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194 lines
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HTML
194 lines
8.3 KiB
HTML
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<title>Concave Upward and Downward</title>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Concave Upward and Downward</h1>
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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> </td>
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<td> </td>
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</tr>
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<tr>
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<td><b>Concave upward</b> is when the slope increases:</td>
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<td> </td>
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<td><img src="images/concave-upward.svg" alt="concave upward slope increases"></td>
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</tr>
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<tr>
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<td><b>Concave downward</b> is when the slope decreases:</td>
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<td> </td>
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<td><img src="images/concave-downward.svg" alt="concave downward slope decreases"></td>
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</tr>
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</tbody></table>
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<p><i>What about when the slope stays the same (straight line)? It could be both! See <a href="#slope">footnote</a>.</i></p>
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<p>Here are some more examples:</p>
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<p class="center"><img src="images/concave-examples.svg" alt="concave upward and downward examples"></p>
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<div class="words">
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<p><b>Concave Upward</b> is also called <b>Convex</b>, or sometimes <b>Convex Downward</b></p>
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<p><b>Concave Downward</b> is also called <b>Concave</b>, or sometimes <b>Convex Upward</b></p>
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</div>
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<h2>Finding where ...</h2>
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<p>Usually our task is to find <b>where</b> a curve is concave upward or concave downward:</p>
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<p class="center"><br>
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<img src="images/concave-sections.svg" alt="concave sections"></p>
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<h2>Definition</h2>
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<p>A line drawn between <b>any</b> two points on the curve won't cross over the curve:</p>
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<p class="center"><img src="images/concave-upward-yes-no.svg" alt="concave upward yes and no examples"></p>
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<p>Let's make a formula for that!</p>
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<p>First, the line: take any two different values <b>a</b> and<b> b</b> (in the interval we are looking at):</p>
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<p class="center"><img src="images/concave-upward-ab.svg" alt="concave upward between a and b"></p>
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<p>Then "slide" between <b>a</b> and <b>b</b> using a value <b>t</b> (which is from 0 to 1):</p>
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<p class="center larger">x = ta + (1−t)b</p>
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<ul>
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<li>When <b>t=0</b> we get <b>x = 0a+1b = b</b></li>
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<li>When <b>t=1</b> we get <b>x = 1a+0b = a</b></li>
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<li>When t is between 0 and 1 we get values between <b>a</b> and <b>b</b></li>
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</ul>
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<p>Now work out the heights at that x-value:</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/concave-line-t.svg" alt="concave line t"></td>
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<td> </td>
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<td>
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<p>When <b>x = ta + (1−t)b</b>:</p>
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<ul>
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<li>The curve is at <b>y = f( ta + (1−t)b )</b></li>
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<li>The line is at <b>y = tf(a) + (1−t)f(b)</b></li>
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</ul></td>
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</tr>
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</tbody></table>
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<p>And (for <b>concave upward</b>) the line should not be below the curve:</p>
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<p class="center"><img src="images/concave-upward-formula.svg" alt="concave upwnward f( ta + (1-t)b ) <= tf(a) + (1-t)f(b)"></p>
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<p>For <b>concave downward</b> the line should not be above the curve (<b>≤</b> becomes <b>≥</b>):</p>
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<p class="center"><img src="images/concave-downward-formula.svg" alt="concave downward f( ta + (1-t)b ) >= tf(a) + (1-t)f(b)"></p>
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<p>And those are the actual definitions of <b>concave upward</b> and <b>concave downward</b>.</p>
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<h2>Remembering</h2>
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<p>Which way is which? Think:</p>
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<p class="center"><img src="images/concave-up-cup.svg" alt="concave up: cup"><br>
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<b>C</b>oncave <b>Up</b>wards = <b>CUP</b></p>
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<h2>Calculus</h2>
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<p><a href="derivatives-introduction.html">Derivatives</a> can help! The derivative of a function gives the slope.</p>
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<ul>
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<li>When the slope continually <b>increases</b>, the function is <b>concave upward</b>.</li>
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<li>When the slope continually <b>decreases</b>, the function is <b>concave downward</b>.</li>
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</ul>
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<p>Taking the <a href="second-derivative.html">second derivative</a> actually tells us if the slope continually increases or decreases.</p>
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<ul>
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<li>When the second derivative is <b>positive</b>, the function is <b>concave upward</b>.</li>
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<li>When the second derivative is<b> negative</b>, the function is <b>concave downward</b>.</li>
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</ul>
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<div class="example">
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<h3>Example: the function x<sup>2</sup></h3>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/x2-concave-upward.svg" alt="x^2 concave upward"></p>
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<p>Its derivative is 2x (see <a href="derivatives-rules.html">Derivative Rules</a>)</p>
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<div class="so">2x continually increases, so the function is <b>concave upward</b>. </div>
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<p>Its second derivative is 2</p>
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<div class="so">2 is <b>positive</b>, so the function is <b>concave upward</b>.</div>
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<p>Both give the correct answer.</p></div>
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<p> </p>
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<div class="example">
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<h3>Example: f(x) = 5x<sup>3</sup> + 2x<sup>2</sup> − 3x</h3>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/5x3-2x2-3x-concave.svg" alt="5x^3 + 2x^2 - 3x inflection point"></p>
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<p>Let's work out the second derivative:</p>
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<ul>
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<li>The derivative is <b>f'(x) = 15x<sup>2</sup> + 4x − 3</b> (using <a href="derivatives-rules.html">Power Rule</a>)</li>
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<li>The second derivative is <b>f''(x) = 30x + 4</b> (using <a href="derivatives-rules.html">Power Rule</a>)</li>
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</ul>
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<p> </p>
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<p>And <b>30x + 4</b> is negative up to x = −4/30 = −2/15, and positive from there onwards. So:</p>
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<div class="so"> f(x) is <b>concave downward</b> up to x = −2/15</div>
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<div class="so">f(x) is <b>concave upward</b> from x = −2/15 on</div>
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<p> </p>
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<p>Note: The point where it changes is called an <a href="inflection-points.html">inflection point</a>.</p>
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</div>
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<p> </p>
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<div class="center80">
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<h3><a name="slope"></a>Footnote: Slope Stays the Same</h3>
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<p>What about when the slope stays the same (straight line)?</p>
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<p>A straight line is acceptable for <b>concave upward</b> or <b>concave downward</b>.</p>
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<p>But when we use the special terms <b>strictly concave upward</b> or <b>strictly concave downward</b> then a straight line is <b>not</b> OK.</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/2x-1.svg" alt="2x+1"></p>
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<h3>Example: y = 2x + 1</h3>
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<p><b>2x + 1</b> is a straight line.</p>
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<p> </p>
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<p>It is <b>concave upward</b>.<br>
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It is also <b>concave downward</b>.</p>
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<p>It is not <b>strictly concave upward</b>.<br>
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And it is not <b>strictly concave downward</b>.</p>
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</div>
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</div>
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<p> </p>
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<div class="related">
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<a href="index.html">Calculus Index</a>
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