{"id":2812,"date":"2024-06-10T14:31:57","date_gmt":"2024-06-10T14:31:57","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=2812"},"modified":"2025-08-17T22:42:19","modified_gmt":"2025-08-17T22:42:19","slug":"defining-the-derivative-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/defining-the-derivative-apply-it\/","title":{"raw":"Defining the Derivative: Apply It","rendered":"Defining the Derivative: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Calculate the slope of a tangent line to a curve and find its equation<\/li>\r\n\t<li>Find the derivative of a function at a given point<\/li>\r\n\t<li>Explain how velocity measures speed over time, and compare average velocity over a period with the exact speed at a specific moment<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Race to the Finish: Using Derivatives to Analyze Drag Racing Speeds<\/h2>\r\n<p>Two cars are drag racing down a long straight highway. A few bystanders have decided to analyze some pieces of the race, including how fast the cars are traveling and who might win.\u00a0<\/p>\r\n\r\n[caption id=\"attachment_2813\" align=\"aligncenter\" width=\"500\"]<img class=\"wp-image-2813\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/10142951\/drag-racing-6406393_960_720-300x200.jpg\" alt=\"Two cars drag racing\" width=\"500\" height=\"333\" \/> Two cars drag racing[\/caption]\r\n\r\n<p>&nbsp;<\/p>\r\n<p>The first car\u2019s position, in feet, after [latex]t[\/latex] seconds can be described by the function [latex]p(t)=25t^{2}[\/latex].\u00a0The second car\u2019s position, in feet, after [latex]t[\/latex] seconds can be described by the graph.<\/p>\r\n\r\n[caption id=\"attachment_4677\" align=\"aligncenter\" width=\"1024\"]<img class=\"wp-image-4677 size-large\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/20184753\/Screenshot-2025-02-20-134727-1-1024x599.png\" alt=\"A graph showing a linear function with a positive slope. The x-axis represents time in seconds, and the y-axis represents the position in feet. The line starts at the origin (0, 0) and increases steadily, indicating constant speed. \" width=\"1024\" height=\"599\" \/> <strong>Graph of the second car\u2019s position, in feet.<\/strong>[\/caption]\r\n\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]287750[\/ohm_question]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]287751[\/ohm_question]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]287752[\/ohm_question]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]287753[\/ohm_question]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Calculate the slope of a tangent line to a curve and find its equation<\/li>\n<li>Find the derivative of a function at a given point<\/li>\n<li>Explain how velocity measures speed over time, and compare average velocity over a period with the exact speed at a specific moment<\/li>\n<\/ul>\n<\/section>\n<h2>Race to the Finish: Using Derivatives to Analyze Drag Racing Speeds<\/h2>\n<p>Two cars are drag racing down a long straight highway. A few bystanders have decided to analyze some pieces of the race, including how fast the cars are traveling and who might win.\u00a0<\/p>\n<figure id=\"attachment_2813\" aria-describedby=\"caption-attachment-2813\" style=\"width: 500px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2813\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/10142951\/drag-racing-6406393_960_720-300x200.jpg\" alt=\"Two cars drag racing\" width=\"500\" height=\"333\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/10142951\/drag-racing-6406393_960_720-300x200.jpg 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/10142951\/drag-racing-6406393_960_720-768x512.jpg 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/10142951\/drag-racing-6406393_960_720-65x43.jpg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/10142951\/drag-racing-6406393_960_720-225x150.jpg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/10142951\/drag-racing-6406393_960_720-350x233.jpg 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/10142951\/drag-racing-6406393_960_720.jpg 960w\" sizes=\"(max-width: 500px) 100vw, 500px\" \/><figcaption id=\"caption-attachment-2813\" class=\"wp-caption-text\">Two cars drag racing<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<p>The first car\u2019s position, in feet, after [latex]t[\/latex] seconds can be described by the function [latex]p(t)=25t^{2}[\/latex].\u00a0The second car\u2019s position, in feet, after [latex]t[\/latex] seconds can be described by the graph.<\/p>\n<figure id=\"attachment_4677\" aria-describedby=\"caption-attachment-4677\" style=\"width: 1024px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4677 size-large\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/20184753\/Screenshot-2025-02-20-134727-1-1024x599.png\" alt=\"A graph showing a linear function with a positive slope. The x-axis represents time in seconds, and the y-axis represents the position in feet. The line starts at the origin (0, 0) and increases steadily, indicating constant speed.\" width=\"1024\" height=\"599\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/20184753\/Screenshot-2025-02-20-134727-1-1024x599.png 1024w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/20184753\/Screenshot-2025-02-20-134727-1-300x175.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/20184753\/Screenshot-2025-02-20-134727-1-768x449.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/20184753\/Screenshot-2025-02-20-134727-1-65x38.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/20184753\/Screenshot-2025-02-20-134727-1-225x132.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/20184753\/Screenshot-2025-02-20-134727-1-350x205.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/20184753\/Screenshot-2025-02-20-134727-1.png 1264w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><figcaption id=\"caption-attachment-4677\" class=\"wp-caption-text\"><strong>Graph of the second car\u2019s position, in feet.<\/strong><\/figcaption><\/figure>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm287750\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=287750&theme=lumen&iframe_resize_id=ohm287750&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm287751\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=287751&theme=lumen&iframe_resize_id=ohm287751&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm287752\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=287752&theme=lumen&iframe_resize_id=ohm287752&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm287753\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=287753&theme=lumen&iframe_resize_id=ohm287753&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n","protected":false},"author":15,"menu_order":8,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":503,"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\/2812"}],"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":12,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/2812\/revisions"}],"predecessor-version":[{"id":4760,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/2812\/revisions\/4760"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/503"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/2812\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=2812"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=2812"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=2812"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=2812"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}