MA 391 Composition and Communication

  Introduction Imagine for a moment that you work for a chocolate factory and help to create new treats. You’re very good at your job and have developed a design for a new candy consisting of a chocolate shell with a caramel-filled cavity. You want the outside of the shell to be 1.5 inches tall and have the same curve as the equation \( f(x)=- \frac{1}{9}\ x^2+ \frac{3}{4}\ \). You also need there to be an opening in the center for the filling, which you want to have the same curve as \( g(x)= \frac{1}{2}\ - \frac{1}{2}\ e^{-x} \). There’s only one problem: before you can begin manufacturing your masterpiece, your boss wants to know what the volume of this candy is so he can order the correct amount of chocolate. Is there a way to calculate the volume of such a complicated shape? Fortunately, there is, with a little help from calculus. What is the Washer Method? As discussed above, we know the wall of our solid will consist of the space between the following t...

When you are making your own blog posts you can copy this page and then edit it so the math feature is already ready to go. Make sure you are in html view - you can change this in the top left corner right under the title. If you want your math to be displayed on its own line you do \[x^2+y^2=1\] If you want your math to appear in the line with your surrounding text you do \(\sin x+\cos y\).

Welcome to MA 391: Composition and Communication at the University of Kentucky! This semester we are going to be practicing the (sometimes frustrating and always ) important skill of communicating mathematics. At the start of the semester we are going to revisit some topics from calculus and for each of those topics you will pick an example you think really helps explain what is going on. You will design and print a 3d model of that example. Then you will write a blog post (here!) and a description card for your model and participate in the class show and tell. After spending some time in calculus we will switch over to topics motivated by geometry and topology. You don't need to know anything about these area yet. We're going to look at them since they will be new to many of you, they lend themselves well to visualization projects and my research is in topology!

This post is from the summer when the innovation lab is not open so the 3d object is a cucumber from my garden rather than a 3d printed object! The main ideas For functions of a single variable the tangent line is the line that is the "closest linear approximation" to the function. For a function \(f(x)\) the tangent to the function at the point \(a,f(a)\) is the line through that point with slope \(f'(a)\). So if we use point slope form the tangent line has the equation $$(y-f(a))=f'(a)(x-a)$$ or in standard form $$y=f'(a)x+(f(a)-f'(a)a).$$ There are many reasons we might be interested in the tangent line to a graph. One of them is to find critical points of that graph. Critical points are points where the tangent line is vertical or horizontal and these are points where we can find local (or global) maximum and minimum values. A tangent line is horizontal if the derivative is zero and it is vertical if the de...