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Posts Tagged ‘Archimedes’


Parabola or catenary?

Parabola or catenary?

In this lecture we will retrace the steps of Archimedes, Newton, Euler, and other great mathematicians and learn about important mathematical functions, their properties, history, and applications. We will look at several interesting competition topics that often show up on AMC 10/12, related to exponential, logarithmic, trigonometric, hyperbolic, and other functions. We will also see how these functions turn up in solutions of some fundamental problems in math, physics, and engineering. We will have fun using them to draw important curves: cicloids, cardioids, catenaries, circles, ellipses, hyperbolas, parabolas, and will discover which one of them is brachistochrone and which one is tautochrone.

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Last year one of our most popular sets of lectures were on Calculus.  Take this opportunity to hear, Dr. Branislav Kisačanin, introduce even our younger students to this beautiful topic.  Here is a description of the talk in his own words:

In this session we will look at calculus and try to dispel the mystery that often surrounds it. We will begin by looking at examples of limiting processes, and show for example that as x approaches 0, the value of \frac{\sin x}{x} approaches 1. Based on several other examples of limits, we will be able to determine line equations for tangents of various curves. This will bring us to the doorstep of the first derivative.

We will look at other interpretations of the first derivative and also look at how it is applied to maximization and minimization, Newton’s method of tangents for solving equations, etc. In the second hour we will look at another major part of calculus, integration, and will show how integration is used to compute areas and volumes. We will also look at the methods used by Archimedes to compute the volumes of solid spheres, which are remarkably similar to modern integration. Of course, there is much more to calculus than we can learn in one day, but this will be a good start!

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Many of our students have started taking Calculus or are curious about what it’s all about.  Take this opportunity to hear one of our favorite lecturers, Dr. Branislav Kisačanin, introduce even our younger students to this beautiful topic.  Here is a description of the talk in his own words:

 

In this session we will look at calculus and try to dispell the mistery that often surrounds it. We will begin by looking at examples of limiting processes, and show for example that as x approaches 0, the value of \frac{\sin x}{x} approaches 1. Based on several other examples of limits, we will be able to determine line equations for tangets of various curves. This will bring us to the doorstep of the first derivative.

We will look at other interpretations of the first derivative and also look at how it is applied to maximization and minimization, Newton’s method of tangents for solving equations, etc. In the second hour we will look at another major part of calculus, integration, and will show how integration is used to compute areas and volumes. We will also look at the methods used by Archimedes to compute the volumes of solid spheres, which are remarkably similar to modern integration. Of course, there is much more to caclulus than we can learn in one day, but this will be a good start!

 

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