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324 lines
15 KiB
HTML
324 lines
15 KiB
HTML
<!DOCTYPE html>
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<title>Introduction to Calculus</title>
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<div id="hdr"></div>
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<div id="cookOK"></div>
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<div id="logo"><a href="../index.html"><img src="../images/style/logo-adv.svg" alt="Math is Fun Advanced"></a></div>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Introduction to Calculus</h1>
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<p class="center">Calculus is all about <i><b>changes</b></i>.</p>
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<table width="100%">
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<tbody>
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<tr>
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<td><img src="images/speedometer-0.jpg" alt="speedometer" height="102" width="150"></td>
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<td>
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<p>Sam and Alex are traveling in the car ... but the speedometer is broken.</p></td>
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</tr>
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</tbody></table>
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<div class="tbl">
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<div class="row">
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<div class="lt">
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Alex:
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</div>
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<div class="rtlt">
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"Hey Sam! How fast are we going now?"
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</div>
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</div>
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<div class="row">
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<div class="lt">
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Sam:
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</div>
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<div class="rtlt">
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<p>"Wait a minute ..."</p>
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<p>"Well in the last minute we went 1.2 km, so we are going:"</p>
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<p class="center large">1.2 km per minute x 60 minutes in an hour = <b>72 km/h</b></p>
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</div>
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</div>
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<div class="row">
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<div class="lt">
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Alex:
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</div>
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<div class="rtlt">
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<p>"No, Sam! Not our <b>average</b> for the last minute, or even the last second, I want to know our speed RIGHT NOW."</p>
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</div>
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</div>
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<div class="row">
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<div class="lt">
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Sam:
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</div>
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<div class="rtlt">
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<p>"OK, let us measure it up here ... at this road sign... NOW!"</p>
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<p class="center"><img src="images/road.jpg" alt="road" height="167" width="360"></p>
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<p>"OK, we were AT the sign for <b>zero seconds</b>, and the distance was ... <b>zero meters</b>!"</p>
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<p class="center">The speed is 0m / 0s = 0/0 = <b>I Don't Know</b>!</p>
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<p>"I can't calculate it, Alex! I need to know <b>some</b> distance over <b>some</b> time, and you are saying the time should be zero? Can't be done."</p>
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</div>
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</div>
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</div>
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<p> </p>
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<div class="center80">
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<p>That is pretty amazing ... you'd think it is easy to work out the speed of a car at any point in time, but it isn't.</p>
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<p>Even the speedometer of a car just shows us an <b>average</b> of how fast we were going for the last (very short) amount of time.</p>
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</div>
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<h2>How About Getting Real Close</h2>
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<p>But our story is not finished yet!</p>
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<p>Sam and Alex get out of the car, because they have arrived on location. Sam is about to do a stunt:</p>
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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><img src="images/jump-1.svg" alt="jump t=1" height="120" width="136"></td>
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<td> </td>
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<td>
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<h3>Sam will do a jump off a 20 m building.</h3>
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<h3>Alex, as photographer, asks:</h3>
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<h3 class="center">"How fast will you be falling after 1 second?"</h3></td>
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</tr>
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</tbody></table>
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<p>Sam uses this simplified formula to find <b>the distance fallen</b>:</p>
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<p class="center"><span class="large">d = 5t<sup>2</sup></span></p>
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<ul>
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<li>d = distance fallen, in meters</li>
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<li>t = time from jump, in seconds</li>
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</ul>
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<p><i>(Note: the formula is a simpler version of falling due to <a href="../physics/gravity.html">gravity</a>: d = ½gt<sup>2</sup>)</i></p>
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<div class="example">
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<p>Example: at 1 second Sam has fallen</p>
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<p class="center"><b>d = 5t<sup>2</sup> = 5 × 1<sup>2</sup> = 5 m</b></p>
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</div>
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<p>But how <b>fast</b> is that? Speed is distance over time:</p>
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<p class="center large">Speed = <span class="intbl"><em>distance</em><strong>time</strong></span></p>
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<p>So at 1 second:</p>
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<p class="center large">Speed = <span class="intbl"><em>5 m</em><strong>1 second</strong></span> = 5 m/s</p>
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<p>"BUT", says Alex, "again that is an <b>average speed</b>, since you started the jump, ... I want to know the speed at <b>exactly</b> 1 second, so I can set up the camera properly."</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/jump-2.svg" alt="jump from t=1 to t=1" height="163" width="212"></p>
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<p> </p>
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<p class="center ">Well ... at <b>exactly 1 second</b> the speed is:</p>
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<p class="center large">Speed = <span class="intbl"><em>5 − 5 m</em><strong>1 − 1 s</strong></span> = <span class="intbl"><em>0 m</em><strong>0 s</strong></span> = ???</p>
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<div style="clear:both"></div>
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<p>So again Sam has a problem.</p>
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<div class="center80">
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<p>Think about it ... how do we figure out a speed at an exact instant in time?</p>
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<p>What is the distance? What is the time difference?</p>
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<p>They are both <b>zero</b>, giving us nothing to calculate with!</p>
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</div>
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<p>But Sam has an idea ... invent a time <b>so short it won't matter</b>.</p>
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<p>Sam won't even give it a value, and will just call it "Δt" (called "delta t").</p>
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<p>So Sam works out the difference in distance between <b>t</b> and <b>t+Δt</b></p>
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<div class="center80">
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<p>At <b>1 second</b> Sam has fallen</p>
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<p class="center large">5t<sup>2</sup> = 5 × (1)<sup>2</sup> = 5 m</p>
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<p> </p>
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<p>At <b>(1+Δt) seconds</b> Sam has fallen</p>
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<p class="center large">5t<sup>2</sup> = 5 × (1+Δt)<sup>2</sup> m</p>
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<p> </p>
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<p>We can <a href="../algebra/expanding.html">expand</a> <b>(1+Δt)<sup>2</sup></b>:</p>
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<div class="tbl">
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<div class="row"><span class="left">(1+Δt)<sup>2</sup></span><span class="right">= (1+Δt)(1+Δt)</span></div>
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<div class="row"><span class="left"> </span><span class="right">= 1 + 2Δt + (Δt)<sup>2</sup></span></div>
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</div>
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<p> </p>
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<p>So at <b>(1+Δt) seconds</b> Sam has fallen</p>
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<div class="tbl">
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<div class="row"><span class="left">d</span><span class="right">= 5 × (1+2Δt+(Δt)<sup>2</sup>) m</span></div>
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<div class="row"><span class="left">d</span><span class="right">= <b>5 + 10Δt + 5(Δt)<sup>2</sup> m</b></span></div>
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</div>
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<p> </p>
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<p class="center"><img src="images/jump-4.svg" alt="jump from t=1 to t=1 + delta t" height="163" width="308"></p>
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<p> </p>
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<p>In Summary:</p>
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<div class="tbl">
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<div class="row"><span class="left">At 1 second:</span><span class="right">d = 5 m</span></div>
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<div class="row"><span class="left">At (1+Δt) seconds:</span><span class="right">d = <b>5 + 10Δt + 5(Δt)<sup>2</sup> m</b></span></div>
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</div>
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<p> </p>
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<p>So between <b>1 second</b> and <b>(1+Δt) seconds</b> we get:</p>
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<div class="tbl">
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<div class="row"><span class="left"><span class="center">Change in d</span></span><span class="right">= 5 + 10Δt + 5(Δt)<sup>2</sup> − 5 m </span></div>
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</div>
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<p> </p>
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<p>Change in distance over time:</p>
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<div class="tbl">
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<div class="row"><span class="left">Speed</span><span class="right">
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= <span class="intbl"><em>5 + 10Δt + 5(Δt)<sup>2</sup> − 5 m</em>
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<strong>Δt s</strong></span>
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</span></div>
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<div class="row"><span class="left"> </span><span class="right">
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= <span class="intbl"><em>10Δt + 5(Δt)<sup>2</sup> m</em>
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<strong>Δt s</strong></span>
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</span></div>
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<div class="row"><span class="left"> </span><span class="right">= <b>10 + 5Δt</b> m/s</span></div>
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</div>
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<p> </p>
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<p>So the speed is <span class="large">10 <span class="center">+ 5Δt </span>m/s</span>, and Sam thinks about that <b>Δt</b> value ... he wants <span class="center"><b>Δt</b></span> to be so small it won't matter ... so he imagines it shrinking towards <b>zero</b> and he gets:</p>
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<p> </p>
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<p class="center large">Speed = 10 m/s</p>
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<p> </p></div>
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<p>Wow! Sam got an answer!</p>
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<p> </p>
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<p><b><i>Sam</i>: "I will be falling at exactly 10 m/s"</b></p>
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<p><b><i>Alex</i>: "I thought you said you couldn't calculate it?"</b></p>
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<p><b><i>Sam</i>: "That was before I used Calculus!"</b></p>
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<p> </p>
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<p class="large">Yes, indeed, that was Calculus.</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/small-stones.jpg" alt="small stones" height="150" width="200"></p>
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<p><b>The word Calculus comes from Latin meaning "small stone".</b></p>
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<p>· <a href="derivatives-introduction.html">Differential Calculus</a> cuts something into small pieces to find how it changes. <b><br>
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</b></p>
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<p>· <a href="integration-introduction.html">Integral Calculus</a> joins (integrates) the small pieces together to find how much there is.</p>
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<p> </p>
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<p>Sam used <b>Differential Calculus</b> to cut time and distance into such small pieces that a pure answer came out.</p>
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<p class="fun">And Differential Calculus and Integral Calculus are like <b>inverses</b> of each other, similar to how multiplication and division are inverses, but that is something for us to discover later!</p>
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<p> </p>
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<p>So ... was Sam's result just luck? Does it work for other things?</p>
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<p><b>Let's try doing this for the function y = x<sup>3</sup></b></p>
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<p>This will be similar to the previous example, but we will just use a slope on a graph, no one has to jump for this one!</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/graph-x3.svg" alt="graph of x^3" height="267" width="223"></p>
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<h3>Example: What is the slope of the function y = x<sup>3</sup> at x=1 ?</h3>
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<p> </p>
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<div class="tbl">
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<div class="row"><span class="left">At x = 1:</span><span class="right">y = 1<sup>3</sup> = 1</span></div>
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<div class="row"><span class="left">At x = (1+Δx):</span><span class="right">y = (1+Δx)<sup>3</sup></span></div>
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</div>
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<p><br>
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We can expand (1+Δx)<sup>3</sup> to 1 + 3Δx + 3(Δx)<sup>2</sup> + (Δx)<sup>3</sup>, and we get:</p>
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<p class="center large">y = 1 + 3Δx + 3(Δx)<sup>2</sup> + (Δx)<sup>3</sup></p>
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<p> </p>
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<p>And the difference between the y values from x = 1 to x = 1+Δx is:</p>
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<div class="tbl">
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<div class="row"><span class="left"><span class="center">Change in y </span></span><span class="right"><span class="center">= 1 + 3Δx + 3(Δx)<sup>2</sup> + (Δx)<sup>3</sup> − 1</span></span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="center">= <b>3Δx + 3(Δx)<sup>2</sup> + (Δx)<sup>3</sup></b></span></span></div>
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</div>
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<p> </p>
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<p>Now we can calculate slope:</p>
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<div class="tbl">
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<div class="row"><span class="left"><span class="large">Slope</span></span><span class="right">= <span class="large intbl">
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<em>3Δx + 3(Δx)<sup>2</sup> + (Δx)<sup>3</sup></em>
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<strong>Δx</strong>
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</span></span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="large">= <b>3 + 3Δx + (Δx)<sup>2</sup></b></span></span></div>
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</div>
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<p> </p>
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<p>Once again, as <b>Δx</b> shrinks towards zero we are left with:</p>
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<p class="center large"><b>Slope = 3</b></p>
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<p> </p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/graph-x3-slope.svg" alt="graph x^3 slope at (1, 1)" height="267" width="223"></p>
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<p class="center">And here we see the graph of <b>y = x<sup>3</sup></b></p>
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<p class="center">The slope is continually changing, but at the<br>
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point<b> (1, 1)</b> we can draw a line tangent to the curve</p>
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<p class="center">and find the slope there <b>really is 3</b>.</p>
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<p class="center">(Count the squares if you want!)</p>
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<div style="clear:both"></div>
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<p>Question for you: what is the slope at the <b>point (2, 8)</b>?</p>
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</div>
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<h2>Try It Yourself!</h2>
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<p>Go to the <a href="slope-function-point.html">Slope of a Function</a> page, put in the formula "x^3", then try to find the slope at the point (1, 1).</p>
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||
<p>Zoom in closer and closer and see what value the slope is heading towards.</p>
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<h2>Conclusion</h2>
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<p>Calculus is about changes.</p>
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<p><b>Differential calculus</b> cuts something into small pieces to find how it changes.</p>
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||
<ul>
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||
<li>Learn more at <a href="derivatives-introduction.html">Introduction to Derivatives</a></li>
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</ul>
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||
<p><b> Integral calculus</b> joins (integrates) the small pieces together to find how much there is.</p>
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||
<ul>
|
||
<li>Learn more at <a href="integration-introduction.html">Introduction to Integration</a></li>
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</ul>
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<p> </p>
|
||
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<div class="questions">6750, 6751, 6752, 6753, 6754, 6755, 6756, 6757, 6758, 6759</div>
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<div class="related">
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<a href="slope-function-point.html">Slope of a Function</a>
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<a href="index.html">Calculus Index</a>
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