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110 lines
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110 lines
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HTML
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<title>Difference Quotient</title>
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Difference Quotient</h1>
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<p>This is the "Difference Quotient":<span class="lim"><strong><br></strong></span></p>
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<p class="center large"><span class="lim"><strong></strong></span><span class="intbl"><em>f(x+Δx) − f(x)</em><strong>Δx</strong></span></p>
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<p>It gives the average slope between two points on a curve <b>f(x) </b>that are <b>Δx</b> apart, and is used with <a href="derivatives-introduction.html">derivatives</a>:</p><br>
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<p class="center"><img src="images/slope-dy-dx2.svg" alt="slope dy/dx"></p><br>
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<div class="example">
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<h3>Example: <b>f(x) = x<sup>2</sup> − 2x + 1</b> at <b>x = 3</b> and <b>Δx = 0.1</b></h3>
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<p>Evaluate f(x) at x=3:</p>
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<p class="center">f(3) = 3<sup>2 </sup>− 2×3 + 1 = <b>4</b></p>
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<p>Now for f(x+Δx):</p>
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<p class="center">f(3.1) = (3.1)<sup>2 </sup>− 2×3.1 + 1 = <b>4.41</b></p>
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<p></p>
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<p>And the Difference Quotient is:</p>
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<p class="center large"><span class="intbl"><em>f(3.1) − f(3)</em><strong>0.1</strong></span> = <span class="intbl"><em>4.41 − 4</em><strong>0.1</strong></span>
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= <span class="intbl"><em>0.41</em><strong>0.1</strong></span>
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= <b>4.1</b>
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</p>
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</div>
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<p> As Δx heads towards 0, the value of the slope heads towards the true slope at x</p>
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<div class="example">
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<h3>Example continued: try Δx = 0.01</h3><br>
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<p class="center">f(3.01) = (3.01)<sup>2 </sup>− 2×3.01 + 1 = <b>4.0401</b></p>
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<p>So we have:</p>
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<p class="center large"><span class="intbl"><em>f(3.01) − f(3)</em><strong>0.01</strong></span> = <span class="intbl"><em>4.0401 − 4</em><strong>0.01</strong></span>
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= <span class="intbl"><em>0.0401</em><strong>0.01</strong></span>
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= <b>4.01</b> </p>
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</div>
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<p>
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As Δx gets smaller the slope <b>seems to be</b> heading towards <b>4</b>, right? That is the idea behind <a href="derivatives-introduction.html">derivatives</a> (which can find the answer exactly).</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-introduction.html">Introduction to Derivatives</a>
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<a href="index.html">Calculus Index</a>
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