{"id":3312,"date":"2025-08-16T01:55:45","date_gmt":"2025-08-16T01:55:45","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3312"},"modified":"2025-10-24T17:10:51","modified_gmt":"2025-10-24T17:10:51","slug":"continuity-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/continuity-apply-it\/","title":{"raw":"Continuity: Apply It","rendered":"Continuity: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Determine whether a function is continuous at a number.<\/li>\r\n \t<li>Determine the input values for which a function is discontinuous.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<div class=\"flex-1 flex flex-col gap-3 px-4 max-w-3xl mx-auto w-full pt-1\">\r\n<div data-test-render-count=\"1\">\r\n<div class=\"group relative pb-3\" data-is-streaming=\"false\">\r\n<div class=\"font-claude-response relative leading-[1.65rem] [&amp;_pre&gt;div]:bg-bg-000\/50 [&amp;_pre&gt;div]:border-0.5 [&amp;_pre&gt;div]:border-border-400 [&amp;_.ignore-pre-bg&gt;div]:bg-transparent [&amp;_.standard-markdown_:is(p,blockquote,h1,h2,h3,h4,h5,h6)]:pl-2 [&amp;_.standard-markdown_:is(p,blockquote,ul,ol,h1,h2,h3,h4,h5,h6)]:pr-8 [&amp;_.progressive-markdown_:is(p,blockquote,h1,h2,h3,h4,h5,h6)]:pl-2 [&amp;_.progressive-markdown_:is(p,blockquote,ul,ol,h1,h2,h3,h4,h5,h6)]:pr-8\">\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 standard-markdown\">\r\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Parking Garage Pricing<\/h2>\r\n<p class=\"whitespace-normal break-words\">Most functions in real life follow a piecewise pattern with jump discontinuities depending on the conditions.<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">A downtown parking garage charges $4 per hour or any fraction of an hour, with a $24 daily maximum. The cost function is:<\/p>\r\n<p class=\"whitespace-normal break-words\">[latex]C(t) = \\begin{cases} 4 &amp; 0 &lt; t \\leq 1 \\\\ 8 &amp; 1 &lt; t \\leq 2 \\\\ 12 &amp; 2 &lt; t \\leq 3 \\\\ 16 &amp; 3 &lt; t \\leq 4 \\\\ 20 &amp; 4 &lt; t \\leq 5 \\\\ 24 &amp; t &gt; 5 \\end{cases}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">where [latex]t[\/latex] is hours parked and [latex]C(t)[\/latex] is cost in dollars.<\/p>\r\nThis function is\u00a0<strong>not continuous\u00a0<\/strong>on the hour as the price \"jumps\" from one level to the next.\r\n\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\">\r\n<p class=\"whitespace-normal break-words\"><strong>How To: Determining Continuity at a Point<\/strong><\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Check Condition 1: Does [latex]f(a)[\/latex] exist?<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Check Condition 2: Does [latex]\\lim_{x \\to a} f(x)[\/latex] exist? (Do left and right limits equal each other?)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Check Condition 3: Does [latex]\\lim_{x \\to a} f(x) = f(a)[\/latex]?<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If all three conditions hold, the function is continuous at [latex]x = a[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If conditions fail, identify the type of discontinuity<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"whitespace-normal break-words\">A laboratory temperature control system is modeled by:<\/p>\r\n<p class=\"whitespace-normal break-words\">[latex]T(t) = \\begin{cases} 4t &amp; t \\leq 3 \\\\ 8 + t &amp; t &gt; 3 \\end{cases}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">where [latex]t[\/latex] is time in hours and [latex]T(t)[\/latex] is temperature in degrees Celsius. Determine whether [latex]T(t)[\/latex] is continuous at [latex]t = 3[\/latex].<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">A cost efficiency model is [latex]E(x) = \\frac{9 - x^2}{x^2 - 3x}[\/latex] where [latex]x[\/latex] represents production level. Determine whether [latex]E(x)[\/latex] is continuous at [latex]x = 3[\/latex]. If not, identify the type of discontinuity.\r\n<p class=\"whitespace-normal break-words\">Is [latex]E(x)[\/latex] continuous at [latex]x = 3[\/latex]? [response area] (yes or no)<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"whitespace-normal break-words\">A data transmission rate is:<\/p>\r\n<p class=\"whitespace-normal break-words\">[latex]T(x) = \\begin{cases} \\sin(x) &amp; x &lt; 0 \\ x^3 &amp; x &gt; 0 \\end{cases}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">where [latex]x[\/latex] is signal strength. What type of discontinuity exists at [latex]x = 0[\/latex]?<\/p>\r\n<p class=\"whitespace-normal break-words\">Type of discontinuity at [latex]x = 0[\/latex]: [response area]<\/p>\r\n\r\n<\/section><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Determine whether a function is continuous at a number.<\/li>\n<li>Determine the input values for which a function is discontinuous.<\/li>\n<\/ul>\n<\/section>\n<div class=\"flex-1 flex flex-col gap-3 px-4 max-w-3xl mx-auto w-full pt-1\">\n<div data-test-render-count=\"1\">\n<div class=\"group relative pb-3\" data-is-streaming=\"false\">\n<div class=\"font-claude-response relative leading-[1.65rem] [&amp;_pre&gt;div]:bg-bg-000\/50 [&amp;_pre&gt;div]:border-0.5 [&amp;_pre&gt;div]:border-border-400 [&amp;_.ignore-pre-bg&gt;div]:bg-transparent [&amp;_.standard-markdown_:is(p,blockquote,h1,h2,h3,h4,h5,h6)]:pl-2 [&amp;_.standard-markdown_:is(p,blockquote,ul,ol,h1,h2,h3,h4,h5,h6)]:pr-8 [&amp;_.progressive-markdown_:is(p,blockquote,h1,h2,h3,h4,h5,h6)]:pl-2 [&amp;_.progressive-markdown_:is(p,blockquote,ul,ol,h1,h2,h3,h4,h5,h6)]:pr-8\">\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 standard-markdown\">\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Parking Garage Pricing<\/h2>\n<p class=\"whitespace-normal break-words\">Most functions in real life follow a piecewise pattern with jump discontinuities depending on the conditions.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">A downtown parking garage charges $4 per hour or any fraction of an hour, with a $24 daily maximum. The cost function is:<\/p>\n<p class=\"whitespace-normal break-words\">[latex]C(t) = \\begin{cases} 4 & 0 < t \\leq 1 \\\\ 8 & 1 < t \\leq 2 \\\\ 12 & 2 < t \\leq 3 \\\\ 16 & 3 < t \\leq 4 \\\\ 20 & 4 < t \\leq 5 \\\\ 24 & t > 5 \\end{cases}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">where [latex]t[\/latex] is hours parked and [latex]C(t)[\/latex] is cost in dollars.<\/p>\n<p>This function is\u00a0<strong>not continuous\u00a0<\/strong>on the hour as the price &#8220;jumps&#8221; from one level to the next.<\/p>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\n<p class=\"whitespace-normal break-words\"><strong>How To: Determining Continuity at a Point<\/strong><\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Check Condition 1: Does [latex]f(a)[\/latex] exist?<\/li>\n<li class=\"whitespace-normal break-words\">Check Condition 2: Does [latex]\\lim_{x \\to a} f(x)[\/latex] exist? (Do left and right limits equal each other?)<\/li>\n<li class=\"whitespace-normal break-words\">Check Condition 3: Does [latex]\\lim_{x \\to a} f(x) = f(a)[\/latex]?<\/li>\n<li class=\"whitespace-normal break-words\">If all three conditions hold, the function is continuous at [latex]x = a[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">If conditions fail, identify the type of discontinuity<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p class=\"whitespace-normal break-words\">A laboratory temperature control system is modeled by:<\/p>\n<p class=\"whitespace-normal break-words\">[latex]T(t) = \\begin{cases} 4t & t \\leq 3 \\\\ 8 + t & t > 3 \\end{cases}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">where [latex]t[\/latex] is time in hours and [latex]T(t)[\/latex] is temperature in degrees Celsius. Determine whether [latex]T(t)[\/latex] is continuous at [latex]t = 3[\/latex].<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">A cost efficiency model is [latex]E(x) = \\frac{9 - x^2}{x^2 - 3x}[\/latex] where [latex]x[\/latex] represents production level. Determine whether [latex]E(x)[\/latex] is continuous at [latex]x = 3[\/latex]. If not, identify the type of discontinuity.<\/p>\n<p class=\"whitespace-normal break-words\">Is [latex]E(x)[\/latex] continuous at [latex]x = 3[\/latex]? [response area] (yes or no)<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p class=\"whitespace-normal break-words\">A data transmission rate is:<\/p>\n<p class=\"whitespace-normal break-words\">[latex]T(x) = \\begin{cases} \\sin(x) & x < 0 \\ x^3 & x > 0 \\end{cases}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">where [latex]x[\/latex] is signal strength. What type of discontinuity exists at [latex]x = 0[\/latex]?<\/p>\n<p class=\"whitespace-normal break-words\">Type of discontinuity at [latex]x = 0[\/latex]: [response area]<\/p>\n<\/section>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":67,"menu_order":22,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":263,"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\/3312"}],"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":6,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3312\/revisions"}],"predecessor-version":[{"id":4896,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3312\/revisions\/4896"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/263"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3312\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3312"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3312"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3312"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3312"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}