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148 lines
5.0 KiB
HTML
148 lines
5.0 KiB
HTML
<!DOCTYPE html>
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<!-- Mirrored from www.mathsisfun.com/calculus/derivative-vs-integral.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:38:50 GMT -->
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<title>Derivative vs Integral</title>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Graphical Intro to<br>Derivatives and Integrals</h1>
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<p class="larger">Derivatives and Integrals have a two-way relationship!</p>
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<p>Let's start by looking at sums and slopes:</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;">
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<img style=" width:180px;" src="../images/art/sage-walks.svg" alt="sage walks"> </p><h3>Example: walking in a straight line
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</h3>
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<ul>
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<li>Walk slow, the distance increases slowly</li>
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<li>Walk fast, the distance increases fast</li>
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<li>Stand still and the distance won't change</li>
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<li>Walk backwards, and you get closer to the start!</li></ul>
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<p>Walking at <b>4 km per hour</b> for 1 hour makes the diistance increase by <b>4 km</b></p>
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<p>A distance increase of <b>4 km</b> in 1 hour gives a speed of <b>4 km per hour</b></p>
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<p class="so">Change in distance is the <b>sum of the speed</b> over time </p>
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<p class="so">Speed is the <b>rate of change (slope)</b> of distance </p>
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</div>
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<p>It will make more sense if you play with it below (drag the distance line at the top or the speed line below to see the other change):</p>
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<div class="script" style="height: 570px;">
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images/deriv-integ.js?mode=1&topic=walking
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</div>
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<p>Play with that a little and get comfortable with the two-way relationship. Try zero speed, or negative
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speed. </p>
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<p>The slope of the distance line gives us the speed line, like this:</p>
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<p> </p>
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<p class="center"><img src="images/area-vs-slope.svg" alt="speed from distance"></p>
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<p>The "area" under the speed line gives us the increase in distance, like this:</p>
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<p class="center"><img src="images/area-vs-slope2.svg" alt="distance from speed"></p>
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<p><b>Many things</b> have that same two-way relationship:</p>
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<ul>
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<li>Wealth and income </li>
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<li>Volume and flow rate</li>
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<li>Energy and power</li>
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<li>lots more!</li></ul>
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<p>Here is the same app as above, but you can choose different topics:</p>
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<div class="script" style="height: 620px;">
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images/deriv-integ.js?mode=multi
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</div>
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<p class="larger">Integrals and Derivatives <b>also </b>have that two-way relationship!</p>
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<p>Try it below, but first note:</p>
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<ul>
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<li>Δx (the gap between x values) only gives an <b>approximate </b>answer</li>
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<li>dx (when Δx approaches zero) gives the actual derivative and integral* </li></ul>
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<div class="script" style="height: 650px;">
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images/deriv-integ.js?mode=fn
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</div>
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<p>*Note: this is a <a href="../algebra/mathematical-models.html">computer model</a> and actually uses a very<b> small Δx</b> to simulate <b>dx</b>, and can make erors.</p>
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<p>For true derivatives refer to <a href="derivatives-rules.html">Derivative Rules</a>, and for integrals refer to
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<a href="integration-introduction.html">Introduction to Integration</a></p>
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<p>
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</p>
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<div class="related">
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<a href="derivatives-rules.html">Derivative Rules</a>
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<a href="integration-introduction.html">Introduction to Integration</a>
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<a href="integral-approximations.html">Integral Approximations</a>
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<a href="index.html">Calculus Index</a>
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</div>
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