e^(Pi) vs (Pi)^e

Posted: February 28, 2016 in Calculus: An Introduction
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Is  e^π > π^e  or  is  e^π < π^e ???

This is a continuation of my Euler-pi theme that seems to have consumed me over the past two days. The answer to the question put forth is easy to determine using technology but little to nothing of value is learned through that process. Natural logarithms, functions and derivatives are once again employed here to arrive at a logical (to me) conclusion.

Click on the link provided here for an exploration of the tangent line as it moves along the function.

This entry opens a discussion on “local” versus “global” extrema as well as indeterminate limits.

Click on L’Hospital’s Rule to see the limit of  x/ln(x)  as  x→∞.

Euler: The Intro Continues

Posted: February 27, 2016 in Calculus: An Introduction
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There are many great explorations to pursue in Math 31 (introduction to Calculus) of which we owe thanks to Leibniz, Newton and many others who followed. With “π” week fast approaching, it is an appropriate time to further introduce my students to the genius of Swiss mathematician Leonhard Euler.

This entry begins with a step function that appears to converge on π. Through our little journey here, students will understand and develop the reasoning used by Euler to show that the infinite series 1/n^2 does in fact converge.

What’s going on here and why is this thing approaching π???

To have any chance of understanding this, we need to build on what students already know. The following set-up is crucial.

The notes directly above are well within the abilities of high school students to conceptualize; these same concepts and procedures will now be transferred to a new context that students are not so familiar with. The Taylor Series allows us to approximate many functions very accurately (within the vicinity of some value of “x”) as polynomial functions. The polynomial representation of f(x)=sin(x) will be exploited here to arrive at our destination.

Taylor Series

Students are well-versed in expressing polynomial functions in factored form as shown below; an appropriate comparison is eventually made to end our little journey.

Reference

Courant, Richard., John, Fritz (1999).  Introduction to Calculus and Analytics: Classics of Mathematics. New York, NY:  Springer-Verlag Berlin Heidelberg.

Posted: February 26, 2016 in Calculus: An Introduction
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In accordance with normal procedures, previously addressed topics are once again reflected on in this entry. Directly below, the derivative of e^x is dealt with from the perspectives of slope fields and differential equations. Beneath that, the derivative of e^x is considered as a Taylor Series.

Slope Field

Taylor Series

It is easy for high school students to determine that dy/dx = y for the Taylor Series representation of e^x above. This sets the stage very nicely for an exploration into power series which, in turn, lead directly to some other very cool “discoveries”. Follow the link provided for a more conventional extraction of the derivative of e^x.