{"id":627,"date":"2025-10-09T15:26:51","date_gmt":"2025-10-09T15:26:51","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=627"},"modified":"2026-04-13T17:28:43","modified_gmt":"2026-04-13T17:28:43","slug":"tangent-planes-and-linear-approximations-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/tangent-planes-and-linear-approximations-apply-it\/","title":{"raw":"Tangent Planes and Linear Approximations: Apply It","rendered":"Tangent Planes and Linear Approximations: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Find equations of tangent planes to surfaces<\/li>\r\n \t<li>Use tangent planes for approximation<\/li>\r\n \t<li>Explain when a function of two variables is differentiable<\/li>\r\n \t<li>Use the total differential to approximate the change in a function of two variables<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Monitoring Water Quality Across a Filtration System<\/h2>\r\nYou are a technician at a municipal water treatment facility responsible for monitoring contaminant concentration levels across a large rectangular filtration bed. The concentration [latex]C[\/latex] of a particular compound (measured in parts per million) varies across the bed and is modeled as a function [latex]C(x, y)[\/latex], where [latex]x[\/latex] and [latex]y[\/latex] represent horizontal distances in meters from the southwest corner. Sensors are placed at specific grid locations, but you often need to estimate concentration values at points between sensors. By finding tangent planes to the concentration surface and using linear approximations, you can make reliable estimates without installing additional equipment. Understanding when these approximations are valid\u2014and how to quantify potential changes\u2014is critical for maintaining water quality standards.\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]322661[\/ohm_question]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]322664[\/ohm_question]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]322665[\/ohm_question]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]322666[\/ohm_question]\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Find equations of tangent planes to surfaces<\/li>\n<li>Use tangent planes for approximation<\/li>\n<li>Explain when a function of two variables is differentiable<\/li>\n<li>Use the total differential to approximate the change in a function of two variables<\/li>\n<\/ul>\n<\/section>\n<h2>Monitoring Water Quality Across a Filtration System<\/h2>\n<p>You are a technician at a municipal water treatment facility responsible for monitoring contaminant concentration levels across a large rectangular filtration bed. The concentration [latex]C[\/latex] of a particular compound (measured in parts per million) varies across the bed and is modeled as a function [latex]C(x, y)[\/latex], where [latex]x[\/latex] and [latex]y[\/latex] represent horizontal distances in meters from the southwest corner. Sensors are placed at specific grid locations, but you often need to estimate concentration values at points between sensors. By finding tangent planes to the concentration surface and using linear approximations, you can make reliable estimates without installing additional equipment. Understanding when these approximations are valid\u2014and how to quantify potential changes\u2014is critical for maintaining water quality standards.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm322661\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=322661&theme=lumen&iframe_resize_id=ohm322661&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm322664\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=322664&theme=lumen&iframe_resize_id=ohm322664&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm322665\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=322665&theme=lumen&iframe_resize_id=ohm322665&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm322666\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=322666&theme=lumen&iframe_resize_id=ohm322666&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\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-627","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\/627","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":5,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/627\/revisions"}],"predecessor-version":[{"id":2859,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/627\/revisions\/2859"}],"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\/627\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=627"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=627"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=627"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=627"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}