{"id":3432,"date":"2024-06-24T16:40:05","date_gmt":"2024-06-24T16:40:05","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=3432"},"modified":"2024-08-05T12:48:02","modified_gmt":"2024-08-05T12:48:02","slug":"linear-approximations-and-differentials-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/linear-approximations-and-differentials-apply-it\/","title":{"raw":"Linear Approximations and Differentials: Apply It","rendered":"Linear Approximations and Differentials: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Explain and use linearization to approximate a function\u2019s value near a specific point<\/li>\r\n\t<li>Calculate and interpret differentials to estimate small changes in function values<\/li>\r\n\t<li>Measure the accuracy of approximations made with differentials by calculating relative and percentage errors<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Differentials in Action: From Medical Dosages to Weather Forecasts<\/h2>\r\n<p class=\"whitespace-pre-wrap break-words\">In this apply-it task, we'll explore how differentials and linearization can be used in medical dosage calculations and weather forecasting. These examples will demonstrate the practical applications of these mathematical concepts in real-world scenarios.<\/p>\r\n<h3 class=\"font-bold\">Part 1: Medication Dosage Calculation<\/h3>\r\n<p class=\"whitespace-pre-wrap break-words\">A doctor is determining the appropriate dosage of a certain medication for patients based on their weight. The recommended dosage [latex]D[\/latex] (in milligrams) of the medication can be modeled by a function of the patient's weight [latex]w[\/latex] (in kilograms):<\/p>\r\n<p style=\"text-align: center;\">[latex]D(w) = 5w^(\\frac{2}{3})[\/latex]<\/p>\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p>[ohm_question hide_question_numbers=1]288238[\/ohm_question]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p>[ohm_question hide_question_numbers=1]288239[\/ohm_question]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p>[ohm_question hide_question_numbers=1]288240[\/ohm_question]<\/p>\r\n<\/section>\r\n<h3 class=\"font-bold\">Part 2: Rainfall Volume Estimation<\/h3>\r\n<p class=\"whitespace-pre-wrap break-words\">A meteorologist is predicting the volume of rainfall over a circular region based on the measured radius of the storm. The radius is measured to be [latex]50[\/latex] km, with a possible error of [latex]\u00b10.5[\/latex] km. The volume [latex]V[\/latex] of rainfall is modeled as the volume of a cylinder, where the height [latex]h[\/latex] represents the average rainfall depth, which is [latex]2[\/latex] cm ([latex]0.02[\/latex] km).<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">The volume [latex]V[\/latex] of the rainfall can be expressed as:<\/p>\r\n<p style=\"text-align: center;\">[latex]V = \u03c0r\u00b2h[\/latex]<\/p>\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]288241[\/ohm_question]<\/section>\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]288242[\/ohm_question]<\/section>\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]288243[\/ohm_question]<\/section>\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]288244[\/ohm_question]<\/section>\r\n<section aria-label=\"Try It\"><\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Explain and use linearization to approximate a function\u2019s value near a specific point<\/li>\n<li>Calculate and interpret differentials to estimate small changes in function values<\/li>\n<li>Measure the accuracy of approximations made with differentials by calculating relative and percentage errors<\/li>\n<\/ul>\n<\/section>\n<h2>Differentials in Action: From Medical Dosages to Weather Forecasts<\/h2>\n<p class=\"whitespace-pre-wrap break-words\">In this apply-it task, we&#8217;ll explore how differentials and linearization can be used in medical dosage calculations and weather forecasting. These examples will demonstrate the practical applications of these mathematical concepts in real-world scenarios.<\/p>\n<h3 class=\"font-bold\">Part 1: Medication Dosage Calculation<\/h3>\n<p class=\"whitespace-pre-wrap break-words\">A doctor is determining the appropriate dosage of a certain medication for patients based on their weight. The recommended dosage [latex]D[\/latex] (in milligrams) of the medication can be modeled by a function of the patient&#8217;s weight [latex]w[\/latex] (in kilograms):<\/p>\n<p style=\"text-align: center;\">[latex]D(w) = 5w^(\\frac{2}{3})[\/latex]<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<iframe loading=\"lazy\" id=\"ohm288238\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288238&theme=lumen&iframe_resize_id=ohm288238&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<iframe loading=\"lazy\" id=\"ohm288239\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288239&theme=lumen&iframe_resize_id=ohm288239&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<iframe loading=\"lazy\" id=\"ohm288240\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288240&theme=lumen&iframe_resize_id=ohm288240&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<h3 class=\"font-bold\">Part 2: Rainfall Volume Estimation<\/h3>\n<p class=\"whitespace-pre-wrap break-words\">A meteorologist is predicting the volume of rainfall over a circular region based on the measured radius of the storm. The radius is measured to be [latex]50[\/latex] km, with a possible error of [latex]\u00b10.5[\/latex] km. The volume [latex]V[\/latex] of rainfall is modeled as the volume of a cylinder, where the height [latex]h[\/latex] represents the average rainfall depth, which is [latex]2[\/latex] cm ([latex]0.02[\/latex] km).<\/p>\n<p class=\"whitespace-pre-wrap break-words\">The volume [latex]V[\/latex] of the rainfall can be expressed as:<\/p>\n<p style=\"text-align: center;\">[latex]V = \u03c0r\u00b2h[\/latex]<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm288241\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288241&theme=lumen&iframe_resize_id=ohm288241&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm288242\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288242&theme=lumen&iframe_resize_id=ohm288242&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm288243\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288243&theme=lumen&iframe_resize_id=ohm288243&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm288244\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288244&theme=lumen&iframe_resize_id=ohm288244&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section aria-label=\"Try It\"><\/section>\n","protected":false},"author":15,"menu_order":13,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":652,"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\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3432"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":7,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3432\/revisions"}],"predecessor-version":[{"id":4525,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3432\/revisions\/4525"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/652"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3432\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=3432"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=3432"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=3432"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=3432"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}