{"id":2133,"date":"2023-04-02T10:02:41","date_gmt":"2023-04-02T02:02:41","guid":{"rendered":"https:\/\/www.appblog.cn\/?p=2133"},"modified":"2023-04-05T20:24:56","modified_gmt":"2023-04-05T12:24:56","slug":"fundamentals-of-advanced-mathematics-application-of-derivatives-taylor-formula","status":"publish","type":"post","link":"https:\/\/www.appblog.cn\/index.php\/2023\/04\/02\/fundamentals-of-advanced-mathematics-application-of-derivatives-taylor-formula\/","title":{"rendered":"\u9ad8\u7b49\u6570\u5b66\u57fa\u7840\uff1a\u5bfc\u6570\u7684\u5e94\u75282\uff1a\u6cf0\u52d2Taylor\u516c\u5f0f"},"content":{"rendered":"

Taylor\u516c\u5f0f<\/h2>\n

Taylor(\u6cf0\u52d2)\u516c\u5f0f<\/h3>\n

Taylor(\u6cf0\u52d2)\u516c\u5f0f\u662f\u7528\u4e00\u4e2a\u51fd\u6570\u5728\u67d0\u70b9\u7684\u4fe1\u606f\u63cf\u8ff0\u5176\u9644\u8fd1\u53d6\u503c\u7684\u516c\u5f0f\u3002\u5982\u679c\u51fd\u6570\u8db3\u591f\u5e73\u6ed1\uff0c\u5728\u5df2\u77e5\u51fd\u6570\u5728\u67d0\u4e00\u70b9\u7684\u5404\u9636\u5bfc\u6570\u503c\u7684\u60c5\u51b5\u4e0b\uff0cTaylor\u516c\u5f0f\u53ef\u4ee5\u5229\u7528\u8fd9\u4e9b\u5bfc\u6570\u503c\u6765\u505a\u7cfb\u6570\u6784\u5efa\u4e00\u4e2a\u591a\u9879\u5f0f\u8fd1\u4f3c\u51fd\u6570\u5728\u8fd9\u4e00\u70b9\u7684\u90bb\u57df\u4e2d\u7684\u503c\u3002<\/p>\n

<\/p>\n

\u82e5\u51fd\u6570f(x)<\/code>\u5728\u5305\u542bx_0<\/code>\u7684\u67d0\u4e2a\u95ed\u533a\u95f4[a,b]<\/code>\u4e0a\u5177\u6709n<\/code>\u9636\u5bfc\u6570\uff0c\u4e14\u5728\u5f00\u533a\u95f4(a,b)<\/code>\u4e0a\u5177\u6709n+1<\/code>\u9636\u5bfc\u6570\uff0c\u5219\u5bf9\u95ed\u533a\u95f4[a,b]<\/code>\u4e0a\u4efb\u610f\u4e00\u70b9x<\/code>\uff0c\u6709Taylor\u516c\u5f0f\u5982\u4e0b\uff1a<\/p>\n

f(x) = \\frac{f(x_0)}{0!} + \\frac{f'(x_0)}{1!}(x-x_0) + \\frac{f''(x_0)}{2!}(x-x_0)^2 + \u00b7\u00b7\u00b7 + \\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n + R_n(x)<\/code><\/pre>\n
\u5176\u4e2df^{(n)}<\/code>\u8868\u793a $f(x) $<\/code>\u7684n<\/code>\u9636\u5bfc\u6570\uff0cR_n(x)<\/code>\u662fTaylor\u516c\u5f0f\u7684\u4f59\u9879\uff0c\u662f(x-x_0)^n<\/code>\u7684\u9ad8\u9636\u65e0\u7a77\u5c0f<\/p>\n
\u4ee4f(x) = a_0 + a_1(x-x_0) + a_2(x-x_0)^2 + a_3(x-x_0)^3 + \u00b7\u00b7\u00b7 + a_n(x-x_0)^n<\/code><\/p>\n
f'(x) = a_1 + 2\u00b71a_2(x-x_0) + 3a_3(x-x_0)^2 + \u00b7\u00b7\u00b7<\/code>\uff0c\u5373a_1 = \\frac{f'(x_0)}{1!}<\/code><\/p>\n
f''(x) = 2\u00b71a_2 + 3\u00b72\u00b71a_3(x-x_0) + \u00b7\u00b7\u00b7<\/code>\uff0c\u5373a_2 = \\frac{f''(x_0)}{2!}<\/code><\/p>\n
f''(x) = 3\u00b72\u00b71a_3 + 4\u00b73\u00b72\u00b71a_4(x-x_0) + \u00b7\u00b7\u00b7<\/code>\uff0c\u5373a_3 = \\frac{f'''(x_0)}{3!}<\/code><\/p>\n
\u9ea6\u514b\u52b3\u6797\u516c\u5f0f<\/h3>\n
\u7279\u6b8a\u5730x_0=0<\/code>\uff0c\u6709<\/p>\n
f(x) = \\frac{f(0)}{0!} + \\frac{f'(0)}{1!}x + \\frac{f''(0)}{2!}x^2 + \u00b7\u00b7\u00b7 + \\frac{f^{(n)}(0)}{n!}x^n + R_n(x)<\/code><\/pre>\n
Taylor\u516c\u5f0f\u4f59\u9879<\/h3>\n
f(x) = \\sum_{k=0}^n\\frac{f^{(k)}(x_0)}{k!}(x-x_0)^k + R_n(x)<\/code><\/pre>\n
\u4f69\u4e9a\u8bfa(Peano)\u4f59\u9879\uff1aR_n(x) = o[(x-x_0)^n]<\/code><\/p>\n
\u62c9\u683c\u6717\u65e5(Lagrange)\u4f59\u9879\uff1aR_n(x) = f^{(n+1)}[x_0 + \\theta(x-x_0)]\\frac{(x-x_0)^{n+1}}{(n+1)!}<\/code><\/p>\n
\u51e0\u4e2a\u5e38\u89c1\u521d\u7b49\u51fd\u6570\u7684\u5e26\u6709\u4f69\u4e9a\u8bfa\u4f59\u9879\u7684\u9ea6\u514b\u52b3\u6797\u516c\u5f0f<\/h3>\n
e^x = 1 + x + \\frac{1}{2!}x^2 + \u00b7\u00b7\u00b7 + \\frac{1}{n!}x^n + o(x^n)<\/code><\/p>\n
\\sin x = x - \\frac{1}{3!}x^3 + \u00b7\u00b7\u00b7 + \\frac{(-1)^{m-1}}{(2m-1)!}x^{2m-1} + o(x^{2m-1})<\/code><\/p>\n
\\cos x = 1 - \\frac{1}{2!}x^2 + \\frac{1}{4!}x^4 - \u00b7\u00b7\u00b7 + \\frac{(-1)^m}{(2m)!}x^{2m} + o(x^{2m})<\/code><\/p>\n
\\operatorname{ln}(1+x) = x - \\frac{1}{2}x^2 + \\frac{1}{3}x^3 - \u00b7\u00b7\u00b7 + \\frac{(-1)^{n-1}}{n}x^{n} + o(x^n)<\/code><\/p>\n
\\frac{1}{1-x} = 1 + x + x^2 + \u00b7\u00b7\u00b7 + x^n + o(x^n)<\/code><\/p>\n
(1+x)^m = 1 + mx + \\frac{m(m-1)}{2!}x^2 + \u00b7\u00b7\u00b7 + \\frac{m(m-1)\u00b7\u00b7\u00b7(m-n+1)}{n!}x^n + o(x^n)<\/code><\/p>\n
\u6b27\u62c9\u516c\u5f0f<\/h2>\n
e^{ix} = \\cos x + i \\sin x<\/code><\/p>\n
\u7b2c\u4e00\u5b87\u5b99\u516c\u5f0f\uff0ci<\/code>\u4e3a\u865a\u6570\u5355\u4f4d<\/p>\n
\u63a8\u8bba\uff1ae^{i\\pi} = -1<\/code><\/p>\n","protected":false},"excerpt":{"rendered":"
Taylor\u516c\u5f0f Taylor(\u6cf0\u52d2)\u516c\u5f0f Taylor(\u6cf0\u52d2)\u516c\u5f0f\u662f\u7528\u4e00\u4e2a\u51fd\u6570\u5728\u67d0\u70b9\u7684\u4fe1\u606f\u63cf\u8ff0\u5176\u9644\u8fd1\u53d6\u503c\u7684\u516c […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[509],"tags":[],"class_list":["post-2133","post","type-post","status-publish","format-standard","hentry","category-mathematics-fundamentals"],"_links":{"self":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts\/2133","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/comments?post=2133"}],"version-history":[{"count":0,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts\/2133\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/media?parent=2133"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/categories?post=2133"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/tags?post=2133"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}