{"id":949,"date":"2025-06-20T17:24:34","date_gmt":"2025-06-20T17:24:34","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=949"},"modified":"2025-12-17T16:47:17","modified_gmt":"2025-12-17T16:47:17","slug":"taylor-and-maclaurin-series-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/taylor-and-maclaurin-series-apply-it\/","title":{"raw":"Taylor and Maclaurin Series: Apply It","rendered":"Taylor and Maclaurin Series: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Learn how to find Taylor polynomials of a given order for a function<\/li>\r\n \t<li>Estimate the remainder when using a Taylor series to approximate a function<\/li>\r\n \t<li>Determine when a Taylor series converges to the original function<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Taylor Polynomials and Remainders<\/h2>\r\n<p class=\"whitespace-normal break-words\">Taylor polynomials aren't just theoretical constructs\u2014they're essential tools used throughout science and technology. From GPS navigation systems calculating satellite positions to computer graphics rendering smooth curves, understanding how to construct Taylor polynomials and estimate their accuracy is crucial for solving real-world problems.<\/p>\r\n<p class=\"whitespace-normal break-words\">Imagine you're part of an engineering team designing a new smartphone app that uses motion sensors to track physical activity. The app needs to calculate trigonometric functions quickly and accurately, but the phone's processor has limited computational power. Your team decides to use Taylor polynomial approximations instead of computing the exact values.<\/p>\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"whitespace-normal break-words\">[ohm_question hide_question_numbers=1]313637[\/ohm_question]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"whitespace-normal break-words\">[ohm_question hide_question_numbers=1]313640[\/ohm_question]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"whitespace-normal break-words\">[ohm_question hide_question_numbers=1]313643[\/ohm_question]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"whitespace-normal break-words\">[ohm_question hide_question_numbers=1]313647[\/ohm_question]<\/p>\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Learn how to find Taylor polynomials of a given order for a function<\/li>\n<li>Estimate the remainder when using a Taylor series to approximate a function<\/li>\n<li>Determine when a Taylor series converges to the original function<\/li>\n<\/ul>\n<\/section>\n<h2>Taylor Polynomials and Remainders<\/h2>\n<p class=\"whitespace-normal break-words\">Taylor polynomials aren&#8217;t just theoretical constructs\u2014they&#8217;re essential tools used throughout science and technology. From GPS navigation systems calculating satellite positions to computer graphics rendering smooth curves, understanding how to construct Taylor polynomials and estimate their accuracy is crucial for solving real-world problems.<\/p>\n<p class=\"whitespace-normal break-words\">Imagine you&#8217;re part of an engineering team designing a new smartphone app that uses motion sensors to track physical activity. The app needs to calculate trigonometric functions quickly and accurately, but the phone&#8217;s processor has limited computational power. Your team decides to use Taylor polynomial approximations instead of computing the exact values.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p class=\"whitespace-normal break-words\"><iframe loading=\"lazy\" id=\"ohm313637\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313637&theme=lumen&iframe_resize_id=ohm313637&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p class=\"whitespace-normal break-words\"><iframe loading=\"lazy\" id=\"ohm313640\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313640&theme=lumen&iframe_resize_id=ohm313640&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p class=\"whitespace-normal break-words\"><iframe loading=\"lazy\" id=\"ohm313643\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313643&theme=lumen&iframe_resize_id=ohm313643&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p class=\"whitespace-normal break-words\"><iframe loading=\"lazy\" id=\"ohm313647\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313647&theme=lumen&iframe_resize_id=ohm313647&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":21,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":673,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/949"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":13,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/949\/revisions"}],"predecessor-version":[{"id":2516,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/949\/revisions\/2516"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/673"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/949\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=949"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=949"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=949"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=949"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}