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<a href="projects.html" onmouseout="MM_swapImgRestore()" onmouseover="MM_swapImage('Image1','','nav/projects2.jpg',1)"> <img src="nav/projects.jpg" alt="AS YouTube Channel" name="Image1" width="205" height="36" border="0" id="Image1"/></a> <a href="conspiracy.html" onmouseout="MM_swapImgRestore()" onmouseover="MM_swapImage('Image2','','nav/conspiracy2.jpg',1)"> <img src="nav/conspiracy.jpg" alt="technology" name="Image2" width="205" height="36" border="0" id="Image2" /></a> <a href="technology.html" onmouseout="MM_swapImgRestore()" onmouseover="MM_swapImage('Image3','','nav/technology2.jpg',1)"> <img src="nav/technology.jpg" alt="technology" name="Image3" width="205" height="36" border="0" id="Image3" /></a> <a href="educational.html" onmouseout="MM_swapImgRestore()" onmouseover="MM_swapImage('Image4','','nav/educational3.jpg',1)"> <img src="nav/educational1.jpg" alt="chemistry" name="Image4" width="205" height="36" border="0" id="Image4" /></a> </td> </tr> </table> <table width="845" border="5" align="center" cellpadding="0" cellspacing="0"> <tr><td height="50" valign="middle" background="nav/blak.jpg"> <a href="physics.html"><img src="nav/physics.jpg" width="160" height="32" border="0"></a> <a href="geometry.html"><img src="nav/geometry.jpg" width="160" height="32" border="0"></a> <a href="vectors.html"><img src="nav/vectors.jpg" width="162" height="32" border="0"></a> <a href="magnetism.html"><img src="nav/magnetism.jpg" width="161" height="32" border="0"></a> <a href="quantum.html"><img src="nav/quantum.jpg" width="162" height="32" border="0"></a> <br> <a href="mathematics.html"><img src="nav/mathbutton.jpg" width="160" height="32" border="0"></a> <a href="algebra.html"><img src="nav/algebra.jpg" width="161" height="32" border="0"></a> <a href="trig.html"><img src="nav/trigonometry.jpg" width="162" height="32" border="0"></a> <a href="calculus.html"><img src="nav/calculus.jpg" width="161" height="32" border="0"></a> <a href="diffeq.html"><img src="nav/diffeq.jpg" width="160" height="32" border="0"></a> </td></tr> </table> <table width="845" border="0" align="center" cellpadding="50" cellspacing="0"> <tr> <td background="nav/hpbg.jpg"> <h1>Calculus</h1> <p> This section is meant to provide the core curriculum for Calculus I, II, III, and IV. We'll start with a short funny video going over some of the basics of calculus. </p> <center> <iframe width="640" height="480" src="http://www.youtube.com/embed/EX_is9LzFSY" frameborder="0" allowfullscreen></iframe> </center> <b>Why is Calculus Important for modeling things?</b> <br> <p> Calculus is the study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. Any problem involving rates of change or variables which don't remain constant require calculus in order to solve. Lets begin with some of the history leading up to the development of calculus. </p> <h2>Xeno's Paradox</h2> <p> The great Greek philosopher Zeno of Elea (born sometime between 495 and 480 B.C.) proposed four paradoxes in an effort to challenge the accepted notions of space and time that he encountered in various philosophical circles. His paradoxes confounded mathematicians for centuries, and it wasn't until Cantor's development (in the 1860's and 1870's) of the theory of infinite sets that the paradoxes could be fully resolved. Zeno's paradoxes focus on the relation of the discrete to the continuous, an issue that is at the very heart of mathematics. </p> <h3>The Arrow Paradox</h3> <table style="margin:auto; border-collapse:collapse; border-style:none; background-color:transparent; width:auto;" class="cquote"> <tr> <td width="20" valign="top" style="color:#B2B7F2;font-size:35px;font-family:'Times New Roman',serif;font-weight:bold;text-align:left;padding:10px 10px;">?</td> <td valign="top" style="padding:4px 10px;"><i>If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless.</i></td> <td width="20" valign="bottom" style="color:#B2B7F2;font-size:35px;font-family:'Times New Roman',serif;font-weight:bold;text-align:right;padding:10px 10px;">?</td> </tr> <tr> <td colspan="3" style="padding-right: 4%"> <p style="font-size:smaller;text-align: right" class="cquotecite"><cite style="font-style:normal;">?<a href="http://en.wikipedia.org/wiki/Aristotle" title="Aristotle">Aristotle</a>, <a href="http://en.wikipedia.org/wiki/Physics_(Aristotle)" title="Physics (Aristotle)"><i>Physics</i></a> VI:9, 239b5</cite></p> </td> </tr> </table> <p>In the arrow paradox (also known as the <b><a href="http://en.wikipedia.org/wiki/Fletching" title="Fletching">fletcher's</a> paradox</b>), Zeno states that for motion to occur, an object must change the position which it occupies. He gives an example of an arrow in flight. He states that in any one (durationless) instant of time, the arrow is neither moving to where it is, nor to where it is not.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span>[</span>11<span>]</span></a></sup> It cannot move to where it is not, because no time elapses for it to move there; it cannot move to where it is, because it is already there. In other words, at every instant of time there is no motion occurring. If everything is motionless at every instant, and time is entirely composed of instants, then motion is impossible. </p> <p> The continuity of space or time, considered by Zeno and others, is represented in mathematics by the continuity of points on a line. As late as the seventeenth century, mathematicians continued to believe, as the ancient Greeks had, that this continuity of points was a simple result of density, meaning that between any two points, no matter how close together, there is always another. This is true, for example, of the rational numbers. However, the rational numbers do not form a continuum, since irrational numbers like ? 2 are missing, leaving holes or discontinuities. The irrational numbers are required to complete the continuum. Together, the rational and irrational numbers do form a continuous set, the set of real numbers. Thus, the continuity of points on a line is ultimately linked to the continuity of the set of real numbers, by establishing a one-to-one correspondence between the two. This approach to continuity was first established in the 1820s, by Augustin-Louis Cauchy, who finally began to solve the problem of handling continuity logically. In Cauchy's view, any line corresponding to the graph of a function is continuous at a point, if the value of the function at x, denoted by f(x), gets arbitrarily close to f(p), when x gets close to a real number p. If f(x) is continuous for all real numbers x contained in a finite interval, then the function is continuous in that interval. If f(x) is continuous for every real number x, then the function is continuous everywhere. </p> <p> Cauchy's definition of continuity is essentially the one we use today, though somewhat more refined versions were developed in the 1850s, and later in the nineteenth century. For example, the concept of continuity is often described in relation to limits. </p> <h2>Limits</h2> <div class="rellink relarticle mainarticle">Main article: <a href="/wiki/Limit_of_a_function" title="Limit of a function">Limit of a function</a></div> <div class="thumb tright"> <div class="thumbinner" style="width:514px;"> <table style="background: transparent; border: 0; padding: 0; margin: 0; width:510px;" cellspacing="0"> <tr style="vertical-align:middle;"> <td style="padding: auto; margin: 0;" class="thumbimage"><a href="/wiki/File:L%C3%ADmite_01.svg" class="image" title="Whenever a point x is within ´ units of c, f(x) is within µ units of L."><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/L%C3%ADmite_01.svg/200px-L%C3%ADmite_01.svg.png" width="200" height="200" /></a></td> <td style="padding: 0; margin: 0; border: 0; width: 2px"></td> <td style="padding: auto; margin: 0;" class="thumbimage"><a href="/wiki/File:Limit-at-infinity-graph.png" class="image" title="For all x &gt; S, f(x) is within µ of L."><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Limit-at-infinity-graph.png/306px-Limit-at-infinity-graph.png" width="306" height="200" /></a></td> </tr> <tr style="vertical-align:top;"> <td style="padding: 0; margin: 0; border: 0;"> <div class="thumbcaption">Whenever a point <span class="texhtml">x</span> is within ´ units of <span class="texhtml">c</span>, <span class="texhtml">f(x)</span> is within µ units of <span class="texhtml">L</span>.</div> </td> <td style="padding: 0; margin: 0; border: 0;"></td> <td style="padding: 0; margin: 0; border: 0;"> <div class="thumbcaption">For all <span class="texhtml">x &gt; S</span>, <span class="texhtml">f(x)</span> is within µ of <span class="texhtml">L</span>.</div> </td> </tr> </table> </div> </div> <p>Suppose <span class="texhtml">f(x)</span> is a <a href="/wiki/Real-valued_function" title="Real-valued function">real-valued function</a> and <span class="texhtml">c</span> is a <a href="/wiki/Real_number" title="Real number">real number</a>. The expression</p> <dl> <dd><img class="tex" alt=" \lim_{x \to c}f(x) = L " src="//upload.wikimedia.org/wikipedia/en/math/e/d/8/ed80e81395fb7b21643891fdd4190429.png" /></dd> </dl> <p>means that <span class="texhtml">f(x)</span> can be made to be as close to <span class="texhtml">L</span> as desired by making <span class="texhtml">x</span> sufficiently close to <span class="texhtml">c</span>. In that case, it can be stated that "the limit of <span class="texhtml">f</span> of <span class="texhtml">x</span>, as <span class="texhtml">x</span> approaches <span class="texhtml">c</span>, is <span class="texhtml">L</span>". Note that this statement can be true even if <span class="texhtml">f(c) `" L</span>. Indeed, the function <span class="texhtml">f(x)</span> need not even be defined at <span class="texhtml">c</span>.</p> <p>For example, if</p> <dl> <dd><img class="tex" alt=" f(x) = \frac{x^2 - 1}{x - 1} " src="//upload.wikimedia.org/wikipedia/en/math/7/a/f/7afed2eefa402af8cbc2eaa3646322ef.png" /></dd> </dl> <p>then <i>f</i>(1) is not defined (see <a href="/wiki/Division_by_zero" title="Division by zero">division by zero</a>), yet as <span class="texhtml">x</span> moved arbitrarily close to 1, <span class="texhtml">f(x)</span> correspondingly approaches 2:</p> <table class="wikitable"> <tr> <td><i>f</i>(0.9)</td> <td><i>f</i>(0.99)</td> <td><i>f</i>(0.999)</td> <td><i>f</i>(1.0)</td> <td><i>f</i>(1.001)</td> <td><i>f</i>(1.01)</td> <td><i>f</i>(1.1)</td> </tr> <tr> <td>1.900</td> <td>1.990</td> <td>1.999</td> <td>Ò! undef Ð!</td> <td>2.001</td> <td>2.010</td> <td>2.100</td> </tr> </table> <p>Thus, <span class="texhtml">f(x)</span> can be made arbitrarily close to the limit of 2 just by making <span class="texhtml">x</span> sufficiently close to 1.</p> <p><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a> in 1821,<sup id="cite_ref-Larson_2-0" class="reference"><a href="#cite_note-Larson-2"><span>[</span>3<span>]</span></a></sup> followed by <a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Karl Weierstrass</a>, formalized the definition of the limit of a function into what became known as the <a href="/wiki/(%CE%B5,_%CE%B4)-definition_of_limit" title="(µ, ´)-definition of limit" class="mw-redirect">(µ, ´)-definition of limit</a> in the 19th century. The definition uses <span class="texhtml"><a href="/wiki/%CE%95" title="•" class="mw-redirect">µ</a></span> (the lowercase Greek letter <i>epsilon</i>) to represent a small positive number, so that "<span class="texhtml">f(x)</span> becomes arbitrarily close to <span class="texhtml">L</span>" means that <span class="texhtml">f(x)</span> lies in the interval <span class="texhtml">(L - µ, L + µ)</span>, which can also be written using absolute value as <span class="texhtml">|f(x) - L| &lt; µ</span>.<sup id="cite_ref-Larson_2-1" class="reference"><a href="#cite_note-Larson-2"><span>[</span>3<span>]</span></a></sup> The statement "<span class="texhtml">x</span> approaches <span class="texhtml">c</span>" then indicates that a positive number <span class="texhtml"><a href="/wiki/%CE%94" title="”" class="mw-redirect">´</a></span> (the lowercase greek letter <i>delta</i>) exists within either <span class="texhtml">(c - ´, c)</span> or <span class="texhtml">(c, c + ´)</span>, which can be expressed with <span class="texhtml">0 &lt; |x - c| &lt; ´</span>. The first inequality means that the distance between <span class="texhtml">x</span> and <span class="texhtml">c</span> is more than 0 and that <span class="texhtml">x `" c</span>, while the second indicates that <span class="texhtml">x</span> is within distance <span class="texhtml">´</span> of <span class="texhtml">c</span>.<sup id="cite_ref-Larson_2-2" class="reference"><a href="#cite_note-Larson-2"><span>[</span>3<span>]</span></a></sup></p> <p>In addition to limits at finite values, functions can also have limits at infinity. For example, consider</p> <dl> <dd><img class="tex" alt="f(x) = {2x-1 \over x}" src="//upload.wikimedia.org/wikipedia/en/math/0/a/2/0a2fb07b22cbdbe68c6342f927312df9.png" /></dd> </dl> <ul> <li><i>f</i>(100) = 1.9900</li> <li><i>f</i>(1000) = 1.9990</li> <li><i>f</i>(10000) = 1.99990</li> </ul> <p>As <span class="texhtml">x</span> becomes extremely large, the value of <span class="texhtml">f(x)</span> approaches 2, and the value of <span class="texhtml">f(x)</span> can be made as close to 2 as one could wish just by picking <span class="texhtml">x</span> sufficiently large. In this case, the limit of <span class="texhtml">f(x)</span> as <span class="texhtml">x</span> approaches infinity is 2. In mathematical notation,</p> <dl> <dd><img class="tex" alt=" \lim_{x \to \infty} f(x) = 2. " src="//upload.wikimedia.org/wikipedia/en/math/e/4/9/e49d9ac64afe08da3ea0563d406d5bc4.png" /></dd> </dl> <br> <h2>Principles</h2> <h3><span class="mw-headline" id="Limits_and_infinitesimals">Limits and infinitesimals</span></h3> <div class="rellink relarticle mainarticle">Main articles: <a href="http://en.wikipedia.org/wiki/Limit_of_a_function">Limit of a function</a>, <a href="http://en.wikipedia.org/wiki/Infinitesimal">Infinitesimal</a>, and <a href="http://en.wikipedia.org/wiki/Infinitesimal_Calculus" class="mw-redirect" title="Infinitesimal Calculus">Infinitesimal Calculus</a></div> <p>Calculus is usually developed by manipulating very small quantities. Historically, the first method of doing so was by <a href="http://en.wikipedia.org/wiki/Infinitesimal" title="Infinitesimal">infinitesimals</a>. These are objects which can be treated like numbers but which are, in some sense, "infinitely small". An infinitesimal number <i>dx</i> could be greater than 0, but less than any number in the sequence 1, 1/2, 1/3, ... and less than any positive <a href="http://en.wikipedia.org/wiki/Real_number">real number</a>. Any integer multiple of an infinitesimal is still infinitely small, i.e., infinitesimals do not satisfy the <a href="http://en.wikipedia.org/wiki/Archimedean_property">Archimedean property</a>. From this point of view, calculus is a collection of techniques for manipulating infinitesimals. This approach fell out of favor in the 19th century because it was difficult to make the notion of an infinitesimal precise. However, the concept was revived in the 20th century with the introduction of <a href="http://en.wikipedia.org/wiki/Non-standard_analysis">non-standard analysis</a> and <a href="http://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis">smooth infinitesimal analysis</a>, which provided solid foundations for the manipulation of infinitesimals.</p> <p>In the 19th century, infinitesimals were replaced by <a href="http://en.wikipedia.org/wiki/Limit_of_a_function" title="Limit of a function">limits</a>. Limits describe the value of a <a href="http://en.wikipedia.org/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> at a certain input in terms of its values at nearby input. They capture small-scale behavior, just like infinitesimals, but use the ordinary <a href="http://en.wikipedia.org/wiki/Real_number" title="Real number">real number system</a>. In this treatment, calculus is a collection of techniques for manipulating certain limits. Infinitesimals get replaced by very small numbers, and the infinitely small behavior of the function is found by taking the limiting behavior for smaller and smaller numbers. Limits are the easiest way to provide rigorous foundations for calculus, and for this reason they are the standard approach.</p> <h3> <span class="mw-headline" id="Differential_calculus">Differential calculus</span></h3> <div class="rellink relarticle mainarticle">Main article: <a href="http://en.wikipedia.org/wiki/Differential_calculus">Differential calculus</a></div> <div class="thumb tright"> <div class="thumbinner" style="width:302px;"><a href="http://en.wikipedia.org/wiki/File:Tangent_derivative_calculusdia.svg" class="image"><img alt="" src="http://upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Tangent_derivative_calculusdia.svg/300px-Tangent_derivative_calculusdia.svg.png" width="300" height="207" class="thumbimage" /></a> <div class="thumbcaption"> <div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Tangent_derivative_calculusdia.svg" class="internal" title="Enlarge"><img src="http://bits.wikimedia.org/skins-1.17/common/images/magnify-clip.png" width="15" height="11" alt="" /></a></div> Tangent line at (<i>x</i>, <i>f</i>(<i>x</i>)). The derivative <i>f'</i>(<i>x</i>) of a curve at a point is the slope (rise over run) of the line tangent to that curve at that point.</div> </div> </div> <p>Differential calculus is the study of the definition, properties, and applications of the <a href="http://en.wikipedia.org/wiki/Derivative">derivative</a> of a function. The process of finding the derivative is called <i>differentiation</i>. Given a function and a point in the domain, the derivative at that point is a way of encoding the small-scale behavior of the function near that point. By finding the derivative of a function at every point in its domain, it is possible to produce a new function, called the <i>derivative function</i> or just the <i>derivative</i> of the original function. In mathematical jargon, the derivative is a <a href="http://en.wikipedia.org/wiki/Linear_operator" class="mw-redirect" title="Linear operator">linear operator</a> which inputs a function and outputs a second function. This is more abstract than many of the processes studied in elementary algebra, where functions usually input a number and output another number. For example, if the doubling function is given the input three, then it outputs six, and if the squaring function is given the input three, then it outputs nine. The derivative, however, can take the squaring function as an input. This means that the derivative takes all the information of the squaring function such as that two is sent to four, three is sent to nine, four is sent to sixteen, and so on and uses this information to produce another function. (The function it produces turns out to be the doubling function.)</p> <p>The most common symbol for a derivative is an apostrophe-like mark called <a href="http://en.wikipedia.org/wiki/Prime_(symbol)" title="Prime (symbol)">prime</a>. Thus, the derivative of the function of <i>f</i> is <i>f'</i>, pronounced "f prime." For instance, if <i>f</i>(<i>x</i>) = <i>x</i><sup>2</sup> is the squaring function, then <i>f'</i>(<i>x</i>) = 2<i>x</i> is its derivative, the doubling function.</p> <p>If the input of the function represents time, then the derivative represents change with respect to time. For example, if <i>f</i> is a function that takes a time as input and gives the position of a ball at that time as output, then the derivative of <i>f</i> is how the position is changing in time, that is, it is the <a href="http://en.wikipedia.org/wiki/Velocity">velocity</a> of the ball.</p> <p>If a function is <a href="http://en.wikipedia.org/wiki/Linear_function" title="Linear function">linear</a> (that is, if the <a href="http://en.wikipedia.org/wiki/Graph_of_a_function" title="Graph of a function">graph</a> of the function is a straight line), then the function can be written as <span style="white-space:nowrap;"><i>y</i> = <i>mx</i> + <i>b</i></span>, where <i>x</i> is the independent variable, <i>y</i> is the dependent variable, <i>b</i> is the <i>y</i>-intercept, and:</p> <dl> <dd><img class="tex" alt="m= \frac{\text{rise}}{\text{run}}= \frac{\text{change in } y}{\text{change in } x} = \frac{\Delta y}{\Delta x}." src="http://upload.wikimedia.org/math/3/0/4/304b59b78bd5e0c3eaa87742fdfe9bbe.png" /></dd> </dl> <p>This gives an exact value for the slope of a straight line. If the graph of the function is not a straight line, however, then the change in <i>y</i> divided by the change in <i>x</i> varies. Derivatives give an exact meaning to the notion of change in output with respect to change in input. To be concrete, let <i>f</i> be a function, and fix a point <i>a</i> in the domain of <i>f</i>. (<i>a</i>, <i>f</i>(<i>a</i>)) is a point on the graph of the function. If <i>h</i> is a number close to zero, then <i>a</i> + <i>h</i> is a number close to <i>a</i>. Therefore (<i>a</i> + <i>h</i>, <i>f</i>(<i>a</i> + <i>h</i>)) is close to (<i>a</i>, <i>f</i>(<i>a</i>)). The slope between these two points is</p> <dl> <dd><img class="tex" alt="m = \frac{f(a+h) - f(a)}{(a+h) - a} = \frac{f(a+h) - f(a)}{h}." src="http://upload.wikimedia.org/math/6/e/3/6e389045846757b62c4fd64b913ac9d4.png" /></dd> </dl> <p>This expression is called a <i>difference quotient</i>. A line through two points on a curve is called a <i>secant line</i>, so <i>m</i> is the slope of the secant line between (<i>a</i>, <i>f</i>(<i>a</i>)) and (<i>a</i> + <i>h</i>, <i>f</i>(<i>a</i> + <i>h</i>)). The secant line is only an approximation to the behavior of the function at the point <i>a</i> because it does not account for what happens between <i>a</i> and <i>a</i> + <i>h</i>. It is not possible to discover the behavior at <i>a</i> by setting <i>h</i> to zero because this would require dividing by zero, which is impossible. The derivative is defined by taking the <a href="http://en.wikipedia.org/wiki/Limit_(mathematics)" title="Limit (mathematics)">limit</a> as <i>h</i> tends to zero, meaning that it considers the behavior of <i>f</i> for all small values of <i>h</i> and extracts a consistent value for the case when <i>h</i> equals zero:</p> <dl> <dd><img class="tex" alt="\lim_{h \to 0}{f(a+h) - f(a)\over{h}}." src="http://upload.wikimedia.org/math/c/e/9/ce948a7cc04a731fcc4c2f977fc5ab80.png" /></dd> </dl> <p>Geometrically, the derivative is the slope of the <a href="http://en.wikipedia.org/wiki/Tangent_line" class="mw-redirect" title="Tangent line">tangent line</a> to the graph of <i>f</i> at <i>a</i>. The tangent line is a limit of secant lines just as the derivative is a limit of difference quotients. For this reason, the derivative is sometimes called the slope of the function <i>f</i>.</p> <p>Here is a particular example, the derivative of the squaring function at the input 3. Let <i>f</i>(<i>x</i>) = <i>x</i><sup>2</sup> be the squaring function.</p> <div class="thumb tright"> <div class="thumbinner" style="width:302px;"><a href="http://en.wikipedia.org/wiki/File:Sec2tan.gif" class="image"><img alt="" src="http://upload.wikimedia.org/wikipedia/commons/thumb/3/34/Sec2tan.gif/300px-Sec2tan.gif" width="300" height="289" class="thumbimage" /></a> <div class="thumbcaption"> <div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Sec2tan.gif" class="internal" title="Enlarge"><img src="http://bits.wikimedia.org/skins-1.17/common/images/magnify-clip.png" width="15" height="11" alt="" /></a></div> The derivative <i>f'</i>(<i>x</i>) of a curve at a point is the slope of the line tangent to that curve at that point. This slope is determined by considering the limiting value of the slopes of secant lines. Here the function involved (drawn in red) is <i>f</i>(<i>x</i>) = <i>x</i><sup>3</sup> - <i>x</i>. The tangent line (in green) which passes through the point (-3/2, -15/8) has a slope of 23/4. Note that the vertical and horizontal scales in this image are different.</div> </div> </div> <dl> <dd><img class="tex" alt="\begin{align}f'(3) &amp;=\lim_{h \to 0}{(3+h)^2 - 3^2\over{h}} \\ &amp;=\lim_{h \to 0}{9 + 6h + h^2 - 9\over{h}} \\ &amp;=\lim_{h \to 0}{6h + h^2\over{h}} \\ &amp;=\lim_{h \to 0} (6 + h) \\ &amp;= 6. \end{align} " src="http://upload.wikimedia.org/math/c/0/2/c02db6c01faf190bf9119e2628ed315d.png" /></dd> </dl> <p>The slope of tangent line to the squaring function at the point (3,9) is 6, that is to say, it is going up six times as fast as it is going to the right. The limit process just described can be performed for any point in the domain of the squaring function. This defines the <i>derivative function</i> of the squaring function, or just the <i>derivative</i> of the squaring function for short. A similar computation to the one above shows that the derivative of the squaring function is the doubling function.</p> <h3> <span class="mw-headline" id="Leibniz_notation">Leibniz notation</span></h3> <div class="rellink relarticle mainarticle">Main article: <a href="http://en.wikipedia.org/wiki/Leibniz%27s_notation">Leibniz's notation</a></div> <p>A common notation, introduced by Leibniz, for the derivative in the example above is</p> <dl> <dd><img class="tex" alt=" \begin{align} y=x^2 \\ \frac{dy}{dx}=2x. \end{align} " src="http://upload.wikimedia.org/math/6/7/a/67ac8d1ed96745357c342c55e979b8a3.png" /></dd> </dl> <p>In an approach based on limits, the symbol <i>dy/dx</i> is to be interpreted not as the quotient of two numbers but as a shorthand for the limit computed above. Leibniz, however, did intend it to represent the quotient of two infinitesimally small numbers, <i>dy</i> being the infinitesimally small change in <i>y</i> caused by an infinitesimally small change <i>dx</i> applied to <i>x</i>. We can also think of <i>d/dx</i> as a differentiation operator, which takes a function as an input and gives another function, the derivative, as the output. For example:</p> <dl> <dd><img class="tex" alt=" \frac{d}{dx}(x^2)=2x. " src="http://upload.wikimedia.org/math/c/0/9/c09dbf4483158a99ca3500cb19203572.png" /></dd> </dl> <p>In this usage, the <i>dx</i> in the denominator is read as "with respect to x". Even when calculus is developed using limits rather than infinitesimals, it is common to manipulate symbols like <i>dx</i> and <i>dy</i> as if they were real numbers; although it is possible to avoid such manipulations, they are sometimes notationally convenient in expressing operations such as the <a href="http://en.wikipedia.org/wiki/Total_derivative">total derivative</a>.</p> <h3> <span class="mw-headline" id="Integral_calculus">Integral calculus</span></h3> <div class="rellink relarticle mainarticle">Main article: <a href="http://en.wikipedia.org/wiki/Integral">Integral</a></div> <p><b>Integral calculus</b> is the study of the definitions, properties, and applications of two related concepts, the <i>indefinite integral</i> and the <i>definite integral</i>. The process of finding the value of an integral is called <i>integration</i>. In technical language, integral calculus studies two related <a href="http://en.wikipedia.org/wiki/Linear_operator" title="Linear operator" class="mw-redirect">linear operators</a>.</p> <p>The <b>indefinite integral</b> is the <i><a href="http://en.wikipedia.org/wiki/Antiderivative">antiderivative</a></i>, the inverse operation to the derivative. <i>F</i> is an indefinite integral of <i>f</i> when <i>f</i> is a derivative of <i>F</i>. (This use of upper- and lower-case letters for a function and its indefinite integral is common in calculus.)</p> <p>The <b>definite integral</b> inputs a function and outputs a number, which gives the area between the graph of the input and the <a href="http://en.wikipedia.org/wiki/X-axis" class="mw-redirect" title="X-axis">x-axis</a>. The technical definition of the definite integral is the <a href="http://en.wikipedia.org/wiki/Limit_(mathematics)" title="Limit (mathematics)">limit</a> of a sum of areas of rectangles, called a <a href="http://en.wikipedia.org/wiki/Riemann_sum">Riemann sum</a>.</p> <p>A motivating example is the distances traveled in a given time.</p> <dl> <dd><img class="tex" alt="\mathrm{Distance} = \mathrm{Speed} \cdot \mathrm{Time}" src="http://upload.wikimedia.org/math/3/9/5/3952cfbfb2edb4d7601d1e1bc8a2481f.png" /></dd> </dl> <p>If the speed is constant, only multiplication is needed, but if the speed changes, then we need a more powerful method of finding the distance. One such method is to approximate the distance traveled by breaking up the time into many short intervals of time, then multiplying the time elapsed in each interval by one of the speeds in that interval, and then taking the sum (a <a href="http://en.wikipedia.org/wiki/Riemann_sum">Riemann sum</a>) of the approximate distance traveled in each interval. The basic idea is that if only a short time elapses, then the speed will stay more or less the same. However, a Riemann sum only gives an approximation of the distance traveled. We must take the limit of all such Riemann sums to find the exact distance traveled.</p> <div class="thumb tleft"> <div class="thumbinner" style="width:282px;"><a href="http://en.wikipedia.org/wiki/File:Integral_as_region_under_curve.svg" class="image"><img alt="" src="http://upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Integral_as_region_under_curve.svg/280px-Integral_as_region_under_curve.svg.png" width="280" height="246" class="thumbimage" /></a> <div class="thumbcaption"> <div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Integral_as_region_under_curve.svg" class="internal" title="Enlarge"><img src="http://bits.wikimedia.org/skins-1.17/common/images/magnify-clip.png" width="15" height="11" alt="" /></a></div> Integration can be thought of as measuring the area under a curve, defined by <i>f</i>(<i>x</i>), between two points (here <i>a</i> and <i>b</i>).</div> </div> </div> <p>If <i>f(x)</i> in the diagram on the left represents speed as it varies over time, the distance traveled (between the times represented by <i>a</i> and <i>b</i>) is the area of the shaded region <i>s</i>.</p> <p>To approximate that area, an intuitive method would be to divide up the distance between <i>a</i> and <i>b</i> into a number of equal segments, the length of each segment represented by the symbol <i>?x</i>. For each small segment, we can choose one value of the function <i>f</i>(<i>x</i>). Call that value <i>h</i>. Then the area of the rectangle with base <i>?x</i> and height <i>h</i> gives the distance (time <i>?x</i> multiplied by speed <i>h</i>) traveled in that segment. Associated with each segment is the average value of the function above it, <i>f(x)</i>=h. The sum of all such rectangles gives an approximation of the area between the axis and the curve, which is an approximation of the total distance traveled. A smaller value for <i>?x</i> will give more rectangles and in most cases a better approximation, but for an exact answer we need to take a limit as <i>?x</i> approaches zero.</p> <p>The symbol of integration is <img class="tex" alt="\int \," src="http://upload.wikimedia.org/math/0/9/b/09b51ce4b5d77f40b7ca997765f9baea.png" />, an elongated <i>S</i> (the S stands for "sum"). The definite integral is written as:</p> <dl> <dd><img class="tex" alt="\int_a^b f(x)\, dx." src="http://upload.wikimedia.org/math/1/9/a/19a0fe4f869ec4b82f44a57d7c4bf714.png" /></dd> </dl> <p>and is read "the integral from <i>a</i> to <i>b</i> of <i>f</i>-of-<i>x</i> with respect to <i>x</i>." The Leibniz notation <i>dx</i> is intended to suggest dividing the area under the curve into an infinite number of rectangles, so that their width <i>?x</i> becomes the infinitesimally small <i>dx</i>. In a formulation of the calculus based on limits, the notation</p> <dl> <dd><img class="tex" alt="\int_a^b \ldots\, dx" src="http://upload.wikimedia.org/math/5/f/f/5ff77e7f7488cf25e06ef817554e288b.png" /></dd> </dl> <p>is to be understood as an operator that takes a function as an input and gives a number, the area, as an output; <i>dx</i> is not a number, and is not being multiplied by <i>f(x)</i>.</p> <p>The indefinite integral, or antiderivative, is written:</p> <dl> <dd><img class="tex" alt="\int f(x)\, dx." src="http://upload.wikimedia.org/math/f/d/0/fd018836770dd637ce06b05c5591b00b.png" /></dd> </dl> <p>Functions differing by only a constant have the same derivative, and therefore the antiderivative of a given function is actually a family of functions differing only by a constant. Since the derivative of the function <i>y</i> = <i>x</i>² + <i>C</i>, where <i>C</i> is any constant, is <i>y'</i> = 2<i>x</i>, the antiderivative of the latter is given by:</p> <dl> <dd><img class="tex" alt="\int 2x\, dx = x^2 + C." src="http://upload.wikimedia.org/math/e/c/9/ec92ae41df80abcfafda1860e6719c25.png" /></dd> </dl> <p>An undetermined constant like <i>C</i> in the antiderivative is known as a <a href="http://en.wikipedia.org/wiki/Constant_of_integration">constant of integration</a>.</p> <h3> <span class="mw-headline" id="Fundamental_theorem">Fundamental theorem</span></h3> <div class="rellink relarticle mainarticle">Main article: <a href="http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus">Fundamental theorem of calculus</a></div> <p>The <a href="http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus">fundamental theorem of calculus</a> states that differentiation and integration are inverse operations. More precisely, it relates the values of antiderivatives to definite integrals. Because it is usually easier to compute an antiderivative than to apply the definition of a definite integral, the Fundamental Theorem of Calculus provides a practical way of computing definite integrals. It can also be interpreted as a precise statement of the fact that differentiation is the inverse of integration.</p> <p>The Fundamental Theorem of Calculus states: If a function <i>f</i> is <a href="http://en.wikipedia.org/wiki/Continuous_function" title="Continuous function">continuous</a> on the interval [<i>a</i>, <i>b</i>] and if <i>F</i> is a function whose derivative is <i>f</i> on the interval (<i>a</i>, <i>b</i>), then</p> <dl> <dd><img class="tex" alt="\int_{a}^{b} f(x)\,dx = F(b) - F(a)." src="http://upload.wikimedia.org/math/6/1/f/61f36d98755743b096c536d6bc5aa726.png" /></dd> </dl> <p>Furthermore, for every <i>x</i> in the interval (<i>a</i>, <i>b</i>),</p> <dl> <dd><img class="tex" alt="\frac{d}{dx}\int_a^x f(t)\, dt = f(x)." src="http://upload.wikimedia.org/math/0/3/0/0302486962619ed1831cc4d8f54235a7.png" /></dd> </dl> <p>This realization, made by both <a href="http://en.wikipedia.org/wiki/Isaac_Newton" title="Isaac Newton">Newton</a> and <a href="http://en.wikipedia.org/wiki/Gottfried_Leibniz" title="Gottfried Leibniz">Leibniz</a>, who based their results on earlier work by <a href="http://en.wikipedia.org/wiki/Isaac_Barrow">Isaac Barrow</a>, was key to the massive proliferation of analytic results after their work became known. The fundamental theorem provides an algebraic method of computing many definite integrals without performing limit processes by finding formulas for <a href="http://en.wikipedia.org/wiki/Antiderivative" title="Antiderivative">antiderivatives</a>. It is also a prototype solution of a <a href="http://en.wikipedia.org/wiki/Differential_equation">differential equation</a>. Differential equations relate an unknown function to its derivatives, and are ubiquitous in the sciences.</p> <hr> <br> <h3>Product Rule</h3> <p>In <a href="/wiki/Calculus" title="Calculus">calculus</a>, the <b>product rule</b> is a formula used to find the <a href="/wiki/Derivative" title="Derivative">derivatives</a> of products of two or more functions. It may be stated thus:</p> <dl> <dd><img class="tex" alt="(f\cdot g)'=f'\cdot g+f\cdot g' \,\! " src="//upload.wikimedia.org/wikipedia/en/math/f/3/4/f3469e50750fb7341c9f20259c8fb695.png" /></dd> </dl> <p>or in the <a href="/wiki/Leibniz_notation" title="Leibniz notation" class="mw-redirect">Leibniz notation</a> thus:</p> <dl> <dd><img class="tex" alt="\dfrac{d}{dx}(u\cdot v)=u\cdot \dfrac{dv}{dx}+v\cdot \dfrac{du}{dx}" src="//upload.wikimedia.org/wikipedia/en/math/b/d/c/bdcb07184715b984c8f7781bc6e30841.png" />.</dd> </dl> <p>The derivative of the product of three functions is:</p> <dl> <dd><img class="tex" alt="\dfrac{d}{dx}(u\cdot v \cdot w)=\dfrac{du}{dx} \cdot v \cdot w + u \cdot \dfrac{dv}{dx} \cdot w + u\cdot v\cdot \dfrac{dw}{dx}" src="//upload.wikimedia.org/wikipedia/en/math/a/7/1/a716467d8f6037aa21e89d0d45b9295b.png" />.</dd> </dl> <h3>Quotient Rule</h3> <p>In <a href="http://en.wikipedia.org/wiki/Calculus" title="Calculus">calculus</a>, the <b>quotient rule</b> is a method of finding the <a href="http://en.wikipedia.org/wiki/Derivative" title="Derivative">derivative</a> of a <a href="http://en.wikipedia.org/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> that is the <a href="http://en.wikipedia.org/wiki/Quotient" title="Quotient">quotient</a> of two other functions for which derivatives exist.<sup id="cite_ref-0" class="reference"><a href="#cite_note-0"><span>[</span>1<span>]</span></a></sup><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span>[</span>2<span>]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span>[</span>3<span>]</span></a></sup></p> <p>If the function one wishes to differentiate, <span class="texhtml" dir="ltr"><i>f</i>(<i>x</i>)</span>, can be written as</p> <dl> <dd><img class="tex" alt="f(x) = \frac{g(x)}{h(x)}" src="//upload.wikimedia.org/wikipedia/en/math/8/5/c/85c4e200964ebf85acaee18f2c1f03f3.png" /></dd> </dl> <p>and <img class="tex" alt="h(x)\not=0" src="//upload.wikimedia.org/wikipedia/en/math/8/0/3/8039e8f6b683a8a8e95c1f0fe5d78b05.png" />, then the rule states that the derivative of <span class="texhtml" dir="ltr"><i>g</i>(<i>x</i>) / <i>h</i>(<i>x</i>)</span> is</p> <dl> <dd><img class="tex" alt="f'(x) = \frac{h(x)g'(x) - h'(x)g(x)}{[h(x)]^2}." src="//upload.wikimedia.org/wikipedia/en/math/3/8/5/3856f28d4b93877f760d4dbd301176cd.png" /></dd> </dl> <p>More precisely, if all <i>x</i> in some <a href="http://en.wikipedia.org/wiki/Open_set" title="Open set">open set</a> containing the number <i>a</i> satisfy <img class="tex" alt="h(x)\not=0" src="//upload.wikimedia.org/wikipedia/en/math/8/0/3/8039e8f6b683a8a8e95c1f0fe5d78b05.png" />, and <span class="texhtml" dir="ltr"><i>g</i>'(<i>a</i>)</span> and <span class="texhtml" dir="ltr"><i>h</i>'(<i>a</i>)</span> both exist, then <span class="texhtml" dir="ltr"><i>f</i>'(<i>a</i>)</span> exists as well and</p> <dl> <dd><img class="tex" alt="f'(a)=\frac{h(a)g'(a) - h'(a)g(a)}{[h(a)]^2}." src="//upload.wikimedia.org/wikipedia/en/math/4/2/3/4235ae32f928cdb31c777115a3c92224.png" /></dd> </dl> <p>And this can be extended to calculate the second derivative as well (you can prove this by taking the derivative of <span class="texhtml" dir="ltr"><i>f</i>(<i>x</i>) = <i>g</i>(<i>x</i>)(<i>h</i>(<i>x</i>)) <sup>" 1</sup></span> twice). The result of this is:</p> <dl> <dd><img class="tex" alt="f''(x)=\frac{g''(x)[h(x)]^2-2g'(x)h(x)h'(x)+g(x)[2[h'(x)]^2-h(x)h''(x)]}{[h(x)]^3}." src="//upload.wikimedia.org/wikipedia/en/math/d/2/9/d298335e3e2777952d00e7c3a0d1f827.png" /></dd> </dl> <p>The quotient rule formula can be derived from the <a href="http://en.wikipedia.org/wiki/Product_rule" title="Product rule">product rule</a> and <a href="http://en.wikipedia.org/wiki/Chain_rule" title="Chain rule">chain rule</a>.</p> <center> <h1>---Under Construction----</h1> </center> <h3>Riemann Sums</h3> <h3>Integral Sums</h3> <h3>Chain Rule</h3> <h3>Intergration by Parts</h3> <h3>Substitution Method</h3> <h2>Series and Expansions</h2> <h3>Taylor Series</h3> <h3>Fourier Series</h3> <h3></h3> </td></tr> </table> </body> </html>