e39465ad2f
new file: Files/flashplayer_32_sa.exe
new file: favicon.ico
new file: globe.gif
new file: imgs/download.png
new file: imgs/zuck.jpg
new file: index.html
new file: other.ico
new file: script.js
new file: site.webmanifest
new file: sitemap.html
new file: styles/backround.css
new file: styles/border.css
new file: styles/fonts/Titillium_Web/OFL.txt
new file: styles/fonts/Titillium_Web/TitilliumWeb-Black.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Bold.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-BoldItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-ExtraLight.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-ExtraLightItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Italic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Light.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-LightItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Regular.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-SemiBold.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-SemiBoldItalic.ttf
new file: styles/fonts/webfontkit-20221027-163353/generator_config.txt
new file: styles/fonts/webfontkit-20221027-163353/specimen_files/grid_12-825-55-15.css
new file: styles/fonts/webfontkit-20221027-163353/specimen_files/specimen_stylesheet.css
new file: styles/fonts/webfontkit-20221027-163353/stylesheet.css
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-demo.html
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff2
new file: styles/fonts/webfontkit-20221027-165950/generator_config.txt
new file: styles/fonts/webfontkit-20221027-165950/specimen_files/grid_12-825-55-15.css
new file: styles/fonts/webfontkit-20221027-165950/specimen_files/specimen_stylesheet.css
new file: styles/fonts/webfontkit-20221027-165950/stylesheet.css
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-demo.html
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff2
new file: styles/style.css
new file: tools/2048/.gitignore
new file: tools/2048/.jshintrc
new file: tools/2048/CONTRIBUTING.md
new file: tools/2048/LICENSE.txt
new file: tools/2048/README.md
new file: tools/2048/Rakefile
new file: tools/2048/favicon.ico
new file: tools/2048/index.html
new file: tools/2048/js/animframe_polyfill.js
new file: tools/2048/js/application.js
new file: tools/2048/js/bind_polyfill.js
new file: tools/2048/js/classlist_polyfill.js
new file: tools/2048/js/game_manager.js
new file: tools/2048/js/grid.js
new file: tools/2048/js/html_actuator.js
new file: tools/2048/js/keyboard_input_manager.js
new file: tools/2048/js/local_storage_manager.js
new file: tools/2048/js/tile.js
new file: tools/2048/meta/apple-touch-icon.png
new file: tools/webretro/cores/neocd_libretro.js
new file: tools/webretro/cores/neocd_libretro.wasm
new file: tools/webretro/cores/nestopia_libretro.js
new file: tools/webretro/cores/nestopia_libretro.wasm
new file: tools/webretro/cores/o2em_libretro.js
new file: tools/webretro/cores/o2em_libretro.wasm
new file: tools/webretro/cores/opera_libretro.js
new file: tools/webretro/cores/opera_libretro.wasm
217 lines
10 KiB
HTML
217 lines
10 KiB
HTML
<!DOCTYPE html>
|
||
<html lang="en"><!-- #BeginTemplate "/Templates/Advanced.dwt" --><!-- DW6 -->
|
||
|
||
<!-- Mirrored from www.mathsisfun.com/calculus/maxima-minima.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:49:05 GMT -->
|
||
<head>
|
||
<meta http-equiv="content-type" content="text/html; charset=UTF-8">
|
||
|
||
|
||
<!-- #BeginEditable "doctitle" -->
|
||
<title>Finding Maxima and Minima using Derivatives</title>
|
||
<meta name="description" content="Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.">
|
||
<script language="JavaScript" type="text/javascript">reSpell = [["traveling", "travelling"]];</script>
|
||
<!-- #EndEditable -->
|
||
<meta name="keywords" content="math, maths, mathematics, school, homework, education">
|
||
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
|
||
<meta name="HandheldFriendly" content="true">
|
||
<meta name="referrer" content="always">
|
||
<link rel="stylesheet" href="../style4.css">
|
||
<script src="../main4.js"></script>
|
||
<script>document.write(gTagHTML())</script>
|
||
</head>
|
||
|
||
<body id="bodybg" class="adv">
|
||
|
||
<div id="stt"></div>
|
||
<div id="adTop"></div>
|
||
<header>
|
||
<div id="hdr"></div>
|
||
<div id="tran"></div>
|
||
<div id="adHide"></div>
|
||
<div id="cookOK"></div>
|
||
</header>
|
||
|
||
<div class="mid">
|
||
|
||
<nav>
|
||
<div id="menuWide" class="menu"></div>
|
||
<div id="logo"><a href="../index.html"><img src="../images/style/logo-adv.svg" alt="Math is Fun"></a></div>
|
||
|
||
<div id="search" role="search"></div>
|
||
<div id="linkto"></div>
|
||
|
||
<div id="menuSlim" class="menu"></div>
|
||
<div id="menuTiny" class="menu"></div>
|
||
</nav>
|
||
|
||
<div id="extra"></div>
|
||
|
||
<article id="content" role="main">
|
||
|
||
<!-- #BeginEditable "Body" -->
|
||
|
||
<h1 class="center">Finding Maxima and Minima using Derivatives</h1>
|
||
|
||
<p class="center">Where is a function at a high or low point? Calculus can help!</p>
|
||
<p>A maximum is a high point and a minimum is a low point:</p>
|
||
<p class="center"><img src="images/function-min-max.svg" alt="function local minimum and maximum"></p>
|
||
<p>In a smoothly changing function a maximum or minimum is always where the function <b>flattens out</b> (except for a <b>saddle point</b>).</p>
|
||
<p><b><i>Where does it flatten out?</i></b> <span class="large">Where the <b>slope is zero</b>.</span></p>
|
||
<p><b><i>Where is the slope zero?</i></b> <span class="large">The <b>Derivative</b> tells us! </span></p>
|
||
<p>Let's dive right in with an example:</p>
|
||
<div class="example">
|
||
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/quadratic-1b.svg" alt="quadratic graph"></p>
|
||
<h3>Example: A ball is thrown in the air. Its height at any time t is given by:</h3>
|
||
<p class="center larger">h = 3 + 14t − 5t<sup>2</sup></p>
|
||
<h3>What is its maximum height?</h3>
|
||
<p> </p>
|
||
<p>Using <a href="derivatives-introduction.html">derivatives</a> we can find the slope of that function:</p>
|
||
<p class="center larger"><span class="intbl"><em>d</em><strong>dt</strong></span>h = 0 + 14 − 5(2t)<br>
|
||
= 14 − 10t</p>
|
||
<p>(See below this example for how we found that derivative.)</p>
|
||
<p> </p>
|
||
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/quadratic-1b-slope.svg" alt="quadratic graph"></p>
|
||
<p>Now find when the <b>slope is zero</b>:</p>
|
||
<div class="so">14 − 10t = 0</div>
|
||
<div class="so">10t = 14</div>
|
||
<div class="so"> t = 14 / 10 = <b>1.4</b></div>
|
||
<p>The slope is zero at <b>t = 1.4 seconds</b></p>
|
||
<p>And the height at that time is:</p>
|
||
<div class="so">h = 3 + 14×1.4 − 5×1.4<sup>2</sup></div>
|
||
<div class="so">h = 3 + 19.6 − 9.8 = <b>12.8</b></div>
|
||
<p>And so:</p>
|
||
<p class="center larger">The maximum height is <b>12.8 m</b> (at t = 1.4 s)</p>
|
||
</div>
|
||
<p> </p>
|
||
<div class="center80">
|
||
<h3>A Quick Refresher on Derivatives</h3>
|
||
<p>A <a href="derivatives-introduction.html">derivative</a> basically finds the slope of a function.</p>
|
||
<p>In the previous example we took this:</p>
|
||
<p class="center larger">h = 3 + 14t − 5t<sup>2</sup></p>
|
||
<p>and came up with this derivative:</p>
|
||
<p class="center larger"><span class="intbl"><em>d</em><strong>dt</strong></span>h = 0 + 14 − 5(2t)<br>
|
||
= 14 − 10t</p>
|
||
<p class="center">Which tells us the <b>slope</b> of the function at any time <b>t</b></p>
|
||
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/slope-examples.svg" alt="slope examples: y=3, slope=0; y=2x, slope=2"></p>
|
||
<p> </p>
|
||
<p>We used these <a href="derivatives-rules.html">Derivative Rules</a>:</p>
|
||
<ul>
|
||
<li>The slope of a <b>constant</b> value (like 3) is 0</li>
|
||
<li>The slope of a <b>line</b> like 2x is 2, so 14t has a slope of 14</li>
|
||
<li>A <b>square</b> function like t<sup>2</sup> has a slope of 2t, so 5t<sup>2</sup> has a slope of 5(2t)</li>
|
||
<li>And then we added them up: <span class="center larger">0 + 14 − 5(2t)</span></li>
|
||
</ul>
|
||
<div style="clear:both"></div>
|
||
</div>
|
||
<p> </p>
|
||
<h2>How Do We Know it is a Maximum (or Minimum)?</h2>
|
||
<p>We saw it on the graph! But otherwise ... derivatives come to the rescue again.</p>
|
||
<p>Take the <b>derivative of the slope</b> (the <a href="second-derivative.html">second derivative</a> of the original function):</p>
|
||
<p class="center larger">The Derivative of 14 − 10t is <b>−10</b></p>
|
||
<p>This means the slope is continually getting smaller (−10): traveling from left to right the slope starts out positive (the function rises), goes through zero (the flat point), and then the slope becomes negative (the function falls):</p>
|
||
<p class="center"><img src="images/function-max.svg" alt="slope positive then zero then negative"><br>
|
||
A slope that gets smaller (and goes though 0) means a maximum.</p>
|
||
<div class="words">
|
||
<p>This is called the <b>Second Derivative Test</b></p>
|
||
</div>
|
||
<p>On the graph above I showed the slope before and after, but in practice we do the test <b>at the point where the slope is zero</b>:</p>
|
||
<div class="center80">
|
||
<h3>Second Derivative Test</h3>
|
||
<p>When a function's <b>slope is zero at x</b>, and the <b>second derivative at x</b> is:</p>
|
||
<ul>
|
||
<li>less than 0, it is a local maximum</li>
|
||
<li>greater than 0, it is a local minimum</li>
|
||
<li>equal to 0, then the test fails (there may be other ways of finding out though)</li>
|
||
</ul>
|
||
</div>
|
||
<p class="center larger"> </p>
|
||
<p class="center larger">"Second Derivative: less than 0 is a maximum, greater than 0 is a minimum"</p>
|
||
<p> </p>
|
||
<div class="example">
|
||
<h3>Example: Find the maxima and minima for:</h3>
|
||
<p class="center larger">y = 5x<sup>3</sup> + 2x<sup>2</sup> − 3x</p>
|
||
<p>The derivative (slope) is:</p>
|
||
<p class="center larger"><span class="intbl"><em>d</em><strong>dx</strong></span>y = 15x<sup>2</sup> + 4x − 3</p>
|
||
<p>Which is <a href="../algebra/quadratic-equation.html">quadratic</a> with zeros at:</p>
|
||
<ul>
|
||
<li>x = −3/5</li>
|
||
<li>x = +1/3</li>
|
||
</ul>
|
||
<p> </p>
|
||
<p>Could they be maxima or minima? (Don't look at the graph yet!)</p>
|
||
<p> </p>
|
||
<p>The <a href="second-derivative.html">second derivative</a> is <b>y'' = 30x + 4</b></p>
|
||
<p>At x = −3/5:</p>
|
||
<div class="so">y'' = 30(−3/5) + 4 = −14</div>
|
||
<div class="so">it is less than 0, so −3/5 is a local maximum</div>
|
||
<p>At x = +1/3:</p>
|
||
<div class="so">y'' = 30(+1/3) + 4 = +14</div>
|
||
<div class="so">it is greater than 0, so +1/3 is a local minimum</div>
|
||
<p>(Now you can look at the graph.)</p>
|
||
<p class="center"><img src="images/5x3-2x2-3x.gif" alt="5x^3 2x^2 3x" height="199" width="196"></p>
|
||
</div>
|
||
<h2>Words</h2>
|
||
<div class="words">
|
||
<p>A high point is called a <b>maximum</b> (plural <i><b>maxima</b></i>).</p>
|
||
<p>A low point is called a <b>minimum</b> (plural <i><b>minima</b></i>).</p>
|
||
<p>The general word for maximum or minimum is <b>extremum</b> (plural <b>extrema</b>).</p>
|
||
<p>We say <b>local</b> maximum (or minimum) when there may be higher (or lower) points elsewhere but not nearby.</p>
|
||
</div>
|
||
<h2>One More Example</h2>
|
||
<div class="example">
|
||
<h3>Example: Find the maxima and minima for:</h3>
|
||
<p class="center larger">y = x<sup>3</sup> − 6x<sup>2</sup> + 12x − 5</p>
|
||
<p>The derivative is:</p>
|
||
<p class="center larger"><span class="intbl"><em>d</em><strong>dx</strong></span>y = 3x<sup>2</sup> − 12x + 12</p>
|
||
<p>Which is <a href="../algebra/quadratic-equation.html">quadratic</a> with only one zero at <b>x = 2</b></p>
|
||
<p>Is it a maximum or minimum?</p>
|
||
<p> </p>
|
||
<p>The <a href="second-derivative.html">second derivative</a> is <b>y'' = 6x − 12</b></p>
|
||
<p>At x = 2:</p>
|
||
<div class="so">y'' = 6(2) − 12 = 0</div>
|
||
<div class="so">it is 0, so the test fails</div>
|
||
<p>And here is why:</p>
|
||
<p class="center"><img src="images/x3-6x2-12x-5.gif" alt="x^3 6x^2 12x 5" height="207" width="171"></p>
|
||
<p>It is an <a href="inflection-points.html">Inflection Point</a> ("saddle point") ... the slope does become zero, but it is neither a maximum nor minimum.</p>
|
||
</div>
|
||
<p> </p>
|
||
<h2>Must Be Differentiable</h2>
|
||
<p>And there is an important technical point:</p>
|
||
<p>The function must be <b><a href="differentiable.html">differentiable</a></b> (the derivative must exist at each point in its domain).</p>
|
||
<div class="example">
|
||
<h3>Example: How about the function f(x) = |x| (<a href="../sets/function-absolute-value.html">absolute value</a>) ?</h3>
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td> </td>
|
||
<td>|x| looks like this:</td>
|
||
<td> </td>
|
||
<td><img src="../sets/images/function-absolute.svg" alt="Absolute Value function"></td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p>At x=0 it has a very pointy change!</p>
|
||
<p>In fact it is not differentiable there (as shown on the <a href="differentiable.html">differentiable</a> page).</p>
|
||
<p>So we can't use the derivative method for the absolute value function.</p>
|
||
</div>
|
||
<p>The function must also be <a href="continuity.html">continuous</a>, but any function that is differentiable is also continuous, so we are covered.</p>
|
||
<p> </p>
|
||
<div class="questions">
|
||
<script>
|
||
getQ(8915, 8916, 8917, 8921, 8922, 8924, 8918, 8919, 8920, 8923);
|
||
</script> </div>
|
||
|
||
<div class="related">
|
||
<a href="index.html">Calculus Index</a>
|
||
</div>
|
||
<!-- #EndEditable -->
|
||
|
||
</article>
|
||
|
||
<div id="adend" class="centerfull noprint"></div>
|
||
<footer id="footer" class="centerfull noprint"></footer>
|
||
<div id="copyrt">Copyright © 2020 MathsIsFun.com</div>
|
||
|
||
</div>
|
||
</body><!-- #EndTemplate -->
|
||
<!-- Mirrored from www.mathsisfun.com/calculus/maxima-minima.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:49:06 GMT -->
|
||
</html> |