{"id":630,"date":"2025-10-09T15:26:42","date_gmt":"2025-10-09T15:26:42","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=630"},"modified":"2026-04-13T17:46:01","modified_gmt":"2026-04-13T17:46:01","slug":"directional-derivatives-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/directional-derivatives-apply-it\/","title":{"raw":"Directional Derivatives and the Gradient: Apply It","rendered":"Directional Derivatives and the Gradient: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Calculate directional derivatives in two dimensions<\/li>\r\n \t<li>Find and interpret the gradient vector<\/li>\r\n \t<li>Use the gradient to find the tangent to a level curve of a given function<\/li>\r\n \t<li>Calculate directional derivatives in three dimensions<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Optimizing Hiking Routes on Mountain Terrain<\/h2>\r\nYou work as a trail guide at a state park, helping hikers plan safe and efficient routes across mountainous terrain. The park uses a mathematical model to represent elevation across the hiking area, and understanding how elevation changes in different directions helps you advise hikers on the steepest climbs, gentlest descents, and paths that maintain constant elevation. Your task is to analyze the terrain using directional derivatives and gradients to answer questions about rate of elevation change, optimal travel directions, and level paths along the mountainside.\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<div>\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0\"><\/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\r\n[ohm_question hide_question_numbers=1]322668[\/ohm_question]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]322670[\/ohm_question]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]322672[\/ohm_question]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]322674[\/ohm_question]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]322675[\/ohm_question]\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Calculate directional derivatives in two dimensions<\/li>\n<li>Find and interpret the gradient vector<\/li>\n<li>Use the gradient to find the tangent to a level curve of a given function<\/li>\n<li>Calculate directional derivatives in three dimensions<\/li>\n<\/ul>\n<\/section>\n<h2>Optimizing Hiking Routes on Mountain Terrain<\/h2>\n<p>You work as a trail guide at a state park, helping hikers plan safe and efficient routes across mountainous terrain. The park uses a mathematical model to represent elevation across the hiking area, and understanding how elevation changes in different directions helps you advise hikers on the steepest climbs, gentlest descents, and paths that maintain constant elevation. Your task is to analyze the terrain using directional derivatives and gradients to answer questions about rate of elevation change, optimal travel directions, and level paths along the mountainside.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<div>\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0\"><\/div>\n<\/div>\n<div>\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0\">\n<p><iframe loading=\"lazy\" id=\"ohm322668\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=322668&theme=lumen&iframe_resize_id=ohm322668&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm322670\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=322670&theme=lumen&iframe_resize_id=ohm322670&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm322672\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=322672&theme=lumen&iframe_resize_id=ohm322672&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm322674\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=322674&theme=lumen&iframe_resize_id=ohm322674&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm322675\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=322675&theme=lumen&iframe_resize_id=ohm322675&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":20,"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-630","chapter","type-chapter","status-publish","hentry"],"part":428,"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\/630","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":6,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/630\/revisions"}],"predecessor-version":[{"id":2862,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/630\/revisions\/2862"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/428"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/630\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=630"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=630"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=630"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=630"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}