{"id":200,"date":"2014-05-22T18:11:58","date_gmt":"2014-05-22T22:11:58","guid":{"rendered":"http:\/\/advancedmathyoungstudents.com\/blog\/?p=200"},"modified":"2018-06-13T16:32:49","modified_gmt":"2018-06-13T20:32:49","slug":"slope-finding-functions","status":"publish","type":"post","link":"https:\/\/advancedmathyoungstudents.com\/blog\/?p=200","title":{"rendered":"Slope-Finding Functions"},"content":{"rendered":"<p><strong><em>By the end of this post, you will know what a derivative is.\u00a0 You won&#8217;t know how to find one yet, but you will know what you have if you are given one!\u00a0<\/em><\/strong><\/p>\n<p>Let\u2019s say that we were given a function such as <strong>f(x) = x<sup>3<\/sup> -4x.<\/strong><\/p>\n<p>We know it isn\u2019t a linear function so when we graph it, we are not surprised to see the curves.\u00a0 And if you can imagine a tangent line \u201csurfing\u201d along the curve, the slope of the tangent line would clearly be changing as you went along.\u00a0 Here&#8217;s what I mean:<\/p>\n<div id=\"kgvid_kgvid_0_wrapper\" class=\"kgvid_wrapper\">\n\t\t\t<div id=\"video_kgvid_0_div\" class=\"fitvidsignore kgvid_videodiv\" data-id=\"kgvid_0\" data-kgvid_video_vars=\"{&quot;id&quot;:&quot;kgvid_0&quot;,&quot;attachment_id&quot;:217,&quot;player_type&quot;:&quot;Video.js&quot;,&quot;width&quot;:&quot;510&quot;,&quot;height&quot;:&quot;498&quot;,&quot;fullwidth&quot;:&quot;false&quot;,&quot;countable&quot;:true,&quot;count_views&quot;:&quot;quarters&quot;,&quot;start&quot;:&quot;&quot;,&quot;autoplay&quot;:&quot;false&quot;,&quot;pauseothervideos&quot;:&quot;false&quot;,&quot;set_volume&quot;:1,&quot;muted&quot;:&quot;false&quot;,&quot;meta&quot;:true,&quot;endofvideooverlay&quot;:false,&quot;resize&quot;:&quot;true&quot;,&quot;auto_res&quot;:&quot;automatic&quot;,&quot;pixel_ratio&quot;:&quot;true&quot;,&quot;right_click&quot;:&quot;on&quot;,&quot;playback_rate&quot;:&quot;false&quot;,&quot;title&quot;:&quot;surfing&quot;,&quot;nativecontrolsfortouch&quot;:&quot;true&quot;,&quot;locale&quot;:&quot;en&quot;,&quot;enable_resolutions_plugin&quot;:false}\" itemprop=\"video\" itemscope itemtype=\"https:\/\/schema.org\/VideoObject\"><meta itemprop=\"embedUrl\" content=\"https:\/\/advancedmathyoungstudents.com\/blog\/?attachment_id=217&#038;kgvid_video_embed%5Benable%5D=true\"><meta itemprop=\"contentUrl\" content=\"https:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2014\/05\/surfing.mp4\"><meta itemprop=\"name\" content=\"surfing\"><meta itemprop=\"description\" content=\"Video\"><meta itemprop=\"uploadDate\" content=\"2014-05-22T17:14:11-04:00\">\n\t\t\t\t<video id=\"video_kgvid_0\" playsinline controls preload=\"metadata\" width=\"510\" height=\"498\" class=\"fitvidsignore video-js kg-video-js-skin\">\n\t\t\t\t\t<source src=\"https:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2014\/05\/surfing.mp4?id=0\" type=\"video\/mp4\" data-res=\"498p\">\n\t\t\t\t<\/video>\n\t\t\t<\/div>\n\t\t\t<div style=\"display:none\" id=\"video_kgvid_0_meta\" class=\"kgvid_video_meta kgvid_video_meta_hover \">\n\t\t\t\t<span class='kgvid_meta_icons'><\/span>\n\t\t\t\t<span id='video_kgvid_0_title' class='kgvid_title'>surfing<\/span>\n<\/div>\n\t\t<\/div>\n<p>&nbsp;<\/p>\n<p>If you wanted to know the slope at a given point, you could estimate it by drawing the tangent line and then finding the slope of that tangent line (as we have seen in the last post).<\/p>\n<p>But there is an alternative to that method that is useful and quick.\u00a0 When you have the rule for a function, there is a way to derive a formula for a new function that will give you the slope of the original function\u2019s tangent line at any x-value you choose.\u00a0 Since it is <i>derived<\/i> from the original function, this new function is called \u201cthe derivative\u201d of the original function.<\/p>\n<p>For example, if the original function is:<\/p>\n<p><strong>f(x) = x<sup>3<\/sup> &#8211; 4x<\/strong><\/p>\n<p>The derivative is written:<\/p>\n<p><strong>f\u2019(x) = 3x<sup>2<\/sup> &#8211; 4<\/strong><\/p>\n<p>The symbol <b>f\u2019(x)<\/b> is pronounced \u201cf-prime of x\u201d and it is the most commonly used symbol to represent the derivative of the function f.<\/p>\n<p><em><strong>TWO THINGS YOU SHOULD NOT WORRY ABOUT (YET)<\/strong><\/em><\/p>\n<p>1. How did we get from the function <b>f(x) = x<sup>3<\/sup> &#8211; 4x<\/b> to its derivative <b>f\u2019(x)<\/b> = <b>3x<sup>2<\/sup> &#8211; 4?<\/b><\/p>\n<p>I will show you how to \u201ctake a derivative\u201d in the next post.\u00a0 Once you learn a handful of rules, the process is not difficult.\u00a0 It\u2019s easier than factoring.\u00a0 It\u2019s easier than long division.<\/p>\n<p>2.\u00a0 Why would anyone want a slope-finding formula ?<\/p>\n<p>Once you know how to take derivatives, I can show you a variety of ways that they are useful.\u00a0 In fact, you\u2019ll even get to use them forward and backwards! (That was supposed to be some fore-shadowing\u2026)<\/p>\n<p>For now, my goal is simpler.\u00a0 I just want to be sure that we understand what it is that the derivative formula tells us.\u00a0 And once again\u2026it\u2019s better with diagrams.<\/p>\n<p>I made the graphs and videos in this post using a web-based applet written by Paul Seeburger, a professor of mathematics at Monroe Community College in New York.\u00a0 You can find the applet at:<\/p>\n<p><a href=\"http:\/\/web.monroecc.edu\/manila\/webfiles\/pseeburger\/JavaCode\/DerivativeDemo2.htm\">http:\/\/web.monroecc.edu\/manila\/webfiles\/pseeburger\/JavaCode\/DerivativeDemo2.htm<\/a><\/p>\n<p><a href=\"http:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2014\/05\/post15pic1.jpg\"><img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-214\" src=\"http:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2014\/05\/post15pic1.jpg\" alt=\"post15pic1\" width=\"748\" height=\"523\" srcset=\"https:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2014\/05\/post15pic1.jpg 748w, https:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2014\/05\/post15pic1-300x209.jpg 300w, https:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2014\/05\/post15pic1-429x300.jpg 429w\" sizes=\"(max-width: 748px) 100vw, 748px\" \/><\/a><\/p>\n<p>The blue line is the original function, <strong>f(x) = x<\/strong><sup>3<\/sup><strong> &#8211; 4x<\/strong><b>.\u00a0 <\/b><\/p>\n<p>The purple line is the derivative, <strong>f\u2019(x) = 3x<sup>2<\/sup><\/strong> <strong>\u2013 4<\/strong><b>.<\/b><\/p>\n<p>The red line is the tangent line.\u00a0 You can calculate the slope of that tangent using rise over run if you want to, but you don\u2019t have to.\u00a0 The applet does the calculation for you \u2013 look in the pink box.<\/p>\n<p>Now let\u2019s pick a point on the graph and examine all of the lines on this diagram.<\/p>\n<p>At x = -2, f(-2) = (-2)<sup>3<\/sup> \u2013 4\u00d7(-2) = -8\u00a0+ 8 = 0.\u00a0 This means that (-2,0) is a point on the graph of f.<\/p>\n<p>At that same point, the tangent line has a slope of 8 (as you can see in the pink box).<\/p>\n<p>If\u00a0we plug x = -2 into the derivative formula, we get f\u2019(-2) = 3(-2)<sup>2<\/sup> \u2013 4 = 12 \u2013 4 = 8.\u00a0 And then, when you look at the graph of the derivative (in purple), you see that (2,8) is a point on that graph.\u00a0 So the derivative formula tells us the slope \u2013 at least at that one point.\u00a0 But does it always work?<\/p>\n<p>Well, let\u2019s look at another point:\u00a0 x = -1.<\/p>\n<p><a href=\"http:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2014\/05\/post15pic2.jpg\"><img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-213\" src=\"http:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2014\/05\/post15pic2.jpg\" alt=\"post15pic2\" width=\"750\" height=\"513\" srcset=\"https:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2014\/05\/post15pic2.jpg 750w, https:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2014\/05\/post15pic2-300x205.jpg 300w, https:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2014\/05\/post15pic2-438x300.jpg 438w\" sizes=\"(max-width: 750px) 100vw, 750px\" \/><\/a><\/p>\n<p>(If you are playing along at home, you can enter any x-value you want into the box on the lower left hand side of the applet where it says \u201cTrace x =\u201d)<\/p>\n<p>At x = -1, f(x) = (-1)<sup>3<\/sup> \u2013 4(-1) = 3.\u00a0 So (-1, 3) is a point on the graph.<\/p>\n<p>Then, f\u2019(x) = 3(-1)<sup>2<\/sup> \u2013 4 = -1.\u00a0 So we expect the slope of the tangent line to be -1, and it is. You can see that (-1,-1) is a point on the graph of the derivative (again, the purple one). So once again, the derivative formula has given the slope of the tangent line.<\/p>\n<p>Here\u2019s one more point that is worth noticing:<\/p>\n<p><a href=\"http:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2014\/05\/post15pic3.jpg\"><img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-212\" src=\"http:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2014\/05\/post15pic3.jpg\" alt=\"post15pic3\" width=\"742\" height=\"515\" srcset=\"https:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2014\/05\/post15pic3.jpg 742w, https:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2014\/05\/post15pic3-300x208.jpg 300w, https:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2014\/05\/post15pic3-432x300.jpg 432w\" sizes=\"(max-width: 742px) 100vw, 742px\" \/><\/a><\/p>\n<p>Look at the value of the derivative.\u00a0 And then look at the shape of the graph.\u00a0 Where the derivative is zero, the graph has a horizontal tangent.\u00a0 That little fact turns out to be a big part of one of the reasons why we care about derivatives. They help us to find places where the graph has a maximum or (as in this case) a minimum.\u00a0 We\u2019ll be hearing more about that later.<\/p>\n<p>So we have confirmed, at least for the points we checked, that for this particular function, the derivative formula does in fact give values that match the slopes of the original function\u2019s tangent lines.\u00a0\u00a0 If you want to, you can check many more points:<\/p>\n<p>As you saw in the video above, Professor Seeburger\u2019s applet lets you trace along the graph.\u00a0 As you do, the tangent line surfs along the curve and the applet keeps track of all of the values.\u00a0 Here is that video again, this time with the derivative function and the values all visible:<\/p>\n<div id=\"kgvid_kgvid_1_wrapper\" class=\"kgvid_wrapper\">\n\t\t\t<div id=\"video_kgvid_1_div\" class=\"fitvidsignore kgvid_videodiv\" data-id=\"kgvid_1\" data-kgvid_video_vars=\"{&quot;id&quot;:&quot;kgvid_1&quot;,&quot;attachment_id&quot;:219,&quot;player_type&quot;:&quot;Video.js&quot;,&quot;width&quot;:&quot;640&quot;,&quot;height&quot;:&quot;445&quot;,&quot;fullwidth&quot;:&quot;false&quot;,&quot;countable&quot;:true,&quot;count_views&quot;:&quot;quarters&quot;,&quot;start&quot;:&quot;&quot;,&quot;autoplay&quot;:&quot;false&quot;,&quot;pauseothervideos&quot;:&quot;false&quot;,&quot;set_volume&quot;:1,&quot;muted&quot;:&quot;false&quot;,&quot;meta&quot;:true,&quot;endofvideooverlay&quot;:false,&quot;resize&quot;:&quot;true&quot;,&quot;auto_res&quot;:&quot;automatic&quot;,&quot;pixel_ratio&quot;:&quot;true&quot;,&quot;right_click&quot;:&quot;on&quot;,&quot;playback_rate&quot;:&quot;false&quot;,&quot;title&quot;:&quot;surfing2&quot;,&quot;nativecontrolsfortouch&quot;:&quot;true&quot;,&quot;locale&quot;:&quot;en&quot;,&quot;enable_resolutions_plugin&quot;:false}\" itemprop=\"video\" itemscope itemtype=\"https:\/\/schema.org\/VideoObject\"><meta itemprop=\"embedUrl\" content=\"https:\/\/advancedmathyoungstudents.com\/blog\/?attachment_id=219&#038;kgvid_video_embed%5Benable%5D=true\"><meta itemprop=\"contentUrl\" content=\"https:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2014\/05\/surfing2.mp4\"><meta itemprop=\"name\" content=\"surfing2\"><meta itemprop=\"description\" content=\"Video\"><meta itemprop=\"uploadDate\" content=\"2014-05-22T18:08:16-04:00\">\n\t\t\t\t<video id=\"video_kgvid_1\" playsinline controls preload=\"metadata\" width=\"640\" height=\"445\" class=\"fitvidsignore video-js kg-video-js-skin\">\n\t\t\t\t\t<source src=\"https:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2014\/05\/surfing2.mp4?id=1\" type=\"video\/mp4\" data-res=\"520p\">\n\t\t\t\t<\/video>\n\t\t\t<\/div>\n\t\t\t<div style=\"display:none\" id=\"video_kgvid_1_meta\" class=\"kgvid_video_meta kgvid_video_meta_hover \">\n\t\t\t\t<span class='kgvid_meta_icons'><\/span>\n\t\t\t\t<span id='video_kgvid_1_title' class='kgvid_title'>surfing2<\/span>\n<\/div>\n\t\t<\/div>\n<p>&nbsp;<\/p>\n<p>So if you are skeptical, run the video and then pause it wherever you want.\u00a0 You\u2019ll see that the slope of the tangent line always matches the value given by the derivative.\u00a0 Derivatives really are slope-finding formulas.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>By the end of this post, you will know what a derivative is.\u00a0 You won&#8217;t know how to find one yet, but you will know what you have if you are given one!\u00a0 Let\u2019s say that we were given a &hellip; <a href=\"https:\/\/advancedmathyoungstudents.com\/blog\/?p=200\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false},"categories":[1],"tags":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4uvY7-3e","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/200"}],"collection":[{"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=200"}],"version-history":[{"count":17,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/200\/revisions"}],"predecessor-version":[{"id":619,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/200\/revisions\/619"}],"wp:attachment":[{"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=200"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=200"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=200"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}