{"id":184,"date":"2021-12-29T23:43:03","date_gmt":"2021-12-29T15:43:03","guid":{"rendered":"https:\/\/memoryshadow.cn\/?p=184"},"modified":"2023-05-31T01:10:12","modified_gmt":"2023-05-30T17:10:12","slug":"advanced-mathematics","status":"publish","type":"post","link":"https:\/\/memoryshadow.cn\/index.php\/2021\/12\/29\/advanced-mathematics\/","title":{"rendered":"\u9ad8\u7b49\u6570\u5b66"},"content":{"rendered":"

\u9ad8\u7b49\u6570\u5b66<\/h1>\n
<\/i> Note<\/div>
<\/p>\n

\u6570\u5b66\u4e0d\u662f\u770b\u51fa\u6765\u7684\uff0c\u662f\u505a\u51fa\u6765\u7684
\n\u4e0d\u8981\u505a\u4e00\u9898\u5c31\u770b\u4e00\u9898\u7684\u7b54\u6848
\n\u5bf9\u4e8e\u4e0d\u4f1a\u7684\u9898\u76ee\uff0c\u8981\u591a\u78e8
\n\u8bb0\u7b54\u6848\u4e0d\u662f\u6709\u6548\u7684\u65b9\u6848\uff0c\u8981\u8bb0\u5f97\u662f\u89e3\u9898\u7684\u65b9\u6cd5
\n\u7b49\u5f0f\u7684\u4e24\u8fb9\u90fd\u8981\u8bb0<\/p>\n

<\/div><\/div>\n

\u5e38\u7528\u516c\u5f0f<\/h2>\n

\u57fa\u7840\u53d8\u5f62<\/h3>\n\n\n\n\n\n\n\n\n\n
\u516c\u5f0f\u540d<\/th>\n\u516c\u5f0f<\/th>\n<\/tr>\n<\/thead>\n
\u5b8c\u5168\u7acb\u65b9\u5dee\u516c\u5f0f<\/td>\n$a^3-b^3 = (a-b)(a^2+ab+b^2)$<\/td>\n<\/tr>\n
\u5b8c\u5168\u7acb\u65b9\u548c\u516c\u5f0f<\/td>\n$a^3+b^3 = (a+b)(a^2-ab+b^2)$<\/td>\n<\/tr>\n
\u5e73\u65b9\u5dee\u516c\u5f0f<\/td>\n$(x_0-x_1)(x_0+x_1) = x_0^2-x_1^2$<\/td>\n<\/tr>\n
\u5b8c\u5168\u5e73\u65b9\u5dee\u516c\u5f0f<\/td>\n$(a+b)^2 = a^2 + 2ab + b^2$<\/td>\n<\/tr>\n
\u5b8c\u5168\u5e73\u65b9\u548c\u516c\u5f0f<\/td>\n$(a-b)^2 = a^2 - 2ab + b^2$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\u521d\u7b49\u51fd\u6570\u7684\u5173\u7cfb<\/h3>\n
    \n
  1. ${ \\lim\\limits_{x \\to 0} \\frac {\\tan x}{x} \\Longrightarrow \\lim\\limits_{x \\to 0} \\frac {\\sin x}{x} \\frac 1{\\cot x} \\Longrightarrow \\lim\\limits_{x \\to 0} \\frac 1{\\cot x} \\Longrightarrow 1}$<\/li>\n<\/ol>\n

    \u5e38\u7528\u4e09\u89d2\u51fd\u6570\u503c<\/h3>\n
      \n
    1. $\\arctan 1 = \\frac {\\pi}4$<\/li>\n
    2. $\\arccos 0= \\frac {\\pi}2$<\/li>\n
    3. $\\arcsin 1= \\frac {\\pi}2$<\/li>\n
    4. $\\arctan 0 = 0$<\/li>\n<\/ol>\n

      \u57fa\u672c\u4e09\u89d2\u51fd\u6570\u7684\u503c<\/h3>\n\n\n\n\n\n\n\n\n\n
      $x$<\/th>\n$\\sin x$<\/th>\n$\\cos x$<\/th>\n$\\tan x$<\/th>\n<\/tr>\n<\/thead>\n
      $0$<\/td>\n$0$<\/td>\n$1$<\/td>\n$0$<\/td>\n<\/tr>\n
      $\\frac {\\pi}2$<\/td>\n$1$<\/td>\n$0$<\/td>\n\u4e0d\u5b58\u5728\/\\infty<\/span><\/td>\n<\/tr>\n
      $\\frac {\\pi}3$<\/td>\n$\\frac {\\sqrt[]{3}}2$<\/td>\n$\\frac 12$<\/td>\n$\\sqrt[]{3}$<\/td>\n<\/tr>\n
      $\\frac {\\pi}4$<\/td>\n$\\frac{\\sqrt[]{2}}2$<\/td>\n$\\frac {\\sqrt[]{2}}2$<\/td>\n$1$<\/td>\n<\/tr>\n
      $\\frac {\\pi}6$<\/td>\n$\\frac 16$<\/td>\n$\\frac {\\sqrt[]{3}}2$<\/td>\n$\\frac {\\sqrt[]{3}}3$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      \u5e38\u7528\u7684\u4e09\u89d2\u51fd\u6570: \\tan x = \\frac {\\sin x}{\\cos x}<\/span><\/h3>\n

      \u5e73\u65b9\u548c\u516c\u5f0f<\/h4>\n
        \n
      1. $\\sin^2x+\\cos^2x = 1$<\/li>\n
      2. $1+\\tan^2x = \\sec^2x$<\/li>\n
      3. $1+\\cot^2x = \\csc^2x$<\/li>\n<\/ol>\n

        \u4e8c\u500d\u89d2<\/h4>\n
          \n
        1. $\\sin2x = 2\\sin x \\cdot \\cos x$<\/li>\n
        2. $\\cos 2x = \\cos^2x - \\sin^2x = 2\\cos^2x - 1 = 1 - 2\\sin^2x$<\/li>\n<\/ol>\n

          \u964d\u6b21<\/h4>\n
            \n
          1. $\\cos^2x = \\frac {1 + \\cos 2x}x$<\/li>\n
          2. $\\sin^2x = \\frac {1 - \\cos 2x}2$<\/li>\n
          3. $\\sec^2x = \\frac 1{\\cos^2x}$<\/li>\n<\/ol>\n

            \u5e38\u7528\u7b49\u4ef7\u65e0\u7a77\u5c0f<\/h3>\n

            \u6b64\u5904\u9700\u8981\u6ce8\u610f: \u7b49\u4ef7\u65e0\u7a77\u5c0f\u8bf7\u4f7f\u7528\u6574\u4f53\u601d\u60f3.\u540c\u65f6\uff0c\u53ea\u6709\u5f53x(\u72d7)=0\u65f6\uff0c\u8fd9\u91cc\u7684\u89c4\u5219\u624d\u80fd\u4f7f\u7528<\/code><\/p>\n

              \n
            1. \u57fa\u672c\u4e09\u89d2\u51fd\u6570<\/li>\n<\/ol>\n

              \\begin{cases}
              \n\\sin x& \\sim x\\
              \n\\arcsin x& \\sim x
              \n\\end{cases}
              \n\\begin{cases}
              \n\\tan x& \\sim x\\
              \n\\arctan x& \\sim x
              \n\\end{cases}
              \n\\begin{cases}
              \ne^x-1&\\sim x\\
              \n\\ln(1 - x)&\\sim x
              \n\\end{cases}<\/span><\/p>\n

                \n
              1. $1 - \\cos x \\sim \\frac 12x^2$<\/li>\n
              2. $\\sqrt[n]{1+x} - 1 \\sim \\frac \\pi n \\iff (1+ax)^b-1 \\sim abx$
                \n\\sqrt[n]{x^m} = x^{\\frac mn}<\/span><\/li>\n
              3. :\n