{"id":33834,"date":"2020-04-06T13:33:04","date_gmt":"2020-04-06T17:33:04","guid":{"rendered":"https:\/\/www.myedme.com\/login\/?p=33834"},"modified":"2020-04-06T14:12:19","modified_gmt":"2020-04-06T18:12:19","slug":"ap-calculus-unit-2-concepts","status":"publish","type":"post","link":"https:\/\/myedme.com\/login\/ap-calculus-unit-2-concepts\/","title":{"rendered":"AP Calculus: Unit 2 Concepts"},"content":{"rendered":"\n<p>The slope of this line is given by an equation in the form of a difference quotient:<\/p>\n\n\n\n<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%F0%9D%91%9A%3D+%5Cfrac%7B%F0%9D%91%93%28%F0%9D%91%A5%29%E2%88%92%F0%9D%91%93%28%F0%9D%91%8E%29%7D%7B%F0%9D%91%A5%E2%88%92%F0%9D%91%8E%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"\ud835\udc5a= &#92;frac{\ud835\udc53(\ud835\udc65)\u2212\ud835\udc53(\ud835\udc4e)}{\ud835\udc65\u2212\ud835\udc4e}\" class=\"latex\" \/>\n\n\n\n<p>We can also calculate the slope of a secant line to a function at a value\u00a0<em>a<\/em>\u00a0by using this equation and replacing \ud835\udc65 with \ud835\udc4e+\u210e, where \u210e is a value close to 0. We can then calculate the slope of the line through the points\u00a0(\ud835\udc4e,\ud835\udc53(\ud835\udc4e))\u00a0and\u00a0(\ud835\udc4e+\u210e,\ud835\udc53(\ud835\udc4e+\u210e)). In this case, we find the secant line has a slope given by the following difference quotient with increment\u00a0\u210e:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"alignright size-large is-resized\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/www.myedme.com\/login\/wp-content\/uploads\/2020\/04\/Note4May_CalcUnit2.png?resize=268%2C254&#038;ssl=1\" alt=\"\" class=\"wp-image-33842\" width=\"268\" height=\"254\" srcset=\"https:\/\/i0.wp.com\/myedme.com\/login\/wp-content\/uploads\/2020\/04\/Note4May_CalcUnit2.png?w=356&amp;ssl=1 356w, https:\/\/i0.wp.com\/myedme.com\/login\/wp-content\/uploads\/2020\/04\/Note4May_CalcUnit2.png?resize=300%2C285&amp;ssl=1 300w, https:\/\/i0.wp.com\/myedme.com\/login\/wp-content\/uploads\/2020\/04\/Note4May_CalcUnit2.png?resize=50%2C47&amp;ssl=1 50w, https:\/\/i0.wp.com\/myedme.com\/login\/wp-content\/uploads\/2020\/04\/Note4May_CalcUnit2.png?resize=100%2C95&amp;ssl=1 100w\" sizes=\"auto, (max-width: 268px) 100vw, 268px\" \/><\/figure><\/div>\n\n\n\n\n\n<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%F0%9D%91%9A%3D+%5Cfrac%7B%F0%9D%91%93%28a%2Bh%29%E2%88%92%F0%9D%91%93%28%F0%9D%91%8E%29%7D%7Ba%2Bh%E2%88%92%F0%9D%91%8E%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"\ud835\udc5a= &#92;frac{\ud835\udc53(a+h)\u2212\ud835\udc53(\ud835\udc4e)}{a+h\u2212\ud835\udc4e}\" class=\"latex\" \/>\n<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%F0%9D%91%9A%3D+%5Cfrac%7B%F0%9D%91%93%28a%2Bh%29%E2%88%92%F0%9D%91%93%28%F0%9D%91%8E%29%7D%7Bh%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"\ud835\udc5a= &#92;frac{\ud835\udc53(a+h)\u2212\ud835\udc53(\ud835\udc4e)}{h}\" class=\"latex\" \/>\n\n\n\n<h3 class=\"wp-block-heading\">DEFINITION<\/h3>\n\n\n\n<p>Let\u00a0\ud835\udc53\u00a0be a function defined on an interval\u00a0containing\u00a0\ud835\udc4e.\u00a0If\u00a0\ud835\udc65\u2260\ud835\udc4e\u00a0is on the interval,\u00a0then<\/p>\n\n\n\n<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%F0%9D%91%84%3D+%5Cfrac%7B%F0%9D%91%93%28%F0%9D%91%A5%29%E2%88%92%F0%9D%91%93%28%F0%9D%91%8E%29%7D%7B%F0%9D%91%A5%E2%88%92%F0%9D%91%8E%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"\ud835\udc44= &#92;frac{\ud835\udc53(\ud835\udc65)\u2212\ud835\udc53(\ud835\udc4e)}{\ud835\udc65\u2212\ud835\udc4e}\" class=\"latex\" \/>\n\n\n\n<p>is a\u00a0<strong>difference quotient<\/strong>. Also, if\u00a0\u210e \u2260 0\u00a0is chosen so that\u00a0\ud835\udc4e+\u210e\u00a0is in\u00a0the interval,\u00a0then<\/p>\n\n\n\n<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%F0%9D%91%84%3D+%5Cfrac%7B%F0%9D%91%93%28%F0%9D%91%8E%2B%E2%84%8E%29%E2%88%92%F0%9D%91%93%28%F0%9D%91%8E%29%7D%7B%E2%84%8E%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"\ud835\udc44= &#92;frac{\ud835\udc53(\ud835\udc4e+\u210e)\u2212\ud835\udc53(\ud835\udc4e)}{\u210e}\" class=\"latex\" \/>\n\n\n\n<p>is a difference quotient with increment\u00a0\u210e.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Defining the Derivative<\/h2>\n\n\n\n<p>Let\u00a0\ud835\udc53(\ud835\udc65)\u00a0be a function defined in an open interval containing\u00a0\ud835\udc4e.\u00a0The derivative of the function\u00a0\ud835\udc53(\ud835\udc65) at\u00a0\ud835\udc4e, denoted by\u00a0\ud835\udc53\u2032(\ud835\udc4e), is defined by<\/p>\n\n\n\n<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%F0%9D%91%93%E2%80%B2%28%F0%9D%91%8E%29%3D+%5Clim%5Climits_%7B%F0%9D%91%A5%E2%86%92%F0%9D%91%8E%7D+%5Cfrac%7B%F0%9D%91%93%28%F0%9D%91%A5%29%E2%88%92%F0%9D%91%93%28%F0%9D%91%8E%29%7D%7B%F0%9D%91%A5%E2%88%92%F0%9D%91%8E%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"\ud835\udc53\u2032(\ud835\udc4e)= &#92;lim&#92;limits_{\ud835\udc65\u2192\ud835\udc4e} &#92;frac{\ud835\udc53(\ud835\udc65)\u2212\ud835\udc53(\ud835\udc4e)}{\ud835\udc65\u2212\ud835\udc4e}\" class=\"latex\" \/>\n\n\n\n<p>provided this limit exists. Alternatively, we may also define the derivative of\u00a0\ud835\udc53(\ud835\udc65)\u00a0at\u00a0\ud835\udc4e\u00a0as<\/p>\n\n\n\n<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%F0%9D%91%93%E2%80%B2%28%F0%9D%91%8E%29%3D+%5Clim%5Climits_%7B%E2%84%8E%E2%86%920%7D+%5Cfrac%7B%F0%9D%91%93%28%F0%9D%91%8E%2B%E2%84%8E%29%E2%88%92%F0%9D%91%93%28%F0%9D%91%8E%29%7D%7B%E2%84%8E%7D.&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"\ud835\udc53\u2032(\ud835\udc4e)= &#92;lim&#92;limits_{\u210e\u21920} &#92;frac{\ud835\udc53(\ud835\udc4e+\u210e)\u2212\ud835\udc53(\ud835\udc4e)}{\u210e}.\" class=\"latex\" \/>\n\n\n\n<h2 class=\"wp-block-heading\">Video Introduction<\/h2>\n\n\n\n<figure class=\"wp-block-embed-youtube wp-block-embed is-type-video is-provider-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<div class=\"epyt-video-wrapper\"><div  id=\"_ytid_31254\"  width=\"980\" height=\"551\"  data-origwidth=\"980\" data-origheight=\"551\" data-facadesrc=\"https:\/\/www.youtube.com\/embed\/N2PpRnFqnqY?enablejsapi=1&#038;autoplay=0&#038;cc_load_policy=0&#038;cc_lang_pref=&#038;iv_load_policy=1&#038;loop=0&#038;rel=1&#038;fs=1&#038;playsinline=0&#038;autohide=2&#038;theme=dark&#038;color=red&#038;controls=1&#038;disablekb=0&#038;\" class=\"__youtube_prefs__ epyt-facade epyt-is-override  no-lazyload\" data-epautoplay=\"1\" ><img data-recalc-dims=\"1\" decoding=\"async\" data-spai-excluded=\"true\" class=\"epyt-facade-poster skip-lazy\" loading=\"lazy\"  alt=\"YouTube player\"  src=\"https:\/\/i0.wp.com\/i.ytimg.com\/vi\/N2PpRnFqnqY\/maxresdefault.jpg?w=980&#038;ssl=1\"  \/><button class=\"epyt-facade-play\" aria-label=\"Play\"><svg data-no-lazy=\"1\" height=\"100%\" version=\"1.1\" viewBox=\"0 0 68 48\" width=\"100%\"><path class=\"ytp-large-play-button-bg\" d=\"M66.52,7.74c-0.78-2.93-2.49-5.41-5.42-6.19C55.79,.13,34,0,34,0S12.21,.13,6.9,1.55 C3.97,2.33,2.27,4.81,1.48,7.74C0.06,13.05,0,24,0,24s0.06,10.95,1.48,16.26c0.78,2.93,2.49,5.41,5.42,6.19 C12.21,47.87,34,48,34,48s21.79-0.13,27.1-1.55c2.93-0.78,4.64-3.26,5.42-6.19C67.94,34.95,68,24,68,24S67.94,13.05,66.52,7.74z\" fill=\"#f00\"><\/path><path d=\"M 45,24 27,14 27,34\" fill=\"#fff\"><\/path><\/svg><\/button><\/div><\/div>\n<\/div><\/figure>\n\n\n\n<hr class=\"wp-block-separator is-style-wide\"\/>\n\n\n\n<p>Want to see these numbers in action? This <a href=\"https:\/\/demonstrations.wolfram.com\/ASnowballsRateOfChange\/\" target=\"_blank\" rel=\"noreferrer noopener\">tool from Wolfram<\/a> uses a &#8220;snowball&#8221; to show the rate of change for different functions. <\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Notes Check<\/h2>\n\n\n\n<p>Which definitions match these images?<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/openstax.org\/resources\/44546d16e90380bcbe1a4470715ba0659039dad2\" alt=\"This figure consists of two graphs labeled a and b. Figure a shows the Cartesian coordinate plane with 0, a, and x marked on the x-axis. There is a curve labeled y = f(x) with points marked (a, f(a)) and (x, f(x)). There is also a straight line that crosses these two points (a, f(a)) and (x, f(x)). At the bottom of the graph, the equation msec = (f(x) - f(a))\/(x - a) is given. Figure b shows a similar graph, but this time a + h is marked on the x-axis instead of x. Consequently, the curve labeled y = f(x) passes through (a, f(a)) and (a + h, f(a + h)) as does the straight line. At the bottom of the graph, the equation msec = (f(a + h) - f(a))\/h is given.\"\/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Video Deep Dives <\/h2>\n\n\n\n<p>NancyPi<\/p>\n\n\n\n<figure class=\"wp-block-embed-youtube is-type-video is-provider-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<div class=\"epyt-video-wrapper\"><div  id=\"_ytid_60217\"  width=\"980\" height=\"551\"  data-origwidth=\"980\" data-origheight=\"551\" data-facadesrc=\"https:\/\/www.youtube.com\/embed\/-ktrtzYVk_I?enablejsapi=1&#038;autoplay=0&#038;cc_load_policy=0&#038;cc_lang_pref=&#038;iv_load_policy=1&#038;loop=0&#038;rel=1&#038;fs=1&#038;playsinline=0&#038;autohide=2&#038;theme=dark&#038;color=red&#038;controls=1&#038;disablekb=0&#038;\" class=\"__youtube_prefs__ epyt-facade epyt-is-override  no-lazyload\" data-epautoplay=\"1\" ><img data-recalc-dims=\"1\" decoding=\"async\" data-spai-excluded=\"true\" class=\"epyt-facade-poster skip-lazy\" loading=\"lazy\"  alt=\"YouTube player\"  src=\"https:\/\/i0.wp.com\/i.ytimg.com\/vi\/-ktrtzYVk_I\/maxresdefault.jpg?w=980&#038;ssl=1\"  \/><button class=\"epyt-facade-play\" aria-label=\"Play\"><svg data-no-lazy=\"1\" height=\"100%\" version=\"1.1\" viewBox=\"0 0 68 48\" width=\"100%\"><path class=\"ytp-large-play-button-bg\" d=\"M66.52,7.74c-0.78-2.93-2.49-5.41-5.42-6.19C55.79,.13,34,0,34,0S12.21,.13,6.9,1.55 C3.97,2.33,2.27,4.81,1.48,7.74C0.06,13.05,0,24,0,24s0.06,10.95,1.48,16.26c0.78,2.93,2.49,5.41,5.42,6.19 C12.21,47.87,34,48,34,48s21.79-0.13,27.1-1.55c2.93-0.78,4.64-3.26,5.42-6.19C67.94,34.95,68,24,68,24S67.94,13.05,66.52,7.74z\" fill=\"#f00\"><\/path><path d=\"M 45,24 27,14 27,34\" fill=\"#fff\"><\/path><\/svg><\/button><\/div><\/div>\n<\/div><\/figure>\n\n\n\n<p>Organic Chemistry Tutor<\/p>\n\n\n\n<figure class=\"wp-block-embed-youtube is-type-video is-provider-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<div class=\"epyt-video-wrapper\"><div  id=\"_ytid_54978\"  width=\"980\" height=\"551\"  data-origwidth=\"980\" data-origheight=\"551\" data-facadesrc=\"https:\/\/www.youtube.com\/embed\/-aTLjoDT1GQ?enablejsapi=1&#038;autoplay=0&#038;cc_load_policy=0&#038;cc_lang_pref=&#038;iv_load_policy=1&#038;loop=0&#038;rel=1&#038;fs=1&#038;playsinline=0&#038;autohide=2&#038;theme=dark&#038;color=red&#038;controls=1&#038;disablekb=0&#038;\" class=\"__youtube_prefs__ epyt-facade epyt-is-override  no-lazyload\" data-epautoplay=\"1\" ><img data-recalc-dims=\"1\" decoding=\"async\" data-spai-excluded=\"true\" class=\"epyt-facade-poster skip-lazy\" loading=\"lazy\"  alt=\"YouTube player\"  src=\"https:\/\/i0.wp.com\/i.ytimg.com\/vi\/-aTLjoDT1GQ\/maxresdefault.jpg?w=980&#038;ssl=1\"  \/><button class=\"epyt-facade-play\" aria-label=\"Play\"><svg data-no-lazy=\"1\" height=\"100%\" version=\"1.1\" viewBox=\"0 0 68 48\" width=\"100%\"><path class=\"ytp-large-play-button-bg\" d=\"M66.52,7.74c-0.78-2.93-2.49-5.41-5.42-6.19C55.79,.13,34,0,34,0S12.21,.13,6.9,1.55 C3.97,2.33,2.27,4.81,1.48,7.74C0.06,13.05,0,24,0,24s0.06,10.95,1.48,16.26c0.78,2.93,2.49,5.41,5.42,6.19 C12.21,47.87,34,48,34,48s21.79-0.13,27.1-1.55c2.93-0.78,4.64-3.26,5.42-6.19C67.94,34.95,68,24,68,24S67.94,13.05,66.52,7.74z\" fill=\"#f00\"><\/path><path d=\"M 45,24 27,14 27,34\" fill=\"#fff\"><\/path><\/svg><\/button><\/div><\/div>\n<\/div><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>The slope of this line is given by an equation in the form of a difference quotient: We can also calculate the slope of a secant line to a function at a value\u00a0a\u00a0by using this equation and replacing \ud835\udc65 with \ud835\udc4e+\u210e, where \u210e is a value close to 0. We can then calculate the slope [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"nf_dc_page":"","_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[1],"tags":[],"class_list":["post-33834","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/myedme.com\/login\/wp-json\/wp\/v2\/posts\/33834","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/myedme.com\/login\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/myedme.com\/login\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/myedme.com\/login\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/myedme.com\/login\/wp-json\/wp\/v2\/comments?post=33834"}],"version-history":[{"count":10,"href":"https:\/\/myedme.com\/login\/wp-json\/wp\/v2\/posts\/33834\/revisions"}],"predecessor-version":[{"id":33849,"href":"https:\/\/myedme.com\/login\/wp-json\/wp\/v2\/posts\/33834\/revisions\/33849"}],"wp:attachment":[{"href":"https:\/\/myedme.com\/login\/wp-json\/wp\/v2\/media?parent=33834"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/myedme.com\/login\/wp-json\/wp\/v2\/categories?post=33834"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/myedme.com\/login\/wp-json\/wp\/v2\/tags?post=33834"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}