# Iteration in First Semester Calculus

### James A. Walsh

Department of Mathematics
Oberlin College
Oberlin, OH 44074
USA
Phone: (216)-775-8387
Fax: (216)-775-8124

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## ABSTRACT

With the advent of powerful personal computers a renewed focus on the ideas of iteration has appeared in the mathematics community. In the early twentieth century the French mathematicians Pierre Fatou and Gaston Julia investigated iterating complex valued functions, but their studies eventually came to a halt because of the amazing complexity of the sets they were considering without the aid of the computer. In recent years beautiful images of Julia sets and the Mandelbrot set seem to be appearing everywhere.

This paper concerns itself with introducing the ideas of iterating a function of a single real variable in the first semester calculus course via the spreadsheet. Students are assigned three computer labs they complete as homework assignments using spreadsheets. The first lab investigates iterating linear maps f(x)=ax+b, where a and b are real parameters. An application is given in terms of a financial model of interest accruing in a savings account.

The second lab investigates the dynamics of the logistic family f_k(x)=kx(1-x). This is a far more complicated family dynamically than the above family of linear maps, yet students investigate this complexity via the spreadsheet. In particular they discover the quadratic bifurcation diagram and learn about Feigenbaum's universal constant. An application is given to population models.

The final lab investigates Newton's method for the family of cubics f_c(x)=(x+2)(x^2+c). Newton's method exhibits chaotic behavior in this setting, and the students are amazed to see the quadratic bifurcation diagram buried within the bifurcation diagram for Newton's method. The students use the spreadsheet to investigate the surprising complexity of Newton's method for this family of cubics.

The above labs consume four class meetings and are introduced just before the appearance of Newton's method in the standard first semester calculus curriculum. It turns out the derivative plays the leading role in determining the dynamics of iterating a function and this, along with Newton's method, forms the connection between iteration and first semester calculus.

Keyword(s): calculus, numerical methods, dynamical systems