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211 lines
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HTML
211 lines
9.7 KiB
HTML
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Second Derivative</h1>
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<p class="center">
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<img src="images/slope-example-2.svg" alt="slope examples"><br>A derivative basically gives you the slope of a function at any point.<br><i>The derivative of <b>2x</b> is <b>2</b></i></p>
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<p>Read more about <a href="derivatives-introduction.html">derivatives</a> if you don't already know what they are!</p>
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<p>The "Second Derivative" is <b>the derivative of</b> the derivative of a function. So:</p>
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<ul>
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<li>Find the derivative of a function</li>
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<li>Then find the derivative of <b>that</b></li>
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</ul>
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<p>A derivative is often shown with a little tick mark: <b>f'(x)</b></p>
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<p><b></b>The second derivative is shown with <b>two</b> tick marks like this: <b>f''(x)</b></p>
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<div class="example">
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<h3>Example: f(x) = x<sup>3</sup></h3>
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<ul>
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<li>Its derivative is <b>f'(x) = 3x<sup>2</sup></b></li>
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<li>The derivative of 3x<sup>2</sup> is 6x<b></b></li></ul>
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<p>So the second derivative of f(x) is <b>6x</b>:</p>
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<ul>
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</ul>
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<p class="center larger">f''(x) = 6x</p>
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</div>
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<p> </p>
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<p class="center">A derivative can also be shown as <b><span class="intbl"><em>dy</em><strong>dx</strong></span></b><br>and the second derivative shown as <b><span class="intbl"><em>d<sup>2</sup>y</em><strong>dx<sup>2</sup></strong></span></b></p>
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<div class="example">
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<h3>Example: (continued)</h3>
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<p>The previous example could be written like this:</p>
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<p class="center"> <b>y = x<sup>3</sup></b></p>
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<p class="center"><b><span class="intbl"><em>dy</em><strong>dx</strong></span> = 3x<sup>2</sup></b></p>
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<p class="center"><b><span class="intbl"><em>d<sup>2</sup>y</em><strong>dx<sup>2</sup></strong></span> = 6x</b> </p>
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</div>
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<h2>Distance, Speed and Acceleration</h2>
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<p>A common real world example of this is distance, speed and acceleration:</p>
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<div class="example">
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<h3>Example: A bike race!</h3>
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<p>You are cruising along in a bike race, going a steady <b>10 m every second</b>.</p>
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<p class="center"><img src="images/speed.svg" alt="speed 10m in 1s"></p>
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<p><b>Distance</b>: is how far you have moved along your path. It is common to use <b>s</b> for distance (from the Latin "spatium").</p>
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<ul>
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</ul>
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<p> </p>
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<p><b>Speed:</b> is how much your distance <b>s</b> changes over time <b>t</b> ...</p>
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<p>... and is the <b>first derivative</b> of distance with respect to time: <span class="large"><span class="intbl"><em>ds</em><strong>dt</strong></span></span></p>
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<p>And we know you are doing 10 m per second, so: <span class="large"><span class="intbl"><em>ds</em><strong>dt</strong></span> = 10 m/s</span></p>
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<p> </p>
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<p><b>Acceleration</b>: Now you start cycling faster! You increase your speed to <b>14 m every second</b> over the next 2 seconds.</p>
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<p class="center"><img src="images/acceleration.svg" alt="acceleration from 10m per 1s to 14m per 1s"></p>
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<p>When you are accelerating your <b>speed is changing</b> over time.</p>
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<p>So <span class="large"><span class="intbl"><em>ds</em><strong>dt</strong> </span></span> is changing over time!</p>
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<table style="border: 0;">
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<tbody>
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<tr>
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<td rowspan="2">We could write it like this: </td>
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<td>
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<table style="border: 0;">
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<tbody>
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<tr>
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<td><b>d</b></td>
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<td><span class="large"><span class="intbl"> <em>ds</em> <strong>dt</strong> </span></span></td>
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</tr>
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</tbody></table></td>
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</tr>
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<tr>
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<td style="border-top: 1px solid;" align="center"><b>dt</b></td>
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</tr>
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</tbody></table>
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<p>But it is usually written <span class="large"><span class="intbl"> <em>d<sup>2</sup>s</em> <strong>dt<sup>2</sup></strong> </span></span></p>
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<p>Your speed <b>increases by 4 m/s</b> over <b>2 seconds</b>, so:</p>
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<p class="center"> <span class="large"><span class="intbl"> <em>d<sup>2</sup>s</em> <strong>dt<sup>2</sup></strong> </span> = <span class="intbl"><em>4</em><strong>2</strong></span> = 2 m/s<sup>2</sup></span></p>
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<p> Your speed changes by <b>2 meters per second</b> <i>per second</i>.</p>
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<p class="center">
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And yes, "per second" is used twice!</p>
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<p>It can be thought of as (m/s)/s but is usually written m/s<sup>2</sup></p>
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<p> </p>
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<p><i>(Note: in the real world your speed and acceleration changes moment to moment, but here we assume you are super steady!)</i></p>
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</div>
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<p>Here it is in one table:</p>
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<div class="beach">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td> </td>
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<td style="text-align:center; width:100px;"> </td>
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<td style="text-align:center; width:120px;"><i>Example<br>
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Measurement</i></td>
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</tr>
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<tr>
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<td style="text-align:right;"><b>Distance</b>:</td>
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<td class="large" align="center">s</td>
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<td style="text-align:center;"><i>100 m</i></td>
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</tr>
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<tr>
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<td style="text-align:right;">First Derivative is <b>Speed</b>:</td>
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<td style="text-align:center;"><span class="large"><span class="intbl"> <em>ds</em> <strong>dt</strong> </span></span></td>
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<td style="text-align:center;"><i>10 m/s</i></td>
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</tr>
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<tr>
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<td style="text-align:right;">Second Derivative is <b>Acceleration</b>:</td>
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<td style="text-align:center;"><span class="large"><span class="intbl"> <em>d<sup>2</sup>s</em> <strong>dt<sup>2</sup></strong> </span></span></td>
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<td style="text-align:center;"><i>2 m/s<sup>2</sup></i></td>
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</tr>
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</tbody></table>
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</div><br>
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<div class="fun">
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<p>But wait, there is more!</p>
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<p>The <b>third</b> derivative of position with respect to time (how acceleration changes over time) is called "Jerk" or "Jolt" !</p>
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<p>We can actually feel Jerk when we start to accelerate, apply brakes or go around corners as our body adjusts to the new forces.</p>
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<p>Engineers try to reduce Jerk when designing elevators, train tracks, etc.</p>
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<p>Also:</p>
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<ul>
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<li>The <b>fourth</b> derivative of position with respect to time is called "Snap" or "Jounce"</li>
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<li>The <b>fifth</b> is "Crackle"</li>
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<li>The <b>sixth</b> is "Pop"</li>
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</ul>
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<p>Yes, really!</p>
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<p class="center larger">They go: distance, speed, acceleration, jerk, snap, crackle and pop</p>
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</div>
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<h2>Play With It</h2>
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<p>Here you can see the derivative f'(x) and the second derivative f''(x) of some common functions.</p>
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<!--<div class="flash">
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<script>putFlash6(620,840,'images/derivative-second.swf','','#FFFFFF');</script>
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<noscript>
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<object type="application/x-shockwave-flash" width="620" height="840" data="images/derivative-second.swf" align="middle">
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<param name="allowScriptAccess" value="sameDomain" /><param name="allowFullScreen" value="true" /><param name="movie" value="images/derivative-second.swf" /><param name="quality" value="high" /><param name="bgcolor" value="#FFFFFF" /></object>
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</noscript>
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</div>-->
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<p>Notice how the <b>slope</b> of each function is the <b>y-value</b> of the derivative plotted below it.</p>
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<p>For example, move to where the sin(x) function slope flattens out (slope=0), then see that the derivative graph is at zero. A similar thing happens between f'(x) and f''(x).
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Try this at different points and other functions.</p>
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<script>functiongraph2derivMain();</script>
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<p> </p>
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<div class="questions">
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<script>getQ(6813, 6814, 6815, 6816, 6817, 6818, 6819, 6820, 6821, 6822, 6823, 15317, 15318, 15319, 15320, 15321, 15322, 15323, 15324, 15325);</script> </div>
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<div class="related">
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<a href="derivatives-introduction.html">Derivatives</a>
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<a href="second-derivative-animation.html">Second Derivatives Animation</a>
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<a href="index.html">Calculus Index</a>
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