{"id":359,"date":"2019-01-30T20:41:35","date_gmt":"2019-01-30T20:41:35","guid":{"rendered":"http:\/\/pressbooks.library.ryerson.ca\/bearguide\/?post_type=chapter&#038;p=359"},"modified":"2021-07-29T12:29:32","modified_gmt":"2021-07-29T12:29:32","slug":"mathematics-bear","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/bearguide\/chapter\/mathematics-bear\/","title":{"raw":"Mathematics Bear","rendered":"Mathematics Bear"},"content":{"raw":"This page contains formulas used by mathematically inclined bears.\u00a0 Very few of these bears are found in the wild.\u00a0 It is believed that a small family live near Meyrin, Switzerland.\r\n\r\nAlbert Einstein's famous formula [latex]E=mc^2[\/latex] can be used ....\r\n<h2>Embedded Geogebra Content (using H5P)<\/h2>\r\n<span>[h5p id=\"16\"]<\/span>\r\n\r\n&nbsp;\r\n<h2>Embedded Geogebra Content (Using an iframe embed)<\/h2>\r\n<iframe style=\"border: 1px solid #e4e4e4; border-radius: 4px;\" src=\"https:\/\/www.geogebra.org\/classic\/ptzufvkv?embed\" width=\"800\" height=\"600\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n\r\n&nbsp;\r\n\r\n<span>At first, we sample [latex]f(x)[\/latex] in the [latex]N[\/latex] ([latex]N[\/latex]is odd) equidistant points around [latex]x^*[\/latex]:<\/span>\r\n<span>\\[<\/span>\r\n<span>\u00a0\u00a0\u00a0f_k = f(x_k),\\: x_k = x^*+kh,\\: k=-\\frac{N-1}{2},\\dots,\\frac{N-1}{2}<\/span>\r\n<span>\\]<\/span>\r\n<span>where [latex]h[\/latex] is some step.<\/span>\r\n<span>Then we interpolate points [latex]\\{(x_k,f_k)\\}[\/latex] by polynomial\u00a0<\/span>\r\n<span>\\begin{equation}\u00a0\\label{eq:poly}<\/span>\r\n<span>\u00a0\u00a0\u00a0P_{N-1}(x)=\\sum_{j=0}^{N-1}{a_jx^j}<\/span>\r\n<span>\\end{equation}<\/span>\r\n<span>Its coefficients [latex]\\{a_j\\}[\/latex] are found as a solution of system of linear equations:<\/span>\r\n<span>\\begin{equation}\u00a0\\label{eq:sys}<\/span>\r\n<span>\u00a0\u00a0\u00a0\\left\\{ P_{N-1}(x_k) = f_k\\right\\},\\quad k=-\\frac{N-1}{2},\\dots,\\frac{N-1}{2}<\/span>\r\n<span>\\end{equation}<\/span>\r\n<span>Here are references to existing equations: ([latex]ref{eq:poly}[\/latex]), ([latex]ref{eq:sys}[\/latex]).<\/span>\r\n<span>Here is reference to non-existing equation ([latex]ref{eq:unknown}[\/latex]).<\/span>\r\n\r\n&nbsp;","rendered":"<p>This page contains formulas used by mathematically inclined bears.\u00a0 Very few of these bears are found in the wild.\u00a0 It is believed that a small family live near Meyrin, Switzerland.<\/p>\n<p>Albert Einstein&#8217;s famous formula [latex]E=mc^2[\/latex] can be used &#8230;.<\/p>\n<h2>Embedded Geogebra Content (using H5P)<\/h2>\n<p><span><\/p>\n<div id=\"h5p-16\">\n<div class=\"h5p-content\" data-content-id=\"16\"><\/div>\n<\/div>\n<p><\/span><\/p>\n<p>&nbsp;<\/p>\n<h2>Embedded Geogebra Content (Using an iframe embed)<\/h2>\n<p><iframe loading=\"lazy\" style=\"border: 1px solid #e4e4e4; border-radius: 4px;\" src=\"https:\/\/www.geogebra.org\/classic\/ptzufvkv?embed\" width=\"800\" height=\"600\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<p><span>At first, we sample [latex]f(x)[\/latex] in the [latex]N[\/latex] ([latex]N[\/latex]is odd) equidistant points around [latex]x^*[\/latex]:<\/span><br \/>\n<span>\\[<\/span><br \/>\n<span>\u00a0\u00a0\u00a0f_k = f(x_k),\\: x_k = x^*+kh,\\: k=-\\frac{N-1}{2},\\dots,\\frac{N-1}{2}<\/span><br \/>\n<span>\\]<\/span><br \/>\n<span>where [latex]h[\/latex] is some step.<\/span><br \/>\n<span>Then we interpolate points [latex]\\{(x_k,f_k)\\}[\/latex] by polynomial\u00a0<\/span><br \/>\n<span>\\begin{equation}\u00a0\\label{eq:poly}<\/span><br \/>\n<span>\u00a0\u00a0\u00a0P_{N-1}(x)=\\sum_{j=0}^{N-1}{a_jx^j}<\/span><br \/>\n<span>\\end{equation}<\/span><br \/>\n<span>Its coefficients [latex]\\{a_j\\}[\/latex] are found as a solution of system of linear equations:<\/span><br \/>\n<span>\\begin{equation}\u00a0\\label{eq:sys}<\/span><br \/>\n<span>\u00a0\u00a0\u00a0\\left\\{ P_{N-1}(x_k) = f_k\\right\\},\\quad k=-\\frac{N-1}{2},\\dots,\\frac{N-1}{2}<\/span><br \/>\n<span>\\end{equation}<\/span><br \/>\n<span>Here are references to existing equations: ([latex]ref{eq:poly}[\/latex]), ([latex]ref{eq:sys}[\/latex]).<\/span><br \/>\n<span>Here is reference to non-existing equation ([latex]ref{eq:unknown}[\/latex]).<\/span><\/p>\n<p>&nbsp;<\/p>\n","protected":false},"author":2,"menu_order":4,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-359","chapter","type-chapter","status-publish","hentry"],"part":115,"_links":{"self":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/bearguide\/wp-json\/pressbooks\/v2\/chapters\/359","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/bearguide\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/bearguide\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/bearguide\/wp-json\/wp\/v2\/users\/2"}],"version-history":[{"count":11,"href":"https:\/\/pressbooks.library.torontomu.ca\/bearguide\/wp-json\/pressbooks\/v2\/chapters\/359\/revisions"}],"predecessor-version":[{"id":535,"href":"https:\/\/pressbooks.library.torontomu.ca\/bearguide\/wp-json\/pressbooks\/v2\/chapters\/359\/revisions\/535"}],"part":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/bearguide\/wp-json\/pressbooks\/v2\/parts\/115"}],"metadata":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/bearguide\/wp-json\/pressbooks\/v2\/chapters\/359\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/bearguide\/wp-json\/wp\/v2\/media?parent=359"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/bearguide\/wp-json\/pressbooks\/v2\/chapter-type?post=359"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/bearguide\/wp-json\/wp\/v2\/contributor?post=359"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/bearguide\/wp-json\/wp\/v2\/license?post=359"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}