{"id":3096,"date":"2025-08-15T21:57:35","date_gmt":"2025-08-15T21:57:35","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3096"},"modified":"2025-08-20T18:36:53","modified_gmt":"2025-08-20T18:36:53","slug":"rates-of-change-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/rates-of-change-apply-it\/","title":{"raw":"Rates of Change: Apply It","rendered":"Rates of Change: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Find the average rate of change<\/li>\r\n \t<li>Use a graph to see where a function is going up, going down, or staying flat<\/li>\r\n \t<li>Use a graph to identify the highest and lowest points<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Climate Research Station<\/h2>\r\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">In calculus, you'll use derivatives to find exact rates of change at any instant, locate precise extrema, and solve optimization problems. Today's work with average rates and graphical analysis builds the foundation for these calculus tools.<\/section>\r\n<p class=\"whitespace-normal break-words\">A climate station recorded atmospheric CO\u2082 levels over a 10-year period:<\/p>\r\n\r\n<table class=\"bg-bg-100 min-w-full border-separate border-spacing-0 text-sm leading-[1.88888] whitespace-normal\" style=\"width: 99.2682%;\">\r\n<thead class=\"border-b-border-100\/50 border-b-[0.5px] text-left\">\r\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\r\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 20.9677%;\">Year<\/th>\r\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 12.3656%;\">2014<\/th>\r\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 12.6344%;\">2016<\/th>\r\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 12.9032%;\">2018<\/th>\r\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 12.0968%;\">2020<\/th>\r\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 11.828%;\">2022<\/th>\r\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 63.7068%;\">2024<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 20.9677%;\">CO\u2082 (ppm)<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 12.3656%;\">398.7<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 12.6344%;\">404.2<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 12.9032%;\">408.5<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 12.0968%;\">414.1<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 11.828%;\">421.0<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 63.7068%;\">424.3<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<section class=\"textbox recall\" aria-label=\"Recall\">Average rate of change is calculated as [latex]\\frac{f(b)-f(a)}{b-a}[\/latex]<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif; background-color: #f4f3ef;\">\u200b<\/span><\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]309543[\/ohm_question]<\/section><section class=\"textbox connectIt\" aria-label=\"Connect It\">\r\n<p class=\"whitespace-normal break-words\">But what was the rate of change in 2014? In calculus, we'll learn to extend the average rate of change and find the instantaneous rate of change at any specific moment using derivatives, giving us more precise information than these averages.<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]309544[\/ohm_question]<\/section>\r\n<p class=\"whitespace-normal break-words\">The NOAA Mauna Loa Observatory (MLO) is a key atmospheric research facility that monitors and collects long-term data on atmospheric composition and climate change. It's known for its continuous measurements of rising carbon dioxide (CO2) concentrations, ozone, and other atmospheric constituents. The data collected at MLO helps scientists understand the impacts of human activities on the atmosphere and climate.<\/p>\r\n<img class=\"alignnone size-full wp-image-3419\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/20164340\/co2_weekly_mlo.png\" alt=\"\" width=\"1000\" height=\"600\" \/>\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]309545[\/ohm_question]<\/section>\r\n\r\n<hr \/>\r\n<p class=\"whitespace-normal break-words\">Your station models the annual CO\u2082 cycle with: [latex]C(d) = 426 + 3.85sin(\\frac{2\u03c0}{365}(d-105))[\/latex] where d = day of year.<\/p>\r\n<p class=\"whitespace-normal break-words\">This function captures how CO\u2082 levels fluctuate seasonally - rising in winter when plants are dormant and falling in summer when photosynthesis is most active.<\/p>\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]309550[\/ohm_question]\r\n\r\n<\/section><section class=\"textbox linkToLearning\" aria-label=\"Link to Learning\">CO\u2082 monitoring is crucial for understanding our climate and how it may be changing over time. Learn more about the NOAA:\u00a0<a href=\"https:\/\/gml.noaa.gov\/about\/networks.html\">https:\/\/gml.noaa.gov\/about\/networks.html<\/a>\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Find the average rate of change<\/li>\n<li>Use a graph to see where a function is going up, going down, or staying flat<\/li>\n<li>Use a graph to identify the highest and lowest points<\/li>\n<\/ul>\n<\/section>\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Climate Research Station<\/h2>\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">In calculus, you&#8217;ll use derivatives to find exact rates of change at any instant, locate precise extrema, and solve optimization problems. Today&#8217;s work with average rates and graphical analysis builds the foundation for these calculus tools.<\/section>\n<p class=\"whitespace-normal break-words\">A climate station recorded atmospheric CO\u2082 levels over a 10-year period:<\/p>\n<table class=\"bg-bg-100 min-w-full border-separate border-spacing-0 text-sm leading-[1.88888] whitespace-normal\" style=\"width: 99.2682%;\">\n<thead class=\"border-b-border-100\/50 border-b-[0.5px] text-left\">\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 20.9677%;\">Year<\/th>\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 12.3656%;\">2014<\/th>\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 12.6344%;\">2016<\/th>\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 12.9032%;\">2018<\/th>\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 12.0968%;\">2020<\/th>\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 11.828%;\">2022<\/th>\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 63.7068%;\">2024<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 20.9677%;\">CO\u2082 (ppm)<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 12.3656%;\">398.7<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 12.6344%;\">404.2<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 12.9032%;\">408.5<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 12.0968%;\">414.1<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 11.828%;\">421.0<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 63.7068%;\">424.3<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<section class=\"textbox recall\" aria-label=\"Recall\">Average rate of change is calculated as [latex]\\frac{f(b)-f(a)}{b-a}[\/latex]<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif; background-color: #f4f3ef;\">\u200b<\/span><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm309543\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=309543&theme=lumen&iframe_resize_id=ohm309543&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">\n<p class=\"whitespace-normal break-words\">But what was the rate of change in 2014? In calculus, we&#8217;ll learn to extend the average rate of change and find the instantaneous rate of change at any specific moment using derivatives, giving us more precise information than these averages.<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm309544\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=309544&theme=lumen&iframe_resize_id=ohm309544&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p class=\"whitespace-normal break-words\">The NOAA Mauna Loa Observatory (MLO) is a key atmospheric research facility that monitors and collects long-term data on atmospheric composition and climate change. It&#8217;s known for its continuous measurements of rising carbon dioxide (CO2) concentrations, ozone, and other atmospheric constituents. The data collected at MLO helps scientists understand the impacts of human activities on the atmosphere and climate.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-3419\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/20164340\/co2_weekly_mlo.png\" alt=\"\" width=\"1000\" height=\"600\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/20164340\/co2_weekly_mlo.png 1000w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/20164340\/co2_weekly_mlo-300x180.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/20164340\/co2_weekly_mlo-768x461.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/20164340\/co2_weekly_mlo-65x39.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/20164340\/co2_weekly_mlo-225x135.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/20164340\/co2_weekly_mlo-350x210.png 350w\" sizes=\"(max-width: 1000px) 100vw, 1000px\" \/><\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm309545\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=309545&theme=lumen&iframe_resize_id=ohm309545&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<hr \/>\n<p class=\"whitespace-normal break-words\">Your station models the annual CO\u2082 cycle with: [latex]C(d) = 426 + 3.85sin(\\frac{2\u03c0}{365}(d-105))[\/latex] where d = day of year.<\/p>\n<p class=\"whitespace-normal break-words\">This function captures how CO\u2082 levels fluctuate seasonally &#8211; rising in winter when plants are dormant and falling in summer when photosynthesis is most active.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm309550\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=309550&theme=lumen&iframe_resize_id=ohm309550&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox linkToLearning\" aria-label=\"Link to Learning\">CO\u2082 monitoring is crucial for understanding our climate and how it may be changing over time. Learn more about the NOAA:\u00a0<a href=\"https:\/\/gml.noaa.gov\/about\/networks.html\">https:\/\/gml.noaa.gov\/about\/networks.html<\/a><\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":24,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":36,"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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3096"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3096\/revisions"}],"predecessor-version":[{"id":3421,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3096\/revisions\/3421"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/36"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3096\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3096"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3096"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3096"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3096"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}