{"id":561,"date":"2025-10-09T15:22:29","date_gmt":"2025-10-09T15:22:29","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=561"},"modified":"2025-12-05T21:26:01","modified_gmt":"2025-12-05T21:26:01","slug":"calculus-of-vector-valued-functions-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/calculus-of-vector-valued-functions-apply-it\/","title":{"raw":"Calculus of Vector-Valued Functions: Apply It","rendered":"Calculus of Vector-Valued Functions: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Find derivatives of vector-valued functions<\/li>\r\n \t<li>Find the tangent vector at a point for a given position vector<\/li>\r\n \t<li>Find and interpret unit tangent vectors<\/li>\r\n \t<li>Calculate the definite integral of a vector-valued function<\/li>\r\n<\/ul>\r\n<\/section>\r\n<div>\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0\">\r\n<h2 class=\"font-claude-response-title mt-1 text-text-100\">Programming a Robotic Arm's Movement<\/h2>\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0\">\r\n<div>\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0\">\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">You're programming a robotic arm for a manufacturing facility. The arm's position at time [latex]t[\/latex] (in seconds) is described by the vector-valued function [latex]\\mathbf{r}(t)[\/latex], where distances are measured in centimeters. To ensure smooth and precise movements, you need to analyze the arm's velocity (the derivative of position), find tangent vectors that show the direction of motion at specific points, calculate unit tangent vectors for directional control, and determine the arm's total displacement using integrals.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0\"><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0\">\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">[ohm_question hide_question_numbers=1]316630[\/ohm_question]<\/p>\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">[ohm_question hide_question_numbers=1]316638[\/ohm_question]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">[ohm_question hide_question_numbers=1]316645[\/ohm_question]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">[ohm_question hide_question_numbers=1]316650[\/ohm_question]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">[ohm_question hide_question_numbers=1]316654[\/ohm_question]<\/p>\r\n\r\n<\/section><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Find derivatives of vector-valued functions<\/li>\n<li>Find the tangent vector at a point for a given position vector<\/li>\n<li>Find and interpret unit tangent vectors<\/li>\n<li>Calculate the definite integral of a vector-valued function<\/li>\n<\/ul>\n<\/section>\n<div>\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0\">\n<h2 class=\"font-claude-response-title mt-1 text-text-100\">Programming a Robotic Arm&#8217;s Movement<\/h2>\n<\/div>\n<\/div>\n<div>\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0\">\n<div>\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0\">\n<p class=\"font-claude-response-body whitespace-normal break-words\">You&#8217;re programming a robotic arm for a manufacturing facility. The arm&#8217;s position at time [latex]t[\/latex] (in seconds) is described by the vector-valued function [latex]\\mathbf{r}(t)[\/latex], where distances are measured in centimeters. To ensure smooth and precise movements, you need to analyze the arm&#8217;s velocity (the derivative of position), find tangent vectors that show the direction of motion at specific points, calculate unit tangent vectors for directional control, and determine the arm&#8217;s total displacement using integrals.<\/p>\n<\/div>\n<\/div>\n<div>\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0\">\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0\">\n<p class=\"font-claude-response-body whitespace-normal break-words\"><iframe loading=\"lazy\" id=\"ohm316630\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=316630&theme=lumen&iframe_resize_id=ohm316630&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p class=\"font-claude-response-body whitespace-normal break-words\"><iframe loading=\"lazy\" id=\"ohm316638\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=316638&theme=lumen&iframe_resize_id=ohm316638&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p class=\"font-claude-response-body whitespace-normal break-words\"><iframe loading=\"lazy\" id=\"ohm316645\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=316645&theme=lumen&iframe_resize_id=ohm316645&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p class=\"font-claude-response-body whitespace-normal break-words\"><iframe loading=\"lazy\" id=\"ohm316650\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=316650&theme=lumen&iframe_resize_id=ohm316650&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p class=\"font-claude-response-body whitespace-normal break-words\"><iframe loading=\"lazy\" id=\"ohm316654\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=316654&theme=lumen&iframe_resize_id=ohm316654&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":15,"menu_order":14,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-561","chapter","type-chapter","status-publish","hentry"],"part":424,"module-header":"apply_it","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\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/561","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":7,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/561\/revisions"}],"predecessor-version":[{"id":2766,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/561\/revisions\/2766"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/424"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/561\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=561"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=561"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=561"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=561"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}