node at ({0.15*cos(\myangle/2)},{0.15*sin(\myangle/2)}) {$\theta$};

      \draw[sinColor, ultra thick] (A) -- (C) node[midway, right, draw, thick, inner sep=1.5pt, xshift=0.2em, yshift=-2ex] {$\sin$};
      \draw[sinColor, ultra thick] (O) -- (G);
      % \draw[gray, dashed, ultra thick] (C) -- (B);
      \draw[dashed, ultra thick] (O) -- (B);
      \draw[dashed, ultra thick] (A) -- (Z1);
      \pic[draw, ultra thick, angle radius=0.2cm] {right angle = O--C--A};
      \pic[draw, ultra thick, angle radius=0.2cm] {right angle = F--A--Z1};
      \pic[draw, ultra thick, angle radius=0.2cm] {right angle = O--G--A};

      \draw[cosColor, ultra thick] (O) -- (C) node[midway, below, inner sep=1.5pt, draw, thick, yshift=-0.5ex] {$\cos$};
      \draw[cosColor, ultra thick] (A) -- (G);

      \draw[ultra thick] (O) -- (A) node[midway, right] {$1$};
      \draw[tanColor, ultra thick] (A) -- (E) node[midway, above, sloped, draw, thick, inner sep=1.5pt, yshift=0.5ex] {$\tan$};

      \draw[secColor, dashed, ultra thick] (O) -- (0, -0.4);
      \draw[secColor, dashed, ultra thick] (E) -- ($(E) + (0, -0.4)$);
      \draw[secColor, ultra thick, |<->|] (-0.008, -0.35) -- ($(E) + (0.008, -0.35)$) node[midway, below, sloped, draw, thick, inner sep=1.5pt, yshift=-0.5ex, xshift=2ex] {$\sec$};

      \draw[cscColor, dashed, ultra thick] (O) -- (Z2);
      \draw[cscColor, dashed, ultra thick] (F) -- ($(F) + (Z2)$);
      \draw[cscColor, ultra thick, |<->|] (-0.35, -0.008) -- ($(F) + (-0.35, 0.008)$) node[midway, right, draw, thick, inner sep=1.5pt, xshift=0.2em] {$\csc$};
      \pic[draw, ultra thick, angle radius=0.2cm] {right angle = G--O--Z2};

      \draw[cotColor, ultra thick, sloped] (A) -- (F) node[midway, above, draw, thick, inner sep=1.5pt, yshift=0.5ex] {$\cot$};

      \draw[versinColor, ultra thick] (C) -- (D) node[midway, below] {$\mathrm{versin}$};

      \draw[exsecColor, ultra thick] (D) -- (E) node[midway, below] {$\mathrm{exsec}$};

      \draw[excscColor, ultra thick] (H) -- (F) node[midway, left] {$\mathrm{excsc}$};

      \draw[cvsColor, ultra thick] (H) -- (G) node[midway, left] {$\mathrm{cvs}$};

      \draw[crdColor, ultra thick] (D) -- (A) node[midway, above, sloped, yshift=-0.3ex] {$\mathrm{crd}$};

      \draw[gray, ultra thick, dashed] (-0.4, 0) -- (K);
      \draw[gray, ultra thick, dashed] (0, -0.4) -- (L);

      \draw[vercosColor, ultra thick, dashed] (K) -- ($(K) + (0, -0.7)$);
      \draw[vercosColor, dashed, ultra thick] (C) -- ($(C) + (0, -0.65)$);
      \draw[gray, dashed, ultra thick] ($(C) + (0, -0.65)$) -- (B);
      \draw[vercosColor, ultra thick, |<->|] ($(K) + (-0.008, -0.65)$) -- ($(C) + (0.008, -0.65)$) node[midway, above] {$\mathrm{vercos}$};

      \draw[covercosColor, ultra thick, dashed] (L) -- ($(L) + (-0.7, 0)$);
      \draw[covercosColor, ultra thick, dashed] (G) -- ($(G) + (-0.7, 0)$);
      \draw[covercosColor, ultra thick, |<->|] ($(G) + (-0.65, 0.008)$) -- ($(L) + (-0.65, -0.008)$) node[midway, yshift=-4ex, right] {$\mathrm{covercos}$};

   其中主要需要掌握的是带框的 6 个三角函数:

   - 正弦函数 :math:`\sin`
   - 余弦函数 :math:`\cos`
   - 正切函数 :math:`\tan`
   - 余切函数 :math:`\cot`
   - 正割函数 :math:`\sec`
   - 余割函数 :math:`\csc`

   其他函数如 :math:`\mathrm{versin}, \mathrm{exsec}` 等大部分已经很少使用, 本课程不做要求.
   这些被称作 Obsolete Trigonometric Functions, 大多出现在中世纪阿拉伯数学和 16–18 世纪欧洲数学中,
   后来因为不如 :math:`\sin,\cos,\tan` 等直观，逐渐被淘汰. 这些函数列举如下:

   - 正矢函数 :math:`\mathrm{versin} \theta = 1 - \cos \theta`
   - 余矢函数 :math:`\mathrm{cvs} \theta = 1 - \sin \theta`
   - 正余矢函数 :math:`\mathrm{vercos} \theta = 1 + \cos \theta`
   - 余余矢函数 :math:`\mathrm{covercos} \theta = 1 + \sin \theta`
   - 外割函数 :math:`\mathrm{exsec} \theta = \sec \theta - 1`
   - 外余割函数 :math:`\mathrm{excsc} \theta = \csc \theta - 1`
   - 弦函数 :math:`\mathrm{crd} \theta = 2 \sin \dfrac{\theta}{2}`