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<h1 align="center">Intermediate Value Theorem</h1>
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<p>The idea behind the Intermediate Value Theorem is this:</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/intermediate-value-a.svg" alt="intermediate value A to B" /></p>
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<p> </p>
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<p>When we have <b>two points</b> connected by a continuous curve:</p>
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<ul>
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<li>one point below the line</li>
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<li>the other point above the line</li>
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</ul>
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<p>... then there will be <b>at least one place</b> where the curve crosses the line!</p>
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<p> </p>
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<p align="center" class="larger">Well <b>of course</b> we must cross the line to get from A to B!</p>
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<p>Now that you know the <b>idea</b>, let's look more closely at the details.</p>
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<h2>Continuous</h2>
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<p> The curve must be <b>continuous</b> ... no gaps or jumps in it. </p>
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<p><a href="../calculus/continuity.html">Continuous</a> is a special term with an exact definition in calculus, but here we will use this simplified definition:</p>
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<div class="center80">
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<p align="center"> <img src="../images/pencil-paper.gif" alt="pencil" width="91" height="77" style="vertical-align:middle;"/><span class="larger">we can draw it without lifting our pen from the paper</span></p>
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</div>
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<h2>More Formal</h2>
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<p>Here is the Intermediate Value Theorem stated more formally:</p>
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<p style="float:right; margin: 0 0 5px 10px;"> </p>
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<span style="float:right; margin: 0 0 5px 10px;"><img src="images/intermediate-value-b.svg" alt="intermediate value A to B crosses at (c,w)" /></span>
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<p> </p>
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<p>When:</p>
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<ul>
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<li>The curve is the function <span class="large">y = f(x)</span>,</li>
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<li>which is <span class="large">continuous</span> on the interval [a, b],</li>
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<li>and <span class="large">w</span> is a number between f(a) and f(b), </li>
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</ul>
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<p>Then ...</p>
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<p align="center" class="larger">... there must be at least one value <b>c</b> within [a, b] such that <b>f(c) = w</b></p>
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<p> </p>
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<p>In other words the function <span class="large">y = f(x)</span> at some point must be <span class="large">w = f(c)</span></p>
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<p>Notice that:</p>
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<ul>
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<li><b>w</b> is between f(a) and f(b), <i>which leads to ...</i></li>
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<li><b>c</b> must be between a and b</li>
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</ul><h2>At Least One</h2>
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<p style="float:right; margin: 0 0 5px 10px;"> </p>
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<span style="float:right; margin: 0 0 5px 10px;"><img src="images/intermediate-value-c.svg" alt="intermediate value A to B crosses at 3 points" /></span>
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<p>It also says "at least one value c", which means we <b>could</b> have more.</p>
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<p> Here, for example, are 3 points where f(x)=w: </p><div style="clear:both"></div>
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<h2>How Is This Useful?</h2>
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<p>Whenever we can show that:</p>
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<ul>
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<li>there is a point above some line</li>
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<li>and a point below that line, and</li>
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<li>that the curve is continuous, </li>
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</ul>
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<p>we can then safely say "yes, there is a value somewhere <b>in between</b> that is on the line".</p>
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<div class="example">
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<h3>Example: is there a solution to x<sup>5</sup> - 2x<sup>3</sup> - 2 = 0
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between x=0 and x=2?</h3>
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<p><b>At x=0:</b></p>
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<p align="center" class="larger">0<sup>5</sup> - 2 × 0<sup>3</sup> - 2 = <b>-2</b></p>
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<p> </p>
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<p><b>At x=2:</b></p>
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<p align="center" class="larger">2<sup>5 </sup>- 2 × 2<sup>3</sup> - 2 = <b>14</b></p>
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<p>Now we know:</p>
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<ul>
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<li>at x=0, the curve is below zero</li>
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<li>at x=2, the curve is above zero</li>
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</ul>
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<p>And, being a polynomial, the curve will be continuous,</p>
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<p>so <b>somewhere in between</b> the curve must cross through y=0</p>
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<p> </p>
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<p align="center" class="larger">Yes, there is a solution to x<sup>5</sup> - 2x<sup>3</sup> - 2 = 0
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in the interval [0, 2]</p>
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</div>
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<h2>An Interesting Thing!</h2>
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<h3>The Intermediate Value Theorem Can Fix a Wobbly Table</h3>
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<table border="0">
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<tr>
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<td><img src="images/intermediate-value-table.gif" alt="intermediate value wobbly table" width="254" height="98" /></td>
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<td><p>If your table is wobbly because of uneven ground ... </p>
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<p align="center" class="larger">... just <b>rotate the table</b> to fix it!</p>
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<p>The ground must be <span class="large">continuous</span> (no steps such as poorly laid tiles).</p></td>
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</tr>
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</table>
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<h3>Why does this work? </h3>
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<p>We can always have 3 legs on the ground, it is the 4th leg that is the trouble.</p>
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<p>Imagine we are <b>rotating the table</b>, and the 4th leg could somehow go into the ground (like sand):</p>
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<ul>
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<li>at some point it will be above the ground</li>
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<li>at another point it will be below the ground</li>
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</ul>
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<p>So there must be some point where the 4th leg <b>perfectly touches the ground</b> and the table won't wobble.</p>
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<p><i>(The famous Martin Gardner wrote about this in Scientific American. There is also a very complicated proof somewhere).</i></p>
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<h2>Another One</h2>
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<p align="center"><img src="images/intermediate-value-trip.gif" alt="country walk" width="410" height="154" /><span class="larger"><br />
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At some point during a round-trip you will be <br />
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exactly as high
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as where you started.</span><br />
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</p>
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<p>(It only works if you don't start at the highest or lowest point.)</p>
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<p>The idea is:</p>
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<ul>
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<li>at some point you will be higher than where you started</li>
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<li>at another point you will be lower than where you started</li>
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</ul>
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<p>So there must be a point in between where you are <b>exactly</b> as high as where you started.</p>
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<p>Oh, and your path must be <b>continuous</b>, no disappearing and reappearing somewhere else.</p>
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<p>The same thing happens with temperature, pressure, and so on.</p>
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<h2>And There's More!</h2>
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<p>If you follow a circular path ... somewhere on that circle there will be points that are:</p>
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<ul>
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<li>directly opposite each other </li>
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<li><b>and</b> at the same height!</li>
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</ul>
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<p align="center"><img src="images/intermediate-value-circle.gif" alt="intermediate value circular path opposites" width="371" height="139" /><br />
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<span class="larger"> two points that are <br />
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directly opposite <b>and</b> at same height</span><br />
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</p>
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<p> </p>
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<p class="large"><i>Can you think of more examples?</i></p>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(573, 574, 1214, 1215, 9076, 9077, 2453, 2454, 9078, 9079);</script> </div>
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<div class="related"><a href="../calculus/continuity.html">Continuity</a><a href="index.html">Algebra Index</a></div>
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