CalculusSolution.com blogs
https://www.calculussolution.com/blog
enCalculus Rhapsody - The Math Behind This Creative Video You're Guaranteed to Be Tested On
https://www.calculussolution.com/blog/calculus-rhapsody-youtube-video
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even" property="content:encoded"><p>If you haven't seen it, the <a href="https://www.youtube.com/watch?v=uqwC41RDPyg" target="_blank">Calculus Rhapsody</a> music video on Youtube is worth watching. It's creative, funny, and educational all at the same time. So what does this video teach us? Did they get anything wrong? What did they get right? Let's go through the video to see what parts of Calculus they're singing about.</p>
<p>First, here's the video:</p>
<center><iframe width="560" height="315" src="//www.youtube.com/embed/uqwC41RDPyg" frameborder="0" allowfullscreen></iframe></center>
<p>At the very beginning they ask, <i>"Is this \(x\) defined?"</i> What they probably mean is, "Is this \(f(x)\) defined?". The variable \(x\) is typically used to denote a real number. For example, \(x=2\). The \(x\)-axis is the set of all real numbers, and the <a href="https://en.wikipedia.org/wiki/Domain_of_a_function" target="_blank">domain</a> of a function is where on the \(x\)-axis the function is defined. A function may or may not be defined for certain values of \(x\). For example, if we know that \(f(2)=17\), then the function is defined at \(x=2\). If we don't know the value of \(f(x)\) at \(x=2\), then the function is undefined there. So the correct way to think about it is "where on the \(x\)-axis is \(f(x)\) defined" and not "is this \(x\) defined."</p>
<h2>Calculus rhapsody quickly moves into important limit concepts you need to know</h2>
<p>Next they say: <i>"Is f continuous? How do you find out? You can use the limit process."</i> This is absolutely correct! To understand what they're talking about you first have to know what a <a href="https://en.wikipedia.org/wiki/Limit_of_a_function" target="_blank">function limit</a> is. It's denoted by </p>
\[\lim_{x\to a}f(x)=L\]
<p>and it means that as \(x\) gets closer and closer to \(a\), then the function get closer and closer to \(L\). </p>
<p>But there's something important that you need to understand here. You see, a function can have a different value at \(a\) than what its function limit is at \(a\). </p>
<!-- <center><img src="/sites/default/files/styles/large/public/function_limit_with_discontinuity.jpg?itok=nAjJMjvu" width="480" height="371" alt="" class="image-large" /></center> -->
<center><img data-src="https://res.cloudinary.com/calculussolution/image/upload/w_480,q_90,f_auto/function_limit_with_discontinuity_b37njy.jpg" alt="Calculus function limit with discontinuity" class="lazyload" /></center>
<p>So the value of the function at \(a\) and it's function limit at \(a\) can be the same number or two different numbers. When they're the same, the function is said to be <a href="https://en.wikipedia.org/wiki/Continuous_function" target="_blank">continuous</a> at that point.</p>
<p>A practical and down-to-earth definition of continuity is that when you draw the function you don't have to pick up your pencil to put it on anther spot. If the function isn't continuous, you'd have to pick your pencil off of the paper and put a dot where the value of the function is at that discontinuous spot as the image above suggests.</p>
<p>So they absolutely got this right. Continuity is defined as \[\lim_{x\to a}f(x)=f(a).\]</p>
<p>Next they say <i>"Approach from both sides. The left and the right meet."</i> This really points out that <a href="/calculus-lesson/28">a limit is defined if the left-sided limit equals the right-sided limit</a>. </p>
<p>Another way to think of it is this way. What if the limit from the left didn't equal the limit from the right? Check out the picture below.</p>
<!-- <center><img src="/sites/default/files/styles/large/public/calculus_two_sided_limit.jpg?itok=8PWFUvS3" width="480" height="371" alt="" class="image-large" /></center> -->
<center><img data-src="https://res.cloudinary.com/calculussolution/image/upload/w_480,q_90,f_auto/calculus_two_sided_limit_svcvxu.jpg" alt="Calculus two-sided limit" class="lazyload" /></center>
<p>When the limit from the left doesn't equal the limit from the left we say that the limit is undefined or does not exist. So they're absolutely correct on this point.</p>
<p><i>"I'm just a limit, defined analytically"</i> True, but what do they mean by this? When we say something is defined analytically we really mean that it has a formal, logical definition. You can check out the analytical definition <a href="/calculus-lesson/9">here</a>.</p>
<p><i>"Function's continuous, there's no holes, no sharp points, or asymptotes"</i> This is not fully correct. It's true that a continuous function will not have holes or <a href="https://en.wikipedia.org/wiki/Asymptote" target="_blank">asymptotes</a>, but it is NOT true there there will be no sharp points. A continuous function can have sharp points. The simplest example is that of the absolute value function which has a sharp point at \(x=0\). It's also continuous there as the formal limit definition shows.</p>
<!-- <center><img src="/sites/default/files/styles/large/public/calculus_of_absolute_value_function.jpg?itok=zM5-iTZD" width="480" height="371" alt="" class="image-large" /></center> -->
<center><img data-src="https://res.cloudinary.com/calculussolution/image/upload/w_480,q_90,f_auto/calculus_of_absolute_value_function_wbod0k.jpg" alt="The absolute value function" class="lazyload" /></center>
<p><i>"Anyway this graph goes, it is differentiable to me."</i> Be careful. I know there is some artistic license going on here, but you can't always differentiate a function. More specifically, a derivative is defined at a specific point and may or may not exist at that point. Let's use the absolute value function again. It is not differentiable at \(x=0\), but it's differentiable everywhere else. </p>
<h2>Calculus rhapsody highlights four derivative rules you will be tested on: power rule, product rule, quotient rule, and chain rule</h2>
<p><i>"\(y\)-prime . . . oooh . . . . is the derivative of \(y\)."</i> This is true as \(y\,'\) is how you can denote the derivative of a function. It's more appropriate to say \(f\,'(x)\), but they get it.</p>
<p><i>"y equal x to the n, dy/dx equals n times x to the n-1."</i> This is true. Remember that we can denote the derivative by \(y\,'\) or by \(dy/dx\). This second method is the <a href="/calculus-lesson/114">Liebniz notation</a>. If \(y=x^n\), then \(y\,'=nx^{n-1}\). This is just a statement of the <a href="https://www.khanacademy.org/math/in-in-grade-11-ncert/in-in-class11-differentiation/copy-of-radical-functions-differentiation-intro-ab/a/power-rule-review" target="_blank">power rule</a>. The power rule is one of the <a href="/calculus-lesson/31730">first four derivatives</a> you must learn.</p>
<p><i>"Other applications of derivatives apply if y is divided or multiplied. You use the quotient and product rules."</i> This is important. When you want to find the derivative of a function that's made up of two functions multiplied by each other, you use the <a href="https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-8/a/product-rule-review" target="_blank">product rule</a> which states that
\[\frac{d}{dx}\big(fg\big) = \frac{df}{dx}g+f\frac{dg}{dx}.\]
And if a function is made up of two functions divided by each other you use the <a href="https://en.wikipedia.org/wiki/Quotient_rule" target="_blank">quotient rule</a> which says that
\[\frac{d}{dx}\Bigg(\frac{f}{g}\Bigg) = \frac{1}{g^2}\Bigg(g\frac{df}{dx}-f\frac{dg}{dx}\Bigg).\]
</p>
<p>The product rule is easy to remember, but the quotient rule is harder to keep in your head. I have an easy way to remember the quotient rule in my book <i>Calculus in 5 Hours</i> which you can get <a href="/where-to-get-calculus-in-5-hours">here</a>.</p>
<p><i>"Also . . . oooh . . . don't forget to do the chain rule. Before you are done you gotta remember to multiply by the chain."</i> The <a href="/calculus-lesson/chain-rule">chain rule</a> states that
\[(f\circ g)\,'(x)=f\,'(g(x))g\,'(x),\]
or in Liebniz notation it's
\[\frac{d}{dx}f\circ g = \frac{df}{dg}\cdot\frac{dg}{dx}.\]
So are they saying that you need to remember to multiply by the \(g\,'(x)\) at the end? Probably.</p>
<h2>Next, they head into integrals, antiderivatives, and the Fundamental Theorem of Integral Calculus</h2>
<p><i>"I need to find the area under a curve. Integrate! Integrate! You can use the integration."</i> Yes, <a href="https://en.wikipedia.org/wiki/Integral" target="_blank">integration is the area under a curve</a>.</p>
<!-- <center><img src="/sites/default/files/node_31818_calculus_integration.jpg" width="480" alt="" /></center> -->
<center><img data-src="https://res.cloudinary.com/calculussolution/image/upload/w_480,q_90,f_auto/node_31818_calculus_integration_wzxgfs.jpg" alt="Integral calculus" class="lazyload" /></center>
<p><i>"Raise exponent by one, multiply the reciprocal."</i> Here they're finding the <a href="https://en.wikipedia.org/wiki/Antiderivative" target="_blank">antiderivative</a> of \(x^n\) which is \[\frac{x^{n+1}}{(n+1)}.\] So to find the antiderivative you add 1 to \(n\) and then multiple \(x^{n+1}\) by \(1/n+1\). The ntiderivative is used to find the value of the integral from the <a href="https://www.dev.calculussolution.com/calculus-lesson/31601">Fundamental Theorem of Integral Calculus</a> which is what they are doing in the next line they sing.</p>
<p>Then the say <i>"Add a constant. Add a constant. Add a constant. Add a constant. Add a constant labeled \(C\)."</i> This is the constant of integration and the fact that they are saying this is telling us that they're doing an indefinite integral as opposed to a definite integral. So to summarize, they're saying \[\int x^n\,dx=\frac{1}{n+1}x^{n+1}+C.\]</p>
<h2>A final barrage of advanced Calculus topics</h2>
<p><i>"Can you find the area between \(f\) and \(g\)? Integrate \(f\) and then integrate \(g\). Then subtract."</i> This is correct. To find the area between \(f(x)\) and \(g(x)\) you do the following: \[\int_a^b f(x)\,dx-\int_a^b g(x)\,dx.\]</p>
<p>Next they say <i>"To revolve around the \(y\)-axis integrate outer radius squared minus inner radius squared multiplied by pi . . . multiply the integral by pi . . ."</i> Here they're finding the volume between two functions by spinning the two functions around the \(y\)-axis.</p>
<p>To do this you first find \(x\) as a function of \(y\). So instead of having \(y=f(x)\) you solve for \(x\) to get \(x=F(y)\), and you do the same thing for \(y=g(x)\) to get \(x=G(y)\). Once you do that your volume will be \[V=\pi\,\int_a^b[F(y)]^2-[G(y)]^2\,dy.\] Go to this <a href="https://en.wikipedia.org/wiki/Disc_integration#Function_of_y" target="_blank">Wikipedia article</a> for more info on this technique.</p>
<p>Then they throw this in: <i>Pre-calculus did not help me to prepare for Calculus, for Calculus, help me!</i> Most student feel this way and it's true. Calculus requires a different way of thinking and there's really nothing that can fully prepare you for that. But don't worry, you can still understand it and do well.</p>
<p><i>"So you think you can find the limit of y . . . so you think you'll find zero and have it defined . . . oh, baby, you can't define that point baby"</i> is probably a reference to the picture in the video of the function \(1/x\) which is not defined at \(x=0\). This is a good reference for them to make. A function limit can exist at a point where a function is not defined. Remember that the function limit is what the function "intends" to be and not what it is at that point.</p>
<!-- <center><img src="/sites/default/files/styles/large/public/node_31818_1_over_x.jpeg?itok=ivi6Do7w" width="480" height="360" alt="" class="image-large" /></center> -->
<center><img data-src="https://res.cloudinary.com/calculussolution/image/upload/w_480,q_90,f_auto/node_31818_1_over_x_xywvku.jpg" alt="The function 1/x" class="lazyload" /></center>
<p>But some functions may have may have points on the \(x\)-axis where they aren't defined <i>and</i> they're function limits aren't defined there either. This is the case for \(1/x\) at \(x=0\). The function is not defined at \(x=0\) because you can't divide a number by zero.</p>
<p>At the same time, it's function limit is not defined at \(x=0\) for two important reasons. First the function goes to infinity - meaning that it is unbounded at \(x=0\). Second, even if it were bounded, it's taking on both negative and positive values at \(x=0\). This means the left-sided and right-sided limits would not be equal and the function limit would not exist. (Remember for a function limit to exist its left-sided and right-sided limits must equal.) </p>
<p>Of course the song continues on to say <i>"it's undefined, goes to positive and negative infinity"</i> which is what I was saying - only they say it better!</p>
<p>Finally, this video follows the progression of a typical Calculus course. You'll learn about function limits first followed by derivatives, rules for calculating derivatives, antiderivatives, and then integrals.</p>
<p>So now that you know the Calculus they're singing about, go back through the video to reinforce your understanding of the concepts. This entertaining video has a lot of Calculus in it. If you understand what they're talking about you'll do well in your class.</p>
<p> Great job, guys, on your Calculus Rhapsody video. I hope you both got an A! </p>
<div class="sixsyc_sharingbuttons">
<!-- Sharingbutton Facebook -->
<a class="resp-sharing-button__link" href="https://facebook.com/sharer/sharer.php?u=http%3A%2F%2Fwww.CalculusSolution.com%2Fblog%2Fcalculus-rhapsody-youtube-video" target="_blank" rel="noopener" aria-label="Facebook"><div class="resp-sharing-button resp-sharing-button--facebook resp-sharing-button--medium"><div aria-hidden="true" class="resp-sharing-button__icon resp-sharing-button__icon--solid"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M18.77 7.46H14.5v-1.9c0-.9.6-1.1 1-1.1h3V.5h-4.33C10.24.5 9.5 3.44 9.5 5.32v2.15h-3v4h3v12h5v-12h3.85l.42-4z"/></svg></div>Facebook</div></a>
<!-- Sharingbutton Twitter -->
<a class="resp-sharing-button__link" href="https://twitter.com/intent/tweet/?text=Love%20the%20Calculus%20Rhapsody%20Youtube%20video.%20Here%27s%20all%20the%20%23calculus%20mentioned%20in%20it%3A%20&url=http%3A%2F%2Fwww.CalculusSolution.com%2Fblog%2Fcalculus-rhapsody-youtube-video" target="_blank" rel="noopener" aria-label="Twitter">
<div class="resp-sharing-button resp-sharing-button--twitter resp-sharing-button--medium"><div aria-hidden="true" class="resp-sharing-button__icon resp-sharing-button__icon--solid">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M23.44 4.83c-.8.37-1.5.38-2.22.02.93-.56.98-.96 1.32-2.02-.88.52-1.86.9-2.9 1.1-.82-.88-2-1.43-3.3-1.43-2.5 0-4.55 2.04-4.55 4.54 0 .36.03.7.1 1.04-3.77-.2-7.12-2-9.36-4.75-.4.67-.6 1.45-.6 2.3 0 1.56.8 2.95 2 3.77-.74-.03-1.44-.23-2.05-.57v.06c0 2.2 1.56 4.03 3.64 4.44-.67.2-1.37.2-2.06.08.58 1.8 2.26 3.12 4.25 3.16C5.78 18.1 3.37 18.74 1 18.46c2 1.3 4.4 2.04 6.97 2.04 8.35 0 12.92-6.92 12.92-12.93 0-.2 0-.4-.02-.6.9-.63 1.96-1.22 2.56-2.14z"/></svg></div>Twitter</div>
</a>
<!-- Sharingbutton Reddit -->
<a class="resp-sharing-button__link" href="https://reddit.com/submit/?url=Love%20the%20Calculus%20Rhapsody%20Youtube%20video.%20Here%27s%20all%20the%20%23calculus%20mentioned%20in%20it%3A%20https%3A%2F%2Fwww.CalculusSolution.com%2Fblog%2Fcalculus-rhapsody-youtube-video&resubmit=true&title=The math behind the Calculus Rhapsody Youtube video" target="_blank" rel="noopener" aria-label="Reddit">
<div class="resp-sharing-button resp-sharing-button--reddit resp-sharing-button--medium"><div aria-hidden="true" class="resp-sharing-button__icon resp-sharing-button__icon--solid">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M24 11.5c0-1.65-1.35-3-3-3-.96 0-1.86.48-2.42 1.24-1.64-1-3.75-1.64-6.07-1.72.08-1.1.4-3.05 1.52-3.7.72-.4 1.73-.24 3 .5C17.2 6.3 18.46 7.5 20 7.5c1.65 0 3-1.35 3-3s-1.35-3-3-3c-1.38 0-2.54.94-2.88 2.22-1.43-.72-2.64-.8-3.6-.25-1.64.94-1.95 3.47-2 4.55-2.33.08-4.45.7-6.1 1.72C4.86 8.98 3.96 8.5 3 8.5c-1.65 0-3 1.35-3 3 0 1.32.84 2.44 2.05 2.84-.03.22-.05.44-.05.66 0 3.86 4.5 7 10 7s10-3.14 10-7c0-.22-.02-.44-.05-.66 1.2-.4 2.05-1.54 2.05-2.84zM2.3 13.37C1.5 13.07 1 12.35 1 11.5c0-1.1.9-2 2-2 .64 0 1.22.32 1.6.82-1.1.85-1.92 1.9-2.3 3.05zm3.7.13c0-1.1.9-2 2-2s2 .9 2 2-.9 2-2 2-2-.9-2-2zm9.8 4.8c-1.08.63-2.42.96-3.8.96-1.4 0-2.74-.34-3.8-.95-.24-.13-.32-.44-.2-.68.15-.24.46-.32.7-.18 1.83 1.06 4.76 1.06 6.6 0 .23-.13.53-.05.67.2.14.23.06.54-.18.67zm.2-2.8c-1.1 0-2-.9-2-2s.9-2 2-2 2 .9 2 2-.9 2-2 2zm5.7-2.13c-.38-1.16-1.2-2.2-2.3-3.05.38-.5.97-.82 1.6-.82 1.1 0 2 .9 2 2 0 .84-.53 1.57-1.3 1.87z"/></svg></div>Reddit</div>
</a>
<!-- Sharingbutton E-Mail -->
<a class="resp-sharing-button__link" href="mailto:?subject=The%20math%20behind%20the%20Calculus%20Rhapsody%20Youtube%20video&body=Love%20the%20Calculus%20Rhapsody%20Youtube%20video.%20Here%27s%20all%20the%20Calculus%20mentioned%20in%20it%3A%20https%3A%2F%2Fwww.CalculusSolution.com%2Fblog%2Fcalculus-rhapsody-youtube-video" target="_self" rel="noopener" aria-label="E-Mail">
<div class="resp-sharing-button resp-sharing-button--email resp-sharing-button--medium"><div aria-hidden="true" class="resp-sharing-button__icon resp-sharing-button__icon--solid">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M22 4H2C.9 4 0 4.9 0 6v12c0 1.1.9 2 2 2h20c1.1 0 2-.9 2-2V6c0-1.1-.9-2-2-2zM7.25 14.43l-3.5 2c-.08.05-.17.07-.25.07-.17 0-.34-.1-.43-.25-.14-.24-.06-.55.18-.68l3.5-2c.24-.14.55-.06.68.18.14.24.06.55-.18.68zm4.75.07c-.1 0-.2-.03-.27-.08l-8.5-5.5c-.23-.15-.3-.46-.15-.7.15-.22.46-.3.7-.14L12 13.4l8.23-5.32c.23-.15.54-.08.7.15.14.23.07.54-.16.7l-8.5 5.5c-.08.04-.17.07-.27.07zm8.93 1.75c-.1.16-.26.25-.43.25-.08 0-.17-.02-.25-.07l-3.5-2c-.24-.13-.32-.44-.18-.68s.44-.32.68-.18l3.5 2c.24.13.32.44.18.68z"/></svg></div>E-Mail</div>
</a>
</div>
</div></div></div>Wed, 17 Sep 2014 10:07:47 +0000Calculus Center31818 at https://www.calculussolution.comThe Top 10 Calculus Websites
https://www.calculussolution.com/blog/top-10-calculus-websites
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even" property="content:encoded"><div id="blog-top-10">
<p>As a new semester begins it is important to start collecting the Calculus resources you will need. In this blog post I want to give you the top 10 Calculus websites. They are not ranked in any particular order. Instead I have categorized them by videos, calculators, textbooks and solutions. This categorization is more art than science as many of these sites have content in more than one category. And what criteria did I judge these sites on? Free, big, useful, and authoritative. Let's get started.</p>
<h3>Videos</h3>
<h4>1) <a href="http://ocw.mit.edu/index.htm">MIT OpenCourseWare</a></h4>
<p><a href="http://ocw.mit.edu/index.htm">MIT OpenCourseWare</a> is a great site with videos and PDFs on a wide range of subjects. The <a href="http://ocw.mit.edu/courses/mathematics/">Mathematics Department</a> has several courses on Calculus. As a first time Calculus student you will be looking for the courses entitled <em>Single Variable Calculus</em>. There are three of them for <a href="http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2005">2005</a>, <a href="http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006">2006</a>, and <a href="http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010">2010</a>. The format of each course is dictated by each course's professor so no strict organization is enforced on the site. A good starting point is the <a href="http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010">2010 Single Variable Calculus course</a>. It is broken into sessions with each session containing several videos, a PDF of the lecture notes, and at the bottom of the page you will find PDFs of homework problems and their solutions.</p>
<p><a href="http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010"><img src="/sites/default/files/styles/large/public/top_10_mitopencourseware.jpeg?itok=e4C9eJBb" width="480" height="222" alt="Calculus Center Top 10 Calculus Websites - MIT OpenCourseWare" title="Calculus Center Top 10 Calculus Websites - MIT OpenCourseWare" class="image-large" /></a></p>
<h4>2) <a href="http://www.khanacademy.org">Khan Academy</a></h4>
<p>Salman Khan began making math videos for his cousin in 2004. He posted them on YouTube and they have become a mainstay for anyone looking for math videos. When you log into the site for the first time you can browse and navigate to a subject via the "Subjects" drop-down at the top left corner next to the Khan Academy logo. If you go to math section you will not find a specific entry for Calculus. Instead you will find Calculus broken down by Differential Calculus, Integral Calculus, Multivariable Calculus, and Differential Equations. As a first time Calculus student you will only want to look at the Differential Calculus and Integral Calculus sections. When you first go to the section on Differential Calculus it may start you off with solving a problem. This is probably not what you want if you are looking for videos. Click on the "View full list of Differential calculus content" at the top right of the page to begin navigating to the videos. Here are the direct links for the <a href="http://www.khanacademy.org/math/differential-calculus">Differential Calculus</a> and <a href="http://www.khanacademy.org/math/integral-calculus">Integral Calculus</a> videos. Finally, if you are not familiar with Khan Academy, then there is a good overview video <a href="https://www.khanacademy.org/talks-and-interviews/other-features/v/overview-of-khanacademy-org">here</a> by Salman Khan himself.</p>
<p><a href="http://www.khanacademy.org/math/integral-calculus"><img src="/sites/default/files/styles/large/public/top_10_khanacademy.jpeg?itok=uWv50SJB" width="480" height="210" alt="Calculus Center Top 10 Calculus Websites - Khan Academy" title="Calculus Center Top 10 Calculus Websites - Khan Academy" class="image-large" /></a></p>
<h4>3) <a href="http://centerofmath.org/">The Worldwide Center of Mathematics</a></h4>
<p>The <a href="http://centerofmath.org/">Worldwide Center of Mathematics</a> is a for-profit site dedicated to Calculus education and research. The site's videos are free so you can check them out <a href="http://centerofmath.org/videos/index.html">here</a>. The lectures are well designed and the writing done on the blackboard is clearly visible. Their collection also includes <a href="http://centerofmath.org/videos/index.html#subject7">pre-calculus videos</a> in case you need a refresher.
</p>
<p><a href="http://centerofmath.org/"><img src="/sites/default/files/styles/large/public/top_10_worldwidecenterofmath.jpeg?itok=euWOWDES" width="480" height="217" alt="Calculus Center Top 10 Calculus Websites - Worldwide Center of Mathematics" title="Calculus Center Top 10 Calculus Websites - Worldwide Center of Mathematics" class="image-large" /></a></p>
<h3>Calculators</h3>
<h4>4) <a href="http://www.wolframalpha.com/">Wofram|Alpha</a></h4>
<p><a href="http://www.wolfram.com">Wolfram Research</a> was started by Stephan Wolfram in 1987 and they are best known for <a href="http://www.wolfram.com/mathematica">Mathematica</a> which specializes in solving math problems symbolically. They provide the power of Mathematica for free via the <a href="http://www.wolframalpha.com/">Wolfram|Alpha</a> website. You can type things like "derivative of x^4+exp(x)" into the text box and it will provide the result, a graph, and other information that may (or may not) be helpful to you. You can also type more general questions like "what is calculus" and be directed to <a href="http://mathworld.wolfram.com">Wolfram MathWorld</a> which is their online mathematics documenation. You can get step-by-step solutions in their "Pro" version which can cost around $3.75 per month with an annual billing.</p>
<p><a href="http://www.wolframalpha.com/"><img src="/sites/default/files/styles/large/public/top_10_wolfram.jpeg?itok=nvIeenSc" width="480" height="233" alt="Calculus Center Top 10 Calculus Websites - Wolfram|Alpha" title="Calculus Center Top 10 Calculus Websites - Wolfram|Alpha" class="image-large" /></a></p>
<h4>5) <a href="http://cloud.sagemath.com/">SageMathCloud</a></h4>
<p><a href="http://www.sagemath.org/">Sage</a> is a FREE software package that combines a lot of other FREE software packages together to solve math problems quickly and easily. Think of it as equivalent to Wolfram Research's Mathematica but for free. More importantly, you don't have to worry about installing it! Sage is available online through <a href="http://cloud.sagemath.com/">SageMathCloud</a> which is supported by the University of Washington, the National Science Foundation and Google. This means you have the smartest people on the planet supporting a platform that you can use for FREE 24/7 to solve almost any math problem. This alone should convince you to check it out. Sage has a steeper learning curve than Wolfram|Alpha, but we have a five part blog series <a href="/blog/make-your-calculus-life-easier-with-sage">here</a> that will get you logged in and solving Calculus problems very quickly.</p>
<p><a href="http://cloud.sagemath.com/"><img src="/sites/default/files/styles/large/public/top_10_sagemathcloud.jpeg?itok=S41jx3SZ" width="480" height="219" alt="Calculus Center Top 10 Calculus Websites - SageMathCloud" title="Calculus Center Top 10 Calculus Websites - SageMathCloud" class="image-large" /></a></p>
<h3>Textbooks
<h4>6) <a href="http://aimath.org/textbooks/approved-textbooks">AIM Open Textbook Initiative</a><a href="http://aimath.org/textbooks/approved-textbooks"></a></h4>
</h3><p><a href="http://en.wikipedia.org/wiki/Open_textbook">Open textbooks</a> are books that are written for the purpose of being freely distributed. Many universities and other organizations promote open textbooks so look to see if your university has a website promoting them. If not then you can head over to the <a href="http://aimath.org/textbooks/approved-textbooks">American Institute of Mathematics Open Textbook Initiative</a> to get their listing of books that they have "approved" as being open textbooks. They have both beginning and advanced texts on Calculus as well as Precalculus textbooks. I recommend downloading them for easy reference throughout the semester.</p>
<p><a href="http://aimath.org/textbooks/approved-textbooks"><img src="/sites/default/files/styles/large/public/top_10_aimopentextbookinitiative.jpeg?itok=1TskIYlD" width="480" height="223" alt="Calculus Center Top 10 Calculus Websites - AIM Open Textbook Initiative" title="Calculus Center Top 10 Calculus Websites - AIM Open Textbook Initiative" class="image-large" /></a></p>
<h4>7) <a href="http://tutorial.math.lamar.edu/">Paul's Online Class Notes</a></h4>
<p><a href="http://www.math.lamar.edu/faculty/dawkins/dawkins.aspx">Paul Dawkins</a> is a <a href="http://www.lamar.edu/">Lamar University</a> professor who has published his class notes online. Sadly, the site looks like it was created in the '90s and has never been updated. Fortunately, the class notes do a great job of explaining things and working out problems in detail. Make sure you grab his 704 page PDF "E-Book Practice Problems Solutions" located on the right side drop-down <a href="http://tutorial.math.lamar.edu/Problems/CalcI/CalcI.aspx">here</a>.</p>
<p><a href="http://tutorial.math.lamar.edu/"><img src="/sites/default/files/styles/large/public/top_10_paulsonlinenotes.jpeg?itok=8TDbr1Fe" width="480" height="222" alt="Calculus Center Top 10 Calculus Websites - Paul's Online Math Notes" title="Calculus Center Top 10 Calculus Websites - Paul's Online Math Notes" class="image-large" /></a></p>
<h4>8) <a href="ttp://en.wikipedia.org/wiki/Calculus">Wikipedia</a></h4>
<p>Wikipedia is another well known site so little introduction is needed. If you are interested in seeing a lot of different Calculus topics without purchasing a ton of books then this is the place to go for both beginner and advanced topics. If you are a first time Calculus student, then you may want to go to some of the other sites on this page before you head down to Wikipedia. There is a lot here, but the presentation is more of a brain dump than a structured approach to teaching Calculus.</p>
<p><a href="ttp://en.wikipedia.org/wiki/Calculus"><img src="/sites/default/files/styles/large/public/top_10_wikipedia.jpeg?itok=osBJWeZM" width="480" height="221" alt="Calculus Center Top 10 Calculus Websites - Wikipedia" title="Calculus Center Top 10 Calculus Websites - Wikipedia" class="image-large" /></a></p>
<h3>Solutions</h3>
<h4>9) <a href="http://www.google.com">Google Search</a></h4>
<p>Got a problem to solve? That's right, Google it! The real advantage of using Google as a problem solver is that it will look for answers across multiple sites (Khan Academy, Stack Exchange, <a href="https://answers.yahoo.com/">Yahoo! Answers</a>, etc.) and across multiple media types (video, blog, PDF, image, etc.) Just make sure that you try different versions of your problem. For example, if you need a solution to a problem with the exponential function in it, then try both "e^x" and "exp(x)" in your Google search.</p>
<p><a href="http://www.google.com"><img src="/sites/default/files/styles/large/public/top_10_google.jpeg?itok=IGI2s2Mu" width="480" height="222" alt="Calculus Center Top 10 Calculus Websites - Google Search" title="Calculus Center Top 10 Calculus Websites - Google Search" class="image-large" /></a></p>
<h4>10) <a href="http://math.stackexchange.com">Stack Exchange</a></h4>
<p><a href="http://stackexchange.com/">Stack Exchange</a> is a network of question and answer communities organized around very specific subjects. For Calculus you want the <a href="http://math.stackexchange.com">Mathematics community</a>. Stack Exchange is simple in that it can be used by anyone to answer any question. However, this site can delve into very complex questions and equally complex answers. Don't be surprised to see posts that look like Egyptian hieroglyphs mixed with Star Trek terminology. If you have a hard problem to solve, then this is a good place to post to after you have exhausted your other resources. Just be sure to tag your problem "calculus" so those wanting to answer Calculus problems will know you posted a question for them.</p>
<p><a href="http://math.stackexchange.com"><img src="/sites/default/files/styles/large/public/top_10_stackexchange.jpeg?itok=ouFHfDY-" width="480" height="223" alt="Calculus Center Top 10 Calculus Websites - Stack Exchange" title="Calculus Center Top 10 Calculus Websites - Stack Exchange" class="image-large" /></a></p>
</div>
</div></div></div>Wed, 03 Sep 2014 10:49:03 +0000Calculus Center31810 at https://www.calculussolution.comThe Calculus Chain Rule
https://www.calculussolution.com/blog/31717
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even" property="content:encoded"><p>The Chain Rule is one of those Calculus rules that you have to always have in the back of your mind when you are doing differentiation. When we have two functions there are several ways that we can combine those two functions to create another function. We can add them, subtract them, multiply them, or divide them. Another way you can combine two functions is by <a href="/node/93">Function composition</a>. Function composition is used to describe and summarize complex dependencies in nature. For example, when sea temperatures rise there are less krill. Because there are less krill there is less food for blue whales to eat. Function composition in this example allows us to directly determine the blue whale population as a function of sea temperature.</p>
<p>Once we have combined two functions by function composition we can then find the derivative. To do this directly is sometimes impossible. The chain rule allows us to differentiate a function composition very quickly and very easily. <a href="/node/110">Our lesson on the Chain Rule is visible for everyone to see so take a moment to check it out for FREE.</a> The great thing about this rule is that it is easy to memorize and can be used to solve very difficult derivatives. This rule is a part of every Calculus class so if you haven't seen it yet then you most certainly will soon.</p>
</div></div></div>Tue, 18 Feb 2014 21:49:41 +0000Calculus Center31717 at https://www.calculussolution.comFinding Critical Points and Limits with Sage!
https://www.calculussolution.com/blog/solving-critical-points-and-limits-with-sage
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even" property="content:encoded"><p>This is the fifth lesson in five where we are discussing <a href="http://www.sagemath.org">Sage</a> and how it can be used to make your Calculus life easier. In the <a href="/make-your-calculus-life-easier-with-sage">first lesson</a> you learned how to create your own account and how to create a worksheet. In the <a href="/plotting-functions-with-sage">second lesson</a> you learned how to plot functions with Sage. In the <a href="/differentiate-with-sage">third lesson</a> you learned how to differentiate using Sage. And in the <a href="/solve-calculus-integrals-with-sage">fourth lesson</a> you learned how to integrate functions with Sage. In this lesson we are going to show how you can find critical points and solve limits. So to get started log into <a href="http://cloud.sagemath.org">your account</a>, create a new worksheet and label it 'critical points and limits'. You can do this by clicking the "New" button, typing "critical points and limits" in the text box and hitting the "Sage Worksheet" button just as we did in the first <a href="/make-your-calculus-life-easier-with-sage">lesson</a>.</p>
<p>
</p><p>First, let's introduce the solve function by solving a <a href="/node/31415">quadratic equation</a>. Create a polynomial as you have done in the other lessons and then type the command "g=solve(f==0,x)" and then "show(g)". Like this:<br /></p><center><img src="/sites/default/files/field/image/sage_article_critical_points_and_limits_1.jpeg" width="939" alt="Solve quadratic equation with Sage" title="Solve quadratic equation with Sage" /></center>You should recognize the two possible solutions to the quadratic equation.
<p>
</p><p>Recall that critical points are where a function's derivative is zero or is not defined. So let's find those points where the derivative is zero using the solve function:<br /></p><center><img src="/sites/default/files/field/image/sage_article_critical_points_and_limits_2.jpeg" width="842" alt="Finding critcal points using solve function in Sage" title="Finding critcal points using solve function in Sage" /></center>This is correct because<br />
\begin{eqnarray}<br />
f(x)&=&ax^2+bx+c,\\<br />
\frac{df}{dx}&=&2ax+b=0,\\<br />
x&=&-\frac{b}{2a}.<br />
\end{eqnarray}
<p>
</p><p>Be careful! This only works for functions that are polynomials. For functions like $1/x$ you will have to do other things like solve(f==oo,x) to solve it. A full discussion of critical points is beyond the scope of today's discussion so we will have to save it for later. Let's move to limits.</p>
<p>
</p><p>Let's solve<br />
\[<br />
\lim_{x\to 2}\,\frac{5x^2-3x-14}{x-2}.<br />
\]<br />
We know from <a href="/node/31615">here</a> that the answer is 17. To do this we use the limit command like so:<br /></p><center><img src="/sites/default/files/field/image/sage_article_critical_points_and_limits_3.jpeg" width="842" alt="Solving limits with Sage" title="\[ \lim_{x\to 2}\,\frac{5x^2-3x-14}{x-2}. \]" /></center>The fact that we are approaching zero in the denominator was handled perfectly by Sage.
<p>
</p><p>We hope that you enjoyed <strong>Sage Week</strong>. You can find more Calculus information on the <a href="http://www.sagemath.org/calctut">Sage Calculus Tutorial page</a>. Finally, if you want to jump to the other blog posts for <strong>Sage Week</strong> then here they are:</p>
<ul><li>Monday: <a href="/make-your-calculus-life-easier-with-sage">Make your Calculus life easier with Sage!</a>
</li><li>Tuesday: <a href="/plotting-functions-with-sage">Plotting functions with Sage!</a>
</li><li>Wednesday: <a href="/differentiate-with-sage">Differentiating functions with Sage!</a>
</li><li>Thursday: <a href="/solve-calculus-integrals-with-sage">Integrating Functions with Sage!</a>
</li><li> Friday/Today: Finding Critical Points and Limits with Sage!
</li></ul></div></div></div>Fri, 07 Feb 2014 11:30:13 +0000Calculus Center31715 at https://www.calculussolution.comIntegrating Functions with Sage!
https://www.calculussolution.com/blog/solve-calculus-integrals-with-sage
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even" property="content:encoded"><p>This is the fourth lesson in five where we are discussing <a href="http://www.sagemath.org">Sage</a> and how it can be used to make your Calculus life easier. In the <a href="/make-your-calculus-life-easier-with-sage">first lesson</a> you learned how to create your own account and how to create a worksheet. In the <a href="/plotting-functions-with-sage">second lesson</a> you learned how to plot functions with Sage. In the <a href="/differentiate-with-sage">third lesson</a> you learned how to differentiate using Sage. In this lesson we are going to learn how to calculate integrals. So to get started log into <a href="http://cloud.sagemath.org">your account</a>, create a new worksheet and label it integrals. You can do this by clicking the "New" button, typing "integrals" in the text box and hitting the "Sage Worksheet" button just as we did in the first <a href="/make-your-calculus-life-easier-with-sage">lesson</a>.</p>
<p>Now you should have a worksheet where you can type in commands. We are going to use sage to solve<br />
\[<br />
\int_0^1x^3+3x^2-8x-2\,dx.<br />
\]<br />
The full solution is <a href="/node/31634">here</a> and we know that the answer is -19/4. Let's solve this with Sage. First, input the function as we did <a href="/differentiate-with-sage">yesterday</a> like so:<br /></p><center><img src="/sites/default/files/field/image/sage_article_integrals_1.jpeg" width="847" alt="" /></center>
<p>
</p><p>To solve the indefinite integral of this function we need to use the command "f.integral(x)" like this:<br /></p><center><img src="/sites/default/files/field/image/sage_article_integrals_2.jpeg" width="865" alt="Solving an indefinite integral with Sage" title="Solving an indefinite integral with Sage" /></center>
<p>
</p><p>To solve the definite integral we use the command "f.integral(x,0,1)". The lower and upper limits of integration come right after the x. This is what we get:<br /></p><center><img src="/sites/default/files/field/image/sage_article_integrals_3.jpeg" width="879" alt="Solving a definite integral with Sage" title="Solving a definite integral with Sage" /></center>
<p>
</p><p>Finally, using our <a href="/plotting-functions-with-sage">plotting skills</a> let's visualize the area we are finding with the following code:<br /></p><center><img src="/sites/default/files/field/image/sage_article_integrals_4.jpeg" width="862" alt="Plotting the definite integral with Sage" title="Plotting the definite integral with Sage" /></center> What we have done is use the fill=True parameter in our plot command to shade in the area we are integrating. To show the function outside of that area we have a second plot that goes beyond our limits of integration.
<p>
</p><p>To summarize, in Sage we solve both indefinite and definite integrals this way:<br />
\begin{eqnarray}<br />
\mbox{f.integral(x)}&=&\int\,f(x)dx\\<br />
\mbox{f.integral(x,a,b)}&=&\int_a^b\, f(x)dx.<br />
\end{eqnarray}
</p>
<p>
</p><p>Tomorrow we will use Sage to find critical points and functions limits. Finally, if you want to jump to the other blog posts for <strong>Sage Week</strong> then here they are:</p>
<ul><li>Monday: <a href="/make-your-calculus-life-easier-with-sage">Make your Calculus life easier with Sage!</a>
</li><li>Tuesday: <a href="/plotting-functions-with-sage">Plotting functions with Sage!</a>
</li><li>Wednesday: <a href="/differentiate-with-sage">Differentiating functions with Sage!</a>
</li><li>Thursday/Today: Integrating Functions with Sage!
</li><li>Friday: <a href="/solving-critical-points-and-limits-with-sage">Finding Critical Points and Limits with Sage!</a>
</li></ul></div></div></div>Thu, 06 Feb 2014 10:49:44 +0000Calculus Center31714 at https://www.calculussolution.comDifferentiating Functions with Sage!
https://www.calculussolution.com/blog/differentiate-with-sage
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even" property="content:encoded"><p>This is the third lesson in five where we are discussing <a href="http://www.sagemath.org">Sage</a> and how it can be used to make your Calculus life easier. In the <a href="/make-your-calculus-life-easier-with-sage">first lesson</a> you learned how to create your own account and how to create a worksheet. In the <a href="/plotting-functions-with-sage">second lesson</a> you learned how to plot functions with Sage. In this lesson we are going to learn how to calculate derivatives. So to get started log into <a href="http://cloud.sagemath.org">your account</a>, create a new worksheet and label it derivatives. You can do this by clicking the "New" button, typing "derivatives" in the text box and hitting the "Sage Worksheet" button just as we did in the first lesson.</p>
<p>Now you should have a worksheet where we can type in commands. Let's get started with finding the derivative of the polynomial $f(x)=ax^2+bx+c$. To do this just type<br /></p><center>var("x a b c")</center><br />
and<br /><center>f=a*x^2+b*x+c</center><br />
into the worksheet. The first line defines variables. Don't worry too much about this. Just trust us that it needs to be done. Now we are going to introduce a new command. Type "show(f)" on the next line. This will give you a pretty representation of the function as you would write it on paper. Now hit the run button or shift+return to see the results. You should have something like this:<br /><center><img src="/sites/default/files/field/image/sage_article_derivatives_2.jpeg" width="837" alt="" /></center>
<p>Now let's differentiate $f(x)$. After show(f) type "dfdx=f.diff()" with "show(dfdx)" below it. Now hit the run button or shift+return to execute. You should get something like this:<br /></p><center></center><img src="/sites/default/files/field/image/sage_article_derivatives_4.jpeg" width="1032" alt="" /><br />
Notice the warning at the bottom which says "ValueError: No differentiation variable specified." Because we have four variables it did not know which one to differentiate the function with. We had $x$ in our heads, but did not specify that. So let's do it properly by typing "f.diff(x)" instead of "f.diff()". Now you should see the correct answer which is $2ax+b$. Here is what you should see in Sage:<br /><center></center><img src="/sites/default/files/field/image/sage_article_derivatives_5.jpeg" width="855" alt="Differentiation with Sage" title="Differentiation with Sage" /><br />
And of course we can find the second derivative by another use of diff(x):<br /><center><img src="/sites/default/files/field/image/sage_article_derivatives_6.jpeg" width="857" alt="Second Derivative with Sage" title="Second Derivative with Sage" /></center>
<p>This has all been symbolic which is great, but what if we have actual numbers instead of the variables $a$, $b$, and $c$? Well that is very easy. Let's say $a=3$, $b=-2$ and $c=1$. This is how you do it:<br /></p><center><img src="/sites/default/files/field/image/sage_article_derivatives_7.jpeg" width="858" alt="Derivaties and replacing Sage variables with numbers" title="Derivaties and replacing Sage variables with numbers" /></center>
<p>We can plot the function and its derivatives:<br /></p><center><img src="/sites/default/files/field/image/sage_article_derivatives_8.jpeg" width="849" alt="Plotting derivatives with Sage" title="Plotting derivatives with Sage" /></center>
<p>Finally, let's plot the function and its tangent line:<br /></p><center><img src="/sites/default/files/field/image/sage_article_derivatives_9.jpeg" width="846" alt="Function and its tangent line with Sage" title="Function and its tangent line with Sage" /></center>
<p><a href="/solve-calculus-integrals-with-sage">Tomorrow</a> we will look at how to integrate functions with Sage. Finally, if you want to jump to the other blog posts for <strong>Sage Week</strong> then here they are:</p>
<ul><li>Monday: <a href="/make-your-calculus-life-easier-with-sage">Make your Calculus life easier with Sage!</a>
</li><li>Tuesday: <a href="/plotting-functions-with-sage">Plotting functions with Sage!</a>
</li><li>Wednesday/Today: Differentiating functions with Sage!
</li><li>Thursday: <a href="/solve-calculus-integrals-with-sage">Integrating Functions with Sage!</a>
</li><li>Friday: <a href="/solving-critical-points-and-limits-with-sage">Finding Critical Points and Limits with Sage!</a>
</li></ul></div></div></div>Wed, 05 Feb 2014 11:49:56 +0000Calculus Center31713 at https://www.calculussolution.comPlotting functions with Sage!
https://www.calculussolution.com/blog/plotting-functions-with-sage
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even" property="content:encoded"><h2 class="h2">Plotting with Sage!</h2>
<p>
<a href="https://www.calculuscenter.com/make-your-calculus-life-easier-with-sage" target="_blank">Yesterday</a> we talked briefly about <a href="http://www.sagemath.org/" target="_blank">Sage</a>, had you create an account and a project called 'Calculus Center'. Today we are going to use that project to plot functions. Let's get started:</p>
<ul><li>Log into your account at <a href="https://cloud.sagemath.com/" target="_blank">https://cloud.sagemath.com/</a>.</li>
<li>When you get in you should see a listing of your projects. Click the 'Calculus Center' project link.</li>
</ul><p><img src="/sites/default/files/field/image/sage10.jpeg" width="1125" alt="Sage Project Selection" title="Sage Project Selection" /></p>
<ul><li>At the top you now want to click the "New" button.</li>
</ul><p>
<img src="/sites/default/files/field/image/sage11.jpeg" width="430" alt="Sage select new " title="Sage select new" /></p>
<ul><li>In the text box put the name plot_example and click the "Sage Worksheet" button. This will create a worksheet called plot_example where we will plot our functions.</li>
</ul><p><img src="/sites/default/files/field/image/sage12.jpeg" width="1098" alt="Sage create worksheet" title="Sage create worksheet" /></p>
<ul><li>Now you will see a screen that looks like this:</li>
</ul><p><img src="/sites/default/files/field/image/sage13.jpeg" width="1436" alt="Sage worksheet" title="Sage worksheet" /></p>
<ul><li>Now we can finally plot something. Type f(x)=x^2 next to the "1" and then plot(f) right under it. These are your plot commands. To have Sage execute your commands you can hit shift+enter or click the right arrow button underneath the "Files" icon. You should get this:</li>
</ul><p><img src="/sites/default/files/field/image/sage14.jpeg" width="841" alt="Sage plot of x-squared" title="Sage plot of x-squared" /><br /><br />
Congratulations! You have now made a plot in Sage. </p>
<p>One of the things to note is that the section with the commands to execute and the output is referred to as a cell. The portion that appears to have a grey, left, open parenthesis is the output or result of running your commands. A worksheet can have multiple cells. Let's demonstrate that by typing more commands below the black horizontal line. This will create a new cell in the worksheet. Now here is a more complex plotting example:<br /><img src="/sites/default/files/field/image/sage15.jpeg" width="823" alt="Sage plot of x-cubed" title="Sage plot of x-cubed" /><br />
In this example we did three additional things. First, we added a second function on the plot. Second, we changed where on the x-axis we make the plot with the domain variable. Third, we changed the colors of the plots to distinguish the two functions from each other.</p>
<p>So let's recap what we learned:</p>
<ul><li>Projects are made up of worksheets and worksheets are made up of cells.</li>
<li>Input commands like f(x)=x^2 and plot(f) into a worksheet cell and hit shift+enter to see the result next to the left, open parenthesis.</li>
<li>More complex graphs can be made with commands like plot(f,domain,color="red)+plot(g,domain,color="green").</li>
</ul><p>
<a href="/differentiate-with-sage">Tomorrow</a> we will use Sage to differentiate a function. Finally, if you want to jump to the other blog posts for <strong>Sage Week</strong> then here they are:</p>
<ul><li>Monday: <a href="/make-your-calculus-life-easier-with-sage">Make your Calculus life easier with Sage!</a>
</li><li>Tuesday/Today: Plotting functions with Sage!
</li><li>Wednesday: <a href="/differentiate-with-sage">Differentiating functions with Sage!</a>
</li><li>Thursday: <a href="/solve-calculus-integrals-with-sage">Integrating Functions with Sage!</a>
</li><li>Friday: <a href="/solving-critical-points-and-limits-with-sage">Finding Critical Points and Limits with Sage!</a>
</li></ul></div></div></div>Tue, 04 Feb 2014 19:17:18 +0000Calculus Center31712 at https://www.calculussolution.comA Calculus Specific Alternative to Khan Academy Videos
https://www.calculussolution.com/blog/a-calculus-specific-alternative-to-khan-academy-videos
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even" property="content:encoded"><p>We make it pretty clear up front that we don't do videos. It's not that we don't like them, it's that there are so many high quality videos out there that we don't feel the need to recreate our own. While everyone knows about <a href="https://www.khanacademy.org/" target="_blank">Khan Academy</a>, very few know about the <a href="http://centerofmath.org/" target="_blank">Worldwide Center of Mathematics</a>.</p>
<p>We recently learned about the Worldwide Center of Mathematics at the <a href="http://jointmathematicsmeetings.org/jmm" target="_blank">Joint Mathematics Meeting</a> in Baltimore earlier this month. They are another for-profit education site, but offer their online videos for free. (You can check them out <a href="http://centerofmath.org/videos/index.html#" target="_blank">here</a>.) </p>
<p>These videos differ from Khan Academy in that they are of David Massey in front of a blackboard doing the lecture as opposed to Sal Khan's method of writing on an electronic tablet. The lectures are well designed and even though they are videos of David lecturing at a blackboard, they are easy to follow and the writing done on the blackboard is clearly visible.</p>
<p>Finally, as a Calculus student you should be aware that this site is focused on Calculus education while Khan Academy is focused on pretty much everything under the sun. You get Calculus, Pre-Calculus and everything related to Calculus. So check out their videos for a great alternative to Khan Academy.</p>
</div></div></div>Mon, 03 Feb 2014 22:34:41 +0000Calculus Center31711 at https://www.calculussolution.comMake Your Calculus Life Easier with Sage!
https://www.calculussolution.com/blog/make-your-calculus-life-easier-with-sage
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even" property="content:encoded"><h3>It's Sage Week!</h3>
<p>This week is <a href="http://www.sagemath.org/">Sage</a> Week at <a href="http://calculuscenter.com">Calculus Center</a>! We are going to spend the week teaching you about <a href="http://www.sagemath.org/">Sage</a> and how it can make learning Calculus a whole lot easier. In fact, after this week we think Sage will become your tool of choice for solving Calculus problems. In this first email we will tell you what Sage is and how to get it. (Don't worry it is 100% FREE!) On Tuesday we will teach you how to plot functions with Sage. On Wednesday we will teach you how to solve derivatives with Sage. On Thursday, we will teach you how to solve integrals with it. Finally, on Friday we are going to list some other things that Sage can do for Calculus students. By the end of the week you will have a working knowledge of Sage and be able to use it to solve homework problems quickly and easily.</p>
<h3>What is Sage?</h3>
<p>Sage is a FREE software package that combines a lot of other FREE software packages together to solve math problems quickly and easily. Think of it as equivalent to Mathematica or any other commercially available software, but for free. More importantly, you don't have to worry about installing it! Sage is available online through <a href="http://cloud.sagemath.com" target="_blank" style="color: #336699;font-weight: normal;text-decoration: none;">SageMathCloud</a> which is supported by the <span class="lighten">University of Washington, the National Science Foundation and Google.</span> This means you have the smartest people on the planet supporting a platform that you can use for FREE 24/7 to solve almost any math problem. This alone should convince you to check it out.</p>
<h3>How do I get started?</h3>
<p>Let's get straight to the point and guide you to creating an account and your first project:</p>
<ul><li>Point your web browser to <a href="https://cloud.sagemath.com/" target="_blank" style="color: #336699;font-weight: normal;text-decoration: none;">https://cloud.sagemath.com/</a>.</li>
</ul><p><img src="/sites/default/files/field/image/sage5.jpeg" width="1334" alt="" /></p>
<ul><li>Click the "create an account" link in the grey box on the right.</li>
<li>Fill out the form and click the green "Create and account for free" button.</li>
</ul><p><img src="/sites/default/files/field/image/sage6.jpeg" width="432" alt="" /></p>
<ul><li>The work that you do will be done in a 'project'. To create your first project click the blue "New Project" button. This will create a project that we will use for the rest of the week.</li>
</ul><p><img src="/sites/default/files/field/image/sage7.jpeg" width="1434" alt="" /></p>
<ul><li>Now a form will appear for you to put in information about your project. In the 'Title' field put 'Calculus Center' so you know this project is to copy what we do this week. You can make the project public or private. It defaults to public so just keep it there for the moment. You can change this later if you want. Once you have put in the title click the blue "Create Project" button.</li>
</ul><p><img src="/sites/default/files/field/image/sage8.jpeg" width="573" alt="" /></p>
<ul><li>You will now be taken back to the projects page. Notice up in the top, right-hand corner a box that says "Creating new project 'Calculus Center'. This takes about 30 seconds.' Once SageMathCloud has done whatever it does behind the scenes to create your project you will see a link with your project's title appear under "Showing projects".</li>
<li>Now you can click on the "Calculus Center" link. This will take you to your project.</li>
</ul><p><img src="/sites/default/files/field/image/sage9.jpeg" width="1126" alt="" /><br /><br /><br />
Congratulations! You have successfully created an account and your first project. </p>
<p><a href="/plotting-functions-with-sage">Tomorrow</a> we will talk about graphing functions with Sage. Finally, if you want to jump to the other blog posts for Sage Week then here they are:</p>
<ul><li>Monday/Today: Make your Calculus life easier with Sage!
</li><li>Tuesday: <a href="/plotting-functions-with-sage">Plotting functions with Sage!</a>
</li><li>Wednesday: <a href="/differentiate-with-sage">Differentiating functions with Sage!</a>
</li><li>Thursday: <a href="/solve-calculus-integrals-with-sage">Integrating Functions with Sage!</a>
</li><li>Friday: <a href="/solving-critical-points-and-limits-with-sage">Finding Critical Points and Limits with Sage!</a>
</li></ul></div></div></div>Mon, 03 Feb 2014 22:00:09 +0000Calculus Center31710 at https://www.calculussolution.comThe Easy Way To Find $(x+y)^n$
https://www.calculussolution.com/blog/easy-way-to-find-powers-of-x-plus-y
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even" property="content:encoded"><p>In many courses in Mathematics we oftentimes have to expand out things like $(x+y)^n$ where $n$ is an <a href="/node/31395">integer</a> and $x$ and $y$ are <a href="/node/31395">real numbers</a>. So many of us will happily expand it out and collect like terms. But this is kind of hard. Isn't it? Well we want to show you an easier and shockingly fast way to just do it.</p>
<p>First, let's do the actual Algebra to show how much of a pain in the butt this is:<br />
\begin{eqnarray}<br />
(x+y)^3&=&(x+y)(x+y)(x+y)\\<br />
&=&(x^2+xy+xy+y^2)(x+y)\\<br />
&=&(x^2+2xy+y^2)(x+y)\\<br />
&=&x^3+x^2y+2x^2y+2xy^2+xy^2+y^3\\<br />
&=&x^3+3x^2y+3xy^2+y^3.<br />
\end{eqnarray}</p>
<p>The quick way to do this is with the formula<br />
\[<br />
(x+y)^n=\sum_{j=0}^nC_nx^{n-j}y^j.<br />
\]<br />
The symbol $\sum$ is the summation formula that tells is to add from $0$ to $n$ and as we go we will have a number $C_n$ multiplied by $x^{n-j}y^j$. In other words,<br />
\[<br />
(x+y)^n=C_0x^ny^0+C_1x^{n-1}y^1+\cdots+C_{n-1}xy^{n-1}+C_nx^0y^n.<br />
\]<br />
The numbers $C_0,\,C_1,\,\cdots$ are referred to as <strong>Binomial Coefficients</strong> and finding them is the hard part in this formula.</p>
<p>It turns out for $n\leq 4$ finding the Binomial Coefficients ($C_n$'s) is really easy with a simple trick. <strong>The trick is to find the number $11^n$</strong>. (We recommend using a calculator since you want to get this done quickly.) It turns out that the digits in $11^n$ are the coefficients $C_n$! In our example above $n=3$. So on your calculator you find that $11^3=1331$. So this means that $C_0=1$, $C_1=3$, $C_2=3$, $C_3=1$. So now using the formula we have<br />
\[<br />
(x+y)^3=C_0x^3y^0+C_1x^2y+C_2xy^2+C_3x^0y^3.<br />
\]Now since $x^0=y^0=1$ and because we found our coefficients from $11^3$ we have<br />
\[<br />
(x+y)^3=x^3+3x^2y+3xy^2+y^3.<br />
\]</p>
<p>So now if you have to find $(x+5)^4$ you will first find that $11^4=14641$ and that<br />
\begin{eqnarray}<br />
(x+5)^4&=&x^4+4x^35^1+6x^25^2+4x5^3+5^4\\<br />
&=&x^4+20x^3+150x^2+500x+625.<br />
\end{eqnarray}</p>
<p>We want to emphasize that this trick is only good for $n$ less than or equal to four. But because $(x+y)^n$ shows up so much when $n\leq 4$ it turns that this is a very useful thing to keep in mind. Finally, this trick is rooted in some pretty deep Mathematics related to Pascal's Triangle, <a href="/node/31652">Factorials</a>, and the <a href="/node/31655">Binomial Theorem</a>. But that's a story for another day.</p>
</div></div></div>Wed, 18 Sep 2013 18:41:54 +0000Calculus Center31664 at https://www.calculussolution.com